diff --git a/parse/train/4qR3coiNaIv/4qR3coiNaIv.md b/parse/train/4qR3coiNaIv/4qR3coiNaIv.md new file mode 100644 index 0000000000000000000000000000000000000000..50e1123d554a799a6858e39cac3674b1ff56eb8d --- /dev/null +++ b/parse/train/4qR3coiNaIv/4qR3coiNaIv.md @@ -0,0 +1,313 @@ +# SCALABLE BAYESIAN INVERSE REINFORCEMENT LEARNING + +Alex J. Chan University of Cambridge, Cambridge, UK alexjchan@maths.cam.ac.uk + +Mihaela van der Schaar +University of Cambridge, Cambridge, UK University of California, Los Angeles, USA Cambridge Centre for AI in Medicine, UK The Alan Turing Institute, London, UK +mv472@cam.ac.uk + +# ABSTRACT + +Bayesian inference over the reward presents an ideal solution to the ill-posed nature of the inverse reinforcement learning problem. Unfortunately current methods generally do not scale well beyond the small tabular setting due to the need for an inner-loop MDP solver, and even non-Bayesian methods that do themselves scale often require extensive interaction with the environment to perform well, being inappropriate for high stakes or costly applications such as healthcare. In this paper we introduce our method, Approximate Variational Reward Imitation Learning (AVRIL), that addresses both of these issues by jointly learning an approximate posterior distribution over the reward that scales to arbitrarily complicated state spaces alongside an appropriate policy in a completely offline manner through a variational approach to said latent reward. Applying our method to real medical data alongside classic control simulations, we demonstrate Bayesian reward inference in environments beyond the scope of current methods, as well as task performance competitive with focused offline imitation learning algorithms. + +# 1 INTRODUCTION + +For applications in complicated and high-stakes environments it can often mean operating in the minimal possible setting - that is with no access to knowledge of the environment dynamics nor intrinsic reward, nor even the ability to interact and test policies. In this case learning and inference must be done solely on the basis of logged trajectories from a competent demonstrator showing only the states visited and the the action taken in each case. + +Clinical decision making is an important example of this, where there is great interest in learning policies from medical professionals but is completely impractical and unethical to deploy policies on patients mid-training. Moreover this is an area where it is not only the policies, but also knowledge of the demonstrator’s preferences and goals, that we are interested in. While imitation learning (IL) generally deals with the problem of producing appropriate policies to match a demonstrator, with the added layer of understanding motivations this would then usually be approached through inverse reinforcement learning (IRL). Here attempting to learn the assumed underlying reward driving the demonstrator, before secondarily learning a policy that is optimal with respect to the reward using some forward reinforcement learning (RL) technique. By composing the RL and IRL procedures in order to perform IL we arrive at apprenticeship learning (AL), which introduces its own challenges, particularly in the offline setting. Notably for any given set of demonstrations there are (infinitely) many rewards for which the actions would be optimal $\mathrm { N g }$ et al., 2000). Max-margin (Abbeel & $\mathrm { N g }$ , 2004) and max-entropy (Ziebart et al., 2008) methods for heuristically differentiating plausible rewards do so at the cost of potentially dismissing the true reward for not possessing desirable qualities. On the other hand a Bayesian approach to IRL (BIRL) is more conceptually satisfying, taking a probabilistic view of the reward, we are interested in the posterior distribution having seen the demonstrations (Ramachandran & Amir, 2007), accounting for all possibilities. BIRL is not without its own drawbacks though, as noted in Brown & Niekum (2019), making it inappropriate for modern complicated environments: assuming linear rewards; small, solvable environments; and repeated, inner-loop, calls to forward RL. + +![](images/d9cfdc5b1bc6f3e16d989317572e562d90d7f13f159f7755e10beb37ec13d22b.jpg) +Figure 1: Overview. AVRIL is a framework for BIRL that works through an approximation in the variational Bayesian framework, considering the reward to be a latent representation of behaviour. A distribution over the reward, which is amortised over the demonstration space, is learnt that then informs an imitator $Q$ -function policy. The dotted line represents a departure from a traditional auto-encoder as the input, alongside the latent reward, informs the decoder. + +The main contribution then of this paper is a method for advancing BIRL beyond these obstacles, allowing for approximate reward inference using an arbitrarily flexible class of functions, in any environment, without costly inner-loop operations, and importantly entirely offline. This leads to our algorithm AVRIL, depicted in figure 1, which represents a framework for jointly learning a variational posterior distribution over the reward alongside an imitator policy in an auto-encoderesque manner. In what follows we review the modern methods for offline IRL/IL (Section 2) with a focus on the approach of Bayesian IRL and the issues it faces when confronted with challenging environments. We then address the above issues by introducing our contributions (Section 3), and demonstrate the gains of our algorithm in real medical data and simulated control environments, notably that it is now possible to achieve Bayesian reward inference in such settings (Section 4). Finally we wrap up with some concluding thoughts and directions (Section 5). Code for AVRIL and our experiments is made available at https://github.com/XanderJC/scalable-birl and https://github.com/vanderschaarlab/mlforhealthlabpub. + +# 2 APPROACHING APPRENTICESHIP AND IMITATION OFFLINE + +Preliminaries. We consider the standard Markov decision process (MDP) environment, with states $s \in \mathcal { S }$ , actions $a \in \mathcal A$ , transitions $T \in \Delta ( { \cal S } ) ^ { s \times { \cal A } }$ , rewards $\vec { R } \in \mathbb { R } ^ { S \times A 1 }$ , and discount $\gamma \in [ 0 , 1 ]$ . For a policy $\pi \in \Delta ( \mathcal { A } ) ^ { s }$ let $\begin{array} { r } { \rho _ { \pi } ( s , a ) = \mathbb { E } _ { \pi , T } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \mathbb { 1 } _ { \{ s _ { t } = s , a _ { t } = a \} } ] } \end{array}$ be the induced unique occupancy measure alongside the state-only occupancy measure $\begin{array} { r } { \rho _ { \pi } ( s ) = \sum _ { a \in \mathcal { A } } \rho _ { \pi } ( s , a ) } \end{array}$ . Despite this full environment model, the only information available to us is the $\mathbf { M D P } \backslash R T$ , in that we have no access to either the underlying reward or the transitions, with our lacking knowledge of the transitions being also strong in the sense that further we are unable to simulate the environment to sample them. The learning signal is then given by access to $m$ -many trajectories of some demonstrator assumed to be acting optimally w.r.t. the MDP, following a policy $\pi _ { D }$ , making up a data set $\mathcal { D } _ { r a w } = \{ ( s _ { 1 } ^ { ( i ) } , a _ { 1 } ^ { ( i ) } , \dotsc , s _ { \tau ^ { ( i ) } } ^ { ( \bar { i } ) } , a _ { \tau ^ { ( i ) } } ^ { ( i ) } ) \} _ { i = 1 } ^ { m }$ where $s _ { t } ^ { ( i ) }$ is the state and $a _ { t } ^ { ( i ) }$ is the action taken at step $t$ during the $i$ th demonstration, and $\tau ^ { ( i ) }$ is the (max) time horizon of the ith demonstration. Given the Markov assumption though it is sufficient and convenient to consider the demonstrations simply as a collection of $n$ -many state, action, next state, next action tuples such that $\mathcal { D } = \{ ( s _ { i } , a _ { i } , s _ { i } ^ { \prime } , a _ { i } ^ { \prime } ) \} _ { i = 1 } ^ { n }$ with $\begin{array} { r } { n = \sum _ { i = 1 } ^ { m } ( \tau ^ { ( i ) } - 1 ) } \end{array}$ . + +Apprenticeship through rewards. Typically AL proceeds by first inferring an appropriate reward function with an IRL procedure $\mathrm { N g }$ et al., 2000; Ramachandran & Amir, 2007; Rothkopf & Dimitrakakis, 2011; Ziebart et al., 2008) before running forward RL to obtain an appropriate policy. This allows for easy mix-and-match procedures, swapping in different standard RL and IRL methods depending on the situation. These algorithms though depend on either knowledge of $T$ in order to solve exactly or the ability to perform roll-outs in the environment, with little previous work focusing on the entirely offline setting. One simple solution is through attempting to learn the dynamics (Herman et al., 2016), though without a large supply of diverse demonstrations or a small environment this becomes impractical given imperfections in the model. Alternatively Klein et al. (2011) and Lee et al. (2019) attempt off-policy feature matching through least-squared temporal difference and deep neural networks to uncover appropriate feature representations. + +Implicit-reward policy learning. Recent work has often forgone an explicit representation of the reward. Moving within the maximum-entropy RL framework (Ziebart, 2010; Levine, 2018), Ho & Ermon (2016) noted that the full procedure ${ \mathrm { ~ R L ~ o ~ I R L } }$ ) can be interpreted equivalently as the minimisation of some divergence between occupancy measures of the imitator and demonstrator: + +$$ +\underset { \pi } { \arg \operatorname* { m i n } } \{ \psi ^ { * } ( \rho _ { \pi } - \rho _ { \pi _ { D } } ) - H ( \pi ) \} , +$$ + +with $H ( \pi )$ being the discounted causal entropy (Bloem & Bambos, 2014) of the policy and $\psi ^ { * }$ the Fenchel conjugate of a chosen regulariser on the form of the reward. These are typically optimised in an adversarial fashion (Goodfellow et al., 2014) and given the focus on evaluating $\rho _ { \pi }$ this often requires extensive interaction with the environment, otherwise banking on approximations over a replay buffer (Kostrikov et al., 2018) or a reformulation of the divergence to allow for off-policy evaluation (Kostrikov et al., 2019). Bear in mind that optimal policies within the maximum-entropy framework are parameterised by a Boltzmann distribution: + +$$ +\pi ( a | s ) = \frac { \exp ( Q ( s , a ) ) } { \sum _ { b \in \cal { A } } \exp ( Q ( s , b ) ) } , +$$ + +with $Q ( s , a )$ the soft $Q$ -function, defined recursively via the soft Bellman-equation: + +$$ +Q ( s , a ) \triangleq R ( s , a ) + \gamma \mathbb { E } _ { s ^ { \prime } \sim \rho _ { \pi } } \Big [ \operatorname { s o f t } _ { a ^ { \prime } } \operatorname { m a x } Q ( s ^ { \prime } , a ^ { \prime } ) ) \Big ] . +$$ + +Then for a learnt parameterised policy given in terms of $Q$ -values from a function approximator $Q _ { \theta }$ we can obtain an implied reward given by: + +$$ +R _ { Q _ { \theta } } ( s , a ) = Q _ { \theta } ( s , a ) - \gamma \mathbb { E } _ { s ^ { \prime } \sim \rho _ { \pi } } \Bigg [ \log \Bigg ( \sum _ { a ^ { \prime } \in \mathcal { A } } \exp ( Q _ { \theta } ( s ^ { \prime } , a ^ { \prime } ) ) \Bigg ) \Bigg ] . +$$ + +A number of algorithms make use of this fact with Piot et al. (2014) and Reddy et al. (2019) working by essentially placing a sparsity prior on this implied reward, encouraging it towards zero, and thus incorporating subsequent state information. Alternatively Jarrett et al. (2020) show that even the simple behavioural cloning (Bain & Sammut, 1995) is implicitly maximising some reward with an approximation that the expectation over states is taken with respect to the demonstrator, not the learnt policy. They then attempt to rectify part of this approximation using the properties of the energy-based model implied by the policy (Grathwohl et al., 2019). + +The problem with learning an implicit reward in an offline setting is that it remains just that, implicit, only able to be evaluated at points seen in the demonstrations, and even then only approximately. Thus even if their consideration improves imitator policies performance they offer no real improvement for interpretation. + +# 2.1 BAYESIAN INVERSE REINFORCEMENT LEARNING + +We are then resigned to directly reason about the underlying reward, bringing us back to the question of IRL, and in particular BIRL for a principled approach to reasoning under uncertainty. Given a prior over possible functions, having seen some demonstrations, we calculate the posterior over the function using a theoretically simple application of Bayes rule. Ramachandran $\&$ Amir (2007) defines the likelihood of an action at a state as a Boltzmann distribution with inverse temperature and respective state-action values, yielding a probabilistic demonstrator policy given by: + +$$ +\pi _ { D } ( a | s , R ) = \frac { \exp ( \beta Q _ { R } ^ { \pi _ { D } } ( s , a ) ) } { \sum _ { b \in \cal { A } } \exp ( \beta Q _ { R } ^ { \pi _ { D } } ( s , b ) ) } , +$$ + +where $\beta \in [ 0 , \infty )$ represents the the confidence in the optimality of the demonstrator. Note that despite similarities, moving forward we are no longer within the maximum-entropy framework and $Q _ { R } ^ { \pi } ( s , a )$ now denotes the traditional, not soft (as in equation 3), state-action value ( $Q$ -value) funcRtion given a reward $R$ and policy $\pi$ such that $\begin{array} { r } { Q _ { R } ^ { \pi } ( s , \bar { a } ) = \mathbb { E } _ { \pi , T } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } R ( s _ { t } ) | s _ { 0 } = s , a _ { 0 } = a ] } \end{array}$ Unsurprisingly this yields an intractable posterior distribution leading to a Markov chain Monte Carlo (MCMC) algorithm based on a random grid-walk to sample from the posterior. + +Issues in complex and unknown environments. This original formulation, alongside extensions that consider maximum-a-posteriori inference (Choi & Kim, 2011) and multiple rewards (Choi & Kim, 2012; Dimitrakakis & Rothkopf, 2011), suffer from three major drawbacks that make them impractical for modern, complicated, and model-free task environments. + +1. The reward is a linear combination of state features. Naturally this is a very restrictive class of functions and assumes access to carefully hand-crafted features of the state space. 2. The cardinality of the state-space is finite, $| S | < \infty$ . Admittedly this can be relaxed in practical terms, although it does mean the rapid-mixing bounds derived by Ramachandran & Amir (2007) do not hold at all in the infinite case. For finite approximations they scale at $\mathcal { O } ( | S | ^ { 2 } )$ , rapidly becoming vacuous and causing BIRL to inherit the usual MCMC difficulties on assessing convergence and sequential computation (Gamerman & Lopes, 2006). 3. The requirement of an inner-loop MDP solve. Most importantly at every step a new reward is sampled and the likelihood of the data must then be evaluated. This requires calculating the $Q$ -values of the policy with respect to the reward, in other words running forward RL. While not an insurmountable problem in the simple cases where everything is known and can be quickly solved with a procedure guaranteed to converge correctly, this becomes an issue in the realm where only deep function approximation works adequately (i.e. the nontabular setting). DQN training for example easily stretches into hours (Mnih et al., 2013) and will have to be repeated thousands of times, making it completely untenable. + +We have seen that even in the most simple setting the problem of exact Bayesian inference over the reward is intractable, and the above limitations of the current MCMC methods are not trivial to overcome. Consequently very little work has been done in the area and there still remain very open challenges. Levine et al. (2011) addressed linearity through a Gaussian process approach, allowing for a significantly more flexible and non-linear representation though introducing issues of its own, namely the computational complexity of inverting large matrices (Rasmussen, 2003). More recently Brown & Niekum (2019) have presented the only current solution to the inner-loop problem by introducing an alternative formulation of the likelihood, one based on human recorded pairwise preferences over demonstrations that significantly reduces the complexity of likelihood. However labelled preferences certainly can’t be assumed always available and while very effective for the given task is not appropriate in the general case. One of the key aspects of our contribution is that we are able to deal with all three of these issues while also not requiring any additional information. + +The usefulness of uncertainty. On top of the philosophical consistency of Bayesian inference there are a number of reasons for wanting a measure of uncertainty over any uncovered reward that are not available from regular IRL algorithms. First that the (epistemic) uncertainty revealed by Bayesian inference tells us a lot about what areas of the state-space we really cannot say anything about because we haven’t seen any demonstrations there - potentially informing future data collection if that is possible (Mindermann et al., 2018). Additionally in the cases we are mostly concerned about (e.g. medicine) we have to be very careful about letting algorithms pick actions in practice and we are interested in performing safe or risk-averse imitation, for which a degree of confidence over learnt rewards is necessary. Brown et al. (2020) for example use a distribution over reward to optimise a conditional value-at-risk instead of expected return so as to bound potential downsides. + +# 3 APPROXIMATE VARIATIONAL REWARD IMITATION LEARNING + +A variational Bayesian approach. In this section we detail our method, AVRIL, for efficiently learning an imitator policy and performing reward inference simultaneously. Unlike the previously mentioned sampling or MAP-based methods, we employ variational inference (Blei et al., 2017) to reason about the posterior. Here we posit a surrogate distribution $q _ { \phi } ( R )$ , parameterised by $\phi$ , and aim to minimise the Kullback-Leibler (KL) divergence to the posterior, resulting in an objective: + +$$ +\operatorname* { m i n } _ { \phi } \{ D _ { \mathrm { K L } } ( q _ { \phi } ( R ) | | p ( R | \mathcal { D } ) ) \} . +$$ + +This divergence is still as troubling as the posterior to evaluate, leading to an auxiliary objective function in the Evidence Lower BOund (ELBO): + +$$ +\mathcal { F } ( \phi ) = \mathbb { E } _ { q _ { \phi } } \big [ \log p ( \mathcal { D } | R ) \big ] - D _ { K L } \big ( q _ { \phi } ( R ) | | p ( R ) \big ) , +$$ + +where it can be seen that maximisation over $\phi$ is equivalent to (6). We are agnostic towards the form of both the prior and variational distribution, for simplicity here we assume a Gaussian process prior with mean zero and unit variance over $R$ alongside the variational posterior distribution given by: + +$$ +q _ { \phi } ( R ) = \mathcal { N } ( R ; \mu , \sigma ^ { 2 } ) , +$$ + +where $\mu , \sigma ^ { 2 }$ are the outputs of an encoder neural network taking $s$ as input and parameterised by $\phi$ . Note that for the algorithm that we will describe these choices are not a necessity and can be easily substituted for more expressive distributions if appropriate. Maintaining the assumption of Boltzmann rationality on the part of the demonstrator, our objective takes the form: + +$$ +\mathcal { F } ( \phi ) = \mathbb { E } _ { q _ { \phi } } \left[ \sum _ { ( s , a ) \in \mathcal { D } } \log \frac { \exp ( \beta Q _ { R } ^ { \pi _ { D } } ( s , a ) ) } { \sum _ { b \in \mathcal { A } } \exp ( \beta Q _ { R } ^ { \pi _ { D } } ( s , b ) ) } \right] - D _ { K L } \big ( q _ { \phi } ( R ) | | p ( R ) \big ) . +$$ + +The most interesting (and problematic) part of this objective as ever centres on the evaluation of $Q _ { R } ^ { \pi _ { D } } ( s , a )$ . Notice that what is really required here is an expression of the $Q$ -values as a smooth function of the reward such that with samples of $R$ we could take gradients w.r.t. $\phi$ . Of course there is little hope of obtaining this simply, by itself it is a harder problem than that of forward RL which only attempts to evaluate the $Q$ -values for a specific $R$ and already in complicated environments has to rely on function approximation and limited guarantees. + +A naive approach would be to sample $\hat { R }$ and then approximate the $Q$ -values with a second neural network, solving offline over the batched data using a least-squared $\mathrm { T D } / Q$ -learning algorithm, as is the approach forced on sampling based BIRL methods. It is in fact though doubly inappropriate for this setting, not only does this require a solve as an inner-loop but importantly differentiating through the solving operation is extremely impractical, it requires backpropagating through a number of gradient updates that are essentially unbounded as the complexity of the environment increases. + +A further approximation. This raises an important question - is it possible to jointly optimise a policy and variational distribution only once instead of requiring a repeated solve? This is theoretically suspect, the $Q$ -values are defined on a singular reward, constrained as $R ( s , a ) \ =$ $\mathbb { E } _ { s ^ { \prime } , a ^ { \prime } \sim \pi , T } [ \bar { Q } _ { R } ^ { \pi } ( { \bar { s , } } a ) - \gamma Q _ { R } ^ { \pi } ( s ^ { \prime } , a ^ { \prime } ) ]$ so we cannot learn a particular standard $Q$ -function that reflects the entire distribution. But can we learn a policy that reflects the expected reward using a second policy neural network $Q _ { \theta } ?$ We can’t simply optimise $\theta$ alongside $\phi$ to maximise the ELBO though as that completely ignores the fact that the learnt policy is intimately related to the distribution over the reward. Our solution to ensure then that they behave as intended is by constraining $q _ { \phi }$ and $Q _ { \theta }$ to be consistent with each other, specifically that the implied reward of the policy is sufficiently likely under the variational posterior (equivalently that the negative log-likelihood is sufficiently low). Thus we arrive at a constrained optimisation objective given by: + +$$ +\operatorname* { m a x } _ { \phi , \theta } \sum _ { ( s , a ) \in \mathcal { D } } \log \frac { \exp ( \beta Q _ { \theta } ( s , a ) ) } { \sum _ { b \in \mathcal { A } } \exp ( \beta Q _ { \theta } ( s , b ) ) } - D _ { K L } \big ( q _ { \phi } ( R ) | | p ( R ) \big ) , +$$ + +with $\epsilon$ reflecting the strength of the constraint. Rewriting (10) as a Lagrangian under the KKT conditions (Karush, 1939; Kuhn $\&$ Tucker, 1951), and given complimentary slackness, we obtain a practical objective function: + +$$ +\begin{array} { r l } { \mathcal { F } ( \phi , \theta , \mathcal { D } ) = \displaystyle \sum _ { ( s , a , s ^ { \prime } , a ^ { \prime } ) \in \mathcal { D } } \log \frac { \exp \beta Q _ { \theta } ( s , a ) ) } { \sum _ { b \in A } \exp ( \beta Q _ { \theta } ( s , b ) ) } - D _ { K L } \big ( q _ { \phi } ( R ( s , a ) ) | | p ( R ( s , a ) ) \big ) } & { } \\ { + \lambda \log q _ { \phi } ( Q _ { \theta } ( s , a ) - \gamma Q _ { \theta } ( s ^ { \prime } , a ^ { \prime } ) ) . } & { } \end{array} +$$ + +Here the KL divergence between processes is approximated over a countable set, and $\lambda$ is introduced to control the strength of constraint. + +On the implementation. Optimisation is simple as both networks are maximising the same objective and gradients can be easily obtained through backpropagation while being amenable to minibatching, allowing you to call your favourite gradient-based stochastic optimisation scheme. We re-iterate though that AVRIL really represents a framework for doing BIRL and not a specific model + +# Algorithm 1: Approximate Variational Reward Imitation Learning (AVRIL) + +Result: Parameters $\phi$ of variational distribution and $\theta$ of policy Q-function +Input: $\mathcal { D } , S , A , \gamma , \lambda$ , learning rate $\eta$ , mini-batch size $b$ ; +Initialise $\phi , \theta$ ; $\triangleright$ Can concatenate into single vector $( \phi , \theta )$ +while not converged do Sample $\mathcal { D } _ { m i n i }$ from $\mathcal { D }$ ; $\begin{array} { r } { \mathcal { F } ( \phi , \theta , \mathcal { D } ) = \mathbb { E } [ \frac { n } { b } \mathcal { F } ( \phi , \theta , \mathcal { D } _ { m i n i } ) ] } \end{array}$ ; . MC estimate total loss $( \phi ^ { \prime } , \theta ^ { \prime } ) ( \phi , \theta ) + \eta \nabla _ { \phi , \theta } \mathcal { F } ( \phi , \theta , \mathcal { D } )$ ; . Gradient step for $\phi , \theta$ $\phi , \theta \phi ^ { \prime } , \theta ^ { \prime }$ +end +Return: $\phi , \theta$ + +since $Q _ { \theta }$ and $q _ { \phi }$ represent arbitrary function approximators. So far we have presented both as neural networks, but this does not have to be the case. Of course the advantage of them is their flexibility and ease of training but they are still inherently black box. It is then perfectly possible to swap in any particular function approximator if the task requires it, using simple linear models for example may slightly hurt performance but allow for more insight. Despite the specific focus on infinite state-spaces, AVRIL can still even be applied in the tabular setting by simply representing the policy and variational distribution with multi-dimensional tensors. Having settled on their forms, equation (11) is calculated simply and the joint gradient with respect to $\theta$ and $\phi$ is straight-forwardly returned using any standard auto-diff package. The whole process is summarised in Algorithm 1. + +We can now see how AVRIL does not suffer the issues outlined in section 2.1. Our form of $q _ { \phi } ( R )$ is flexible and easily accommodates a non-linear form of the reward given a neural architecture - this also removes any restriction on $s$ , or at least allows for any state space that is commonly tackled within the IL/RL literature. Additionally we have a single objective for which all parameters are maximised simultaneously - there are no inner-loops, costly or otherwise, meaning training is faster than the MCMC methods by a factor equal roughly to the number of samples they would require. + +The generative model view. Ultimately a policy represents a generative model for the behavioural data we see. Ho & Ermon (2016) explicitly make use of this fact by casting the problem in the GAN framework (Goodfellow et al., 2014). Our method is more analogous to a VAE (Kingma & Welling, 2013), though not exactly, where given the graphical model in figure 2 the reward can be seen as a latent representation of the policy. Our approach takes the seen data and amortises the inference, encoding over the state space. The policy does not act as a decoder in precisely taking any given encoded reward and outputting a policy, but it does take the whole reward posterior and translate it into actions and therefore behaviour. This approach has its advantages, in both meaningful interpretation of the latent reward (which is non-existent in adversarial methods), and that we forgo the practical difficulties of alternating min-max optimisation (Kodali et al., 2017) while maintaining a generative view of the policy. + +![](images/787d6d634246cb213461ef2627f55b1feb1315b21e4be42cc9a37cfd4f67f240.jpg) +Figure 2: Graphical model for Bayesian IRL + +Temporal consistency through reward regularisation. Considering only the first term of (11) yields the standard behavioural cloning setup (where the logits output can be interpreted as the $Q$ - values) as it removes the reward from the equation and just focuses on matching actions to states. AVRIL can then be seen as a policy-learning method regularised by the need for the implied reward to be consistent. Note that this does not induce any necessary bias since the logits normally contain an extra degree of freedom allowing them to arbitrarily shift by some scale factor. This factor is now explicitly constrained by giving the logits additional meaning in that they represent $Q$ -values. This places great importance on the KL term, since every parameterisation of a policy will have an associated implied reward, the KL regularises these to be not so far from the prior and preventing the reward from overfitting to the policy and becoming pointless. It also is able to double as a regularising term in a similar manor to previous reward-regularisation methods (Piot et al., 2014; Reddy et al., 2019) depending on the chosen prior, encouraging the reward to be close to zero: + +Proposition 1 (Reward Regularisation) Assume that the constraint in (10) is satisfied in that $\begin{array} { r } { { \mathbb E } _ { q _ { \phi } } [ R ( s , a ) ] = { \mathbb E } _ { \pi , T } [ Q _ { \theta } ( s , a ) - \gamma Q _ { \theta } ( s ^ { \prime } , a ^ { \prime } ) ] } \end{array}$ , then given a standard normal prior $p ( R ) = \mathcal { N } ( R ; 0 , 1 )$ the $\mathrm { K L }$ divergence yields a sparsity regulator on the implied reward: + +$$ +\mathcal { L } _ { r e g } = \sum _ { ( s , a , s ^ { \prime } , a ^ { \prime } ) \in \mathcal { D } } \frac { 1 } { 2 } \big ( Q _ { \theta } ( s , a ) - \gamma Q _ { \theta } ( s ^ { \prime } , a ^ { \prime } ) \big ) ^ { 2 } + g ( \mathrm { V a r } _ { q _ { \phi } } [ R ( s , a ) ] ) . +$$ + +Proof. Appendix.  This follows immediately from the fact that the divergence evaluates as DK $\begin{array} { r } { \overset { \vartriangle } { \boldsymbol { \mathrm { \iota } } } \big ( q _ { \phi } ( \hat { R } ( s , a ) ) | | p ( R ( s , a ) ) \big ) = \frac { 1 } { 2 } ( - \log ( \operatorname { V a r } _ { q _ { \phi } } [ R ( s , a ) ] ) - 1 + \operatorname { V a r } _ { q _ { \phi } } [ R ( s , a ) ] + \mathbb { E } _ { q _ { \phi } } [ R ( s , a ) ] ^ { 2 } ) } \end{array}$ + +This then allows AVRIL to inherit the benefit of these methods while also explicitly learning a reward that can be queried at any point. We are also allowed the choice of whether it is state-only or state-action. This has so far been arbitrary, but it is important to consider that a state-only reward is a necessary and sufficient condition for a reward that is fully disentangled from the dynamics $\operatorname { F u }$ et al., 2018). Thus by learning such a reward and given the final term of (11) that directly connects one-step rewards in terms of the policy, this forces the policy (not the reward) to account for the dynamics of the system ensuring temporal consistency in a way that BC for example simply can’t. Alternatively using a state-action reward means that inevitably some of the temporal information leaks out of the policy and into the reward - ultimately to the detriment of the policy but potentially allowing for a more interpretable (or useful) form of reward depending on the task at hand. + +# 4 EXPERIMENTS + +Experimental setup. We are primarily concerned with the case of medical environments, which is exactly where the issue of learning without interaction is most crucial, you just cannot let a policy sample treatments for a patient to try to learn more about the dynamics. It is also where a level of interpretability in what has been learnt is important, since the consequence of actions are potentially very impactful on human lives. As such we focus our evaluation on learning on a real-life healthcare problem, with demonstrations taken from the Medical Information Mart for Intensive Care (MIMIC-III) dataset (Johnson et al., 2016). The data contains trajectories of patients in intensive care recording their condition and theraputic interventions at one day intervals. We evaluate the ability of the methods to learn a medical policy in both the two and four action setting - specifically whether the patient should be placed on a ventilator, and the decision for ventilation in combination with antibiotic treatment. These represent the two most common, and important, clinical interventions recorded in the data. Without a recorded notion of reward, performance is measured with respect to action matching against a held out test set of demonstrations with cross-validation. + +Alongside the healthcare data and for the purposes of demonstrating generalisability, we provide additional results on standard environments of varying complexity in the RL literature, the standard control problems of: CartPole, a classic control environment aiming to swing up and balance a pendulum; Acrobot, which aims to maintain a sequence of joints above a given height; and LunarLander, guiding a landing module to a safe touchdown on the moon surface. In these settings given sufficient demonstration data all benchmarks are very much capable of reaching demonstrator level performance, so we test the algorithms on their ability to handle sample complexity in the low data regime by testing their performance when given access to a select number of trajectories which we adjust, replicating the setup in Jarrett et al. (2020). With access to a simulation through the OpenAI gym (Brockman et al., 2016), we measure performance by deploying the learnt policies live and calculating their average return over 300 episodes. + +Benchmarks. We test our method (AVRIL) against a number of benchmarks from the offline IRL/IL setting: Deep Successor Feature Network (DSFN) (Lee et al., 2019), an offline adaptation of max-margin IRL that generalises past the linear methods using a deep network with leastsquares temporal-difference learning, the only other method that produces both a reward and policy; Reward-regularized Classification for Apprenticeship Learning (RCAL) (Piot et al., 2014), where an explicit regulariser on the sparsity of the implied reward is introduced in order to account for the dynamics information; ValueDICE (VDICE) (Kostrikov et al., 2019), an adversarial imitation learning, adapted for the offline setting by removing the replay regularisation; Energy-based Distribution Matching (EDM) (Jarrett et al., 2020), the state-of-the-art in offline imitation learning; and finally the standard example of Behavioural Cloning (BC). To provide evidence that we are indeed learning an appropriate reward we show an ablation of our method on the MIMIC data: we take the reward learnt by AVRIL and use it as the ‘true’ reward used to train a $Q$ -network offline to learn a policy (A-RL). Note that we have not included previous BIRL methods for the reasons explained in section 2.1, training a network just once in these environments takes in the order of minutes and repeating this sequentially thousands of times is just not practical. For aid in comparison all methods share the same network architecture of two hidden layers of 64 units with ELU activation functions and are trained using Adam (Kingma & Ba, 2014) with learning rates individually tuned. Further details on experimental setup and the implementation of benchmarks can be found in the appendix. + +![](images/99c662ff38503bdc43e89701fae5833c8fdd14b9ef6e894424ecac8256a1baf1.jpg) +Figure 3: Control environments performance. We plot the average returns received by the policies when deployed live in the environment against the number of trajectories seen during training. + +Table 1: Healthcare performance. Comparison of methods on the MIMIC-III dataset. Performance of the policy is evaluated on the quality of action matching against a held out test set of demonstrations. We report the accuracy (ACC), area under the receiving operator characteristic curve (AUC) and average precision score (APS). + +
VentilatorVentilator + Antibiotics
MetricACCAUCAPSACCAUCAPS
BC0.873 ± 0.0070.916 ± 0.0020.904±0.0030.700±0.0090.864 ± 0.0030.665 ± 0.009
VDICE0.879 ±0.0020.915 ± 0.0020.904±0.0030.710 ±0.0050.863 ±0.0020.675 ±0.004
RCAL0.870 ± 0.0120.916 ± 0.0030.904±0.0050.702 ±0.0080.865 ± 0.0040.669 ± 0.006
DSFN0.869 ± 0.0050.905 ± 0.0030.885 ±0.0010.683 ± 0.0070.856 ±0.0020.670 ± 0.004
EDM0.882 ± 0.0110.920±0.0020.909 ± 0.0030.716 ±0.0080.873±0.0020.682 ± 0.004
A-RL10.875±0.0100.904±0.0020.927±0.0020.718 ± 0.0100.864±0.0020.665±0.005
AVRIL0.891±0.002 0.917 ± 0.0010.940±0.0010.754± 0.001 0.884 ± 0.000 0.708 ± 0.002
+ +Evaluation. We see for all tasks AVRIL learns an appropriate policy that performs strongly across the board, being competitive in all cases and in places beating out all of the other benchmarks. The results for our healthcare example are given in table 1, with AVRIL performing very strongly, having the highest accuracy and precision score in both tasks. The results for the control environments are shown in figure 3. AVRIL performs competitively and is easily capable of reaching demonstrator level performance in the samples given for these tasks, though not always as quickly as some of the dedicated offline IL methods. + +Reward insight. Remember though that task performance is not exactly our goal. Rather the key aspect of AVRIL is the inference over the unseen reward in order to gain information about the preferences of the agent that other black-box policy methods can’t. In the previous experiments our reward encoder was a neural network for maximum flexibility and we can see from the performance of A-RL we learn a representation of the reward that can be used to relearn in the environment very effectively, albeit not quite to the same standard of AVRIL. Note this also reflects an original motivation for AVRIL in that offpolicy RL on top of a learnt reward suffers. In figure 4 we explore how to gain more insight from the learnt reward using different parameterisations of the reward. The top graph shows how a learnt state-action reward changes as a function of blood-oxygen level for an otherwise healthy patient, and it can be seen that as it drops below average the reward for ventilating the patient becomes much higher (note this is average for patients in the ICU, not across the general population). While this is intuitive we still have to query a neural network repeatedly over the state space to gain insight, the bottom graph of figure 4 presents then a simpler but perhaps more useful representation. In this case we learn a state-only reward as before but as a linear model. This is not as strong a constraint on the policy since that is still free to be non-linear as a neural network but simultaneously allows us the insight of what our model considers high value in the environment as we plot the relative model coefficients for each covariate. We can see here for example that the biggest impact on the overall estimated quality of a state is given by blood pressure, well known as an important indicator of health (Hepworth et al., 1994), strongly impacted by trauma and infection. + +![](images/ca7c32b8888060a1a5cb258f469a662fd368830830a97e61c43ea9064d01b67d.jpg) +Figure 5: Gridworld example. Scaled heat-maps of: the ground truth reward; the relative state occupancy of the expert demonstrations; the reward posterior mean; and reward standard deviation. + +Gridworld ground-truth comparison While environments like MIMIC are the main focus of this work they do not lend them selves to inspection of the uncovered reward as the ground truth simply is not available to us. We thus demonstrate on a toy gridworld environment, in order to clearly see the effect of learning a posterior distribution over the reward. In this (finite) example both the encoder and decoder are represented by tensors but otherwise the procedure remains the same. Figure 5 plots scaled heat-maps of: a) the ground truth reward; b) the relative state occupancy of the expert demonstrations, obtained using value-iteration; c) the reward posterior mean; and d) the reward standard deviation. The interesting thing to note is that the standard deviation of the learnt reward essentially resembles the complement of the state occupancy - revealing the epistemic uncertainty around that part of the state-space given we haven’t seen any demonstrations there. + +# 5 CONCLUSIONS + +We have presented a novel algorithm, Approximate Variational Reward Imitation Learning, for addressing the scalability issues that prevent current Bayesian IRL methods being used in large and unknown environments. We show that this performs strongly on real and toy data for learning imitation policies completely offline and importantly recovers a reward that is both effective for retraining policies but also offers useful insight into the preferences of the demonstrator. Of course this still represents an approximation, and there is room for further, more exact methods or else guarantees on the maximum divergence. We have focused on simply obtaining the appropriate uncertainty over reward as well as imitation in high stakes environments - in these settings it is crucial that learnt policies avoid catastrophic failure and so how exactly to use the uncertainty in order to achieve truly safe imitation (or indeed better-that-demonstrator apprenticeship) is increasingly of interest. + +![](images/10c9875b3e37b7a44561ac5529ce0724c415439487c6fd0baa51655f4911daf4.jpg) +Figure 4: (Top) A state-action reward is learnt and plotted for an otherwise average patient as their blood oxygen level changes. (Bottom) The associated weights given a state-only reward as a linear function of the state-space. + +# ACKNOWLEDGEMENTS + +AJC would like to acknowledge and thank Microsoft Research for its support through its PhD Scholarship Program with the EPSRC. This work was additionally supported by the Office of Naval Research (ONR) and the NSF (Grant number: 1722516). We would like to thank all of the anonymous reviewers on OpenReview, alongside the many members of the van der Schaar lab, for their input, comments, and suggestions at various stages that have ultimately improved the manuscript. + +# REFERENCES + +Pieter Abbeel and Andrew $\mathrm { ~ Y ~ N ~ g ~ }$ . Apprenticeship learning via inverse reinforcement learning. In Proceedings of the twenty-first international conference on Machine learning, pp. 1, 2004. + +Michael Bain and Claude Sammut. A framework for behavioural cloning. In Machine Intelligence 15, pp. 103–129, 1995. + +David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians. Journal of the American statistical Association, 112(518):859–877, 2017. + +Michael Bloem and Nicholas Bambos. Infinite time horizon maximum causal entropy inverse reinforcement learning. In 53rd IEEE Conference on Decision and Control, pp. 4911–4916. 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In Proceedings of the 2014 international conference on Autonomous agents and multi-agent systems, pp. 1249–1256. International Foundation for Autonomous Agents and Multiagent Systems, 2014. + +Antonin Raffin. Rl baselines zoo. https://github.com/araffin/rl-baselines-zoo, 2018. + +Deepak Ramachandran and Eyal Amir. Bayesian inverse reinforcement learning. In IJCAI, volume 7, pp. 2586–2591, 2007. + +Carl Edward Rasmussen. Gaussian processes in machine learning. In Summer School on Machine Learning, pp. 63–71. Springer, 2003. + +Siddharth Reddy, Anca D Dragan, and Sergey Levine. Sqil: Imitation learning via reinforcement learning with sparse rewards. In International Conference on Learning Representations, 2019. + +Constantin A Rothkopf and Christos Dimitrakakis. Preference elicitation and inverse reinforcement learning. In Joint European conference on machine learning and knowledge discovery in databases, pp. 34–48. Springer, 2011. + +John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017. + +Brian D Ziebart. Modeling Purposeful Adaptive Behavior with the Principle of Maximum Causal Entropy. PhD thesis, University of Washington, 2010. + +Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, and Anind K Dey. Maximum entropy inverse reinforcement learning. In Aaai, volume 8, pp. 1433–1438. Chicago, IL, USA, 2008. + +# A EXPERIMENTAL SETUP + +Expert Demonstrators. Demonstrations are produced by running pre-trained and hyperparmeteroptimised agents taken from the RL Baselines Zoo (Raffin, 2018) in OpenAI Stable Baselines (Hill et al., 2018). For Acrobot and LunarLander these are DQNs (Mnih et al., 2013), while CartPole uses PPO2 (Schulman et al., 2017). Trajectories were then sub-sampled for every 20th step in Acrobot and CartPole, and every 5th step in LunarLander. + +Testing setup. For control environments algorithms were presented with (1,3,7,10,15) trajectories uniformly sampled from a pool of 1000 expert trajectories. Each algorithm was then trained until convergence and tested by performing 300 live roll-outs in the simulated environment and recording the average accumulated reward received in each episode. This whole process was then repeated 10 times, consequently with different initialisations and seen trajectories. + +Implementations. All methods are neural network based and so in experiments they share the same architecture of 2 hidden layers of 64 units each connected by exponential linear unit (ELU) activation functions. + +Publicly available code was used in the implementations of a number of the benchmarks, specifically: + +• VDICE (Kostrikov et al., 2019): https://github.com/google-research/google-research/tree/ master/value_dice +• DSFN (Lee et al., 2019): https://github.com/dtak/batch-apprenticeship-learning +• EDM (Jarrett et al., 2020): https://github.com/wgrathwohl/JEM + +Note that VDICE was originally designed for continuous actions with a Normal distribution output which we adapt for the experiments by replacing with a Gumbel-softmax. + +# B PROOFS + +Proof of proposition 1. Assuming the constraint is satisfied, we are maximising the following objective: + +$$ +\mathcal { F } ( \phi , \theta ) = \sum _ { ( s , a , s ^ { \prime } , a ^ { \prime } ) \in \mathcal { D } } \log \frac { \exp \beta Q _ { \theta } ( s , a ) ) } { \sum _ { b \in \mathcal { A } } \exp ( \beta Q _ { \theta } ( s , b ) ) } - D _ { K L } \big ( q _ { \phi } ( R ( s , a ) ) | | p ( R ( s , a ) ) \big ) +$$ + +Which is equivalent to minimising the negative value + +$$ +\mathcal { F } ( \phi , \theta ) = \sum _ { ( s , a , s ^ { \prime } , a ^ { \prime } ) \in \mathcal { D } } - \log \frac { \exp \beta Q _ { \theta } ( s , a ) ) } { \sum _ { b \in \mathcal { A } } \exp ( \beta Q _ { \theta } ( s , b ) ) } + \underbrace { D _ { K L } \big ( q _ { \phi } ( R ( s , a ) ) | | p ( R ( s , a ) ) \big ) } _ { \mathcal { L } _ { r e g } } , +$$ + +with the first term $\mathcal { L } _ { B C }$ being the negative log-likelihood of the data and the classic behavioural cloning objective. Now given a standard Gaussian prior then the $\mathrm { K L }$ divergence of a Gaussian with mean $\mu$ and variance $\sigma ^ { 2 }$ from the prior is given by $\begin{array} { r } { \frac { 1 } { 2 } ( - \log ( \sigma ^ { 2 } ) + \sigma ^ { 2 } - 1 + \bar { \mu } ^ { 2 } ) } \end{array}$ (Kingma & Welling, + +2013). Then given our prior $p ( R ( s , a ) ) = \mathcal { N } ( R ; 0 , 1 )$ , the KL evaluates as: + +$$ +\begin{array} { l } { { \displaystyle { \mathcal L } _ { r e g } = \sum _ { ( s , a , s ^ { \prime } , a ^ { \prime } ) \in \mathcal D } _ { K L } \big ( q _ { \phi } ( R ( s , a ) ) | | p ( R ( s , a ) ) \big ) } } \\ { { = \sum _ { ( s , a , s ^ { \prime } , a ^ { \prime } ) \in \mathcal D } \frac { 1 } { 2 } \big ( - \log ( \operatorname { V a r } _ { q _ { \phi } } [ R ( s , a ) ] ) + \operatorname { V a r } _ { q _ { \phi } } [ R ( s , a ) ] - 1 + \mathbb { E } _ { q _ { \phi } } [ R ( s , a ) ] ^ { 2 } \big ) } } \\ { { = \sum _ { ( s , a , s ^ { \prime } , a ^ { \prime } ) \in \mathcal D } \frac { 1 } { 2 } \big ( \mathbb { E } _ { q _ { \phi } } [ R ( s , a ) ] ^ { 2 } ) + g ( \operatorname { V a r } _ { q _ { \phi } } [ R ( s , a ) ] ) } } \\ { { = \sum _ { ( s , a , s ^ { \prime } , a ^ { \prime } ) \in \mathcal D } \frac { 1 } { 2 } \big ( Q _ { \theta } ( s , a ) - \gamma Q _ { \theta } ( s ^ { \prime } , a ^ { \prime } ) \big ) ^ { 2 } + g ( \operatorname { V a r } _ { q _ { \phi } } [ R ( s , a ) ] ) } } \end{array} +$$ + +Since by assumption $\mathbb { E } _ { q _ { \phi } } [ R ( s , a ) ] ~ = ~ \mathbb { E } _ { \pi , T } [ Q _ { \theta } ( s , a ) ~ - ~ \gamma Q _ { \theta } ( s ^ { \prime } , a ^ { \prime } ) ]$ with the expectation approximated over samples in the data and considering a definition of the function $g$ to be $\begin{array} { r } { \dot { g } ( \mathrm { V a r } _ { q _ { \phi } } [ R ( s , a ) ] ) = \frac { 1 } { 2 } ( - \log ( \mathrm { V a r } _ { q _ { \phi } } [ R ( s , a ) ] ) + \mathrm { V a r } _ { q _ { \phi } } [ \bar { R } ( s , a ) ] - 1 ) . } \end{array}$ .  \ No newline at end of file diff --git a/parse/train/4qR3coiNaIv/4qR3coiNaIv_content_list.json b/parse/train/4qR3coiNaIv/4qR3coiNaIv_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..61e4807409de298f579eeeed8e8fb0cbad689e1d --- /dev/null +++ b/parse/train/4qR3coiNaIv/4qR3coiNaIv_content_list.json @@ -0,0 +1,1599 @@ +[ + { + "type": "text", + "text": "SCALABLE BAYESIAN INVERSE REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 99, + 821, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Alex J. Chan University of Cambridge, Cambridge, UK alexjchan@maths.cam.ac.uk ", + "bbox": [ + 184, + 170, + 460, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Mihaela van der Schaar \nUniversity of Cambridge, Cambridge, UK University of California, Los Angeles, USA Cambridge Centre for AI in Medicine, UK The Alan Turing Institute, London, UK \nmv472@cam.ac.uk ", + "bbox": [ + 501, + 170, + 789, + 253 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 290, + 544, + 305 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Bayesian inference over the reward presents an ideal solution to the ill-posed nature of the inverse reinforcement learning problem. Unfortunately current methods generally do not scale well beyond the small tabular setting due to the need for an inner-loop MDP solver, and even non-Bayesian methods that do themselves scale often require extensive interaction with the environment to perform well, being inappropriate for high stakes or costly applications such as healthcare. In this paper we introduce our method, Approximate Variational Reward Imitation Learning (AVRIL), that addresses both of these issues by jointly learning an approximate posterior distribution over the reward that scales to arbitrarily complicated state spaces alongside an appropriate policy in a completely offline manner through a variational approach to said latent reward. Applying our method to real medical data alongside classic control simulations, we demonstrate Bayesian reward inference in environments beyond the scope of current methods, as well as task performance competitive with focused offline imitation learning algorithms. ", + "bbox": [ + 233, + 320, + 764, + 513 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 539, + 336, + 554 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "For applications in complicated and high-stakes environments it can often mean operating in the minimal possible setting - that is with no access to knowledge of the environment dynamics nor intrinsic reward, nor even the ability to interact and test policies. In this case learning and inference must be done solely on the basis of logged trajectories from a competent demonstrator showing only the states visited and the the action taken in each case. ", + "bbox": [ + 174, + 569, + 823, + 638 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Clinical decision making is an important example of this, where there is great interest in learning policies from medical professionals but is completely impractical and unethical to deploy policies on patients mid-training. Moreover this is an area where it is not only the policies, but also knowledge of the demonstrator’s preferences and goals, that we are interested in. While imitation learning (IL) generally deals with the problem of producing appropriate policies to match a demonstrator, with the added layer of understanding motivations this would then usually be approached through inverse reinforcement learning (IRL). Here attempting to learn the assumed underlying reward driving the demonstrator, before secondarily learning a policy that is optimal with respect to the reward using some forward reinforcement learning (RL) technique. By composing the RL and IRL procedures in order to perform IL we arrive at apprenticeship learning (AL), which introduces its own challenges, particularly in the offline setting. Notably for any given set of demonstrations there are (infinitely) many rewards for which the actions would be optimal $\\mathrm { N g }$ et al., 2000). Max-margin (Abbeel & $\\mathrm { N g }$ , 2004) and max-entropy (Ziebart et al., 2008) methods for heuristically differentiating plausible rewards do so at the cost of potentially dismissing the true reward for not possessing desirable qualities. On the other hand a Bayesian approach to IRL (BIRL) is more conceptually satisfying, taking a probabilistic view of the reward, we are interested in the posterior distribution having seen the demonstrations (Ramachandran & Amir, 2007), accounting for all possibilities. BIRL is not without its own drawbacks though, as noted in Brown & Niekum (2019), making it inappropriate for modern complicated environments: assuming linear rewards; small, solvable environments; and repeated, inner-loop, calls to forward RL. ", + "bbox": [ + 174, + 646, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/d9cfdc5b1bc6f3e16d989317572e562d90d7f13f159f7755e10beb37ec13d22b.jpg", + "image_caption": [ + "Figure 1: Overview. AVRIL is a framework for BIRL that works through an approximation in the variational Bayesian framework, considering the reward to be a latent representation of behaviour. A distribution over the reward, which is amortised over the demonstration space, is learnt that then informs an imitator $Q$ -function policy. The dotted line represents a departure from a traditional auto-encoder as the input, alongside the latent reward, informs the decoder. " + ], + "image_footnote": [], + "bbox": [ + 186, + 98, + 797, + 234 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The main contribution then of this paper is a method for advancing BIRL beyond these obstacles, allowing for approximate reward inference using an arbitrarily flexible class of functions, in any environment, without costly inner-loop operations, and importantly entirely offline. This leads to our algorithm AVRIL, depicted in figure 1, which represents a framework for jointly learning a variational posterior distribution over the reward alongside an imitator policy in an auto-encoderesque manner. In what follows we review the modern methods for offline IRL/IL (Section 2) with a focus on the approach of Bayesian IRL and the issues it faces when confronted with challenging environments. We then address the above issues by introducing our contributions (Section 3), and demonstrate the gains of our algorithm in real medical data and simulated control environments, notably that it is now possible to achieve Bayesian reward inference in such settings (Section 4). Finally we wrap up with some concluding thoughts and directions (Section 5). Code for AVRIL and our experiments is made available at https://github.com/XanderJC/scalable-birl and https://github.com/vanderschaarlab/mlforhealthlabpub. ", + "bbox": [ + 173, + 325, + 825, + 506 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 APPROACHING APPRENTICESHIP AND IMITATION OFFLINE", + "text_level": 1, + "bbox": [ + 176, + 529, + 691, + 545 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Preliminaries. We consider the standard Markov decision process (MDP) environment, with states $s \\in \\mathcal { S }$ , actions $a \\in \\mathcal A$ , transitions $T \\in \\Delta ( { \\cal S } ) ^ { s \\times { \\cal A } }$ , rewards $\\vec { R } \\in \\mathbb { R } ^ { S \\times A 1 }$ , and discount $\\gamma \\in [ 0 , 1 ]$ . For a policy $\\pi \\in \\Delta ( \\mathcal { A } ) ^ { s }$ let $\\begin{array} { r } { \\rho _ { \\pi } ( s , a ) = \\mathbb { E } _ { \\pi , T } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\mathbb { 1 } _ { \\{ s _ { t } = s , a _ { t } = a \\} } ] } \\end{array}$ be the induced unique occupancy measure alongside the state-only occupancy measure $\\begin{array} { r } { \\rho _ { \\pi } ( s ) = \\sum _ { a \\in \\mathcal { A } } \\rho _ { \\pi } ( s , a ) } \\end{array}$ . Despite this full environment model, the only information available to us is the $\\mathbf { M D P } \\backslash R T$ , in that we have no access to either the underlying reward or the transitions, with our lacking knowledge of the transitions being also strong in the sense that further we are unable to simulate the environment to sample them. The learning signal is then given by access to $m$ -many trajectories of some demonstrator assumed to be acting optimally w.r.t. the MDP, following a policy $\\pi _ { D }$ , making up a data set $\\mathcal { D } _ { r a w } = \\{ ( s _ { 1 } ^ { ( i ) } , a _ { 1 } ^ { ( i ) } , \\dotsc , s _ { \\tau ^ { ( i ) } } ^ { ( \\bar { i } ) } , a _ { \\tau ^ { ( i ) } } ^ { ( i ) } ) \\} _ { i = 1 } ^ { m }$ where $s _ { t } ^ { ( i ) }$ is the state and $a _ { t } ^ { ( i ) }$ is the action taken at step $t$ during the $i$ th demonstration, and $\\tau ^ { ( i ) }$ is the (max) time horizon of the ith demonstration. Given the Markov assumption though it is sufficient and convenient to consider the demonstrations simply as a collection of $n$ -many state, action, next state, next action tuples such that $\\mathcal { D } = \\{ ( s _ { i } , a _ { i } , s _ { i } ^ { \\prime } , a _ { i } ^ { \\prime } ) \\} _ { i = 1 } ^ { n }$ with $\\begin{array} { r } { n = \\sum _ { i = 1 } ^ { m } ( \\tau ^ { ( i ) } - 1 ) } \\end{array}$ . ", + "bbox": [ + 173, + 561, + 825, + 768 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Apprenticeship through rewards. Typically AL proceeds by first inferring an appropriate reward function with an IRL procedure $\\mathrm { N g }$ et al., 2000; Ramachandran & Amir, 2007; Rothkopf & Dimitrakakis, 2011; Ziebart et al., 2008) before running forward RL to obtain an appropriate policy. This allows for easy mix-and-match procedures, swapping in different standard RL and IRL methods depending on the situation. These algorithms though depend on either knowledge of $T$ in order to solve exactly or the ability to perform roll-outs in the environment, with little previous work focusing on the entirely offline setting. One simple solution is through attempting to learn the dynamics (Herman et al., 2016), though without a large supply of diverse demonstrations or a small environment this becomes impractical given imperfections in the model. Alternatively Klein et al. (2011) and Lee et al. (2019) attempt off-policy feature matching through least-squared temporal difference and deep neural networks to uncover appropriate feature representations. ", + "bbox": [ + 174, + 785, + 825, + 882 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Implicit-reward policy learning. Recent work has often forgone an explicit representation of the reward. Moving within the maximum-entropy RL framework (Ziebart, 2010; Levine, 2018), Ho & Ermon (2016) noted that the full procedure ${ \\mathrm { ~ R L ~ o ~ I R L } }$ ) can be interpreted equivalently as the minimisation of some divergence between occupancy measures of the imitator and demonstrator: ", + "bbox": [ + 173, + 174, + 825, + 229 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/870742468cae69ab150cfe440ab1ebd1f7bf665c0794f41299537990da853ad6.jpg", + "text": "$$\n\\underset { \\pi } { \\arg \\operatorname* { m i n } } \\{ \\psi ^ { * } ( \\rho _ { \\pi } - \\rho _ { \\pi _ { D } } ) - H ( \\pi ) \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 232, + 612, + 257 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "with $H ( \\pi )$ being the discounted causal entropy (Bloem & Bambos, 2014) of the policy and $\\psi ^ { * }$ the Fenchel conjugate of a chosen regulariser on the form of the reward. These are typically optimised in an adversarial fashion (Goodfellow et al., 2014) and given the focus on evaluating $\\rho _ { \\pi }$ this often requires extensive interaction with the environment, otherwise banking on approximations over a replay buffer (Kostrikov et al., 2018) or a reformulation of the divergence to allow for off-policy evaluation (Kostrikov et al., 2019). Bear in mind that optimal policies within the maximum-entropy framework are parameterised by a Boltzmann distribution: ", + "bbox": [ + 173, + 258, + 825, + 356 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0c74caa5f9bc1d89acad1468e1f149a3ed4c26b48ade8e568c23ddc47fd8e772.jpg", + "text": "$$\n\\pi ( a | s ) = \\frac { \\exp ( Q ( s , a ) ) } { \\sum _ { b \\in \\cal { A } } \\exp ( Q ( s , b ) ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 393, + 358, + 602, + 393 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "with $Q ( s , a )$ the soft $Q$ -function, defined recursively via the soft Bellman-equation: ", + "bbox": [ + 176, + 395, + 718, + 410 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0aa78fb99ec235065b0573f570f97b7482c82125adfc93974b6d64143c688551.jpg", + "text": "$$\nQ ( s , a ) \\triangleq R ( s , a ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim \\rho _ { \\pi } } \\Big [ \\operatorname { s o f t } _ { a ^ { \\prime } } \\operatorname { m a x } Q ( s ^ { \\prime } , a ^ { \\prime } ) ) \\Big ] .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 411, + 671, + 439 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Then for a learnt parameterised policy given in terms of $Q$ -values from a function approximator $Q _ { \\theta }$ we can obtain an implied reward given by: ", + "bbox": [ + 178, + 440, + 820, + 469 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1d7ed7ec38bc7068c7c374fe0563075cd630f77ba98fea016777db39e6364b88.jpg", + "text": "$$\nR _ { Q _ { \\theta } } ( s , a ) = Q _ { \\theta } ( s , a ) - \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim \\rho _ { \\pi } } \\Bigg [ \\log \\Bigg ( \\sum _ { a ^ { \\prime } \\in \\mathcal { A } } \\exp ( Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) ) \\Bigg ) \\Bigg ] .\n$$", + "text_format": "latex", + "bbox": [ + 279, + 470, + 718, + 513 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A number of algorithms make use of this fact with Piot et al. (2014) and Reddy et al. (2019) working by essentially placing a sparsity prior on this implied reward, encouraging it towards zero, and thus incorporating subsequent state information. Alternatively Jarrett et al. (2020) show that even the simple behavioural cloning (Bain & Sammut, 1995) is implicitly maximising some reward with an approximation that the expectation over states is taken with respect to the demonstrator, not the learnt policy. They then attempt to rectify part of this approximation using the properties of the energy-based model implied by the policy (Grathwohl et al., 2019). ", + "bbox": [ + 174, + 515, + 825, + 613 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The problem with learning an implicit reward in an offline setting is that it remains just that, implicit, only able to be evaluated at points seen in the demonstrations, and even then only approximately. Thus even if their consideration improves imitator policies performance they offer no real improvement for interpretation. ", + "bbox": [ + 174, + 619, + 823, + 676 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 BAYESIAN INVERSE REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 173, + 691, + 555, + 705 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We are then resigned to directly reason about the underlying reward, bringing us back to the question of IRL, and in particular BIRL for a principled approach to reasoning under uncertainty. Given a prior over possible functions, having seen some demonstrations, we calculate the posterior over the function using a theoretically simple application of Bayes rule. Ramachandran $\\&$ Amir (2007) defines the likelihood of an action at a state as a Boltzmann distribution with inverse temperature and respective state-action values, yielding a probabilistic demonstrator policy given by: ", + "bbox": [ + 173, + 717, + 825, + 801 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/eaead0b2596f07403155ca11eab69b0df05df986ed1d025b26750cb7161f5e84.jpg", + "text": "$$\n\\pi _ { D } ( a | s , R ) = \\frac { \\exp ( \\beta Q _ { R } ^ { \\pi _ { D } } ( s , a ) ) } { \\sum _ { b \\in \\cal { A } } \\exp ( \\beta Q _ { R } ^ { \\pi _ { D } } ( s , b ) ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 803, + 632, + 838 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\beta \\in [ 0 , \\infty )$ represents the the confidence in the optimality of the demonstrator. Note that despite similarities, moving forward we are no longer within the maximum-entropy framework and $Q _ { R } ^ { \\pi } ( s , a )$ now denotes the traditional, not soft (as in equation 3), state-action value ( $Q$ -value) funcRtion given a reward $R$ and policy $\\pi$ such that $\\begin{array} { r } { Q _ { R } ^ { \\pi } ( s , \\bar { a } ) = \\mathbb { E } _ { \\pi , T } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } R ( s _ { t } ) | s _ { 0 } = s , a _ { 0 } = a ] } \\end{array}$ Unsurprisingly this yields an intractable posterior distribution leading to a Markov chain Monte Carlo (MCMC) algorithm based on a random grid-walk to sample from the posterior. ", + "bbox": [ + 173, + 840, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Issues in complex and unknown environments. This original formulation, alongside extensions that consider maximum-a-posteriori inference (Choi & Kim, 2011) and multiple rewards (Choi & Kim, 2012; Dimitrakakis & Rothkopf, 2011), suffer from three major drawbacks that make them impractical for modern, complicated, and model-free task environments. ", + "bbox": [ + 176, + 103, + 825, + 160 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1. The reward is a linear combination of state features. Naturally this is a very restrictive class of functions and assumes access to carefully hand-crafted features of the state space. 2. The cardinality of the state-space is finite, $| S | < \\infty$ . Admittedly this can be relaxed in practical terms, although it does mean the rapid-mixing bounds derived by Ramachandran & Amir (2007) do not hold at all in the infinite case. For finite approximations they scale at $\\mathcal { O } ( | S | ^ { 2 } )$ , rapidly becoming vacuous and causing BIRL to inherit the usual MCMC difficulties on assessing convergence and sequential computation (Gamerman & Lopes, 2006). 3. The requirement of an inner-loop MDP solve. Most importantly at every step a new reward is sampled and the likelihood of the data must then be evaluated. This requires calculating the $Q$ -values of the policy with respect to the reward, in other words running forward RL. While not an insurmountable problem in the simple cases where everything is known and can be quickly solved with a procedure guaranteed to converge correctly, this becomes an issue in the realm where only deep function approximation works adequately (i.e. the nontabular setting). DQN training for example easily stretches into hours (Mnih et al., 2013) and will have to be repeated thousands of times, making it completely untenable. ", + "bbox": [ + 210, + 171, + 825, + 388 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We have seen that even in the most simple setting the problem of exact Bayesian inference over the reward is intractable, and the above limitations of the current MCMC methods are not trivial to overcome. Consequently very little work has been done in the area and there still remain very open challenges. Levine et al. (2011) addressed linearity through a Gaussian process approach, allowing for a significantly more flexible and non-linear representation though introducing issues of its own, namely the computational complexity of inverting large matrices (Rasmussen, 2003). More recently Brown & Niekum (2019) have presented the only current solution to the inner-loop problem by introducing an alternative formulation of the likelihood, one based on human recorded pairwise preferences over demonstrations that significantly reduces the complexity of likelihood. However labelled preferences certainly can’t be assumed always available and while very effective for the given task is not appropriate in the general case. One of the key aspects of our contribution is that we are able to deal with all three of these issues while also not requiring any additional information. ", + "bbox": [ + 173, + 400, + 825, + 565 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The usefulness of uncertainty. On top of the philosophical consistency of Bayesian inference there are a number of reasons for wanting a measure of uncertainty over any uncovered reward that are not available from regular IRL algorithms. First that the (epistemic) uncertainty revealed by Bayesian inference tells us a lot about what areas of the state-space we really cannot say anything about because we haven’t seen any demonstrations there - potentially informing future data collection if that is possible (Mindermann et al., 2018). Additionally in the cases we are mostly concerned about (e.g. medicine) we have to be very careful about letting algorithms pick actions in practice and we are interested in performing safe or risk-averse imitation, for which a degree of confidence over learnt rewards is necessary. Brown et al. (2020) for example use a distribution over reward to optimise a conditional value-at-risk instead of expected return so as to bound potential downsides. ", + "bbox": [ + 173, + 580, + 825, + 719 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 APPROXIMATE VARIATIONAL REWARD IMITATION LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 739, + 717, + 756 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A variational Bayesian approach. In this section we detail our method, AVRIL, for efficiently learning an imitator policy and performing reward inference simultaneously. Unlike the previously mentioned sampling or MAP-based methods, we employ variational inference (Blei et al., 2017) to reason about the posterior. Here we posit a surrogate distribution $q _ { \\phi } ( R )$ , parameterised by $\\phi$ , and aim to minimise the Kullback-Leibler (KL) divergence to the posterior, resulting in an objective: ", + "bbox": [ + 174, + 770, + 825, + 840 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/469374a8cd250696c803b1c0a945b8a40c02cba5529457319872279071ab9709.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\phi } \\{ D _ { \\mathrm { K L } } ( q _ { \\phi } ( R ) | | p ( R | \\mathcal { D } ) ) \\} .\n$$", + "text_format": "latex", + "bbox": [ + 398, + 843, + 598, + 868 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This divergence is still as troubling as the posterior to evaluate, leading to an auxiliary objective function in the Evidence Lower BOund (ELBO): ", + "bbox": [ + 174, + 872, + 826, + 900 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/722a4c9f996fb1f0dcf1ce1851d1761a6df383a55436cc191dd6fbada90e1fd0.jpg", + "text": "$$\n\\mathcal { F } ( \\phi ) = \\mathbb { E } _ { q _ { \\phi } } \\big [ \\log p ( \\mathcal { D } | R ) \\big ] - D _ { K L } \\big ( q _ { \\phi } ( R ) | | p ( R ) \\big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 330, + 902, + 666, + 922 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where it can be seen that maximisation over $\\phi$ is equivalent to (6). We are agnostic towards the form of both the prior and variational distribution, for simplicity here we assume a Gaussian process prior with mean zero and unit variance over $R$ alongside the variational posterior distribution given by: ", + "bbox": [ + 173, + 103, + 826, + 146 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1532b16207f68ec3cfe9b06cfe011627db774328a75ce3351dc6541f8afdee54.jpg", + "text": "$$\nq _ { \\phi } ( R ) = \\mathcal { N } ( R ; \\mu , \\sigma ^ { 2 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 421, + 151, + 575, + 170 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mu , \\sigma ^ { 2 }$ are the outputs of an encoder neural network taking $s$ as input and parameterised by $\\phi$ . Note that for the algorithm that we will describe these choices are not a necessity and can be easily substituted for more expressive distributions if appropriate. Maintaining the assumption of Boltzmann rationality on the part of the demonstrator, our objective takes the form: ", + "bbox": [ + 174, + 178, + 825, + 234 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f161d3dcacc8d099ea4aae0724813e18fd50a03c53b7ef7c02012c5149826eb6.jpg", + "text": "$$\n\\mathcal { F } ( \\phi ) = \\mathbb { E } _ { q _ { \\phi } } \\left[ \\sum _ { ( s , a ) \\in \\mathcal { D } } \\log \\frac { \\exp ( \\beta Q _ { R } ^ { \\pi _ { D } } ( s , a ) ) } { \\sum _ { b \\in \\mathcal { A } } \\exp ( \\beta Q _ { R } ^ { \\pi _ { D } } ( s , b ) ) } \\right] - D _ { K L } \\big ( q _ { \\phi } ( R ) | | p ( R ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 246, + 239, + 750, + 285 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The most interesting (and problematic) part of this objective as ever centres on the evaluation of $Q _ { R } ^ { \\pi _ { D } } ( s , a )$ . Notice that what is really required here is an expression of the $Q$ -values as a smooth function of the reward such that with samples of $R$ we could take gradients w.r.t. $\\phi$ . Of course there is little hope of obtaining this simply, by itself it is a harder problem than that of forward RL which only attempts to evaluate the $Q$ -values for a specific $R$ and already in complicated environments has to rely on function approximation and limited guarantees. ", + "bbox": [ + 173, + 291, + 825, + 376 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "A naive approach would be to sample $\\hat { R }$ and then approximate the $Q$ -values with a second neural network, solving offline over the batched data using a least-squared $\\mathrm { T D } / Q$ -learning algorithm, as is the approach forced on sampling based BIRL methods. It is in fact though doubly inappropriate for this setting, not only does this require a solve as an inner-loop but importantly differentiating through the solving operation is extremely impractical, it requires backpropagating through a number of gradient updates that are essentially unbounded as the complexity of the environment increases. ", + "bbox": [ + 173, + 383, + 825, + 468 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "A further approximation. This raises an important question - is it possible to jointly optimise a policy and variational distribution only once instead of requiring a repeated solve? This is theoretically suspect, the $Q$ -values are defined on a singular reward, constrained as $R ( s , a ) \\ =$ $\\mathbb { E } _ { s ^ { \\prime } , a ^ { \\prime } \\sim \\pi , T } [ \\bar { Q } _ { R } ^ { \\pi } ( { \\bar { s , } } a ) - \\gamma Q _ { R } ^ { \\pi } ( s ^ { \\prime } , a ^ { \\prime } ) ]$ so we cannot learn a particular standard $Q$ -function that reflects the entire distribution. But can we learn a policy that reflects the expected reward using a second policy neural network $Q _ { \\theta } ?$ We can’t simply optimise $\\theta$ alongside $\\phi$ to maximise the ELBO though as that completely ignores the fact that the learnt policy is intimately related to the distribution over the reward. Our solution to ensure then that they behave as intended is by constraining $q _ { \\phi }$ and $Q _ { \\theta }$ to be consistent with each other, specifically that the implied reward of the policy is sufficiently likely under the variational posterior (equivalently that the negative log-likelihood is sufficiently low). Thus we arrive at a constrained optimisation objective given by: ", + "bbox": [ + 173, + 483, + 825, + 637 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ab6bb01b3d406269eacf9275cd5158b503c48ae031c27f3ab6ca365465af1c90.jpg", + "text": "$$\n\\operatorname* { m a x } _ { \\phi , \\theta } \\sum _ { ( s , a ) \\in \\mathcal { D } } \\log \\frac { \\exp ( \\beta Q _ { \\theta } ( s , a ) ) } { \\sum _ { b \\in \\mathcal { A } } \\exp ( \\beta Q _ { \\theta } ( s , b ) ) } - D _ { K L } \\big ( q _ { \\phi } ( R ) | | p ( R ) \\big ) ,\n$$", + "text_format": "latex", + "bbox": [ + 285, + 642, + 710, + 684 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "with $\\epsilon$ reflecting the strength of the constraint. Rewriting (10) as a Lagrangian under the KKT conditions (Karush, 1939; Kuhn $\\&$ Tucker, 1951), and given complimentary slackness, we obtain a practical objective function: ", + "bbox": [ + 176, + 708, + 820, + 751 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2d42ccb3925545343a792675f7aacd47709b0e80950a893270372c9b3f1092f8.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathcal { F } ( \\phi , \\theta , \\mathcal { D } ) = \\displaystyle \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal { D } } \\log \\frac { \\exp \\beta Q _ { \\theta } ( s , a ) ) } { \\sum _ { b \\in A } \\exp ( \\beta Q _ { \\theta } ( s , b ) ) } - D _ { K L } \\big ( q _ { \\phi } ( R ( s , a ) ) | | p ( R ( s , a ) ) \\big ) } & { } \\\\ { + \\lambda \\log q _ { \\phi } ( Q _ { \\theta } ( s , a ) - \\gamma Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) ) . } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 209, + 757, + 787, + 820 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Here the KL divergence between processes is approximated over a countable set, and $\\lambda$ is introduced to control the strength of constraint. ", + "bbox": [ + 173, + 824, + 826, + 853 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "On the implementation. Optimisation is simple as both networks are maximising the same objective and gradients can be easily obtained through backpropagation while being amenable to minibatching, allowing you to call your favourite gradient-based stochastic optimisation scheme. We re-iterate though that AVRIL really represents a framework for doing BIRL and not a specific model ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1: Approximate Variational Reward Imitation Learning (AVRIL) ", + "text_level": 1, + "bbox": [ + 178, + 107, + 673, + 122 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Result: Parameters $\\phi$ of variational distribution and $\\theta$ of policy Q-function \nInput: $\\mathcal { D } , S , A , \\gamma , \\lambda$ , learning rate $\\eta$ , mini-batch size $b$ ; \nInitialise $\\phi , \\theta$ ; $\\triangleright$ Can concatenate into single vector $( \\phi , \\theta )$ \nwhile not converged do Sample $\\mathcal { D } _ { m i n i }$ from $\\mathcal { D }$ ; $\\begin{array} { r } { \\mathcal { F } ( \\phi , \\theta , \\mathcal { D } ) = \\mathbb { E } [ \\frac { n } { b } \\mathcal { F } ( \\phi , \\theta , \\mathcal { D } _ { m i n i } ) ] } \\end{array}$ ; . MC estimate total loss $( \\phi ^ { \\prime } , \\theta ^ { \\prime } ) ( \\phi , \\theta ) + \\eta \\nabla _ { \\phi , \\theta } \\mathcal { F } ( \\phi , \\theta , \\mathcal { D } )$ ; . Gradient step for $\\phi , \\theta$ $\\phi , \\theta \\phi ^ { \\prime } , \\theta ^ { \\prime }$ \nend \nReturn: $\\phi , \\theta$ ", + "bbox": [ + 173, + 125, + 807, + 265 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "since $Q _ { \\theta }$ and $q _ { \\phi }$ represent arbitrary function approximators. So far we have presented both as neural networks, but this does not have to be the case. Of course the advantage of them is their flexibility and ease of training but they are still inherently black box. It is then perfectly possible to swap in any particular function approximator if the task requires it, using simple linear models for example may slightly hurt performance but allow for more insight. Despite the specific focus on infinite state-spaces, AVRIL can still even be applied in the tabular setting by simply representing the policy and variational distribution with multi-dimensional tensors. Having settled on their forms, equation (11) is calculated simply and the joint gradient with respect to $\\theta$ and $\\phi$ is straight-forwardly returned using any standard auto-diff package. The whole process is summarised in Algorithm 1. ", + "bbox": [ + 174, + 303, + 825, + 429 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We can now see how AVRIL does not suffer the issues outlined in section 2.1. Our form of $q _ { \\phi } ( R )$ is flexible and easily accommodates a non-linear form of the reward given a neural architecture - this also removes any restriction on $s$ , or at least allows for any state space that is commonly tackled within the IL/RL literature. Additionally we have a single objective for which all parameters are maximised simultaneously - there are no inner-loops, costly or otherwise, meaning training is faster than the MCMC methods by a factor equal roughly to the number of samples they would require. ", + "bbox": [ + 174, + 435, + 825, + 520 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The generative model view. Ultimately a policy represents a generative model for the behavioural data we see. Ho & Ermon (2016) explicitly make use of this fact by casting the problem in the GAN framework (Goodfellow et al., 2014). Our method is more analogous to a VAE (Kingma & Welling, 2013), though not exactly, where given the graphical model in figure 2 the reward can be seen as a latent representation of the policy. Our approach takes the seen data and amortises the inference, encoding over the state space. The policy does not act as a decoder in precisely taking any given encoded reward and outputting a policy, but it does take the whole reward posterior and translate it into actions and therefore behaviour. This approach has its advantages, in both meaningful interpretation of the latent reward (which is non-existent in adversarial methods), and that we forgo the practical difficulties of alternating min-max optimisation (Kodali et al., 2017) while maintaining a generative view of the policy. ", + "bbox": [ + 174, + 541, + 614, + 693 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/787d6d634246cb213461ef2627f55b1feb1315b21e4be42cc9a37cfd4f67f240.jpg", + "image_caption": [ + "Figure 2: Graphical model for Bayesian IRL " + ], + "image_footnote": [], + "bbox": [ + 632, + 544, + 813, + 640 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 694, + 821, + 736 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Temporal consistency through reward regularisation. Considering only the first term of (11) yields the standard behavioural cloning setup (where the logits output can be interpreted as the $Q$ - values) as it removes the reward from the equation and just focuses on matching actions to states. AVRIL can then be seen as a policy-learning method regularised by the need for the implied reward to be consistent. Note that this does not induce any necessary bias since the logits normally contain an extra degree of freedom allowing them to arbitrarily shift by some scale factor. This factor is now explicitly constrained by giving the logits additional meaning in that they represent $Q$ -values. This places great importance on the KL term, since every parameterisation of a policy will have an associated implied reward, the KL regularises these to be not so far from the prior and preventing the reward from overfitting to the policy and becoming pointless. It also is able to double as a regularising term in a similar manor to previous reward-regularisation methods (Piot et al., 2014; Reddy et al., 2019) depending on the chosen prior, encouraging the reward to be close to zero: ", + "bbox": [ + 173, + 757, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 1 (Reward Regularisation) Assume that the constraint in (10) is satisfied in that $\\begin{array} { r } { { \\mathbb E } _ { q _ { \\phi } } [ R ( s , a ) ] = { \\mathbb E } _ { \\pi , T } [ Q _ { \\theta } ( s , a ) - \\gamma Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) ] } \\end{array}$ , then given a standard normal prior $p ( R ) = \\mathcal { N } ( R ; 0 , 1 )$ the $\\mathrm { K L }$ divergence yields a sparsity regulator on the implied reward: ", + "bbox": [ + 171, + 103, + 826, + 146 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/8c1fb4ca7a6b5f8b5fda4b490f7a45c3bb86bcc967bc98e1f7e8f7e2af6b9568.jpg", + "text": "$$\n\\mathcal { L } _ { r e g } = \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal { D } } \\frac { 1 } { 2 } \\big ( Q _ { \\theta } ( s , a ) - \\gamma Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) \\big ) ^ { 2 } + g ( \\mathrm { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] ) .\n$$", + "text_format": "latex", + "bbox": [ + 266, + 152, + 732, + 193 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proof. Appendix. \u0003 This follows immediately from the fact that the divergence evaluates as DK $\\begin{array} { r } { \\overset { \\vartriangle } { \\boldsymbol { \\mathrm { \\iota } } } \\big ( q _ { \\phi } ( \\hat { R } ( s , a ) ) | | p ( R ( s , a ) ) \\big ) = \\frac { 1 } { 2 } ( - \\log ( \\operatorname { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] ) - 1 + \\operatorname { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] + \\mathbb { E } _ { q _ { \\phi } } [ R ( s , a ) ] ^ { 2 } ) } \\end{array}$ ", + "bbox": [ + 174, + 200, + 825, + 231 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "This then allows AVRIL to inherit the benefit of these methods while also explicitly learning a reward that can be queried at any point. We are also allowed the choice of whether it is state-only or state-action. This has so far been arbitrary, but it is important to consider that a state-only reward is a necessary and sufficient condition for a reward that is fully disentangled from the dynamics $\\operatorname { F u }$ et al., 2018). Thus by learning such a reward and given the final term of (11) that directly connects one-step rewards in terms of the policy, this forces the policy (not the reward) to account for the dynamics of the system ensuring temporal consistency in a way that BC for example simply can’t. Alternatively using a state-action reward means that inevitably some of the temporal information leaks out of the policy and into the reward - ultimately to the detriment of the policy but potentially allowing for a more interpretable (or useful) form of reward depending on the task at hand. ", + "bbox": [ + 174, + 234, + 825, + 375 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 396, + 326, + 411 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Experimental setup. We are primarily concerned with the case of medical environments, which is exactly where the issue of learning without interaction is most crucial, you just cannot let a policy sample treatments for a patient to try to learn more about the dynamics. It is also where a level of interpretability in what has been learnt is important, since the consequence of actions are potentially very impactful on human lives. As such we focus our evaluation on learning on a real-life healthcare problem, with demonstrations taken from the Medical Information Mart for Intensive Care (MIMIC-III) dataset (Johnson et al., 2016). The data contains trajectories of patients in intensive care recording their condition and theraputic interventions at one day intervals. We evaluate the ability of the methods to learn a medical policy in both the two and four action setting - specifically whether the patient should be placed on a ventilator, and the decision for ventilation in combination with antibiotic treatment. These represent the two most common, and important, clinical interventions recorded in the data. Without a recorded notion of reward, performance is measured with respect to action matching against a held out test set of demonstrations with cross-validation. ", + "bbox": [ + 174, + 428, + 825, + 608 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Alongside the healthcare data and for the purposes of demonstrating generalisability, we provide additional results on standard environments of varying complexity in the RL literature, the standard control problems of: CartPole, a classic control environment aiming to swing up and balance a pendulum; Acrobot, which aims to maintain a sequence of joints above a given height; and LunarLander, guiding a landing module to a safe touchdown on the moon surface. In these settings given sufficient demonstration data all benchmarks are very much capable of reaching demonstrator level performance, so we test the algorithms on their ability to handle sample complexity in the low data regime by testing their performance when given access to a select number of trajectories which we adjust, replicating the setup in Jarrett et al. (2020). With access to a simulation through the OpenAI gym (Brockman et al., 2016), we measure performance by deploying the learnt policies live and calculating their average return over 300 episodes. ", + "bbox": [ + 174, + 616, + 825, + 767 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Benchmarks. We test our method (AVRIL) against a number of benchmarks from the offline IRL/IL setting: Deep Successor Feature Network (DSFN) (Lee et al., 2019), an offline adaptation of max-margin IRL that generalises past the linear methods using a deep network with leastsquares temporal-difference learning, the only other method that produces both a reward and policy; Reward-regularized Classification for Apprenticeship Learning (RCAL) (Piot et al., 2014), where an explicit regulariser on the sparsity of the implied reward is introduced in order to account for the dynamics information; ValueDICE (VDICE) (Kostrikov et al., 2019), an adversarial imitation learning, adapted for the offline setting by removing the replay regularisation; Energy-based Distribution Matching (EDM) (Jarrett et al., 2020), the state-of-the-art in offline imitation learning; and finally the standard example of Behavioural Cloning (BC). To provide evidence that we are indeed learning an appropriate reward we show an ablation of our method on the MIMIC data: we take the reward learnt by AVRIL and use it as the ‘true’ reward used to train a $Q$ -network offline to learn a policy (A-RL). Note that we have not included previous BIRL methods for the reasons explained in section 2.1, training a network just once in these environments takes in the order of minutes and repeating this sequentially thousands of times is just not practical. For aid in comparison all methods share the same network architecture of two hidden layers of 64 units with ELU activation functions and are trained using Adam (Kingma & Ba, 2014) with learning rates individually tuned. Further details on experimental setup and the implementation of benchmarks can be found in the appendix. ", + "bbox": [ + 174, + 784, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/99c662ff38503bdc43e89701fae5833c8fdd14b9ef6e894424ecac8256a1baf1.jpg", + "image_caption": [ + "Figure 3: Control environments performance. We plot the average returns received by the policies when deployed live in the environment against the number of trajectories seen during training. " + ], + "image_footnote": [], + "bbox": [ + 176, + 112, + 805, + 229 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/67e92acc9b38e72efd681d6f45c1970e9fcc5cfb81050ddb8eaaeab98d4fc5ca.jpg", + "table_caption": [ + "Table 1: Healthcare performance. Comparison of methods on the MIMIC-III dataset. Performance of the policy is evaluated on the quality of action matching against a held out test set of demonstrations. We report the accuracy (ACC), area under the receiving operator characteristic curve (AUC) and average precision score (APS). " + ], + "table_footnote": [], + "table_body": "
VentilatorVentilator + Antibiotics
MetricACCAUCAPSACCAUCAPS
BC0.873 ± 0.0070.916 ± 0.0020.904±0.0030.700±0.0090.864 ± 0.0030.665 ± 0.009
VDICE0.879 ±0.0020.915 ± 0.0020.904±0.0030.710 ±0.0050.863 ±0.0020.675 ±0.004
RCAL0.870 ± 0.0120.916 ± 0.0030.904±0.0050.702 ±0.0080.865 ± 0.0040.669 ± 0.006
DSFN0.869 ± 0.0050.905 ± 0.0030.885 ±0.0010.683 ± 0.0070.856 ±0.0020.670 ± 0.004
EDM0.882 ± 0.0110.920±0.0020.909 ± 0.0030.716 ±0.0080.873±0.0020.682 ± 0.004
A-RL10.875±0.0100.904±0.0020.927±0.0020.718 ± 0.0100.864±0.0020.665±0.005
AVRIL0.891±0.002 0.917 ± 0.0010.940±0.0010.754± 0.001 0.884 ± 0.000 0.708 ± 0.002
", + "bbox": [ + 174, + 347, + 823, + 489 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 502, + 825, + 616 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Evaluation. We see for all tasks AVRIL learns an appropriate policy that performs strongly across the board, being competitive in all cases and in places beating out all of the other benchmarks. The results for our healthcare example are given in table 1, with AVRIL performing very strongly, having the highest accuracy and precision score in both tasks. The results for the control environments are shown in figure 3. AVRIL performs competitively and is easily capable of reaching demonstrator level performance in the samples given for these tasks, though not always as quickly as some of the dedicated offline IL methods. ", + "bbox": [ + 173, + 630, + 825, + 727 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Reward insight. Remember though that task performance is not exactly our goal. Rather the key aspect of AVRIL is the inference over the unseen reward in order to gain information about the preferences of the agent that other black-box policy methods can’t. In the previous experiments our reward encoder was a neural network for maximum flexibility and we can see from the performance of A-RL we learn a representation of the reward that can be used to relearn in the environment very effectively, albeit not quite to the same standard of AVRIL. Note this also reflects an original motivation for AVRIL in that offpolicy RL on top of a learnt reward suffers. In figure 4 we explore how to gain more insight from the learnt reward using different parameterisations of the reward. The top graph shows how a learnt state-action reward changes as a function of blood-oxygen level for an otherwise healthy patient, and it can be seen that as it drops below average the reward for ventilating the patient becomes much higher (note this is average for patients in the ICU, not across the general population). While this is intuitive we still have to query a neural network repeatedly over the state space to gain insight, the bottom graph of figure 4 presents then a simpler but perhaps more useful representation. In this case we learn a state-only reward as before but as a linear model. This is not as strong a constraint on the policy since that is still free to be non-linear as a neural network but simultaneously allows us the insight of what our model considers high value in the environment as we plot the relative model coefficients for each covariate. We can see here for example that the biggest impact on the overall estimated quality of a state is given by blood pressure, well known as an important indicator of health (Hepworth et al., 1994), strongly impacted by trauma and infection. ", + "bbox": [ + 174, + 743, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/ca7c32b8888060a1a5cb258f469a662fd368830830a97e61c43ea9064d01b67d.jpg", + "image_caption": [ + "Figure 5: Gridworld example. Scaled heat-maps of: the ground truth reward; the relative state occupancy of the expert demonstrations; the reward posterior mean; and reward standard deviation. " + ], + "image_footnote": [], + "bbox": [ + 192, + 97, + 816, + 204 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 258, + 825, + 343 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Gridworld ground-truth comparison While environments like MIMIC are the main focus of this work they do not lend them selves to inspection of the uncovered reward as the ground truth simply is not available to us. We thus demonstrate on a toy gridworld environment, in order to clearly see the effect of learning a posterior distribution over the reward. In this (finite) example both the encoder and decoder are represented by tensors but otherwise the procedure remains the same. Figure 5 plots scaled heat-maps of: a) the ground truth reward; b) the relative state occupancy of the expert demonstrations, obtained using value-iteration; c) the reward posterior mean; and d) the reward standard deviation. The interesting thing to note is that the standard deviation of the learnt reward essentially resembles the complement of the state occupancy - revealing the epistemic uncertainty around that part of the state-space given we haven’t seen any demonstrations there. ", + "bbox": [ + 174, + 359, + 581, + 579 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 602, + 328, + 617 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have presented a novel algorithm, Approximate Variational Reward Imitation Learning, for addressing the scalability issues that prevent current Bayesian IRL methods being used in large and unknown environments. We show that this performs strongly on real and toy data for learning imitation policies completely offline and importantly recovers a reward that is both effective for retraining policies but also offers useful insight into the preferences of the demonstrator. Of course this still represents an approximation, and there is room for further, more exact methods or else guarantees on the maximum divergence. We have focused on simply obtaining the appropriate uncertainty over reward as well as imitation in high stakes environments - in these settings it is crucial that learnt policies avoid catastrophic failure and so how exactly to use the uncertainty in order to achieve truly safe imitation (or indeed better-that-demonstrator apprenticeship) is increasingly of interest. ", + "bbox": [ + 174, + 635, + 581, + 704 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/10c9875b3e37b7a44561ac5529ce0724c415439487c6fd0baa51655f4911daf4.jpg", + "image_caption": [ + "Figure 4: (Top) A state-action reward is learnt and plotted for an otherwise average patient as their blood oxygen level changes. (Bottom) The associated weights given a state-only reward as a linear function of the state-space. " + ], + "image_footnote": [], + "bbox": [ + 604, + 366, + 815, + 582 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 704, + 825, + 801 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 823, + 366, + 838 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "AJC would like to acknowledge and thank Microsoft Research for its support through its PhD Scholarship Program with the EPSRC. This work was additionally supported by the Office of Naval Research (ONR) and the NSF (Grant number: 1722516). We would like to thank all of the anonymous reviewers on OpenReview, alongside the many members of the van der Schaar lab, for their input, comments, and suggestions at various stages that have ultimately improved the manuscript. 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", + "bbox": [ + 174, + 306, + 823, + 335 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 102, + 403, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Expert Demonstrators. Demonstrations are produced by running pre-trained and hyperparmeteroptimised agents taken from the RL Baselines Zoo (Raffin, 2018) in OpenAI Stable Baselines (Hill et al., 2018). For Acrobot and LunarLander these are DQNs (Mnih et al., 2013), while CartPole uses PPO2 (Schulman et al., 2017). Trajectories were then sub-sampled for every 20th step in Acrobot and CartPole, and every 5th step in LunarLander. ", + "bbox": [ + 173, + 138, + 825, + 208 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Testing setup. For control environments algorithms were presented with (1,3,7,10,15) trajectories uniformly sampled from a pool of 1000 expert trajectories. Each algorithm was then trained until convergence and tested by performing 300 live roll-outs in the simulated environment and recording the average accumulated reward received in each episode. This whole process was then repeated 10 times, consequently with different initialisations and seen trajectories. ", + "bbox": [ + 174, + 232, + 825, + 303 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Implementations. All methods are neural network based and so in experiments they share the same architecture of 2 hidden layers of 64 units each connected by exponential linear unit (ELU) activation functions. ", + "bbox": [ + 174, + 325, + 825, + 368 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Publicly available code was used in the implementations of a number of the benchmarks, specifically: ", + "bbox": [ + 173, + 375, + 825, + 404 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• VDICE (Kostrikov et al., 2019): https://github.com/google-research/google-research/tree/ master/value_dice \n• DSFN (Lee et al., 2019): https://github.com/dtak/batch-apprenticeship-learning \n• EDM (Jarrett et al., 2020): https://github.com/wgrathwohl/JEM ", + "bbox": [ + 212, + 421, + 784, + 574 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Note that VDICE was originally designed for continuous actions with a Normal distribution output which we adapt for the experiments by replacing with a Gumbel-softmax. ", + "bbox": [ + 173, + 592, + 825, + 621 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B PROOFS ", + "text_level": 1, + "bbox": [ + 174, + 650, + 276, + 665 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof of proposition 1. Assuming the constraint is satisfied, we are maximising the following objective: ", + "bbox": [ + 173, + 685, + 825, + 715 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/6cc3bd8ee994e7067352680b1c0f4e7bf683ad55e33076f9db973e6035afd169.jpg", + "text": "$$\n\\mathcal { F } ( \\phi , \\theta ) = \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal { D } } \\log \\frac { \\exp \\beta Q _ { \\theta } ( s , a ) ) } { \\sum _ { b \\in \\mathcal { A } } \\exp ( \\beta Q _ { \\theta } ( s , b ) ) } - D _ { K L } \\big ( q _ { \\phi } ( R ( s , a ) ) | | p ( R ( s , a ) ) \\big )\n$$", + "text_format": "latex", + "bbox": [ + 205, + 728, + 763, + 770 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Which is equivalent to minimising the negative value ", + "bbox": [ + 174, + 784, + 522, + 799 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/6e760f31446e4ca138988bf7ec1f10f266bed97384094a2066f4c54c88f1c129.jpg", + "text": "$$\n\\mathcal { F } ( \\phi , \\theta ) = \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal { D } } - \\log \\frac { \\exp \\beta Q _ { \\theta } ( s , a ) ) } { \\sum _ { b \\in \\mathcal { A } } \\exp ( \\beta Q _ { \\theta } ( s , b ) ) } + \\underbrace { D _ { K L } \\big ( q _ { \\phi } ( R ( s , a ) ) | | p ( R ( s , a ) ) \\big ) } _ { \\mathcal { L } _ { r e g } } ,\n$$", + "text_format": "latex", + "bbox": [ + 196, + 811, + 772, + 868 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "with the first term $\\mathcal { L } _ { B C }$ being the negative log-likelihood of the data and the classic behavioural cloning objective. Now given a standard Gaussian prior then the $\\mathrm { K L }$ divergence of a Gaussian with mean $\\mu$ and variance $\\sigma ^ { 2 }$ from the prior is given by $\\begin{array} { r } { \\frac { 1 } { 2 } ( - \\log ( \\sigma ^ { 2 } ) + \\sigma ^ { 2 } - 1 + \\bar { \\mu } ^ { 2 } ) } \\end{array}$ (Kingma & Welling, ", + "bbox": [ + 176, + 881, + 823, + 925 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "2013). Then given our prior $p ( R ( s , a ) ) = \\mathcal { N } ( R ; 0 , 1 )$ , the KL evaluates as: ", + "bbox": [ + 173, + 102, + 665, + 119 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/3982fadd32581fcf7ff439784c25ae2e8dbbf7258f95998ae859909bd77b402a.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\mathcal L } _ { r e g } = \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal D } _ { K L } \\big ( q _ { \\phi } ( R ( s , a ) ) | | p ( R ( s , a ) ) \\big ) } } \\\\ { { = \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal D } \\frac { 1 } { 2 } \\big ( - \\log ( \\operatorname { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] ) + \\operatorname { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] - 1 + \\mathbb { E } _ { q _ { \\phi } } [ R ( s , a ) ] ^ { 2 } \\big ) } } \\\\ { { = \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal D } \\frac { 1 } { 2 } \\big ( \\mathbb { E } _ { q _ { \\phi } } [ R ( s , a ) ] ^ { 2 } ) + g ( \\operatorname { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] ) } } \\\\ { { = \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal D } \\frac { 1 } { 2 } \\big ( Q _ { \\theta } ( s , a ) - \\gamma Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) \\big ) ^ { 2 } + g ( \\operatorname { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 126, + 766, + 284 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Since by assumption $\\mathbb { E } _ { q _ { \\phi } } [ R ( s , a ) ] ~ = ~ \\mathbb { E } _ { \\pi , T } [ Q _ { \\theta } ( s , a ) ~ - ~ \\gamma Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) ]$ with the expectation approximated over samples in the data and considering a definition of the function $g$ to be $\\begin{array} { r } { \\dot { g } ( \\mathrm { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] ) = \\frac { 1 } { 2 } ( - \\log ( \\mathrm { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] ) + \\mathrm { V a r } _ { q _ { \\phi } } [ \\bar { R } ( s , a ) ] - 1 ) . } \\end{array}$ . \u0003 ", + "bbox": [ + 173, + 290, + 821, + 335 + ], + "page_idx": 13 + } +] \ No newline at end of file diff --git a/parse/train/4qR3coiNaIv/4qR3coiNaIv_middle.json b/parse/train/4qR3coiNaIv/4qR3coiNaIv_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..e8315fadfc2ce6d13976550691c898472e16489b --- /dev/null +++ b/parse/train/4qR3coiNaIv/4qR3coiNaIv_middle.json @@ -0,0 +1,35914 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 79, + 503, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 506, + 98 + ], + "score": 1.0, + "content": "SCALABLE BAYESIAN INVERSE REINFORCEMENT", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 189, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 189, + 118 + ], + "score": 1.0, + "content": "LEARNING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 135, + 282, + 168 + ], + "lines": [ + { + "bbox": [ + 112, + 135, + 170, + 147 + ], + "spans": [ + { + "bbox": [ + 112, + 135, + 170, + 147 + ], + "score": 1.0, + "content": "Alex J. 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Unfortunately current meth-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 276, + 469, + 288 + ], + "spans": [ + { + "bbox": [ + 142, + 276, + 469, + 288 + ], + "score": 1.0, + "content": "ods generally do not scale well beyond the small tabular setting due to the need", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 288, + 469, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 288, + 469, + 299 + ], + "score": 1.0, + "content": "for an inner-loop MDP solver, and even non-Bayesian methods that do themselves", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 298, + 470, + 310 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 470, + 310 + ], + "score": 1.0, + "content": "scale often require extensive interaction with the environment to perform well,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 309, + 470, + 321 + ], + "spans": [ + { + "bbox": [ + 141, + 309, + 470, + 321 + ], + "score": 1.0, + "content": "being inappropriate for high stakes or costly applications such as healthcare. 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In this case learning and inference", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "must be done solely on the basis of logged trajectories from a competent demonstrator showing only", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 325, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 325, + 508 + ], + "score": 1.0, + "content": "the states visited and the the action taken in each case.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 526 + ], + "score": 1.0, + "content": "Clinical decision making is an important example of this, where there is great interest in learning", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "policies from medical professionals but is completely impractical and unethical to deploy policies on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "patients mid-training. 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While imitation learning (IL)", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "generally deals with the problem of producing appropriate policies to match a demonstrator, with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "the added layer of understanding motivations this would then usually be approached through inverse", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "reinforcement learning (IRL). 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Notably for any given set of demonstrations there are (infinitely)", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 333, + 645 + ], + "score": 1.0, + "content": "many rewards for which the actions would be optimal", + "type": "text" + }, + { + "bbox": [ + 334, + 633, + 348, + 644 + ], + "score": 0.42, + "content": "\\mathrm { N g }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "et al., 2000). 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While imitation learning (IL)", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "generally deals with the problem of producing appropriate policies to match a demonstrator, with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "the added layer of understanding motivations this would then usually be approached through inverse", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "reinforcement learning (IRL). Here attempting to learn the assumed underlying reward driving the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "score": 1.0, + "content": "demonstrator, before secondarily learning a policy that is optimal with respect to the reward using", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "some forward reinforcement learning (RL) technique. By composing the RL and IRL procedures in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "order to perform IL we arrive at apprenticeship learning (AL), which introduces its own challenges,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "particularly in the offline setting. Notably for any given set of demonstrations there are (infinitely)", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 333, + 645 + ], + "score": 1.0, + "content": "many rewards for which the actions would be optimal", + "type": "text" + }, + { + "bbox": [ + 334, + 633, + 348, + 644 + ], + "score": 0.42, + "content": "\\mathrm { N g }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "et al., 2000). Max-margin (Abbeel &", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 120, + 655 + ], + "score": 0.29, + "content": "\\mathrm { N g }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 644, + 505, + 657 + ], + "score": 1.0, + "content": ", 2004) and max-entropy (Ziebart et al., 2008) methods for heuristically differentiating plausible", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "rewards do so at the cost of potentially dismissing the true reward for not possessing desirable", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 665, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 680 + ], + "score": 1.0, + "content": "qualities. On the other hand a Bayesian approach to IRL (BIRL) is more conceptually satisfying,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "taking a probabilistic view of the reward, we are interested in the posterior distribution having seen", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "the demonstrations (Ramachandran & Amir, 2007), accounting for all possibilities. BIRL is not", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "without its own drawbacks though, as noted in Brown & Niekum (2019), making it inappropriate", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "for modern complicated environments: assuming linear rewards; small, solvable environments; and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 274, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 274, + 732 + ], + "score": 1.0, + "content": "repeated, inner-loop, calls to forward RL.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 511, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 114, + 78, + 488, + 186 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 78, + 488, + 186 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 78, + 488, + 186 + ], + "spans": [ + { + "bbox": [ + 114, + 78, + 488, + 186 + ], + "score": 0.968, + "type": "image", + "image_path": "d9cfdc5b1bc6f3e16d989317572e562d90d7f13f159f7755e10beb37ec13d22b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 114, + 78, + 488, + 114.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 114, + 114.0, + 488, + 150.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 114, + 150.0, + 488, + 186.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 191, + 505, + 247 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "Figure 1: Overview. AVRIL is a framework for BIRL that works through an approximation in the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "variational Bayesian framework, considering the reward to be a latent representation of behaviour.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "A distribution over the reward, which is amortised over the demonstration space, is learnt that then", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 189, + 237 + ], + "score": 1.0, + "content": "informs an imitator", + "type": "text" + }, + { + "bbox": [ + 190, + 225, + 199, + 236 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "-function policy. The dotted line represents a departure from a traditional", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 235, + 408, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 408, + 248 + ], + "score": 1.0, + "content": "auto-encoder as the input, alongside the latent reward, informs the decoder.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "The main contribution then of this paper is a method for advancing BIRL beyond these obstacles,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 268, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 283 + ], + "score": 1.0, + "content": "allowing for approximate reward inference using an arbitrarily flexible class of functions, in any", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "environment, without costly inner-loop operations, and importantly entirely offline. This leads to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 305 + ], + "score": 1.0, + "content": "our algorithm AVRIL, depicted in figure 1, which represents a framework for jointly learning a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "variational posterior distribution over the reward alongside an imitator policy in an auto-encoder-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "esque manner. In what follows we review the modern methods for offline IRL/IL (Section 2) with", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 322, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 506, + 338 + ], + "score": 1.0, + "content": "a focus on the approach of Bayesian IRL and the issues it faces when confronted with challenging", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "environments. We then address the above issues by introducing our contributions (Section 3), and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 345, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 359 + ], + "score": 1.0, + "content": "demonstrate the gains of our algorithm in real medical data and simulated control environments,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "notably that it is now possible to achieve Bayesian reward inference in such settings (Section 4).", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "Finally we wrap up with some concluding thoughts and directions (Section 5). Code for AVRIL and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "our experiments is made available at https://github.com/XanderJC/scalable-birl", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 390, + 439, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 439, + 402 + ], + "score": 1.0, + "content": "and https://github.com/vanderschaarlab/mlforhealthlabpub.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 419, + 423, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 425, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 425, + 434 + ], + "score": 1.0, + "content": "2 APPROACHING APPRENTICESHIP AND IMITATION OFFLINE", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "Preliminaries. We consider the standard Markov decision process (MDP) environment, with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 453, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 104, + 453, + 132, + 469 + ], + "score": 1.0, + "content": "states", + "type": "text" + }, + { + "bbox": [ + 132, + 456, + 162, + 466 + ], + "score": 0.89, + "content": "s \\in \\mathcal { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 453, + 199, + 469 + ], + "score": 1.0, + "content": ", actions", + "type": "text" + }, + { + "bbox": [ + 200, + 456, + 231, + 466 + ], + "score": 0.89, + "content": "a \\in \\mathcal A", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 453, + 282, + 469 + ], + "score": 1.0, + "content": ", transitions", + "type": "text" + }, + { + "bbox": [ + 282, + 455, + 349, + 468 + ], + "score": 0.91, + "content": "T \\in \\Delta ( { \\cal S } ) ^ { s \\times { \\cal A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 453, + 389, + 469 + ], + "score": 1.0, + "content": ", rewards", + "type": "text" + }, + { + "bbox": [ + 389, + 456, + 445, + 466 + ], + "score": 0.92, + "content": "\\vec { R } \\in \\mathbb { R } ^ { S \\times A 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 453, + 506, + 469 + ], + "score": 1.0, + "content": ", and discount", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 460, + 509, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 149, + 479 + ], + "score": 0.92, + "content": "\\gamma \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 460, + 209, + 486 + ], + "score": 1.0, + "content": ". For a policy", + "type": "text" + }, + { + "bbox": [ + 209, + 467, + 261, + 479 + ], + "score": 0.93, + "content": "\\pi \\in \\Delta ( \\mathcal { A } ) ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 460, + 277, + 486 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 277, + 467, + 440, + 480 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\rho _ { \\pi } ( s , a ) = \\mathbb { E } _ { \\pi , T } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\mathbb { 1 } _ { \\{ s _ { t } = s , a _ { t } = a \\} } ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 460, + 509, + 486 + ], + "score": 1.0, + "content": "be the induced", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 478, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 478, + 400, + 493 + ], + "score": 1.0, + "content": "unique occupancy measure alongside the state-only occupancy measure", + "type": "text" + }, + { + "bbox": [ + 400, + 479, + 501, + 492 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\rho _ { \\pi } ( s ) = \\sum _ { a \\in \\mathcal { A } } \\rho _ { \\pi } ( s , a ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 478, + 505, + 493 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 491, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 429, + 503 + ], + "score": 1.0, + "content": "Despite this full environment model, the only information available to us is the", + "type": "text" + }, + { + "bbox": [ + 429, + 492, + 472, + 503 + ], + "score": 0.7, + "content": "\\mathbf { M D P } \\backslash R T", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 491, + 506, + 503 + ], + "score": 1.0, + "content": ", in that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "we have no access to either the underlying reward or the transitions, with our lacking knowledge of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 513, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 524 + ], + "score": 1.0, + "content": "the transitions being also strong in the sense that further we are unable to simulate the environment", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 525, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 354, + 536 + ], + "score": 1.0, + "content": "to sample them. 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Code for AVRIL and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "our experiments is made available at https://github.com/XanderJC/scalable-birl", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 390, + 439, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 439, + 402 + ], + "score": 1.0, + "content": "and https://github.com/vanderschaarlab/mlforhealthlabpub.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 257, + 506, + 402 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 419, + 423, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 425, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 425, + 434 + ], + "score": 1.0, + "content": "2 APPROACHING APPRENTICESHIP AND IMITATION OFFLINE", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "Preliminaries. 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For a policy", + "type": "text" + }, + { + "bbox": [ + 209, + 467, + 261, + 479 + ], + "score": 0.93, + "content": "\\pi \\in \\Delta ( \\mathcal { A } ) ^ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 460, + 277, + 486 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 277, + 467, + 440, + 480 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\rho _ { \\pi } ( s , a ) = \\mathbb { E } _ { \\pi , T } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\mathbb { 1 } _ { \\{ s _ { t } = s , a _ { t } = a \\} } ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 460, + 509, + 486 + ], + "score": 1.0, + "content": "be the induced", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 478, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 478, + 400, + 493 + ], + "score": 1.0, + "content": "unique occupancy measure alongside the state-only occupancy measure", + "type": "text" + }, + { + "bbox": [ + 400, + 479, + 501, + 492 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\rho _ { \\pi } ( s ) = \\sum _ { a \\in \\mathcal { A } } \\rho _ { \\pi } ( s , a ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 478, + 505, + 493 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 491, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 429, + 503 + ], + "score": 1.0, + "content": "Despite this full environment model, the only information available to us is the", + "type": "text" + }, + { + "bbox": [ + 429, + 492, + 472, + 503 + ], + "score": 0.7, + "content": "\\mathbf { M D P } \\backslash R T", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 491, + 506, + 503 + ], + "score": 1.0, + "content": ", in that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "we have no access to either the underlying reward or the transitions, with our lacking knowledge of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 513, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 524 + ], + "score": 1.0, + "content": "the transitions being also strong in the sense that further we are unable to simulate the environment", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 525, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 354, + 536 + ], + "score": 1.0, + "content": "to sample them. The learning signal is then given by access to", + "type": "text" + }, + { + "bbox": [ + 355, + 526, + 365, + 534 + ], + "score": 0.78, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 525, + 505, + 536 + ], + "score": 1.0, + "content": "-many trajectories of some demon-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 534, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 104, + 534, + 401, + 549 + ], + "score": 1.0, + "content": "strator assumed to be acting optimally w.r.t. the MDP, following a policy", + "type": "text" + }, + { + "bbox": [ + 402, + 537, + 415, + 546 + ], + "score": 0.85, + "content": "\\pi _ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 534, + 506, + 549 + ], + "score": 1.0, + "content": ", making up a data set", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 544, + 504, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 271, + 562 + ], + "score": 0.92, + "content": "\\mathcal { D } _ { r a w } = \\{ ( s _ { 1 } ^ { ( i ) } , a _ { 1 } ^ { ( i ) } , \\dotsc , s _ { \\tau ^ { ( i ) } } ^ { ( \\bar { i } ) } , a _ { \\tau ^ { ( i ) } } ^ { ( i ) } ) \\} _ { i = 1 } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 544, + 299, + 564 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 300, + 546, + 315, + 561 + ], + "score": 0.92, + "content": "s _ { t } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 544, + 378, + 564 + ], + "score": 1.0, + "content": "is the state and", + "type": "text" + }, + { + "bbox": [ + 379, + 546, + 394, + 561 + ], + "score": 0.92, + "content": "a _ { t } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 544, + 498, + 564 + ], + "score": 1.0, + "content": "is the action taken at step", + "type": "text" + }, + { + "bbox": [ + 498, + 550, + 504, + 559 + ], + "score": 0.63, + "content": "t", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 149, + 575 + ], + "score": 1.0, + "content": "during the", + "type": "text" + }, + { + "bbox": [ + 150, + 564, + 154, + 572 + ], + "score": 0.29, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 560, + 242, + 575 + ], + "score": 1.0, + "content": "th demonstration, and", + "type": "text" + }, + { + "bbox": [ + 242, + 562, + 258, + 573 + ], + "score": 0.9, + "content": "\\tau ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "is the (max) time horizon of the ith demonstration. Given the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "Markov assumption though it is sufficient and convenient to consider the demonstrations simply as", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 583, + 504, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 583, + 165, + 599 + ], + "score": 1.0, + "content": "a collection of", + "type": "text" + }, + { + "bbox": [ + 166, + 586, + 172, + 594 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 583, + 403, + 599 + ], + "score": 1.0, + "content": "-many state, action, next state, next action tuples such that", + "type": "text" + }, + { + "bbox": [ + 404, + 584, + 504, + 597 + ], + "score": 0.91, + "content": "\\mathcal { D } = \\{ ( s _ { i } , a _ { i } , s _ { i } ^ { \\prime } , a _ { i } ^ { \\prime } ) \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 592, + 218, + 613 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 127, + 613 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 595, + 211, + 610 + ], + "score": 0.93, + "content": "\\begin{array} { r } { n = \\sum _ { i = 1 } ^ { m } ( \\tau ^ { ( i ) } - 1 ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 592, + 218, + 613 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 444, + 509, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 636 + ], + "score": 1.0, + "content": "Apprenticeship through rewards. 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This", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "allows for easy mix-and-match procedures, swapping in different standard RL and IRL methods", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 450, + 678 + ], + "score": 1.0, + "content": "depending on the situation. These algorithms though depend on either knowledge of", + "type": "text" + }, + { + "bbox": [ + 450, + 667, + 459, + 676 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "in order to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "solve exactly or the ability to perform roll-outs in the environment, with little previous work focus-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "ing on the entirely offline setting. One simple solution is through attempting to learn the dynamics", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "(Herman et al., 2016), though without a large supply of diverse demonstrations or a small environ-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "ment this becomes impractical given imperfections in the model. Alternatively Klein et al. (2011)", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "and Lee et al. 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Alternatively Klein et al. (2011)", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "and Lee et al. (2019) attempt off-policy feature matching through least-squared temporal difference", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 399, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 399, + 128 + ], + "score": 1.0, + "content": "and deep neural networks to uncover appropriate feature representations.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "Implicit-reward policy learning. Recent work has often forgone an explicit representation of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 505, + 161 + ], + "score": 1.0, + "content": "reward. 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(2014) and Reddy et al. (2019) working", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "by essentially placing a sparsity prior on this implied reward, encouraging it towards zero, and thus", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "incorporating subsequent state information. Alternatively Jarrett et al. (2020) show that even the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "simple behavioural cloning (Bain & Sammut, 1995) is implicitly maximising some reward with an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "approximation that the expectation over states is taken with respect to the demonstrator, not the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "learnt policy. They then attempt to rectify part of this approximation using the properties of the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 474, + 378, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 378, + 487 + ], + "score": 1.0, + "content": "energy-based model implied by the policy (Grathwohl et al., 2019).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 504, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "The problem with learning an implicit reward in an offline setting is that it remains just that, implicit,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "only able to be evaluated at points seen in the demonstrations, and even then only approximately.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "Thus even if their consideration improves imitator policies performance they offer no real improve-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 524, + 202, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 202, + 536 + ], + "score": 1.0, + "content": "ment for interpretation.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 106, + 548, + 340, + 559 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 343, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 343, + 561 + ], + "score": 1.0, + "content": "2.1 BAYESIAN INVERSE REINFORCEMENT LEARNING", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "We are then resigned to directly reason about the underlying reward, bringing us back to the question", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "of IRL, and in particular BIRL for a principled approach to reasoning under uncertainty. Given a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "prior over possible functions, having seen some demonstrations, we calculate the posterior over", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 441, + 614 + ], + "score": 1.0, + "content": "the function using a theoretically simple application of Bayes rule. 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(2014) and Reddy et al. (2019) working", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "by essentially placing a sparsity prior on this implied reward, encouraging it towards zero, and thus", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "incorporating subsequent state information. Alternatively Jarrett et al. 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They then attempt to rectify part of this approximation using the properties of the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 474, + 378, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 378, + 487 + ], + "score": 1.0, + "content": "energy-based model implied by the policy (Grathwohl et al., 2019).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 407, + 505, + 487 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 504, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "The problem with learning an implicit reward in an offline setting is that it remains just that, implicit,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "only able to be evaluated at points seen in the demonstrations, and even then only approximately.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "Thus even if their consideration improves imitator policies performance they offer no real improve-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 524, + 202, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 202, + 536 + ], + "score": 1.0, + "content": "ment for interpretation.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 491, + 505, + 536 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 548, + 340, + 559 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 343, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 343, + 561 + ], + "score": 1.0, + "content": "2.1 BAYESIAN INVERSE REINFORCEMENT LEARNING", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 505, + 635 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "We are then resigned to directly reason about the underlying reward, bringing us back to the question", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "of IRL, and in particular BIRL for a principled approach to reasoning under uncertainty. Given a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "prior over possible functions, having seen some demonstrations, we calculate the posterior over", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 441, + 614 + ], + "score": 1.0, + "content": "the function using a theoretically simple application of Bayes rule. 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Note that", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "despite similarities, moving forward we are no longer within the maximum-entropy framework and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 144, + 700 + ], + "score": 0.93, + "content": "Q _ { R } ^ { \\pi } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 686, + 443, + 702 + ], + "score": 1.0, + "content": "now denotes the traditional, not soft (as in equation 3), state-action value (", + "type": "text" + }, + { + "bbox": [ + 443, + 688, + 452, + 699 + ], + "score": 0.76, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "-value) func-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 103, + 694, + 502, + 716 + ], + "spans": [ + { + "bbox": [ + 103, + 694, + 189, + 716 + ], + "score": 1.0, + "content": "Rtion given a reward", + "type": "text" + }, + { + "bbox": [ + 189, + 699, + 198, + 709 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 694, + 245, + 716 + ], + "score": 1.0, + "content": "and policy", + "type": "text" + }, + { + "bbox": [ + 246, + 702, + 253, + 709 + ], + "score": 0.73, + "content": "\\pi", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 694, + 294, + 716 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 295, + 698, + 502, + 712 + ], + "score": 0.86, + "content": "\\begin{array} { r } { Q _ { R } ^ { \\pi } ( s , \\bar { a } ) = \\mathbb { E } _ { \\pi , T } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } R ( s _ { t } ) | s _ { 0 } = s , a _ { 0 } = a ] } \\end{array}", + "type": "inline_equation" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Unsurprisingly this yields an intractable posterior distribution leading to a Markov chain Monte", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 720, + 449, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 449, + 733 + ], + "score": 1.0, + "content": "Carlo (MCMC) algorithm based on a random grid-walk to sample from the posterior.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5, + "bbox_fs": [ + 103, + 664, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Issues in complex and unknown environments. This original formulation, alongside extensions", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "that consider maximum-a-posteriori inference (Choi & Kim, 2011) and multiple rewards (Choi &", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "Kim, 2012; Dimitrakakis & Rothkopf, 2011), suffer from three major drawbacks that make them", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 397, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 397, + 127 + ], + "score": 1.0, + "content": "impractical for modern, complicated, and model-free task environments.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 129, + 136, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 129, + 135, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 129, + 135, + 505, + 149 + ], + "score": 1.0, + "content": "1. The reward is a linear combination of state features. Naturally this is a very restrictive", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 146, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 141, + 146, + 504, + 160 + ], + "score": 1.0, + "content": "class of functions and assumes access to carefully hand-crafted features of the state space.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 128, + 160, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 128, + 160, + 318, + 174 + ], + "score": 1.0, + "content": "2. The cardinality of the state-space is finite,", + "type": "text" + }, + { + "bbox": [ + 318, + 161, + 358, + 173 + ], + "score": 0.92, + "content": "| S | < \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 160, + 506, + 174 + ], + "score": 1.0, + "content": ". Admittedly this can be relaxed in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 140, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 140, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "practical terms, although it does mean the rapid-mixing bounds derived by Ramachandran", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 142, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 142, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "& Amir (2007) do not hold at all in the infinite case. For finite approximations they scale", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 141, + 194, + 152, + 207 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 153, + 194, + 186, + 206 + ], + "score": 0.93, + "content": "\\mathcal { O } ( | S | ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 194, + 505, + 207 + ], + "score": 1.0, + "content": ", rapidly becoming vacuous and causing BIRL to inherit the usual MCMC diffi-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 205, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 142, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "culties on assessing convergence and sequential computation (Gamerman & Lopes, 2006).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 130, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 130, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "3. The requirement of an inner-loop MDP solve. Most importantly at every step a new reward", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 140, + 229, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 140, + 229, + 506, + 244 + ], + "score": 1.0, + "content": "is sampled and the likelihood of the data must then be evaluated. This requires calculating", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 141, + 241, + 157, + 254 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 157, + 242, + 166, + 253 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "-values of the policy with respect to the reward, in other words running forward RL.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 142, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "While not an insurmountable problem in the simple cases where everything is known and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 142, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "can be quickly solved with a procedure guaranteed to converge correctly, this becomes an", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "issue in the realm where only deep function approximation works adequately (i.e. the non-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "tabular setting). DQN training for example easily stretches into hours (Mnih et al., 2013)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 297, + 467, + 309 + ], + "spans": [ + { + "bbox": [ + 142, + 297, + 467, + 309 + ], + "score": 1.0, + "content": "and will have to be repeated thousands of times, making it completely untenable.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "We have seen that even in the most simple setting the problem of exact Bayesian inference over", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 329, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 339 + ], + "score": 1.0, + "content": "the reward is intractable, and the above limitations of the current MCMC methods are not trivial", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "to overcome. Consequently very little work has been done in the area and there still remain very", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "open challenges. Levine et al. (2011) addressed linearity through a Gaussian process approach,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "allowing for a significantly more flexible and non-linear representation though introducing issues of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "its own, namely the computational complexity of inverting large matrices (Rasmussen, 2003). More", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 384, + 504, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 504, + 395 + ], + "score": 1.0, + "content": "recently Brown & Niekum (2019) have presented the only current solution to the inner-loop problem", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "score": 1.0, + "content": "by introducing an alternative formulation of the likelihood, one based on human recorded pairwise", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "preferences over demonstrations that significantly reduces the complexity of likelihood. However", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "score": 1.0, + "content": "labelled preferences certainly can’t be assumed always available and while very effective for the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 427, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 438 + ], + "score": 1.0, + "content": "given task is not appropriate in the general case. One of the key aspects of our contribution is that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "we are able to deal with all three of these issues while also not requiring any additional information.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 506, + 473 + ], + "score": 1.0, + "content": "The usefulness of uncertainty. On top of the philosophical consistency of Bayesian inference", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "there are a number of reasons for wanting a measure of uncertainty over any uncovered reward that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 481, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 481, + 505, + 496 + ], + "score": 1.0, + "content": "are not available from regular IRL algorithms. First that the (epistemic) uncertainty revealed by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 492, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 507 + ], + "score": 1.0, + "content": "Bayesian inference tells us a lot about what areas of the state-space we really cannot say anything", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "about because we haven’t seen any demonstrations there - potentially informing future data collec-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "tion if that is possible (Mindermann et al., 2018). Additionally in the cases we are mostly concerned", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 526, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 539 + ], + "score": 1.0, + "content": "about (e.g. medicine) we have to be very careful about letting algorithms pick actions in practice", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "and we are interested in performing safe or risk-averse imitation, for which a degree of confidence", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "over learnt rewards is necessary. Brown et al. (2020) for example use a distribution over reward to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 559, + 500, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 500, + 572 + ], + "score": 1.0, + "content": "optimise a conditional value-at-risk instead of expected return so as to bound potential downsides.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 107, + 586, + 439, + 599 + ], + "lines": [ + { + "bbox": [ + 104, + 585, + 441, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 585, + 441, + 601 + ], + "score": 1.0, + "content": "3 APPROXIMATE VARIATIONAL REWARD IMITATION LEARNING", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "A variational Bayesian approach. In this section we detail our method, AVRIL, for efficiently", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "learning an imitator policy and performing reward inference simultaneously. Unlike the previously", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "mentioned sampling or MAP-based methods, we employ variational inference (Blei et al., 2017) to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 374, + 657 + ], + "score": 1.0, + "content": "reason about the posterior. 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This original formulation, alongside extensions", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "that consider maximum-a-posteriori inference (Choi & Kim, 2011) and multiple rewards (Choi &", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "Kim, 2012; Dimitrakakis & Rothkopf, 2011), suffer from three major drawbacks that make them", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 397, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 397, + 127 + ], + "score": 1.0, + "content": "impractical for modern, complicated, and model-free task environments.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 505, + 127 + ] + }, + { + "type": "text", + "bbox": [ + 129, + 136, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 129, + 135, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 129, + 135, + 505, + 149 + ], + "score": 1.0, + "content": "1. The reward is a linear combination of state features. Naturally this is a very restrictive", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 146, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 141, + 146, + 504, + 160 + ], + "score": 1.0, + "content": "class of functions and assumes access to carefully hand-crafted features of the state space.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 128, + 160, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 128, + 160, + 318, + 174 + ], + "score": 1.0, + "content": "2. The cardinality of the state-space is finite,", + "type": "text" + }, + { + "bbox": [ + 318, + 161, + 358, + 173 + ], + "score": 0.92, + "content": "| S | < \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 160, + 506, + 174 + ], + "score": 1.0, + "content": ". Admittedly this can be relaxed in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 140, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 140, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "practical terms, although it does mean the rapid-mixing bounds derived by Ramachandran", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 142, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 142, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "& Amir (2007) do not hold at all in the infinite case. For finite approximations they scale", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 141, + 194, + 152, + 207 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 153, + 194, + 186, + 206 + ], + "score": 0.93, + "content": "\\mathcal { O } ( | S | ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 194, + 505, + 207 + ], + "score": 1.0, + "content": ", rapidly becoming vacuous and causing BIRL to inherit the usual MCMC diffi-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 205, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 142, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "culties on assessing convergence and sequential computation (Gamerman & Lopes, 2006).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 130, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 130, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "3. The requirement of an inner-loop MDP solve. Most importantly at every step a new reward", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 140, + 229, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 140, + 229, + 506, + 244 + ], + "score": 1.0, + "content": "is sampled and the likelihood of the data must then be evaluated. This requires calculating", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 141, + 241, + 157, + 254 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 157, + 242, + 166, + 253 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "-values of the policy with respect to the reward, in other words running forward RL.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 142, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "While not an insurmountable problem in the simple cases where everything is known and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 142, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "can be quickly solved with a procedure guaranteed to converge correctly, this becomes an", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 141, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "issue in the realm where only deep function approximation works adequately (i.e. the non-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "tabular setting). DQN training for example easily stretches into hours (Mnih et al., 2013)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 297, + 467, + 309 + ], + "spans": [ + { + "bbox": [ + 142, + 297, + 467, + 309 + ], + "score": 1.0, + "content": "and will have to be repeated thousands of times, making it completely untenable.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11, + "bbox_fs": [ + 128, + 135, + 506, + 309 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "We have seen that even in the most simple setting the problem of exact Bayesian inference over", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 329, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 339 + ], + "score": 1.0, + "content": "the reward is intractable, and the above limitations of the current MCMC methods are not trivial", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 352 + ], + "score": 1.0, + "content": "to overcome. Consequently very little work has been done in the area and there still remain very", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "open challenges. Levine et al. (2011) addressed linearity through a Gaussian process approach,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "allowing for a significantly more flexible and non-linear representation though introducing issues of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "its own, namely the computational complexity of inverting large matrices (Rasmussen, 2003). More", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 384, + 504, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 504, + 395 + ], + "score": 1.0, + "content": "recently Brown & Niekum (2019) have presented the only current solution to the inner-loop problem", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 407 + ], + "score": 1.0, + "content": "by introducing an alternative formulation of the likelihood, one based on human recorded pairwise", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "preferences over demonstrations that significantly reduces the complexity of likelihood. However", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 427 + ], + "score": 1.0, + "content": "labelled preferences certainly can’t be assumed always available and while very effective for the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 427, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 438 + ], + "score": 1.0, + "content": "given task is not appropriate in the general case. One of the key aspects of our contribution is that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "we are able to deal with all three of these issues while also not requiring any additional information.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 316, + 506, + 450 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 506, + 473 + ], + "score": 1.0, + "content": "The usefulness of uncertainty. On top of the philosophical consistency of Bayesian inference", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "there are a number of reasons for wanting a measure of uncertainty over any uncovered reward that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 481, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 104, + 481, + 505, + 496 + ], + "score": 1.0, + "content": "are not available from regular IRL algorithms. First that the (epistemic) uncertainty revealed by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 492, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 507 + ], + "score": 1.0, + "content": "Bayesian inference tells us a lot about what areas of the state-space we really cannot say anything", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "about because we haven’t seen any demonstrations there - potentially informing future data collec-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "tion if that is possible (Mindermann et al., 2018). Additionally in the cases we are mostly concerned", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 526, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 539 + ], + "score": 1.0, + "content": "about (e.g. medicine) we have to be very careful about letting algorithms pick actions in practice", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "and we are interested in performing safe or risk-averse imitation, for which a degree of confidence", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "over learnt rewards is necessary. Brown et al. (2020) for example use a distribution over reward to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 559, + 500, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 500, + 572 + ], + "score": 1.0, + "content": "optimise a conditional value-at-risk instead of expected return so as to bound potential downsides.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 460, + 506, + 572 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 586, + 439, + 599 + ], + "lines": [ + { + "bbox": [ + 104, + 585, + 441, + 601 + ], + "spans": [ + { + "bbox": [ + 104, + 585, + 441, + 601 + ], + "score": 1.0, + "content": "3 APPROXIMATE VARIATIONAL REWARD IMITATION LEARNING", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "A variational Bayesian approach. In this section we detail our method, AVRIL, for efficiently", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "learning an imitator policy and performing reward inference simultaneously. Unlike the previously", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "mentioned sampling or MAP-based methods, we employ variational inference (Blei et al., 2017) to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 374, + 657 + ], + "score": 1.0, + "content": "reason about the posterior. Here we posit a surrogate distribution", + "type": "text" + }, + { + "bbox": [ + 374, + 644, + 400, + 656 + ], + "score": 0.92, + "content": "q _ { \\phi } ( R )", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 644, + 477, + 657 + ], + "score": 1.0, + "content": ", parameterised by", + "type": "text" + }, + { + "bbox": [ + 477, + 644, + 484, + 655 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 644, + 505, + 657 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 493, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 493, + 668 + ], + "score": 1.0, + "content": "aim to minimise the Kullback-Leibler (KL) divergence to the posterior, resulting in an objective:", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 611, + 506, + 668 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 668, + 366, + 688 + ], + "lines": [ + { + "bbox": [ + 244, + 668, + 366, + 688 + ], + "spans": [ + { + "bbox": [ + 244, + 668, + 366, + 688 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\phi } \\{ D _ { \\mathrm { K L } } ( q _ { \\phi } ( R ) | | p ( R | \\mathcal { D } ) ) \\} .", + "type": "interline_equation", + "image_path": "469374a8cd250696c803b1c0a945b8a40c02cba5529457319872279071ab9709.jpg" + } + ] + } + ], + "index": 47, + "virtual_lines": [ + { + "bbox": [ + 244, + 668, + 366, + 688 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 691, + 506, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 690, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 704 + ], + "score": 1.0, + "content": "This divergence is still as troubling as the posterior to evaluate, leading to an auxiliary objective", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 701, + 304, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 304, + 714 + ], + "score": 1.0, + "content": "function in the Evidence Lower BOund (ELBO):", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 690, + 505, + 714 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 715, + 408, + 731 + ], + "lines": [ + { + "bbox": [ + 202, + 715, + 408, + 731 + ], + "spans": [ + { + "bbox": [ + 202, + 715, + 408, + 731 + ], + "score": 0.91, + "content": "\\mathcal { F } ( \\phi ) = \\mathbb { E } _ { q _ { \\phi } } \\big [ \\log p ( \\mathcal { D } | R ) \\big ] - D _ { K L } \\big ( q _ { \\phi } ( R ) | | p ( R ) \\big ) ,", + "type": "interline_equation", + "image_path": "722a4c9f996fb1f0dcf1ce1851d1761a6df383a55436cc191dd6fbada90e1fd0.jpg" + } + ] + } + ], + "index": 50, + "virtual_lines": [ + { + "bbox": [ + 202, + 715, + 408, + 731 + ], + "spans": [], + "index": 50 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 282, + 95 + ], + "score": 1.0, + "content": "where it can be seen that maximisation over", + "type": "text" + }, + { + "bbox": [ + 282, + 83, + 290, + 94 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "is equivalent to (6). We are agnostic towards the form", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "of both the prior and variational distribution, for simplicity here we assume a Gaussian process prior", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 497, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 261, + 117 + ], + "score": 1.0, + "content": "with mean zero and unit variance over", + "type": "text" + }, + { + "bbox": [ + 262, + 105, + 270, + 114 + ], + "score": 0.85, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 104, + 497, + 117 + ], + "score": 1.0, + "content": "alongside the variational posterior distribution given by:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 120, + 352, + 135 + ], + "lines": [ + { + "bbox": [ + 258, + 120, + 352, + 135 + ], + "spans": [ + { + "bbox": [ + 258, + 120, + 352, + 135 + ], + "score": 0.93, + "content": "q _ { \\phi } ( R ) = \\mathcal { N } ( R ; \\mu , \\sigma ^ { 2 } ) ,", + "type": "interline_equation", + "image_path": "1532b16207f68ec3cfe9b06cfe011627db774328a75ce3351dc6541f8afdee54.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 258, + 120, + 352, + 135 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 141, + 505, + 186 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 134, + 154 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 141, + 156, + 153 + ], + "score": 0.92, + "content": "\\mu , \\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 140, + 372, + 154 + ], + "score": 1.0, + "content": "are the outputs of an encoder neural network taking", + "type": "text" + }, + { + "bbox": [ + 372, + 144, + 379, + 151 + ], + "score": 0.71, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 140, + 505, + 154 + ], + "score": 1.0, + "content": "as input and parameterised by", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 152, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 107, + 153, + 114, + 164 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 152, + 505, + 165 + ], + "score": 1.0, + "content": ". Note that for the algorithm that we will describe these choices are not a necessity and can be", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "easily substituted for more expressive distributions if appropriate. Maintaining the assumption of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 441, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 441, + 187 + ], + "score": 1.0, + "content": "Boltzmann rationality on the part of the demonstrator, our objective takes the form:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 190, + 459, + 226 + ], + "lines": [ + { + "bbox": [ + 151, + 190, + 459, + 226 + ], + "spans": [ + { + "bbox": [ + 151, + 190, + 459, + 226 + ], + "score": 0.94, + "content": "\\mathcal { F } ( \\phi ) = \\mathbb { E } _ { q _ { \\phi } } \\left[ \\sum _ { ( s , a ) \\in \\mathcal { D } } \\log \\frac { \\exp ( \\beta Q _ { R } ^ { \\pi _ { D } } ( s , a ) ) } { \\sum _ { b \\in \\mathcal { A } } \\exp ( \\beta Q _ { R } ^ { \\pi _ { D } } ( s , b ) ) } \\right] - D _ { K L } \\big ( q _ { \\phi } ( R ) | | p ( R ) \\big ) .", + "type": "interline_equation", + "image_path": "f161d3dcacc8d099ea4aae0724813e18fd50a03c53b7ef7c02012c5149826eb6.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 151, + 190, + 459, + 202.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 151, + 202.0, + 459, + 214.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 151, + 214.0, + 459, + 226.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "The most interesting (and problematic) part of this objective as ever centres on the evaluation of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 239, + 507, + 259 + ], + "spans": [ + { + "bbox": [ + 107, + 243, + 149, + 255 + ], + "score": 0.91, + "content": "Q _ { R } ^ { \\pi _ { D } } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 239, + 414, + 259 + ], + "score": 1.0, + "content": ". Notice that what is really required here is an expression of the", + "type": "text" + }, + { + "bbox": [ + 414, + 243, + 423, + 254 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 239, + 507, + 259 + ], + "score": 1.0, + "content": "-values as a smooth", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 300, + 266 + ], + "score": 1.0, + "content": "function of the reward such that with samples of", + "type": "text" + }, + { + "bbox": [ + 301, + 254, + 309, + 263 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 254, + 429, + 266 + ], + "score": 1.0, + "content": "we could take gradients w.r.t.", + "type": "text" + }, + { + "bbox": [ + 430, + 254, + 437, + 265 + ], + "score": 0.82, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 254, + 505, + 266 + ], + "score": 1.0, + "content": ". Of course there", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "is little hope of obtaining this simply, by itself it is a harder problem than that of forward RL which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 222, + 288 + ], + "score": 1.0, + "content": "only attempts to evaluate the", + "type": "text" + }, + { + "bbox": [ + 222, + 276, + 231, + 287 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 276, + 314, + 288 + ], + "score": 1.0, + "content": "-values for a specific", + "type": "text" + }, + { + "bbox": [ + 314, + 276, + 323, + 285 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "and already in complicated environments has", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 287, + 339, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 339, + 298 + ], + "score": 1.0, + "content": "to rely on function approximation and limited guarantees.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 263, + 318 + ], + "score": 1.0, + "content": "A naive approach would be to sample", + "type": "text" + }, + { + "bbox": [ + 263, + 303, + 272, + 315 + ], + "score": 0.86, + "content": "\\hat { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 305, + 379, + 318 + ], + "score": 1.0, + "content": "and then approximate the", + "type": "text" + }, + { + "bbox": [ + 379, + 306, + 388, + 316 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "-values with a second neural", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 378, + 329 + ], + "score": 1.0, + "content": "network, solving offline over the batched data using a least-squared", + "type": "text" + }, + { + "bbox": [ + 378, + 316, + 404, + 327 + ], + "score": 0.33, + "content": "\\mathrm { T D } / Q", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "-learning algorithm, as is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "the approach forced on sampling based BIRL methods. It is in fact though doubly inappropriate for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "this setting, not only does this require a solve as an inner-loop but importantly differentiating through", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "the solving operation is extremely impractical, it requires backpropagating through a number of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 490, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 490, + 373 + ], + "score": 1.0, + "content": "gradient updates that are essentially unbounded as the complexity of the environment increases.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "A further approximation. This raises an important question - is it possible to jointly optimise", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "a policy and variational distribution only once instead of requiring a repeated solve? This is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 404, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 404, + 213, + 419 + ], + "score": 1.0, + "content": "theoretically suspect, the", + "type": "text" + }, + { + "bbox": [ + 213, + 406, + 222, + 416 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 404, + 459, + 419 + ], + "score": 1.0, + "content": "-values are defined on a singular reward, constrained as", + "type": "text" + }, + { + "bbox": [ + 459, + 405, + 504, + 417 + ], + "score": 0.92, + "content": "R ( s , a ) \\ =", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 415, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 254, + 429 + ], + "score": 0.9, + "content": "\\mathbb { E } _ { s ^ { \\prime } , a ^ { \\prime } \\sim \\pi , T } [ \\bar { Q } _ { R } ^ { \\pi } ( { \\bar { s , } } a ) - \\gamma Q _ { R } ^ { \\pi } ( s ^ { \\prime } , a ^ { \\prime } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 415, + 425, + 430 + ], + "score": 1.0, + "content": "so we cannot learn a particular standard", + "type": "text" + }, + { + "bbox": [ + 425, + 416, + 434, + 428 + ], + "score": 0.84, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 415, + 506, + 430 + ], + "score": 1.0, + "content": "-function that re-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 425, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 506, + 441 + ], + "score": 1.0, + "content": "flects the entire distribution. But can we learn a policy that reflects the expected reward using a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 227, + 451 + ], + "score": 1.0, + "content": "second policy neural network", + "type": "text" + }, + { + "bbox": [ + 227, + 438, + 244, + 450 + ], + "score": 0.59, + "content": "Q _ { \\theta } ?", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 438, + 352, + 451 + ], + "score": 1.0, + "content": "We can’t simply optimise", + "type": "text" + }, + { + "bbox": [ + 353, + 439, + 359, + 448 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 438, + 401, + 451 + ], + "score": 1.0, + "content": "alongside", + "type": "text" + }, + { + "bbox": [ + 401, + 439, + 408, + 450 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "to maximise the ELBO", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "though as that completely ignores the fact that the learnt policy is intimately related to the distri-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 458, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 458, + 506, + 474 + ], + "score": 1.0, + "content": "bution over the reward. Our solution to ensure then that they behave as intended is by constraining", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 118, + 483 + ], + "score": 0.84, + "content": "q _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 470, + 137, + 484 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 138, + 471, + 151, + 482 + ], + "score": 0.89, + "content": "Q _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "to be consistent with each other, specifically that the implied reward of the policy is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "sufficiently likely under the variational posterior (equivalently that the negative log-likelihood is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 492, + 434, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 434, + 506 + ], + "score": 1.0, + "content": "sufficiently low). Thus we arrive at a constrained optimisation objective given by:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28 + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 509, + 435, + 542 + ], + "lines": [ + { + "bbox": [ + 175, + 509, + 435, + 542 + ], + "spans": [ + { + "bbox": [ + 175, + 509, + 435, + 542 + ], + "score": 0.62, + "content": "\\operatorname* { m a x } _ { \\phi , \\theta } \\sum _ { ( s , a ) \\in \\mathcal { D } } \\log \\frac { \\exp ( \\beta Q _ { \\theta } ( s , a ) ) } { \\sum _ { b \\in \\mathcal { A } } \\exp ( \\beta Q _ { \\theta } ( s , b ) ) } - D _ { K L } \\big ( q _ { \\phi } ( R ) | | p ( R ) \\big ) ,", + "type": "interline_equation", + "image_path": "ab6bb01b3d406269eacf9275cd5158b503c48ae031c27f3ab6ca365465af1c90.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 175, + 509, + 435, + 520.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 175, + 520.0, + 435, + 531.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 175, + 531.0, + 435, + 542.0 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 561, + 502, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 504, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 127, + 574 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 128, + 564, + 134, + 572 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 561, + 504, + 574 + ], + "score": 1.0, + "content": "reflecting the strength of the constraint. Rewriting (10) as a Lagrangian under the KKT", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 237, + 586 + ], + "score": 1.0, + "content": "conditions (Karush, 1939; Kuhn", + "type": "text" + }, + { + "bbox": [ + 238, + 574, + 247, + 583 + ], + "score": 0.28, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "Tucker, 1951), and given complimentary slackness, we obtain a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 584, + 221, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 221, + 596 + ], + "score": 1.0, + "content": "practical objective function:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 128, + 600, + 482, + 650 + ], + "lines": [ + { + "bbox": [ + 128, + 600, + 482, + 650 + ], + "spans": [ + { + "bbox": [ + 128, + 600, + 482, + 650 + ], + "score": 0.94, + "content": "\\begin{array} { r l } { \\mathcal { F } ( \\phi , \\theta , \\mathcal { D } ) = \\displaystyle \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal { D } } \\log \\frac { \\exp \\beta Q _ { \\theta } ( s , a ) ) } { \\sum _ { b \\in A } \\exp ( \\beta Q _ { \\theta } ( s , b ) ) } - D _ { K L } \\big ( q _ { \\phi } ( R ( s , a ) ) | | p ( R ( s , a ) ) \\big ) } & { } \\\\ { + \\lambda \\log q _ { \\phi } ( Q _ { \\theta } ( s , a ) - \\gamma Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) ) . } & { } \\end{array}", + "type": "interline_equation", + "image_path": "2d42ccb3925545343a792675f7aacd47709b0e80950a893270372c9b3f1092f8.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 128, + 600, + 482, + 616.6666666666666 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 128, + 616.6666666666666, + 482, + 633.3333333333333 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 128, + 633.3333333333333, + 482, + 649.9999999999999 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 653, + 506, + 676 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 443, + 666 + ], + "score": 1.0, + "content": "Here the KL divergence between processes is approximated over a countable set, and", + "type": "text" + }, + { + "bbox": [ + 443, + 654, + 450, + 663 + ], + "score": 0.81, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "is introduced", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 664, + 252, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 252, + 676 + ], + "score": 1.0, + "content": "to control the strength of constraint.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "On the implementation. Optimisation is simple as both networks are maximising the same ob-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "jective and gradients can be easily obtained through backpropagation while being amenable to mini-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "batching, allowing you to call your favourite gradient-based stochastic optimisation scheme. We", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 104, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "re-iterate though that AVRIL really represents a framework for doing BIRL and not a specific model", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "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": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 282, + 95 + ], + "score": 1.0, + "content": "where it can be seen that maximisation over", + "type": "text" + }, + { + "bbox": [ + 282, + 83, + 290, + 94 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "is equivalent to (6). We are agnostic towards the form", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "of both the prior and variational distribution, for simplicity here we assume a Gaussian process prior", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 497, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 261, + 117 + ], + "score": 1.0, + "content": "with mean zero and unit variance over", + "type": "text" + }, + { + "bbox": [ + 262, + 105, + 270, + 114 + ], + "score": 0.85, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 104, + 497, + 117 + ], + "score": 1.0, + "content": "alongside the variational posterior distribution given by:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 83, + 505, + 117 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 120, + 352, + 135 + ], + "lines": [ + { + "bbox": [ + 258, + 120, + 352, + 135 + ], + "spans": [ + { + "bbox": [ + 258, + 120, + 352, + 135 + ], + "score": 0.93, + "content": "q _ { \\phi } ( R ) = \\mathcal { N } ( R ; \\mu , \\sigma ^ { 2 } ) ,", + "type": "interline_equation", + "image_path": "1532b16207f68ec3cfe9b06cfe011627db774328a75ce3351dc6541f8afdee54.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 258, + 120, + 352, + 135 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 141, + 505, + 186 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 134, + 154 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 141, + 156, + 153 + ], + "score": 0.92, + "content": "\\mu , \\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 140, + 372, + 154 + ], + "score": 1.0, + "content": "are the outputs of an encoder neural network taking", + "type": "text" + }, + { + "bbox": [ + 372, + 144, + 379, + 151 + ], + "score": 0.71, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 140, + 505, + 154 + ], + "score": 1.0, + "content": "as input and parameterised by", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 152, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 107, + 153, + 114, + 164 + ], + "score": 0.83, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 152, + 505, + 165 + ], + "score": 1.0, + "content": ". Note that for the algorithm that we will describe these choices are not a necessity and can be", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 506, + 176 + ], + "score": 1.0, + "content": "easily substituted for more expressive distributions if appropriate. Maintaining the assumption of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 441, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 441, + 187 + ], + "score": 1.0, + "content": "Boltzmann rationality on the part of the demonstrator, our objective takes the form:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 140, + 506, + 187 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 190, + 459, + 226 + ], + "lines": [ + { + "bbox": [ + 151, + 190, + 459, + 226 + ], + "spans": [ + { + "bbox": [ + 151, + 190, + 459, + 226 + ], + "score": 0.94, + "content": "\\mathcal { F } ( \\phi ) = \\mathbb { E } _ { q _ { \\phi } } \\left[ \\sum _ { ( s , a ) \\in \\mathcal { D } } \\log \\frac { \\exp ( \\beta Q _ { R } ^ { \\pi _ { D } } ( s , a ) ) } { \\sum _ { b \\in \\mathcal { A } } \\exp ( \\beta Q _ { R } ^ { \\pi _ { D } } ( s , b ) ) } \\right] - D _ { K L } \\big ( q _ { \\phi } ( R ) | | p ( R ) \\big ) .", + "type": "interline_equation", + "image_path": "f161d3dcacc8d099ea4aae0724813e18fd50a03c53b7ef7c02012c5149826eb6.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 151, + 190, + 459, + 202.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 151, + 202.0, + 459, + 214.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 151, + 214.0, + 459, + 226.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "The most interesting (and problematic) part of this objective as ever centres on the evaluation of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 239, + 507, + 259 + ], + "spans": [ + { + "bbox": [ + 107, + 243, + 149, + 255 + ], + "score": 0.91, + "content": "Q _ { R } ^ { \\pi _ { D } } ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 239, + 414, + 259 + ], + "score": 1.0, + "content": ". Notice that what is really required here is an expression of the", + "type": "text" + }, + { + "bbox": [ + 414, + 243, + 423, + 254 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 239, + 507, + 259 + ], + "score": 1.0, + "content": "-values as a smooth", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 300, + 266 + ], + "score": 1.0, + "content": "function of the reward such that with samples of", + "type": "text" + }, + { + "bbox": [ + 301, + 254, + 309, + 263 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 254, + 429, + 266 + ], + "score": 1.0, + "content": "we could take gradients w.r.t.", + "type": "text" + }, + { + "bbox": [ + 430, + 254, + 437, + 265 + ], + "score": 0.82, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 254, + 505, + 266 + ], + "score": 1.0, + "content": ". Of course there", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "is little hope of obtaining this simply, by itself it is a harder problem than that of forward RL which", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 222, + 288 + ], + "score": 1.0, + "content": "only attempts to evaluate the", + "type": "text" + }, + { + "bbox": [ + 222, + 276, + 231, + 287 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 276, + 314, + 288 + ], + "score": 1.0, + "content": "-values for a specific", + "type": "text" + }, + { + "bbox": [ + 314, + 276, + 323, + 285 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "and already in complicated environments has", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 287, + 339, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 339, + 298 + ], + "score": 1.0, + "content": "to rely on function approximation and limited guarantees.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 231, + 507, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 263, + 318 + ], + "score": 1.0, + "content": "A naive approach would be to sample", + "type": "text" + }, + { + "bbox": [ + 263, + 303, + 272, + 315 + ], + "score": 0.86, + "content": "\\hat { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 305, + 379, + 318 + ], + "score": 1.0, + "content": "and then approximate the", + "type": "text" + }, + { + "bbox": [ + 379, + 306, + 388, + 316 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "-values with a second neural", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 378, + 329 + ], + "score": 1.0, + "content": "network, solving offline over the batched data using a least-squared", + "type": "text" + }, + { + "bbox": [ + 378, + 316, + 404, + 327 + ], + "score": 0.33, + "content": "\\mathrm { T D } / Q", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "-learning algorithm, as is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "the approach forced on sampling based BIRL methods. It is in fact though doubly inappropriate for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "this setting, not only does this require a solve as an inner-loop but importantly differentiating through", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "the solving operation is extremely impractical, it requires backpropagating through a number of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 360, + 490, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 490, + 373 + ], + "score": 1.0, + "content": "gradient updates that are essentially unbounded as the complexity of the environment increases.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 303, + 506, + 373 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "A further approximation. This raises an important question - is it possible to jointly optimise", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "a policy and variational distribution only once instead of requiring a repeated solve? This is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 404, + 504, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 404, + 213, + 419 + ], + "score": 1.0, + "content": "theoretically suspect, the", + "type": "text" + }, + { + "bbox": [ + 213, + 406, + 222, + 416 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 404, + 459, + 419 + ], + "score": 1.0, + "content": "-values are defined on a singular reward, constrained as", + "type": "text" + }, + { + "bbox": [ + 459, + 405, + 504, + 417 + ], + "score": 0.92, + "content": "R ( s , a ) \\ =", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 415, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 254, + 429 + ], + "score": 0.9, + "content": "\\mathbb { E } _ { s ^ { \\prime } , a ^ { \\prime } \\sim \\pi , T } [ \\bar { Q } _ { R } ^ { \\pi } ( { \\bar { s , } } a ) - \\gamma Q _ { R } ^ { \\pi } ( s ^ { \\prime } , a ^ { \\prime } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 415, + 425, + 430 + ], + "score": 1.0, + "content": "so we cannot learn a particular standard", + "type": "text" + }, + { + "bbox": [ + 425, + 416, + 434, + 428 + ], + "score": 0.84, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 415, + 506, + 430 + ], + "score": 1.0, + "content": "-function that re-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 425, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 506, + 441 + ], + "score": 1.0, + "content": "flects the entire distribution. But can we learn a policy that reflects the expected reward using a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 227, + 451 + ], + "score": 1.0, + "content": "second policy neural network", + "type": "text" + }, + { + "bbox": [ + 227, + 438, + 244, + 450 + ], + "score": 0.59, + "content": "Q _ { \\theta } ?", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 438, + 352, + 451 + ], + "score": 1.0, + "content": "We can’t simply optimise", + "type": "text" + }, + { + "bbox": [ + 353, + 439, + 359, + 448 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 438, + 401, + 451 + ], + "score": 1.0, + "content": "alongside", + "type": "text" + }, + { + "bbox": [ + 401, + 439, + 408, + 450 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "to maximise the ELBO", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "though as that completely ignores the fact that the learnt policy is intimately related to the distri-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 458, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 458, + 506, + 474 + ], + "score": 1.0, + "content": "bution over the reward. Our solution to ensure then that they behave as intended is by constraining", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 118, + 483 + ], + "score": 0.84, + "content": "q _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 470, + 137, + 484 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 138, + 471, + 151, + 482 + ], + "score": 0.89, + "content": "Q _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "to be consistent with each other, specifically that the implied reward of the policy is", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "sufficiently likely under the variational posterior (equivalently that the negative log-likelihood is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 492, + 434, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 434, + 506 + ], + "score": 1.0, + "content": "sufficiently low). 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So far we have presented both as neural", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "networks, but this does not have to be the case. Of course the advantage of them is their flexibility", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "and ease of training but they are still inherently black box. It is then perfectly possible to swap in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "score": 1.0, + "content": "any particular function approximator if the task requires it, using simple linear models for example", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "may slightly hurt performance but allow for more insight. 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Having settled on their forms, equation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 318, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 353, + 329 + ], + "score": 1.0, + "content": "(11) is calculated simply and the joint gradient with respect to", + "type": "text" + }, + { + "bbox": [ + 353, + 318, + 360, + 327 + ], + "score": 0.77, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 318, + 377, + 329 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 378, + 318, + 385, + 329 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 318, + 505, + 329 + ], + "score": 1.0, + "content": "is straight-forwardly returned", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 329, + 460, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 460, + 341 + ], + "score": 1.0, + "content": "using any standard auto-diff package. The whole process is summarised in Algorithm 1.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 345, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 468, + 358 + ], + "score": 1.0, + "content": "We can now see how AVRIL does not suffer the issues outlined in section 2.1. 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Additionally we have a single objective for which all parameters are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "maximised simultaneously - there are no inner-loops, costly or otherwise, meaning training is faster", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 399, + 496, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 496, + 413 + ], + "score": 1.0, + "content": "than the MCMC methods by a factor equal roughly to the number of samples they would require.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 429, + 376, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 377, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 377, + 442 + ], + "score": 1.0, + "content": "The generative model view. 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Our method is more", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 472, + 376, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 376, + 484 + ], + "score": 1.0, + "content": "analogous to a VAE (Kingma & Welling, 2013), though not ex-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 484, + 376, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 376, + 496 + ], + "score": 1.0, + "content": "actly, where given the graphical model in figure 2 the reward can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 494, + 376, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 376, + 507 + ], + "score": 1.0, + "content": "be seen as a latent representation of the policy. 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This approach has its advantages, in both meaningful interpretation of the latent reward", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "(which is non-existent in adversarial methods), and that we forgo the practical difficulties of alter-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "nating min-max optimisation (Kodali et al., 2017) while maintaining a generative view of the policy.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Temporal consistency through reward regularisation. 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So far we have presented both as neural", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "networks, but this does not have to be the case. Of course the advantage of them is their flexibility", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "and ease of training but they are still inherently black box. It is then perfectly possible to swap in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "score": 1.0, + "content": "any particular function approximator if the task requires it, using simple linear models for example", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "may slightly hurt performance but allow for more insight. Despite the specific focus on infinite", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "state-spaces, AVRIL can still even be applied in the tabular setting by simply representing the policy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "and variational distribution with multi-dimensional tensors. Having settled on their forms, equation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 318, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 353, + 329 + ], + "score": 1.0, + "content": "(11) is calculated simply and the joint gradient with respect to", + "type": "text" + }, + { + "bbox": [ + 353, + 318, + 360, + 327 + ], + "score": 0.77, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 318, + 377, + 329 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 378, + 318, + 385, + 329 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 318, + 505, + 329 + ], + "score": 1.0, + "content": "is straight-forwardly returned", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 329, + 460, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 460, + 341 + ], + "score": 1.0, + "content": "using any standard auto-diff package. The whole process is summarised in Algorithm 1.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 241, + 506, + 341 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 345, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 468, + 358 + ], + "score": 1.0, + "content": "We can now see how AVRIL does not suffer the issues outlined in section 2.1. Our form of", + "type": "text" + }, + { + "bbox": [ + 469, + 345, + 495, + 357 + ], + "score": 0.93, + "content": "q _ { \\phi } ( R )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "flexible and easily accommodates a non-linear form of the reward given a neural architecture - this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 236, + 379 + ], + "score": 1.0, + "content": "also removes any restriction on", + "type": "text" + }, + { + "bbox": [ + 236, + 368, + 244, + 377 + ], + "score": 0.72, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 367, + 506, + 379 + ], + "score": 1.0, + "content": ", or at least allows for any state space that is commonly tackled", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "score": 1.0, + "content": "within the IL/RL literature. Additionally we have a single objective for which all parameters are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "maximised simultaneously - there are no inner-loops, costly or otherwise, meaning training is faster", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 399, + 496, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 496, + 413 + ], + "score": 1.0, + "content": "than the MCMC methods by a factor equal roughly to the number of samples they would require.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 345, + 506, + 413 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 429, + 376, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 377, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 377, + 442 + ], + "score": 1.0, + "content": "The generative model view. Ultimately a policy represents a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 439, + 377, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 377, + 451 + ], + "score": 1.0, + "content": "generative model for the behavioural data we see. Ho & Ermon", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 451, + 376, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 376, + 462 + ], + "score": 1.0, + "content": "(2016) explicitly make use of this fact by casting the problem in the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 461, + 376, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 376, + 473 + ], + "score": 1.0, + "content": "GAN framework (Goodfellow et al., 2014). Our method is more", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 472, + 376, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 376, + 484 + ], + "score": 1.0, + "content": "analogous to a VAE (Kingma & Welling, 2013), though not ex-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 484, + 376, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 376, + 496 + ], + "score": 1.0, + "content": "actly, where given the graphical model in figure 2 the reward can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 494, + 376, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 376, + 507 + ], + "score": 1.0, + "content": "be seen as a latent representation of the policy. Our approach takes", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 505, + 376, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 376, + 518 + ], + "score": 1.0, + "content": "the seen data and amortises the inference, encoding over the state", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 516, + 376, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 376, + 529 + ], + "score": 1.0, + "content": "space. The policy does not act as a decoder in precisely taking any", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 527, + 377, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 377, + 540 + ], + "score": 1.0, + "content": "given encoded reward and outputting a policy, but it does take the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 538, + 376, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 376, + 550 + ], + "score": 1.0, + "content": "whole reward posterior and translate it into actions and therefore", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "behaviour. This approach has its advantages, in both meaningful interpretation of the latent reward", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "(which is non-existent in adversarial methods), and that we forgo the practical difficulties of alter-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "nating min-max optimisation (Kodali et al., 2017) while maintaining a generative view of the policy.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 428, + 377, + 550 + ] + }, + { + "type": "image", + "bbox": [ + 387, + 431, + 498, + 507 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 387, + 431, + 498, + 507 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 387, + 431, + 498, + 507 + ], + "spans": [ + { + "bbox": [ + 387, + 431, + 498, + 507 + ], + "score": 0.96, + "type": "image", + "image_path": "787d6d634246cb213461ef2627f55b1feb1315b21e4be42cc9a37cfd4f67f240.jpg" + } + ] + } + ], + "index": 37.5, + "virtual_lines": [ + { + "bbox": [ + 387, + 431, + 498, + 469.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 387, + 469.0, + 498, + 507.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 384, + 520, + 504, + 542 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 383, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 383, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "Figure 2: Graphical model", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 383, + 531, + 456, + 543 + ], + "spans": [ + { + "bbox": [ + 383, + 531, + 456, + 543 + ], + "score": 1.0, + "content": "for Bayesian IRL", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 503, + 583 + ], + "lines": [], + "index": 42, + "bbox_fs": [ + 105, + 549, + 506, + 585 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Temporal consistency through reward regularisation. Considering only the first term of (11)", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 491, + 624 + ], + "score": 1.0, + "content": "yields the standard behavioural cloning setup (where the logits output can be interpreted as the", + "type": "text" + }, + { + "bbox": [ + 491, + 612, + 500, + 622 + ], + "score": 0.8, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "values) as it removes the reward from the equation and just focuses on matching actions to states.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "AVRIL can then be seen as a policy-learning method regularised by the need for the implied reward", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "to be consistent. Note that this does not induce any necessary bias since the logits normally contain", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "an extra degree of freedom allowing them to arbitrarily shift by some scale factor. This factor is", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 666, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 463, + 678 + ], + "score": 1.0, + "content": "now explicitly constrained by giving the logits additional meaning in that they represent", + "type": "text" + }, + { + "bbox": [ + 464, + 666, + 473, + 677 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 666, + 504, + 678 + ], + "score": 1.0, + "content": "-values.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "This places great importance on the KL term, since every parameterisation of a policy will have an", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 104, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 505, + 701 + ], + "score": 1.0, + "content": "associated implied reward, the KL regularises these to be not so far from the prior and preventing", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "the reward from overfitting to the policy and becoming pointless. It also is able to double as a", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "regularising term in a similar manor to previous reward-regularisation methods (Piot et al., 2014;", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 721, + 484, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 484, + 733 + ], + "score": 1.0, + "content": "Reddy et al., 2019) depending on the chosen prior, encouraging the reward to be close to zero:", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 49.5, + "bbox_fs": [ + 104, + 600, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 506, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Proposition 1 (Reward Regularisation) Assume that the constraint in (10) is satisfied in that", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 91, + 504, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 286, + 106 + ], + "score": 0.91, + "content": "\\begin{array} { r } { { \\mathbb E } _ { q _ { \\phi } } [ R ( s , a ) ] = { \\mathbb E } _ { \\pi , T } [ Q _ { \\theta } ( s , a ) - \\gamma Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 91, + 425, + 109 + ], + "score": 1.0, + "content": ", then given a standard normal prior", + "type": "text" + }, + { + "bbox": [ + 425, + 93, + 504, + 106 + ], + "score": 0.93, + "content": "p ( R ) = \\mathcal { N } ( R ; 0 , 1 )", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 380, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 121, + 117 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 106, + 136, + 115 + ], + "score": 0.25, + "content": "\\mathrm { K L }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 105, + 380, + 117 + ], + "score": 1.0, + "content": "divergence yields a sparsity regulator on the implied reward:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 121, + 448, + 153 + ], + "lines": [ + { + "bbox": [ + 163, + 121, + 448, + 153 + ], + "spans": [ + { + "bbox": [ + 163, + 121, + 448, + 153 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { r e g } = \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal { D } } \\frac { 1 } { 2 } \\big ( Q _ { \\theta } ( s , a ) - \\gamma Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) \\big ) ^ { 2 } + g ( \\mathrm { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] ) .", + "type": "interline_equation", + "image_path": "8c1fb4ca7a6b5f8b5fda4b490f7a45c3bb86bcc967bc98e1f7e8f7e2af6b9568.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 163, + 121, + 448, + 131.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 163, + 131.66666666666666, + 448, + 142.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 163, + 142.33333333333331, + 448, + 152.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 159, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 106, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "Proof. Appendix. \u0003 This follows immediately from the fact that the divergence evaluates as", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 124, + 183 + ], + "score": 1.0, + "content": "DK", + "type": "text" + }, + { + "bbox": [ + 125, + 169, + 505, + 184 + ], + "score": 0.63, + "content": "\\begin{array} { r } { \\overset { \\vartriangle } { \\boldsymbol { \\mathrm { \\iota } } } \\big ( q _ { \\phi } ( \\hat { R } ( s , a ) ) | | p ( R ( s , a ) ) \\big ) = \\frac { 1 } { 2 } ( - \\log ( \\operatorname { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] ) - 1 + \\operatorname { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] + \\mathbb { E } _ { q _ { \\phi } } [ R ( s , a ) ] ^ { 2 } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 186, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "This then allows AVRIL to inherit the benefit of these methods while also explicitly learning a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "reward that can be queried at any point. We are also allowed the choice of whether it is state-only", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 209, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 504, + 221 + ], + "score": 1.0, + "content": "or state-action. This has so far been arbitrary, but it is important to consider that a state-only reward", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 491, + 232 + ], + "score": 1.0, + "content": "is a necessary and sufficient condition for a reward that is fully disentangled from the dynamics", + "type": "text" + }, + { + "bbox": [ + 492, + 220, + 505, + 230 + ], + "score": 0.45, + "content": "\\operatorname { F u }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "et al., 2018). Thus by learning such a reward and given the final term of (11) that directly connects", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "one-step rewards in terms of the policy, this forces the policy (not the reward) to account for the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "dynamics of the system ensuring temporal consistency in a way that BC for example simply can’t.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 277 + ], + "score": 1.0, + "content": "Alternatively using a state-action reward means that inevitably some of the temporal information", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "leaks out of the policy and into the reward - ultimately to the detriment of the policy but potentially", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 471, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 471, + 298 + ], + "score": 1.0, + "content": "allowing for a more interpretable (or useful) form of reward depending on the task at hand.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 314, + 200, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 201, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 201, + 328 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "Experimental setup. We are primarily concerned with the case of medical environments, which", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 349, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 364 + ], + "score": 1.0, + "content": "is exactly where the issue of learning without interaction is most crucial, you just cannot let a policy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "sample treatments for a patient to try to learn more about the dynamics. It is also where a level", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "of interpretability in what has been learnt is important, since the consequence of actions are poten-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "tially very impactful on human lives. As such we focus our evaluation on learning on a real-life", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "healthcare problem, with demonstrations taken from the Medical Information Mart for Intensive", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 405, + 504, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 504, + 417 + ], + "score": 1.0, + "content": "Care (MIMIC-III) dataset (Johnson et al., 2016). The data contains trajectories of patients in inten-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "sive care recording their condition and theraputic interventions at one day intervals. We evaluate the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 504, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 504, + 440 + ], + "score": 1.0, + "content": "ability of the methods to learn a medical policy in both the two and four action setting - specifically", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "whether the patient should be placed on a ventilator, and the decision for ventilation in combination", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "with antibiotic treatment. These represent the two most common, and important, clinical interven-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "tions recorded in the data. Without a recorded notion of reward, performance is measured with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 471, + 479, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 479, + 483 + ], + "score": 1.0, + "content": "respect to action matching against a held out test set of demonstrations with cross-validation.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 488, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "Alongside the healthcare data and for the purposes of demonstrating generalisability, we provide", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "additional results on standard environments of varying complexity in the RL literature, the stan-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 523 + ], + "score": 1.0, + "content": "dard control problems of: CartPole, a classic control environment aiming to swing up and balance", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "a pendulum; Acrobot, which aims to maintain a sequence of joints above a given height; and Lu-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "score": 1.0, + "content": "narLander, guiding a landing module to a safe touchdown on the moon surface. In these settings", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 543, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 555 + ], + "score": 1.0, + "content": "given sufficient demonstration data all benchmarks are very much capable of reaching demonstrator", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 554, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 566 + ], + "score": 1.0, + "content": "level performance, so we test the algorithms on their ability to handle sample complexity in the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "low data regime by testing their performance when given access to a select number of trajectories", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "which we adjust, replicating the setup in Jarrett et al. (2020). With access to a simulation through", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 600 + ], + "score": 1.0, + "content": "the OpenAI gym (Brockman et al., 2016), we measure performance by deploying the learnt policies", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 597, + 343, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 343, + 610 + ], + "score": 1.0, + "content": "live and calculating their average return over 300 episodes.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "Benchmarks. We test our method (AVRIL) against a number of benchmarks from the offline", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "IRL/IL setting: Deep Successor Feature Network (DSFN) (Lee et al., 2019), an offline adapta-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "tion of max-margin IRL that generalises past the linear methods using a deep network with least-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "squares temporal-difference learning, the only other method that produces both a reward and policy;", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "Reward-regularized Classification for Apprenticeship Learning (RCAL) (Piot et al., 2014), where", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "an explicit regulariser on the sparsity of the implied reward is introduced in order to account for", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "the dynamics information; ValueDICE (VDICE) (Kostrikov et al., 2019), an adversarial imitation", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "learning, adapted for the offline setting by removing the replay regularisation; Energy-based Distri-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "bution Matching (EDM) (Jarrett et al., 2020), the state-of-the-art in offline imitation learning; and", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "finally the standard example of Behavioural Cloning (BC). To provide evidence that we are indeed", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 47.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 506, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Proposition 1 (Reward Regularisation) Assume that the constraint in (10) is satisfied in that", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 91, + 504, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 286, + 106 + ], + "score": 0.91, + "content": "\\begin{array} { r } { { \\mathbb E } _ { q _ { \\phi } } [ R ( s , a ) ] = { \\mathbb E } _ { \\pi , T } [ Q _ { \\theta } ( s , a ) - \\gamma Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 91, + 425, + 109 + ], + "score": 1.0, + "content": ", then given a standard normal prior", + "type": "text" + }, + { + "bbox": [ + 425, + 93, + 504, + 106 + ], + "score": 0.93, + "content": "p ( R ) = \\mathcal { N } ( R ; 0 , 1 )", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 380, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 121, + 117 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 106, + 136, + 115 + ], + "score": 0.25, + "content": "\\mathrm { K L }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 105, + 380, + 117 + ], + "score": 1.0, + "content": "divergence yields a sparsity regulator on the implied reward:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 82, + 505, + 117 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 121, + 448, + 153 + ], + "lines": [ + { + "bbox": [ + 163, + 121, + 448, + 153 + ], + "spans": [ + { + "bbox": [ + 163, + 121, + 448, + 153 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { r e g } = \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal { D } } \\frac { 1 } { 2 } \\big ( Q _ { \\theta } ( s , a ) - \\gamma Q _ { \\theta } ( s ^ { \\prime } , a ^ { \\prime } ) \\big ) ^ { 2 } + g ( \\mathrm { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] ) .", + "type": "interline_equation", + "image_path": "8c1fb4ca7a6b5f8b5fda4b490f7a45c3bb86bcc967bc98e1f7e8f7e2af6b9568.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 163, + 121, + 448, + 131.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 163, + 131.66666666666666, + 448, + 142.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 163, + 142.33333333333331, + 448, + 152.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 159, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 106, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "Proof. Appendix. \u0003 This follows immediately from the fact that the divergence evaluates as", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 124, + 183 + ], + "score": 1.0, + "content": "DK", + "type": "text" + }, + { + "bbox": [ + 125, + 169, + 505, + 184 + ], + "score": 0.63, + "content": "\\begin{array} { r } { \\overset { \\vartriangle } { \\boldsymbol { \\mathrm { \\iota } } } \\big ( q _ { \\phi } ( \\hat { R } ( s , a ) ) | | p ( R ( s , a ) ) \\big ) = \\frac { 1 } { 2 } ( - \\log ( \\operatorname { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] ) - 1 + \\operatorname { V a r } _ { q _ { \\phi } } [ R ( s , a ) ] + \\mathbb { E } _ { q _ { \\phi } } [ R ( s , a ) ] ^ { 2 } ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 158, + 505, + 184 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 186, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "This then allows AVRIL to inherit the benefit of these methods while also explicitly learning a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "reward that can be queried at any point. We are also allowed the choice of whether it is state-only", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 209, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 504, + 221 + ], + "score": 1.0, + "content": "or state-action. This has so far been arbitrary, but it is important to consider that a state-only reward", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 491, + 232 + ], + "score": 1.0, + "content": "is a necessary and sufficient condition for a reward that is fully disentangled from the dynamics", + "type": "text" + }, + { + "bbox": [ + 492, + 220, + 505, + 230 + ], + "score": 0.45, + "content": "\\operatorname { F u }", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "et al., 2018). Thus by learning such a reward and given the final term of (11) that directly connects", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "one-step rewards in terms of the policy, this forces the policy (not the reward) to account for the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "dynamics of the system ensuring temporal consistency in a way that BC for example simply can’t.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 277 + ], + "score": 1.0, + "content": "Alternatively using a state-action reward means that inevitably some of the temporal information", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 287 + ], + "score": 1.0, + "content": "leaks out of the policy and into the reward - ultimately to the detriment of the policy but potentially", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 471, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 471, + 298 + ], + "score": 1.0, + "content": "allowing for a more interpretable (or useful) form of reward depending on the task at hand.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 186, + 506, + 298 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 314, + 200, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 201, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 201, + 328 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "Experimental setup. We are primarily concerned with the case of medical environments, which", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 349, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 364 + ], + "score": 1.0, + "content": "is exactly where the issue of learning without interaction is most crucial, you just cannot let a policy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "sample treatments for a patient to try to learn more about the dynamics. It is also where a level", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "of interpretability in what has been learnt is important, since the consequence of actions are poten-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "tially very impactful on human lives. As such we focus our evaluation on learning on a real-life", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "healthcare problem, with demonstrations taken from the Medical Information Mart for Intensive", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 405, + 504, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 504, + 417 + ], + "score": 1.0, + "content": "Care (MIMIC-III) dataset (Johnson et al., 2016). The data contains trajectories of patients in inten-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "sive care recording their condition and theraputic interventions at one day intervals. We evaluate the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 504, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 504, + 440 + ], + "score": 1.0, + "content": "ability of the methods to learn a medical policy in both the two and four action setting - specifically", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "whether the patient should be placed on a ventilator, and the decision for ventilation in combination", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "with antibiotic treatment. These represent the two most common, and important, clinical interven-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "tions recorded in the data. Without a recorded notion of reward, performance is measured with", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 471, + 479, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 479, + 483 + ], + "score": 1.0, + "content": "respect to action matching against a held out test set of demonstrations with cross-validation.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 340, + 506, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 488, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "Alongside the healthcare data and for the purposes of demonstrating generalisability, we provide", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "additional results on standard environments of varying complexity in the RL literature, the stan-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 509, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 523 + ], + "score": 1.0, + "content": "dard control problems of: CartPole, a classic control environment aiming to swing up and balance", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "a pendulum; Acrobot, which aims to maintain a sequence of joints above a given height; and Lu-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "score": 1.0, + "content": "narLander, guiding a landing module to a safe touchdown on the moon surface. In these settings", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 543, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 555 + ], + "score": 1.0, + "content": "given sufficient demonstration data all benchmarks are very much capable of reaching demonstrator", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 554, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 566 + ], + "score": 1.0, + "content": "level performance, so we test the algorithms on their ability to handle sample complexity in the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "low data regime by testing their performance when given access to a select number of trajectories", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "which we adjust, replicating the setup in Jarrett et al. (2020). With access to a simulation through", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 586, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 600 + ], + "score": 1.0, + "content": "the OpenAI gym (Brockman et al., 2016), we measure performance by deploying the learnt policies", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 597, + 343, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 343, + 610 + ], + "score": 1.0, + "content": "live and calculating their average return over 300 episodes.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 488, + 506, + 610 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "Benchmarks. 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To provide evidence that we are indeed", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "learning an appropriate reward we show an ablation of our method on the MIMIC data: we take the", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 389, + 422 + ], + "score": 1.0, + "content": "reward learnt by AVRIL and use it as the ‘true’ reward used to train a", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 390, + 411, + 399, + 421 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 399, + 410, + 506, + 422 + ], + "score": 1.0, + "content": "-network offline to learn a", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "policy (A-RL). Note that we have not included previous BIRL methods for the reasons explained", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "in section 2.1, training a network just once in these environments takes in the order of minutes and", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "repeating this sequentially thousands of times is just not practical. For aid in comparison all methods", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "share the same network architecture of two hidden layers of 64 units with ELU activation functions", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "and are trained using Adam (Kingma & Ba, 2014) with learning rates individually tuned. Further", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 476, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 504, + 488 + ], + "score": 1.0, + "content": "details on experimental setup and the implementation of benchmarks can be found in the appendix.", + "type": "text", + "cross_page": true + } + ], + "index": 19 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 622, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 89, + 493, + 182 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 89, + 493, + 182 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 89, + 493, + 182 + ], + "spans": [ + { + "bbox": [ + 108, + 89, + 493, + 182 + ], + "score": 0.963, + "type": "image", + "image_path": "99c662ff38503bdc43e89701fae5833c8fdd14b9ef6e894424ecac8256a1baf1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 89, + 493, + 120.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 120.0, + 493, + 151.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 151.0, + 493, + 182.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 197, + 504, + 220 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "Figure 3: Control environments performance. We plot the average returns received by the poli-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 207, + 502, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 502, + 222 + ], + "score": 1.0, + "content": "cies when deployed live in the environment against the number of trajectories seen during training.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "table", + "bbox": [ + 107, + 275, + 504, + 388 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 228, + 505, + 272 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 228, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 241 + ], + "score": 1.0, + "content": "Table 1: Healthcare performance. Comparison of methods on the MIMIC-III dataset. Performance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "of the policy is evaluated on the quality of action matching against a held out test set of demonstra-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 249, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 249, + 506, + 264 + ], + "score": 1.0, + "content": "tions. We report the accuracy (ACC), area under the receiving operator characteristic curve (AUC)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 261, + 249, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 249, + 274 + ], + "score": 1.0, + "content": "and average precision score (APS).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 275, + 504, + 388 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 275, + 504, + 388 + ], + "spans": [ + { + "bbox": [ + 107, + 275, + 504, + 388 + ], + "score": 0.984, + "html": "
VentilatorVentilator + Antibiotics
MetricACCAUCAPSACCAUCAPS
BC0.873 ± 0.0070.916 ± 0.0020.904±0.0030.700±0.0090.864 ± 0.0030.665 ± 0.009
VDICE0.879 ±0.0020.915 ± 0.0020.904±0.0030.710 ±0.0050.863 ±0.0020.675 ±0.004
RCAL0.870 ± 0.0120.916 ± 0.0030.904±0.0050.702 ±0.0080.865 ± 0.0040.669 ± 0.006
DSFN0.869 ± 0.0050.905 ± 0.0030.885 ±0.0010.683 ± 0.0070.856 ±0.0020.670 ± 0.004
EDM0.882 ± 0.0110.920±0.0020.909 ± 0.0030.716 ±0.0080.873±0.0020.682 ± 0.004
A-RL10.875±0.0100.904±0.0020.927±0.0020.718 ± 0.0100.864±0.0020.665±0.005
AVRIL0.891±0.002 0.917 ± 0.0010.940±0.0010.754± 0.001 0.884 ± 0.000 0.708 ± 0.002
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Note that we have not included previous BIRL methods for the reasons explained", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "in section 2.1, training a network just once in these environments takes in the order of minutes and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "repeating this sequentially thousands of times is just not practical. For aid in comparison all methods", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "share the same network architecture of two hidden layers of 64 units with ELU activation functions", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "and are trained using Adam (Kingma & Ba, 2014) with learning rates individually tuned. Further", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 476, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 504, + 488 + ], + "score": 1.0, + "content": "details on experimental setup and the implementation of benchmarks can be found in the appendix.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 499, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 513 + ], + "score": 1.0, + "content": "Evaluation. We see for all tasks AVRIL learns an appropriate policy that performs strongly across", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "the board, being competitive in all cases and in places beating out all of the other benchmarks. 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AVRIL performs competitively and is easily capable of reaching demonstrator", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "level performance in the samples given for these tasks, though not always as quickly as some of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 565, + 226, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 226, + 577 + ], + "score": 1.0, + "content": "dedicated offline IL methods.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "Reward insight. Remember though that task performance is not exactly our goal. Rather the key", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "score": 1.0, + "content": "aspect of AVRIL is the inference over the unseen reward in order to gain information about the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "score": 1.0, + "content": "preferences of the agent that other black-box policy methods can’t. In the previous experiments our", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "reward encoder was a neural network for maximum flexibility and we can see from the performance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "of A-RL we learn a representation of the reward that can be used to relearn in the environment", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "very effectively, albeit not quite to the same standard of AVRIL. Note this also reflects an original", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "motivation for AVRIL in that offpolicy RL on top of a learnt reward suffers. In figure 4 we explore", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "how to gain more insight from the learnt reward using different parameterisations of the reward. 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We plot the average returns received by the poli-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 207, + 502, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 502, + 222 + ], + "score": 1.0, + "content": "cies when deployed live in the environment against the number of trajectories seen during training.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "table", + "bbox": [ + 107, + 275, + 504, + 388 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 228, + 505, + 272 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 228, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 241 + ], + "score": 1.0, + "content": "Table 1: Healthcare performance. Comparison of methods on the MIMIC-III dataset. Performance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "of the policy is evaluated on the quality of action matching against a held out test set of demonstra-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 249, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 104, + 249, + 506, + 264 + ], + "score": 1.0, + "content": "tions. We report the accuracy (ACC), area under the receiving operator characteristic curve (AUC)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 261, + 249, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 249, + 274 + ], + "score": 1.0, + "content": "and average precision score (APS).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "table_body", + "bbox": [ + 107, + 275, + 504, + 388 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 275, + 504, + 388 + ], + "spans": [ + { + "bbox": [ + 107, + 275, + 504, + 388 + ], + "score": 0.984, + "html": "
VentilatorVentilator + Antibiotics
MetricACCAUCAPSACCAUCAPS
BC0.873 ± 0.0070.916 ± 0.0020.904±0.0030.700±0.0090.864 ± 0.0030.665 ± 0.009
VDICE0.879 ±0.0020.915 ± 0.0020.904±0.0030.710 ±0.0050.863 ±0.0020.675 ±0.004
RCAL0.870 ± 0.0120.916 ± 0.0030.904±0.0050.702 ±0.0080.865 ± 0.0040.669 ± 0.006
DSFN0.869 ± 0.0050.905 ± 0.0030.885 ±0.0010.683 ± 0.0070.856 ±0.0020.670 ± 0.004
EDM0.882 ± 0.0110.920±0.0020.909 ± 0.0030.716 ±0.0080.873±0.0020.682 ± 0.004
A-RL10.875±0.0100.904±0.0020.927±0.0020.718 ± 0.0100.864±0.0020.665±0.005
AVRIL0.891±0.002 0.917 ± 0.0010.940±0.0010.754± 0.001 0.884 ± 0.000 0.708 ± 0.002
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We see for all tasks AVRIL learns an appropriate policy that performs strongly across", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "the board, being competitive in all cases and in places beating out all of the other benchmarks. The", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 520, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 104, + 520, + 505, + 535 + ], + "score": 1.0, + "content": "results for our healthcare example are given in table 1, with AVRIL performing very strongly, having", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 532, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "the highest accuracy and precision score in both tasks. The results for the control environments are", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "shown in figure 3. AVRIL performs competitively and is easily capable of reaching demonstrator", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "level performance in the samples given for these tasks, though not always as quickly as some of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 565, + 226, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 226, + 577 + ], + "score": 1.0, + "content": "dedicated offline IL methods.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 498, + 506, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "Reward insight. Remember though that task performance is not exactly our goal. Rather the key", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 614 + ], + "score": 1.0, + "content": "aspect of AVRIL is the inference over the unseen reward in order to gain information about the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "score": 1.0, + "content": "preferences of the agent that other black-box policy methods can’t. In the previous experiments our", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "reward encoder was a neural network for maximum flexibility and we can see from the performance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "of A-RL we learn a representation of the reward that can be used to relearn in the environment", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "very effectively, albeit not quite to the same standard of AVRIL. Note this also reflects an original", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "motivation for AVRIL in that offpolicy RL on top of a learnt reward suffers. In figure 4 we explore", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "how to gain more insight from the learnt reward using different parameterisations of the reward. 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While this is intuitive we still have to query a neural network repeatedly over the state", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "space to gain insight, the bottom graph of figure 4 presents then a simpler but perhaps more useful", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 104, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "representation. In this case we learn a state-only reward as before but as a linear model. This is", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 217, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 228 + ], + "score": 1.0, + "content": "not as strong a constraint on the policy since that is still free to be non-linear as a neural network", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 240 + ], + "score": 1.0, + "content": "but simultaneously allows us the insight of what our model considers high value in the environment", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 239, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 250 + ], + "score": 1.0, + "content": "as we plot the relative model coefficients for each covariate. 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Scaled heat-maps of: the ground truth reward; the relative state", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "occupancy of the expert demonstrations; the reward posterior mean; and reward standard deviation.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 205, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 104, + 205, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 104, + 205, + 506, + 217 + ], + "score": 1.0, + "content": "representation. In this case we learn a state-only reward as before but as a linear model. 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We can see here for example that the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "biggest impact on the overall estimated quality of a state is given by blood pressure, well known as", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 260, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 273 + ], + "score": 1.0, + "content": "an important indicator of health (Hepworth et al., 1994), strongly impacted by trauma and infection.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 356, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 356, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 356, + 296 + ], + "score": 1.0, + "content": "Gridworld ground-truth comparison While environments", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 295, + 357, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 357, + 308 + ], + "score": 1.0, + "content": "like MIMIC are the main focus of this work they do not", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 306, + 357, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 357, + 318 + ], + "score": 1.0, + "content": "lend them selves to inspection of the uncovered reward as the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 318, + 357, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 357, + 329 + ], + "score": 1.0, + "content": "ground truth simply is not available to us. We thus demonstrate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 329, + 356, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 356, + 340 + ], + "score": 1.0, + "content": "on a toy gridworld environment, in order to clearly see the ef-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 339, + 357, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 357, + 352 + ], + "score": 1.0, + "content": "fect of learning a posterior distribution over the reward. In this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 350, + 357, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 357, + 362 + ], + "score": 1.0, + "content": "(finite) example both the encoder and decoder are represented", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 362, + 356, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 356, + 373 + ], + "score": 1.0, + "content": "by tensors but otherwise the procedure remains the same. 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We have focused on simply obtaining the appropriate uncertainty over reward as", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 613 + ], + "score": 1.0, + "content": "well as imitation in high stakes environments - in these settings it is crucial that learnt policies avoid", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "catastrophic failure and so how exactly to use the uncertainty in order to achieve truly safe imitation", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 623, + 418, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 418, + 636 + ], + "score": 1.0, + "content": "(or indeed better-that-demonstrator apprenticeship) is increasingly of interest.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44 + }, + { + "type": "title", + "bbox": [ + 108, + 652, + 224, + 664 + ], + "lines": [ + { + "bbox": [ + 107, + 651, + 226, + 667 + ], + "spans": [ + { + "bbox": [ + 107, + 651, + 226, + 667 + ], + "score": 1.0, + "content": "ACKNOWLEDGEMENTS", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "AJC would like to acknowledge and thank Microsoft Research for its support through its PhD Schol-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "arship Program with the EPSRC. This work was additionally supported by the Office of Naval Re-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "search (ONR) and the NSF (Grant number: 1722516). 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This work was additionally supported by the Office of Naval Re-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "search (ONR) and the NSF (Grant number: 1722516). We would like to thank all of the anonymous", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "reviewers on OpenReview, alongside the many members of the van der Schaar lab, for their input,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 472, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 472, + 734 + ], + "score": 1.0, + "content": "comments, and suggestions at various stages that have ultimately improved the manuscript.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 677, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 176, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 123 + ], + "lines": [ + { + "bbox": [ + 106, + 100, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 217, + 113 + ], + "score": 1.0, + "content": "Pieter Abbeel and Andrew", + "type": "text" + }, + { + "bbox": [ + 217, + 100, + 241, + 112 + ], + "score": 0.28, + "content": "\\mathrm { ~ Y ~ N ~ g ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 100, + 505, + 113 + ], + "score": 1.0, + "content": ". 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Assuming the constraint is satisfied, we are maximising the following", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 555, + 149, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 149, + 568 + ], + "score": 1.0, + "content": "objective:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 541, + 506, + 568 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 126, + 577, + 467, + 610 + ], + "lines": [ + { + "bbox": [ + 126, + 577, + 467, + 610 + ], + "spans": [ + { + "bbox": [ + 126, + 577, + 467, + 610 + ], + "score": 0.92, + "content": "\\mathcal { F } ( \\phi , \\theta ) = \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal { D } } \\log \\frac { \\exp \\beta Q _ { \\theta } ( s , a ) ) } { \\sum _ { b \\in \\mathcal { A } } \\exp ( \\beta Q _ { \\theta } ( s , b ) ) } - D _ { K L } \\big ( q _ { \\phi } ( R ( s , a ) ) | | p ( R ( s , a ) ) \\big )", + "type": "interline_equation", + "image_path": "6cc3bd8ee994e7067352680b1c0f4e7bf683ad55e33076f9db973e6035afd169.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 126, + 577, + 467, + 588.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 126, + 588.0, + 467, + 599.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 126, + 599.0, + 467, + 610.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 320, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 619, + 321, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 321, + 635 + ], + "score": 1.0, + "content": "Which is equivalent to minimising the negative value", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 619, + 321, + 635 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 643, + 473, + 688 + ], + "lines": [ + { + "bbox": [ + 120, + 643, + 473, + 688 + ], + "spans": [ + { + "bbox": [ + 120, + 643, + 473, + 688 + ], + "score": 0.93, + "content": "\\mathcal { F } ( \\phi , \\theta ) = \\sum _ { ( s , a , s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal { D } } - \\log \\frac { \\exp \\beta Q _ { \\theta } ( s , a ) ) } { \\sum _ { b \\in \\mathcal { A } } \\exp ( \\beta Q _ { \\theta } ( s , b ) ) } + \\underbrace { D _ { K L } \\big ( q _ { \\phi } ( R ( s , a ) ) | | p ( R ( s , a ) ) \\big ) } _ { \\mathcal { L } _ { r e g } } ,", + "type": "interline_equation", + "image_path": "6e760f31446e4ca138988bf7ec1f10f266bed97384094a2066f4c54c88f1c129.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 120, + 643, + 473, + 658.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 120, + 658.0, + 473, + 673.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 120, + 673.0, + 473, + 688.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 184, + 712 + ], + "score": 1.0, + "content": "with the first term", + "type": "text" + }, + { + "bbox": [ + 184, + 699, + 205, + 710 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { B C }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "being the negative log-likelihood of the data and the classic behavioural", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 366, + 723 + ], + "score": 1.0, + "content": "cloning objective. 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VentilatorVentilator + Antibiotics
MetricACCAUCAPSACCAUCAPS
BC0.873 ± 0.0070.916 ± 0.0020.904±0.0030.700±0.0090.864 ± 0.0030.665 ± 0.009
VDICE0.879 ±0.0020.915 ± 0.0020.904±0.0030.710 ±0.0050.863 ±0.0020.675 ±0.004
RCAL0.870 ± 0.0120.916 ± 0.0030.904±0.0050.702 ±0.0080.865 ± 0.0040.669 ± 0.006
DSFN0.869 ± 0.0050.905 ± 0.0030.885 ±0.0010.683 ± 0.0070.856 ±0.0020.670 ± 0.004
EDM0.882 ± 0.0110.920±0.0020.909 ± 0.0030.716 ±0.0080.873±0.0020.682 ± 0.004
A-RL10.875±0.0100.904±0.0020.927±0.0020.718 ± 0.0100.864±0.0020.665±0.005
AVRIL0.891±0.002 0.917 ± 0.0010.940±0.0010.754± 0.001 0.884 ± 0.000 0.708 ± 0.002
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0000000000000000000000000000000000000000..3438973f37db03b5c8996bd2cfecc29c6988d649 --- /dev/null +++ b/parse/train/BJIgi_eCZ/BJIgi_eCZ_content_list.json @@ -0,0 +1,3287 @@ +[ + { + "type": "text", + "text": "FUSIONNET: FUSING VIA FULLY-AWARE ATTENTION WITH APPLICATION TO MACHINE COMPREHENSION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Hsin-Yuan Huang\\*1,2, Chenguang $\\mathbf { Z } \\mathbf { h } \\mathbf { u } ^ { \\mathbf { 1 } }$ , Yelong Shen1, Weizhu Chen1 ", + "bbox": [ + 186, + 169, + 668, + 185 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1Microsoft Business AI and Research \n2National Taiwan University \nmomohuang@gmail.com, {chezhu,yeshen,wzchen}@microsoft.com ", + "bbox": [ + 181, + 186, + 740, + 228 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 265, + 544, + 280 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This paper introduces a new neural structure called FusionNet, which extends existing attention approaches from three perspectives. First, it puts forward a novel concept of “history of word” to characterize attention information from the lowest word-level embedding up to the highest semantic-level representation. Second, it identifies an attention scoring function that better utilizes the “history of word” concept. Third, it proposes a fully-aware multi-level attention mechanism to capture the complete information in one text (such as a question) and exploit it in its counterpart (such as context or passage) layer by layer. We apply FusionNet to the Stanford Question Answering Dataset (SQuAD) and it achieves the first position for both single and ensemble model on the official SQuAD leaderboard at the time of writing (Oct. 4th, 2017). Meanwhile, we verify the generalization of FusionNet with two adversarial SQuAD datasets and it sets up the new state-of-the-art on both datasets: on AddSent, FusionNet increases the best F1 metric from $4 6 . 6 \\%$ to $5 1 . 4 \\%$ ; on AddOneSent, FusionNet boosts the best F1 metric from $5 6 . 0 \\%$ to $6 0 . 7 \\%$ . ", + "bbox": [ + 233, + 296, + 766, + 502 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 530, + 336, + 546 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Teaching machines to read, process and comprehend text and then answer questions is one of key problems in artificial intelligence. Figure 1 gives an example of the machine reading comprehension task. It feeds a machine with a piece of context and a question and teaches it to find a correct answer to the question. This requires the machine to possess high capabilities in comprehension, inference and reasoning. This is considered a challenging task in artificial intelligence and has already attracted numerous research efforts from the neural network and natural language processing communities. Many neural network models have been proposed for this challenge and they generally frame this problem as a machine reading comprehension (MRC) task (Hochreiter & Schmidhuber, 1997; Wang et al., 2017; Seo et al., 2017; Shen et al., 2017; Xiong et al., 2017; Weissenborn et al., 2017; Chen et al., 2017a). ", + "bbox": [ + 174, + 561, + 516, + 810 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Context: The Alpine Rhine is part of the Rhine, a famous European river. The Alpine Rhine begins in the most western part of the Swiss canton of Graubünden, and later forms the border between Switzerland to the West and Liechtenstein and later Austria to the East. On the other hand, the Danube separates Romania and Bulgaria. ", + "bbox": [ + 544, + 587, + 808, + 674 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Question: What is the other country the Rhine separates Switzerland to? ", + "bbox": [ + 544, + 681, + 789, + 705 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Answer: Liechtenstein ", + "bbox": [ + 542, + 713, + 665, + 723 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The key innovation in recent models lies in how to ingest information in the question and characterize it in the context, in order to provide an accurate answer to the question. This is often modeled as attention in the neural network community, which is a mechanism to attend the question into the context so as to find the answer related to the question. Some (Chen et al., 2017a; Weissenborn et al., 2017) attend the word-level embedding from the question to context, while some (Wang et al., 2017) attend the high-level representation in the question to augment the context. However we observed that none of the existing approaches has captured the full information in the context or the question, which could be vital for complete information comprehension. Taking image recognition as an example, information in various levels of representations can capture different aspects of details in an image: pixel, stroke and shape. We argue that this hypothesis also holds in language understanding and MRC. In other words, an approach that utilizes all the information from the word embedding level up to the highest level representation would be substantially beneficial for understanding both the question and the context, hence yielding more accurate answers. ", + "bbox": [ + 173, + 818, + 825, + 901 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 200 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "However, the ability to consider all layers of representation is often limited by the difficulty to make the neural model learn well, as model complexity will surge beyond capacity. We conjectured this is why previous literature tailored their models to only consider partial information. To alleviate this challenge, we identify an attention scoring function utilizing all layers of representation with less training burden. This leads to an attention that thoroughly captures the complete information between the question and the context. With this fully-aware attention, we put forward a multi-level attention mechanism to understand the information in the question, and exploit it layer by layer on the context side. All of these innovations are integrated into a new end-to-end structure called FusionNet in Figure 4, with details described in Section 3. ", + "bbox": [ + 174, + 208, + 825, + 333 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We submitted FusionNet to SQuAD (Rajpurkar et al., 2016), a machine reading comprehension dataset. At the time of writing (Oct. 4th, 2017), our model ranked in the first place in both single model and ensemble model categories. The ensemble model achieves an exact match (EM) score of $78 . 8 \\%$ and F1 score of $8 5 . 9 \\%$ . Furthermore, we have tested FusionNet against adversarial SQuAD datasets (Jia & Liang, 2017). Results show that FusionNet outperforms existing state-of-the-art architectures in both datasets: on AddSent, FusionNet increases the best F1 metric from $4 6 . 6 \\%$ to $5 1 . 4 \\%$ ; on AddOneSent, FusionNet boosts the best F1 metric from $5 6 . 0 \\%$ to $6 0 . 7 \\%$ . In Appendix D, we also applied to natural language inference task and shown decent improvement. This demonstrated the exceptional performance of FusionNet. An open-source implementation of FusionNet can be found at https://github.com/momohuang/FusionNet-NLI. ", + "bbox": [ + 174, + 339, + 825, + 478 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 MACHINE COMPREHENSION & FULLY-AWARE ATTENTION ", + "text_level": 1, + "bbox": [ + 176, + 501, + 692, + 517 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this section, we briefly introduce the task of machine comprehension as well as a conceptual architecture that summarizes recent advances in machine reading comprehension. Then, we introduce a novel concept called history-of-word. History-of-word can capture different levels of contextual information to fully understand the text. Finally, a light-weight implementation for history-of-word, Fully-Aware Attention, is proposed. ", + "bbox": [ + 174, + 532, + 825, + 603 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 TASK DESCRIPTION ", + "text_level": 1, + "bbox": [ + 176, + 621, + 352, + 635 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In machine comprehension, given a context and a question, the machine needs to read and understand the context, and then find the answer to the question. The context is described as a sequence of word tokens: $C ~ = ~ \\{ w _ { 1 } ^ { C } , \\ldots , w _ { m } ^ { C } \\}$ , and the question as: $\\textit { \\textbf { Q } } = \\ \\{ w _ { 1 } ^ { Q } , \\ldots , w _ { n } ^ { Q } \\}$ , where $m$ is the number of words in the context, and $n$ is the number of words in the question. In general, $m \\gg n$ . The answer Ans can have different forms depending on the task. In the SQuAD dataset (Rajpurkar et al., 2016), the answer Ans is guaranteed to be a contiguous span in the context $C$ , e.g., $\\mathbf { A n s } = \\{ w _ { i } ^ { C } , \\dots , w _ { i + k } ^ { C } \\}$ , where $k$ is the number of words in the answer and $k \\leq m$ . ", + "bbox": [ + 174, + 647, + 825, + 747 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 CONCEPTUAL ARCHITECTURE FOR MACHINE READING COMPREHENSION ", + "text_level": 1, + "bbox": [ + 176, + 765, + 727, + 780 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In all state-of-the-art architectures for machine reading comprehension, a recurring pattern is the following process. Given two sets of vectors, A and B, we enhance or modify every single vector in set A with the information from set B. We call this a fusion process, where set B is fused into set A. Fusion processes are commonly based on attention (Bahdanau et al., 2015), but some are not. Major improvements in recent MRC work lie in how the fusion process is designed. ", + "bbox": [ + 174, + 791, + 825, + 861 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A conceptual architecture illustrating state-of-the-art architectures is shown in Figure 2, which consists of three components. ", + "bbox": [ + 173, + 868, + 823, + 897 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• Input vectors: Embedding vectors for each word in the context and the question. ", + "bbox": [ + 214, + 909, + 754, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/ef53a9d62835839db03dd60fa5bf44c53d378371d9b1d0cebb4360838ecc5ed3.jpg", + "image_caption": [ + "Table 1: A summarized view on the fusion processes used in several state-of-the-art architectures. ", + "Figure 2: A conceptual architecture illustrating recent advances in MRC. " + ], + "image_footnote": [], + "bbox": [ + 173, + 119, + 828, + 309 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Integration components: The rectangular box. It is usually implemented using an RNN such as an LSTM (Hochreiter & Schmidhuber, 1997) or a GRU (Cho et al., 2014). • Fusion processes: The numbered arrows (1), (2), (2’), (3), $( 3 ^ { \\circ } )$ . The set pointing outward is fused into the set being pointed to. ", + "bbox": [ + 214, + 373, + 825, + 435 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "There are three main types of fusion processes in recent advanced architectures. Table 1 shows what fusion processes are used in different state-of-the-art architectures. We now discuss them in detail. ", + "bbox": [ + 176, + 446, + 823, + 474 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "(1) Word-level fusion. By providing the direct word information in question to the context, we can quickly zoom in to more related regions in the context. However, it may not be helpful if a word has different semantic meaning based on the context. Many word-level fusions are not based on attention, e.g., (Hu et al., 2017; Chen et al., 2017a) appends binary features to context words, indicating whether each context word appears in the question. ", + "bbox": [ + 174, + 482, + 825, + 553 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "(2) High-level fusion. Informing the context about the semantic information in the question could help us find the correct answer. But high-level information is more imprecise than word information, which may cause models to be less aware of details. ", + "bbox": [ + 174, + 559, + 825, + 601 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "(2’) High-level fusion (Alternative). Similarly, we could also fuse high-level concept of $Q$ into the word-level of $C$ . ", + "bbox": [ + 176, + 607, + 821, + 636 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "(3) Self-boosted fusion. Since the context can be long and distant parts of text may rely on each other to fully understand the content, recent advances have proposed to fuse the context into itself. As the context contains excessive information, one common choice is to perform self-boosted fusion after fusing the question $Q$ . This allows us to be more aware of the regions related to the question. ", + "bbox": [ + 174, + 643, + 825, + 699 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "(3’) Self-boosted fusion (Alternative). Another choice is to directly condition the self-boosted fusion process on the question $Q$ , such as the coattention mechanism proposed in (Xiong et al., 2017). Then we can perform self-boosted fusion before fusing question information. ", + "bbox": [ + 174, + 705, + 825, + 748 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A common trait of existing fusion mechanisms is that none of them employs all levels of representation jointly. In the following, we claim that employing all levels of representation is crucial to achieving better text understanding. ", + "bbox": [ + 174, + 755, + 825, + 797 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.3 FULLY-AWARE ATTENTION ON HISTORY OF WORD ", + "text_level": 1, + "bbox": [ + 174, + 814, + 566, + 829 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Consider the illustration shown in Figure 3. As we read through the context, each input word will gradually transform into a more abstract representation, e.g., from low-level to high-level concepts. Altogether, they form the history of each word in our mental flow. For a human, we utilize the history-of-word so frequently but we often neglect its importance. For example, to answer the question in Figure 3 correctly, we need to focus on both the high-level concept of forms the border and the word-level information of Alpine Rhine. If we focus only on the high-level concepts, we will ", + "bbox": [ + 174, + 839, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Context: The Alpine Rhine is part of the Rhine, a famous European river. The Alpine Rhine begins in the most western part of the Swiss canton of Graubünden, and later forms the border between Switzerland to the West and Liechtenstein and later Austria to the East. On the other hand, the Danube separates Romania and Bulgaria. ", + "bbox": [ + 184, + 109, + 450, + 195 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Question: What is the other country the Rhine separates Switzerland to? ", + "bbox": [ + 186, + 202, + 429, + 226 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Answer: Liechtenstein ", + "bbox": [ + 186, + 232, + 308, + 243 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/d3b58b3a8ba2ca6de01ca0430c65b35e60ff963f937f77dd27b865c4cf91802f.jpg", + "image_caption": [ + "Figure 3: Illustrations of the history-of-word for the example shown in Figure 1. Utilizing the entire history-of-word is crucial for the full understanding of the context. " + ], + "image_footnote": [], + "bbox": [ + 457, + 103, + 816, + 244 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "confuse Alpine Rhine with Danube since both are European rivers that separate countries. Therefore we hypothesize that the entire history-of-word is important to fully understand the text. ", + "bbox": [ + 171, + 314, + 823, + 342 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In neural architectures, we define the history of the $i$ -th word, $\\mathrm { H o W } _ { i }$ , to be the concatenation of all the representations generated for this word. This may include word embedding, multiple intermediate and output hidden vectors in RNN, and corresponding representation vectors in any further layers. To incorporate history-of-word into a wide range of neural models, we present a lightweight implementation we call Fully-Aware Attention. ", + "bbox": [ + 173, + 348, + 825, + 419 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Attention can be applied to different scenarios. To be more conclusive, we focus on attention applied to fusing informatiotext bodies A and B: $\\{ h _ { 1 } ^ { A } , \\ldots , h _ { m } ^ { A } \\}$ y, $\\{ h _ { 1 } ^ { B } , \\ldots , h _ { n } ^ { B } \\} \\subset \\mathbb { R } ^ { d }$ two sets of hidden vectors for words in. Their associated history-of-word are, $\\{ \\mathrm { H o W } _ { 1 } ^ { A } , \\dots , \\mathrm { H o W } _ { m } ^ { A } \\} , ~ \\{ \\mathrm { H o W } _ { 1 } ^ { B } , \\dots , \\mathrm { H o W } _ { n } ^ { B } \\} \\subset \\mathbb { R } ^ { d _ { h } } ,$ ", + "bbox": [ + 174, + 425, + 826, + 491 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $d _ { h } \\gg d$ . Fusing body $\\mathbf { B }$ to body A via standard attention means for every $ { \\boldsymbol { h } } _ { i } ^ { A }$ in body A, ", + "bbox": [ + 171, + 494, + 797, + 510 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1. Compute an attention score $S _ { i j } = S ( \\pmb { h } _ { i } ^ { A } , \\pmb { h } _ { j } ^ { B } ) \\in \\mathbb { R }$ for each $h _ { j } ^ { B }$ in body $\\mathbf { B }$ . \n2. Form the attention weight $\\alpha _ { i j }$ through softmax: $\\begin{array} { r } { \\alpha _ { i j } = \\exp ( S _ { i j } ) / \\sum _ { k } \\exp ( S _ { i k } ) } \\end{array}$ . \n3. Concatenate $ { \\boldsymbol { h } } _ { i } ^ { A }$ with the summarized information, $\\begin{array} { r } { \\hat { \\pmb { h } } _ { i } ^ { A } = \\sum _ { j } \\alpha _ { i j } \\pmb { h } _ { j } ^ { B } } \\end{array}$ . ", + "bbox": [ + 207, + 517, + 758, + 582 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In fully-aware attention, we replace attention score computation with the history-of-word. ", + "bbox": [ + 169, + 589, + 759, + 604 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8f754747bf68aa99deea077be3cd6b21b01eee49ec5f520f0d6c69357909b636.jpg", + "text": "$$\nS ( h _ { i } ^ { A } , h _ { j } ^ { B } ) \\implies S ( \\mathrm { H o W } _ { i } ^ { A } , \\mathrm { H o W } _ { j } ^ { B } ) .\n$$", + "text_format": "latex", + "bbox": [ + 370, + 606, + 620, + 626 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This allows us to be fully aware of the complete understanding of each word. The ablation study in Section 4.4 demonstrates that this lightweight enhancement offers a decent improvement in performance. ", + "bbox": [ + 176, + 627, + 823, + 669 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To fully utilize history-of-word in attention, we need a suitable attention scoring function $S ( { \\pmb x } , { \\pmb y } )$ . A commonly used function is multiplicative attention (Britz et al., 2017): $\\pmb { x } ^ { T } \\pmb { U } ^ { T } \\pmb { V } \\pmb { y }$ , leading to ", + "bbox": [ + 171, + 675, + 823, + 704 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e01cb8af491fe3abe8c117ad5a86b394bd740741198cac28d6b2c308c41f6ef1.jpg", + "text": "$$\n\\begin{array} { r } { S _ { i j } = ( \\mathrm { H o W } _ { i } ^ { A } ) ^ { T } U ^ { T } V ( \\mathrm { H o W } _ { j } ^ { B } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 388, + 705, + 606, + 726 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $U , V ~ \\in ~ \\mathbb { R } ^ { k \\times d _ { h } }$ , and $k$ is the attention hidden size. However, we suspect that two large matrices interacting directly will make the neural model harder to train. Therefore we propose to constrain the matrix $U ^ { T } V$ to be symmetric. A symmetric matrix can always be decomposed into $U ^ { T } D U$ , thus ", + "bbox": [ + 173, + 727, + 825, + 782 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b5ead9f5bb6b356668e104a50bcd1d982c4e29334a47948b75e3b3b7c14fb1ce.jpg", + "text": "$$\n\\begin{array} { r } { S _ { i j } = ( \\mathrm { H o W } _ { i } ^ { A } ) ^ { T } U ^ { T } D U ( \\mathrm { H o W } _ { j } ^ { B } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 382, + 780, + 614, + 801 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $U \\in \\mathbb { R } ^ { k \\times d _ { h } }$ , $D \\in \\mathbb { R } ^ { k \\times k }$ and $D$ is a diagonal matrix. The symmetric form retains the ability to give high attention score between dissimilar $\\mathrm { \\bar { H } o W } _ { i } ^ { A } , \\mathrm { H o W } _ { j } ^ { B }$ . Additionally, we marry nonlinearity with the symmetric form to provide richer interaction among different parts of the history-of-word. The final formulation for attention score is ", + "bbox": [ + 173, + 801, + 825, + 859 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/82ab4e2788c5f3d7d10613ad43f37e3f3850b709a609d4a3926df5ac3fa925b3.jpg", + "text": "$$\nS _ { i j } = f ( U ( \\mathrm { H o W } _ { i } ^ { A } ) ) ^ { T } D ~ f ( U ( \\mathrm { H o W } _ { j } ^ { B } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 362, + 859, + 632, + 880 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $f ( x )$ is an activation function applied element-wise. In the following context, we employ $f ( x ) = \\operatorname* { m a x } ( 0 , x )$ . A detailed ablation study in Section 4 demonstrates its advantage over many alternatives. ", + "bbox": [ + 174, + 882, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/0257ec1f4f3ec3d036b2ec9c1a8204d14a2dfc7469e37a1de56bf42b7d5b2f0c.jpg", + "image_caption": [ + "Figure 4: An illustration of FusionNet architecture. Each upward arrow represents one layer of BiLSTM. Each circle to the right is a detailed illustration of the corresponding component in FusionNet. Circle 1: Fully-aware attention between $C$ and $Q$ . Illustration of Equation (C1) in Section 3.1. Circle 2: Concatenate all concepts in $C$ with multi-level $Q$ information, then pass through BiLSTM. Illustration of Equation (C2) in Section 3.1. " + ], + "image_footnote": [], + "bbox": [ + 176, + 101, + 823, + 422 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Circle 3: Fully-aware attention on the context $C$ itself. Illustration of Equation (C3) in Section 3.1. Circle 4: Concatenate the understanding vector of $C$ with self-attention information, then pass through BiLSTM. Illustration of Equation (C4) in Section 3.1. ", + "bbox": [ + 174, + 506, + 823, + 547 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 FULLY-AWARE FUSION NETWORK ", + "text_level": 1, + "bbox": [ + 174, + 574, + 490, + 590 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 END-TO-END ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 606, + 421, + 621 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Based on fully-aware attention, we propose an end-to-end architecture: the fully-aware fusion network (FusionNet). Given text A and B, FusionNet fuses information from text $\\mathbf { B }$ to text A and generates two set of vectors ", + "bbox": [ + 173, + 632, + 825, + 675 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/32b0f646184d694bd256b68772f99e90452dfdcdf3152b72857259601481d57c.jpg", + "text": "$$\nU _ { A } = \\{ \\pmb { u } _ { 1 } ^ { A } , \\ldots , \\pmb { u } _ { m } ^ { A } \\} , \\quad U _ { B } = \\{ \\pmb { u } _ { 1 } ^ { B } , \\ldots , \\pmb { u } _ { n } ^ { B } \\} .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 683, + 655, + 702 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In the following, we consider the special case where text A is context $C$ and text $\\mathbf { B }$ is question $Q$ . \nAn illustration for FusionNet is shown in Figure 4. It consists of the following components. ", + "bbox": [ + 174, + 709, + 823, + 738 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Input Vectors. First, each word in $C$ and $Q$ is transformed into an input vector $\\textbf { \\em w }$ . We utilize the 300-dim GloVe embedding (Pennington et al., 2014) and 600-dim contextualized vector (McCann et al., 2017). In the SQuAD task, we also include 12-dim POS embedding, 8-dim NER embedding and a normalized term frequency for context $C$ as suggested in (Chen et al., 2017a). Together $\\{ \\pmb { w } _ { 1 } ^ { C } , \\ldots , \\pmb { w } _ { m } ^ { C } \\} \\subset \\mathbb { R } ^ { 9 0 0 + 2 0 + 1 }$ , and $\\{ \\pmb { w } _ { 1 } ^ { Q } , \\ldots , \\pmb { w } _ { n } ^ { Q } \\} \\subset \\mathbb { R } ^ { 9 0 0 }$ . ", + "bbox": [ + 173, + 743, + 825, + 819 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Fully-Aware Multi-level Fusion: Word-level. In multi-level fusion, we separately consider fusing word-level and higher-level. Word-level fusion informs $C$ about what kind of words are in $Q$ . It is illustrated as arrow (1) in Figure 2. For this component, we follow the approach in (Chen et al., 2017a) First, a feature vector $\\mathrm { e m } _ { i }$ is created for each word in $C$ to indicate whether the word occurs in the question $Q$ . Second, attention-based fusion on GloVe embedding $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { g } } _ { i } }$ is used ", + "bbox": [ + 173, + 823, + 825, + 893 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/99dc2ff21714f858c7d0fbf27c6f23cc2ca2e5bae434cba56a9ca5f79f5b4ec2.jpg", + "text": "$$\n\\hat { g } _ { i } ^ { C } = \\sum _ { j } \\alpha _ { i j } g _ { j } ^ { Q } , \\quad \\alpha _ { i j } \\propto \\exp ( S ( g _ { i } ^ { C } , g _ { j } ^ { Q } ) ) , \\quad S ( x , y ) = \\mathrm { R e L U } ( W x ) ^ { T } \\mathrm { R e L U } ( W y ) ,\n$$", + "text_format": "latex", + "bbox": [ + 207, + 901, + 787, + 929 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $W \\in \\mathbb { R } ^ { 3 0 0 \\times 3 0 0 }$ . Since history-of-word is the input vector itself, fully-aware attention is not employed here. The enhanced input vector for context is $\\mathbf { \\tilde { w } } _ { i } ^ { C } = [ \\mathbf { w } _ { i } ^ { C } ; \\mathrm { e m } _ { i } ; \\bar { \\mathbf { g } } _ { i } ^ { C } ]$ . ", + "bbox": [ + 171, + 102, + 825, + 135 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Reading. In the reading component, we use a separate bidirectional LSTM (BiLSTM) to form low-level and high-level concepts for $C$ and $Q$ . ", + "bbox": [ + 169, + 140, + 825, + 170 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/d1fdcaed4618992b1457bd199f2eb1bcec31665ed1d897bac0ee6a19d9a967f2.jpg", + "text": "$$\n{ h } _ { 1 } ^ { C l } , \\dots , { h } _ { m } ^ { C l } = \\mathrm { B i L S T M } ( \\tilde { w } _ { 1 } ^ { C } , \\dots , \\tilde { w } _ { m } ^ { C } ) , \\quad { h } _ { 1 } ^ { Q l } , \\dots , { h } _ { n } ^ { Q l } = \\mathrm { B i L S T M } ( w _ { 1 } ^ { Q } , \\dots , w _ { n } ^ { Q } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 210, + 176, + 784, + 198 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/580300d31cea37ed2418c9aff7009501f7fa5b3971052292a011b0bb5746ae00.jpg", + "text": "$$\n{ h } _ { 1 } ^ { C h } , \\ldots , { h } _ { m } ^ { C h } = \\mathrm { B i L S T M } ( { h } _ { 1 } ^ { C l } , \\ldots , { h } _ { m } ^ { C l } ) , \\quad { h } _ { 1 } ^ { Q h } , \\ldots , { h } _ { n } ^ { Q h } = \\mathrm { B i L S T M } ( { h } _ { 1 } ^ { Q l } , \\ldots , { h } _ { n } ^ { Q l } ) .\n$$", + "text_format": "latex", + "bbox": [ + 204, + 205, + 790, + 227 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Hence low-level and high-level concepts $h ^ { l } , h ^ { h } \\in \\mathbb { R } ^ { 2 5 0 }$ are created for each word. ", + "bbox": [ + 174, + 231, + 714, + 248 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Question Understanding. In the Question Understanding component, we apply a new BiLSTM taking in both ${ h ^ { Q l } , h ^ { Q h } }$ to obtain the final question representation $U _ { Q }$ : ", + "bbox": [ + 173, + 253, + 823, + 286 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/4add0719b763e3c73ec77b317812917d3a81a8ba156c40742973cc6b52a597b3.jpg", + "text": "$$\n\\begin{array} { r } { U _ { Q } = \\{ \\boldsymbol { { u } } _ { 1 } ^ { Q } , \\ldots , \\boldsymbol { { u } } _ { n } ^ { Q } \\} = \\mathrm { { B i L S T M } } ( [ \\boldsymbol { h } _ { 1 } ^ { Q l } ; \\boldsymbol { h } _ { 1 } ^ { Q h } ] , \\ldots , [ \\boldsymbol { h } _ { n } ^ { Q l } ; \\boldsymbol { h } _ { n } ^ { Q h } ] ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 292, + 710, + 314 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\{ \\boldsymbol { u } _ { i } ^ { Q } \\in \\mathbb { R } ^ { 2 5 0 } \\} _ { i = 1 } ^ { n }$ are the understanding vectors for $Q$ ", + "bbox": [ + 174, + 323, + 573, + 340 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Fully-Aware Multi-level Fusion: Higher-level. This component fuses all higher-level information in the question $Q$ to the context $C$ through fully-aware attention on history-of-word. Since the proposed attention scoring function for fully-aware attention is constrained to be symmetric, we need to identify the common history-of-word for both $C , Q$ . This yields ", + "bbox": [ + 173, + 344, + 826, + 402 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/d809ab78d0fe1e1f9c75886019e208f842da6ed276613a87869bb720655286a0.jpg", + "text": "$$\n\\mathrm { H o W } _ { i } ^ { C } = [ \\pmb { g } _ { i } ^ { C } ; \\pmb { c } _ { i } ^ { C } ; \\pmb { h } _ { i } ^ { C l } ; \\pmb { h } _ { i } ^ { C h } ] , ~ \\mathrm { H o W } _ { i } ^ { Q } = [ \\pmb { g } _ { i } ^ { Q } ; \\pmb { c } _ { i } ^ { Q } ; \\pmb { h } _ { i } ^ { Q l } ; \\pmb { h } _ { i } ^ { Q h } ] \\in \\mathbb { R } ^ { 1 4 0 0 } ,\n$$", + "text_format": "latex", + "bbox": [ + 259, + 410, + 735, + 430 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\mathbf { \\pmb { g } } _ { i }$ is the GloVe embedding and $c _ { i }$ is the CoVe embedding. Then we fuse low, high, and understanding-level information from $Q$ to $C$ via fully-aware attention. Different sets of attention weights are calculated through attention function $S ^ { \\tilde { l } } ( x , y ) , S ^ { h } ( x , y ) , S ^ { u } ( x , y )$ to combine low, high, and understanding-level of concepts. All three functions are the proposed symmetric form with nonlinearity in Section 2.3, but are parametrized by independent parameters to attend to different regions for different level. Attention hidden size is set to be $k = 2 5 0$ . ", + "bbox": [ + 173, + 436, + 826, + 522 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/4988641819727045586758b826efd980f8a73a8d9185a60e97e39f222ad37989.jpg", + "text": "$$\n\\begin{array} { r } { \\hat { \\boldsymbol { h } } _ { i } ^ { C l } = \\sum _ { j } \\alpha _ { i j } ^ { l } \\boldsymbol { h } _ { j } ^ { Q l } , \\quad \\alpha _ { i j } ^ { l } \\propto \\exp ( S ^ { l } ( \\mathrm { H o W } _ { i } ^ { C } , \\mathrm { H o W } _ { j } ^ { Q } ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 351, + 534, + 720, + 558 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "2. High-level fusion: $\\begin{array} { r } { \\hat { \\boldsymbol { \\mathsf { h } } } _ { i } ^ { C h } = \\sum _ { j } \\alpha _ { i j } ^ { h } \\boldsymbol { h } _ { j } ^ { Q h } , \\quad \\alpha _ { i j } ^ { h } \\propto \\exp ( S ^ { h } ( \\mathrm { H o W } _ { i } ^ { C } , \\mathrm { H o W } _ { j } ^ { Q } ) ) . } \\end{array}$ ", + "bbox": [ + 210, + 561, + 743, + 585 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This multi-level attention mechanism captures different levels of information independently, while taking all levels of information into account. A new BiLSTM is applied to obtain the representation for $C$ fully fused with information in the question $Q$ : ", + "bbox": [ + 176, + 621, + 821, + 664 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/8dcce15acb993a32a6ffe25437c7b96c47d8ea9ad9c8a3427ed97b9b25ec9d52.jpg", + "text": "$$\n\\{ \\pmb { v } _ { 1 } ^ { C } , \\ldots , \\pmb { v } _ { m } ^ { C } \\} = \\mathrm { B i L S T M } ( [ { \\pmb { h } } _ { 1 } ^ { C l } ; { \\pmb { h } } _ { 1 } ^ { C h } ; { \\hat { \\pmb { h } } } _ { 1 } ^ { C l } ; { \\hat { \\pmb { h } } } _ { 1 } ^ { C h } ; { \\hat { \\pmb { u } } } _ { 1 } ^ { C l } ] , \\ldots , [ { \\pmb { h } } _ { m } ^ { C l } ; { \\pmb { h } } _ { m } ^ { C h } ; { \\hat { \\pmb { h } } } _ { m } ^ { C l } ; { \\hat { \\pmb { h } } } _ { m } ^ { C h } ; { \\hat { \\pmb { u } } } _ { m } ^ { C } ] ) .\n$$", + "text_format": "latex", + "bbox": [ + 194, + 670, + 772, + 695 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Fully-Aware Self-Boosted Fusion. We now use self-boosted fusion to consider distant parts in the context, as illustrated by arrow (3) in Figure 2. Again, we achieve this via fully-aware attention on history-of-word. We identify the history-of-word to be ", + "bbox": [ + 173, + 709, + 825, + 752 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/810e1435335c87870a4c10de901f452b589e89a6645d884b19249ece23a8494f.jpg", + "text": "$$\n\\operatorname { H o W } _ { i } ^ { C } = [ \\pmb { g } _ { i } ^ { C } ; \\pmb { c } _ { i } ^ { C } ; \\pmb { h } _ { i } ^ { C l } ; \\pmb { h } _ { i } ^ { C h } ; \\hat { \\pmb { h } } _ { i } ^ { C l } ; \\hat { \\pmb { h } } _ { i } ^ { C h } ; \\hat { \\pmb { u } } _ { i } ^ { C } ; \\pmb { v } _ { i } ^ { C } ] \\in \\mathbb { R } ^ { 2 4 0 0 } .\n$$", + "text_format": "latex", + "bbox": [ + 302, + 758, + 692, + 782 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We then perform fully-aware attention, $\\begin{array} { r } { \\hat { \\pmb v } _ { i } ^ { C } = \\sum _ { j } \\alpha _ { i j } ^ { s } \\pmb { v } _ { j } ^ { C } , \\alpha _ { i j } ^ { s } \\propto \\exp ( S ^ { s } ( \\mathrm { H o W } _ { i } ^ { C } , \\mathrm { H o W } _ { j } ^ { C } ) ) . } \\end{array}$ The final context representation is obtained by ", + "bbox": [ + 178, + 790, + 777, + 823 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/946826f16b3340087bf4395eda5a620cee8ce4793b8ecacc4fcdb326ce7cd59c.jpg", + "text": "$$\nU _ { C } = \\{ \\pmb { u } _ { 1 } ^ { C } , \\ldots , \\pmb { u } _ { m } ^ { C } \\} = \\mathrm { B i L S T M } ( [ \\pmb { v } _ { 1 } ^ { C } ; \\hat { \\pmb { v } } _ { 1 } ^ { C } ] , \\ldots , [ \\pmb { v } _ { m } ^ { C } ; \\hat { \\pmb { v } } _ { m } ^ { C } ] ) .\n$$", + "text_format": "latex", + "bbox": [ + 299, + 830, + 699, + 851 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$\\{ \\boldsymbol { u } _ { i } ^ { C } \\in \\mathbb { R } ^ { 2 5 0 } \\} _ { i = 1 } ^ { m }$ are the understanding vectors for $C$ ", + "bbox": [ + 174, + 859, + 573, + 876 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "After these components in FusionNet, we have created the understanding vectors, $U _ { C }$ , for the context $C$ , which are fully fused with the question $Q$ . We also have the understanding vectors, $U _ { Q }$ , for the question $Q$ . ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 APPLICATION IN MACHINE COMPREHENSION ", + "text_level": 1, + "bbox": [ + 176, + 103, + 527, + 118 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We focus particularly on the output format in $\\mathrm { S Q u A D }$ (Rajpurkar et al., 2016) where the answer is always a span in the context. The output of FusionNet are the understanding vectors for both $C$ and $Q$ , $U _ { C } = \\{ \\mathbf { { u } } _ { 1 } ^ { C } , \\dots , \\mathbf { { u } } _ { m } ^ { C } \\}$ , $U _ { Q } = \\{ \\bar { \\pmb { u } } _ { 1 } ^ { Q } , \\dots , \\pmb { u } _ { n } ^ { Q } \\}$ . ", + "bbox": [ + 174, + 128, + 825, + 176 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We then use them to find the answer span in the context. Firstly, a single summarized question understanding vector is obtained through $\\begin{array} { r } { { \\pmb u } ^ { Q } = \\sum _ { i } \\beta _ { i } { \\pmb u } _ { i } ^ { Q } } \\end{array}$ , where $\\beta _ { i } \\propto \\mathrm { { e x p } } ( { \\pmb w } ^ { T } { \\pmb u } _ { i } ^ { Q } )$ and $\\pmb { w }$ is a trainable vector. Then we attend for the span start using the summarized question understanding vector $\\pmb { u } ^ { Q }$ , ", + "bbox": [ + 174, + 180, + 825, + 238 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/fa311c803fdd350b2b7ef382f167828161ad73a10229d5b0a705805e938d606a.jpg", + "text": "$$\nP _ { i } ^ { S } \\propto \\exp ( ( \\boldsymbol { \\mathbf { \\mathit { u } } } ^ { Q } ) ^ { T } W _ { S } \\boldsymbol { \\mathbf { \\mathit { u } } } _ { i } ^ { C } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 406, + 236, + 588, + 255 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $W _ { S } \\in \\mathbb { R } ^ { d \\times d }$ is a trainable matrix. To use the information of the span start when we attend for the span end, we combine the context understanding vector for the span start with $\\pmb { u } ^ { Q }$ through a GRU (Cho et al., 2014), $\\begin{array} { r } { { \\pmb v } ^ { Q } = \\mathrm { G R U } ( { \\pmb u } ^ { Q } , \\sum _ { i } P _ { i } ^ { S } { \\pmb u } _ { i } ^ { \\top } ) } \\end{array}$ , where $\\pmb { u } ^ { Q }$ is taken as the memory and $\\textstyle \\sum _ { i } P _ { i } ^ { S } { \\boldsymbol { u } } _ { i } ^ { C }$ as the input in GRU. Finally we attend for the end of the span using $v ^ { Q }$ , ", + "bbox": [ + 174, + 257, + 825, + 316 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/508dd248555ec57d70c929cd23cac8f188f98a94877f19eeb584f1a0c82b2c21.jpg", + "text": "$$\nP _ { i } ^ { E } \\propto \\exp ( ( \\pmb { v } ^ { Q } ) ^ { T } W _ { E } \\pmb { u } _ { i } ^ { C } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 406, + 323, + 589, + 343 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $W _ { E } \\in \\mathbb { R } ^ { d \\times d }$ is another trainable matrix. ", + "bbox": [ + 176, + 348, + 482, + 364 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Training. During training, we maximize the log probabilities of the ground truth span start and end, $\\begin{array} { r } { \\sum _ { k } ( \\log \\mathsf { \\bar { ( } } P _ { i _ { k } ^ { s } } ^ { S } ) + \\mathsf { \\bar { l o g } } ( P _ { i _ { k } ^ { e } } ^ { E } ) ) } \\end{array}$ , where $i _ { k } ^ { s } , i _ { k } ^ { e }$ are the answer span for the $k$ -th instance. ", + "bbox": [ + 173, + 371, + 823, + 401 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Prediction. We predict the answer span to be $i ^ { s } , i ^ { e }$ with the maximum $P _ { i ^ { s } } ^ { S } P _ { i ^ { e } } ^ { E }$ under the constraint $0 \\leq i ^ { e } - i ^ { s } \\leq 1 5$ . ", + "bbox": [ + 173, + 410, + 823, + 439 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 459, + 326, + 476 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In this section, we first present the datasets used for evaluation. Then we compare our end-toend FusionNet model with existing machine reading models. Finally, we conduct experiments to validate the effectiveness of our proposed components. Additional ablation study on input vectors can be found in Appendix C. Detailed experimental settings can be found in Appendix E. ", + "bbox": [ + 174, + 491, + 825, + 547 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 564, + 287, + 579 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We focus on the SQuAD dataset (Rajpurkar et al., 2016) to train and evaluate our model. SQuAD is a popular machine comprehension dataset consisting of $1 0 0 { , } 0 0 0 { + }$ questions created by crowd workers on 536 Wikipedia articles. Each context is a paragraph from an article and the answer to each question is guaranteed to be a span in the context. ", + "bbox": [ + 174, + 590, + 825, + 646 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "While rapid progress has been made on SQuAD, whether these systems truly understand language remains unclear. In a recent paper, Jia & Liang (2017) proposed several adversarial schemes to test the understanding of the systems. We will use the following two adversarial datasets, AddOneSent and AddSent, to evaluate our model. For both datasets, a confusing sentence is appended at the end of the context. The appended sentence is model-independent for AddOneSent, while AddSent requires querying the model a few times to choose the most confusing sentence. ", + "bbox": [ + 174, + 652, + 825, + 737 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 MAIN RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 753, + 323, + 768 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We submitted our model to SQuAD for evaluation on the hidden test set. We also tested the model on the adversarial SQuAD datasets. Two official evaluation criteria are used: Exact Match (EM) and F1 score. EM measures how many predicted answers exactly match the correct answer, while F1 score measures the weighted average of the precision and recall at token level. The evaluation results for our model and other competing approaches are shown in Table 2.1 Additional comparisons with state-of-the-art models in the literature can be found in Appendix A. ", + "bbox": [ + 174, + 780, + 825, + 864 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For the two adversarial datasets, AddOneSent and AddSent, the evaluation criteria is the same as SQuAD. However, all models are trained only on the original SQuAD, so the model never sees the ", + "bbox": [ + 176, + 871, + 820, + 900 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/275b7fdadeeb779db9084052916419020474f591904dbce78928b78ef9a75837.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
AddSentEM/F1
LRBaseline17.0/23.2
Match-LSTM (E) BiDAF (E)24.3 /34.2 29.6 /34.2
SEDT (E) Mnemonic Reader (S)30.0 /35.0
Mnemonic Reader (E)39.8/46.6
40.7 / 46.2
ReasoNet (E) FusionNet (E)34.6 /39.4 46.2 / 51.4
", + "bbox": [ + 553, + 103, + 812, + 232 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/e8605ba131ed5f1d19bd85c55d9fa59ceaa5cc4b5a45aff4f8cb39bb7fd203b1.jpg", + "table_caption": [ + "Table 2: The performance of FusionNet and competing approaches on SQuAD hidden test set at the time of writing (Oct. 4th, 2017). " + ], + "table_footnote": [], + "table_body": "
AddOneSentEM/F1
LRBaseline Match-LSTM (E)22.3/30.4 34.8 / 41.8
BiDAF (E) SEDT (E)40.7 /46.9 40.0 / 46.5
Mnemonic Reader (S)48.5 /56.0
Mnemonic Reader (E)48.7 / 55.3
ReasoNet (E) FusionNet (E)43.6 /49.8
", + "bbox": [ + 553, + 286, + 812, + 415 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/eda0852d2edf6c5400f30b997799aaada7526459d1f832c670df1f8416d97005.jpg", + "table_caption": [ + "Table 3: Comparison on AddSent. (S: Single model, E: Ensemble) " + ], + "table_footnote": [ + "Table 4: Comparison on AddOneSent. (S: Single model, E: Ensemble) " + ], + "table_body": "
Single ModelTest Set EM/F1
LR Baseline (Rajpurkar et al., 2016) Match-LSTM (Wang & Jiang,2016) BiDAF (Seo et al., 2017) SEDT (Liu et al., 2017) RaSoR (Lee et al., 2016) DrQA (Chen et al., 2017a) ReasoNet (Shen et al.,2017) R.Mnemonic Reader (Hu et al., 2017) DCN+ R-nett FusionNet40.4 /51.0 64.7/73.7 68.0 / 77.3 68.2 /77.5 70.8/78.7 70.7 /79.4 70.6 / 79.4 73.2/81.8 74.9 / 82.8 75.7 /83.5 76.0 / 83.9
EnsembleModel ReasoNet (Shen et al., 2017) MEMEN (Pan et al., 2017) R.Mnemonic Reader (Hu et al., 2017) R-nett DCN+ FusionNet75.0 / 82.3 75.4/82.7 77.7 / 84.9 78.2/85.2 78.7 / 85.6
Human (Rajpurkar et al., 2016)78.8 / 85.9 82.3/91.2
", + "bbox": [ + 173, + 101, + 537, + 401 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "adversarial datasets during training. The results for AddSent and AddOneSent are shown in Table 3 \nand Table 4, respectively.2 ", + "bbox": [ + 171, + 481, + 823, + 511 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "From the results, we can see that our models not only perform well on the original SQuAD dataset, but also outperform all previous models by more than $5 \\%$ in EM score on the adversarial datasets. This shows that FusionNet is better at language understanding of both the context and question. ", + "bbox": [ + 174, + 516, + 823, + 559 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 COMPARISON ON ATTENTION FUNCTION ", + "text_level": 1, + "bbox": [ + 176, + 577, + 496, + 590 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this experiment, we compare the performance of different attention scoring functions $S ( { \\pmb x } , { \\pmb y } )$ for fully-aware attention. We utilize the end-to-end architecture presented in Section 3.1. Fully-aware attention is used in two places, fully-aware multi-level fusion: higher level and fully-aware selfboosted fusion. Word-level fusion remains unchanged. Based on the discussion in Section 2.3, we consider the following formulations for comparison: ", + "bbox": [ + 174, + 603, + 825, + 674 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "1. Additive attention (MLP) (Bahdanau et al., 2015): $\\pmb { s } ^ { T } \\operatorname { t a n h } ( W _ { 1 } \\pmb { x } + W _ { 2 } \\pmb { y } ) .$ 1 \n2. Multiplicative attention: $\\pmb { x } ^ { T } \\pmb { U } ^ { T } \\pmb { V } \\pmb { y }$ . \n3. Scaled multiplicative attention: $\\scriptstyle { \\frac { 1 } { \\sqrt { k } } } x ^ { T } U ^ { T } V y$ , where $k$ is the attention hidden size. It is \nproposed in (Vaswani et al., 2017). \n4. Scaled multiplicative with nonlinearity: $\\begin{array} { r } { \\frac { 1 } { \\sqrt { k } } f ( U \\pmb { x } ) ^ { T } f ( V \\pmb { y } ) } \\end{array}$ . \n5. Our proposed symmetric form: $\\pmb { x } ^ { T } \\pmb { U } ^ { T } \\pmb { D } \\pmb { U } \\pmb { y }$ , where $D$ is diagonal. \n6. Proposed symmetric form with nonlinearity: $f ( U \\mathbf { x } ) ^ { T } D f ( U \\mathbf { y } )$ . ", + "bbox": [ + 210, + 684, + 825, + 820 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We consider the activation function $f ( x )$ to be $\\operatorname* { m a x } ( 0 , x )$ . The results of various attention functions on SQuAD development set are shown in Table 5. It is clear that the symmetric form consistently outperforms all alternatives. We attribute this gain to the fact that symmetric form has a single large matrix $U$ . All other alternatives have two large parametric matrices. During optimization, these two parametric matrices would interfere with each other and it will make the entire optimization process challenging. Besides, by constraining $U ^ { T } V$ to be a symmetric matrix $U ^ { T } D U$ , we retain the ability for $_ { \\textbf { \\em x } }$ to attend to dissimilar $\\textbf { { y } }$ . Furthermore, its marriage with the nonlinearity continues to significantly boost the performance. ", + "bbox": [ + 174, + 830, + 825, + 875 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/9a01a3abf6d707818a30ce5be329406b39e3a98464be7429964467add6593dd1.jpg", + "table_caption": [ + "Table 5: Comparison of different attention functions $S ( { \\pmb x } , { \\pmb y } )$ on SQuAD dev set. " + ], + "table_footnote": [], + "table_body": "
Attention FunctionEM/F1
Additive (MLP)71.8/ 80.172.1 / 80.672.4 / 80.772.6 / 80.8
MultiplicativeScaled MultiplicativeScaled Multiplicative + ReLU
Symmetric FormSymmetric Form + ReLU73.1 /81.575.3 / 83.6
Previous SotA (Hu et al., 2017)72.1/ 81.6
", + "bbox": [ + 176, + 116, + 496, + 250 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/7556abd7cea9b5c9f22f8b2015963a515d607babc94889bd999cebb19e4a9de3.jpg", + "table_caption": [ + "Table 6: Comparison of different configurations demonstrates the effectiveness of history-of-word. " + ], + "table_footnote": [], + "table_body": "
ConfigurationC,Q Fusion Self CEM/F1
High-LevelFA High-LevelFA All-LevelFA Multi-LevelNone64.6/73.273.3 /81.472.3 / 80.774.6 / 82.7
FA Multi-LevelNormalFA74.4 / 82.675.3 / 83.6
Previous SotA (Hu et al., 2017)72.1/ 81.6
", + "bbox": [ + 509, + 101, + 830, + 250 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 321, + 825, + 392 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.4 EFFECTIVENESS OF HISTORY-OF-WORD ", + "text_level": 1, + "bbox": [ + 176, + 416, + 491, + 431 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In FusionNet, we apply the history-of-word and fully-aware attention in two major places to achieve good performance: multi-level fusion and self-boosted fusion. In this section, we present experiments to demonstrate the effectiveness of our application. In the experiments, we fix the attention function to be our proposed symmetric form with nonlinearity due to its good performance shown in Section 4.3. The results are shown in Table 6, and the details for each configuration can be found in Appendix B. ", + "bbox": [ + 174, + 445, + 825, + 530 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "High-Level is a vanilla model where only the high-level information is fused from $Q$ to $C$ via standard attention. When placed in the conceptual architecture (Figure 2), it only contains arrow (2) without any other fusion processes. ", + "bbox": [ + 174, + 536, + 825, + 579 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "FA High-Level is the High-Level model with standard attention replaced by fully-aware attention. ", + "bbox": [ + 174, + 585, + 812, + 601 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "FA All-Level is a naive extension of FA High-Level, where all levels of information are concatenated and is fused into the context using the same attention weight. ", + "bbox": [ + 174, + 607, + 818, + 636 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "FA Multi-Level is our proposed Fully-aware Multi-level fusion, where different levels of information are attended under separate attention weight. ", + "bbox": [ + 176, + 642, + 821, + 671 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Self $C = \\mathbf { N o n e }$ means we do not make use of self-boosted fusion. ", + "bbox": [ + 173, + 678, + 604, + 691 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Self $C = \\mathrm { { N o r m a l } }$ means we employ a standard attention-based self-boosted fusion after fusing question to context. This is illustrated as arrow (3) in the conceptual architecture (Figure 2). ", + "bbox": [ + 173, + 699, + 823, + 728 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Self $C = \\mathbf { F } \\mathbf { A }$ means we enhance the self-boosted fusion with fully-aware attention. ", + "bbox": [ + 174, + 734, + 714, + 748 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "High-Level vs. FA High-Level. From Table 6, we can see that High-Level performs poorly as expected. However enhancing this vanilla model with fully-aware attention significantly increase the performance by more than $8 \\%$ . The performance of FA High-Level already outperforms many state-of-the-art MRC models. This clearly demonstrates the power of fully-aware attention. ", + "bbox": [ + 174, + 756, + 825, + 811 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "FA All-Level vs. FA Multi-Level. Next, we consider models that fuse all levels of information from question $Q$ to context $C$ . FA All-Level is a naive extension of FA High-Level, but its performance is actually worse than $F A$ High-Level. However, by fusing different parts of history-of-word in $Q$ independently as in $F A$ Multi-Level, we are able to further improve the performance. ", + "bbox": [ + 174, + 818, + 823, + 875 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Self $C$ options. We have achieved decent performance without self-boosted fusion. Now, we compare adding normal and fully-aware self-boosted fusion into the architecture. Comparing None and Normal in Table 6, we can see that the use of normal self-boosted fusion is not very effective under our improved $C , Q$ Fusion. Then by comparing with $F A$ , it is clear that through the enhancement of fully-aware attention, the enhanced self-boosted fusion can provide considerable improvement. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Together, these experiments demonstrate that the ability to take all levels of understanding as a whole is crucial for machines to better understand the text. ", + "bbox": [ + 171, + 138, + 823, + 166 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 186, + 328, + 203 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we describe a new deep learning model called FusionNet with its application to machine comprehension. FusionNet proposes a novel attention mechanism with following three contributions: 1. the concept of history-of-word to build the attention using complete information from the lowest word-level embedding up to the highest semantic-level representation; 2. an attention scoring function to effectively and efficiently utilize history-of-word; 3. a fully-aware multi-level fusion to exploit information layer by layer discriminatingly. We applied FusionNet to MRC task and experimental results show that FusionNet outperforms existing machine reading models on both the SQuAD dataset and the adversarial SQuAD dataset. We believe FusionNet is a general and improved attention mechanism and can be applied to many tasks. Our future work is to study its capability in other NLP problems. ", + "bbox": [ + 173, + 218, + 825, + 356 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 373, + 326, + 386 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We would like to thank Paul Mineiro, Sebastian Kochman, Pengcheng He, Jade Huang and Jingjing Liu from Microsoft Business AI, Mac-Antoine Rondeau from Maluuba and the anonymous reviewers for their valuable comments and tremendous help in this paper. ", + "bbox": [ + 174, + 396, + 825, + 438 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 459, + 285, + 473 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. 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Learning recurrent span representations for extractive question answering. arXiv preprint arXiv:1611.01436, 2016. ", + "bbox": [ + 173, + 881, + 823, + 922 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Rui Liu, Junjie Hu, Wei Wei, Zi Yang, and Eric Nyberg. Structural embedding of syntactic trees for machine comprehension. arXiv preprint arXiv:1703.00572, 2017. \nB. McCann, J. Bradbury, C. Xiong, and R. Socher. Learned in Translation: Contextualized Word Vectors. arXiv preprint arXiv:1708.00107, 2017. \nBoyuan Pan, Hao Li, Zhou Zhao, Bin Cao, Deng Cai, and Xiaofei He. Memen: Multi-layer embedding with memory networks for machine comprehension. arXiv preprint arXiv:1707.09098, 2017. \nJeffrey Pennington, Richard Socher, and Christopher Manning. Glove: Global vectors for word representation. In EMNLP, 2014. \nPranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. Squad: $1 0 0 { , } 0 0 0 { + }$ questions for machine comprehension of text. EMNLP, 2016. \nMinjoon Seo, Aniruddha Kembhavi, Ali Farhadi, and Hannaneh Hajishirzi. Bidirectional attention flow for machine comprehension. In ICLR, 2017. \nYelong Shen, Po-Sen Huang, Jianfeng Gao, and Weizhu Chen. Reasonet: Learning to stop reading in machine comprehension. In KDD, 2017. \nNitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. JMLR, 2014. \nAshish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. 2017. \nShuohang Wang and Jing Jiang. Machine comprehension using match-lstm and answer pointer. arXiv preprint arXiv:1608.07905, 2016. \nWenhui Wang, Nan Yang, Furu Wei, Baobao Chang, and Ming Zhou. Gated self-matching networks for reading comprehension and question answering. In ACL, 2017. \nZhiguo Wang, Haitao Mi, Wael Hamza, and Radu Florian. Multi-perspective context matching for machine comprehension. arXiv preprint arXiv:1612.04211, 2016. \nDirk Weissenborn, Georg Wiese, and Laura Seiffe. Making neural qa as simple as possible but not simpler. In CoNLL, 2017. \nAdina Williams, Nikita Nangia, and Samuel R Bowman. A broad-coverage challenge corpus for sentence understanding through inference. arXiv preprint arXiv:1704.05426, 2017. \nCaiming Xiong, Victor Zhong, and Richard Socher. Dynamic coattention networks for question answering. ICLR, 2017. ", + "bbox": [ + 169, + 99, + 826, + 712 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A COMPARISON WITH PUBLISHED MODELS ", + "text_level": 1, + "bbox": [ + 174, + 731, + 553, + 747 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In this appendix, we compare with published state-of-the-art architectures on the SQuAD dev set. The comparison is shown in Figure 5 and 6 for EM and F1 score respectively. The performance of FusionNet is shown under different training epochs. Each epoch loops through all the examples in the training set once. On a single NVIDIA GeForce GTX Titan X GPU, each epoch took roughly 20 minutes when batch size 32 is used. ", + "bbox": [ + 173, + 763, + 825, + 833 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The state-of-the-art models compared in this experiment include: \n1. Published version of R-net in their technical report (Wang et al., 2017), \n2. Reinforced Mnemonic Reader (Hu et al., 2017), 3. MEMEN (Pan et al., 2017), \n4. ReasoNet (Shen et al., 2017), 5. Document reader (DrQA) (Chen et al., 2017a), \n6. DCN (Xiong et al., 2017), 7. DCN $^ +$ character embedding (Char) $^ +$ CoVe (McCann et al., 2017), \n8. BiDAF (Seo et al., 2017), 9. the best-performing variant of Match-LSTM (Wang & Jiang, 2016). ", + "bbox": [ + 173, + 840, + 825, + 924 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/83cd5a9f9a0174460ffcd39b92ddb060f758e9749cab54075cd31bb5c5f087b2.jpg", + "image_caption": [ + "Figure 5: EM score on the SQuAD dev set under different training epoch. " + ], + "image_footnote": [], + "bbox": [ + 173, + 98, + 826, + 348 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/d8a5b20b41be47ba0875b1e75dbb679e81f4e2b06dd71346b750da38f888986b.jpg", + "image_caption": [ + "Figure 6: F1 score on the SQuAD dev set under different training epoch. " + ], + "image_footnote": [], + "bbox": [ + 174, + 388, + 826, + 632 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B DETAILED CONFIGURATIONS IN THE ABLATION STUDY ", + "text_level": 1, + "bbox": [ + 173, + 685, + 669, + 703 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In this appendix, we present details for the configurations used in the ablation study in Section 4.4. For all configurations, the understanding vectors for both the context $C$ and the question $Q$ will be generated, then we follow the same output architecture in Section 3.2 to apply them to machine reading comprehension problem. ", + "bbox": [ + 173, + 718, + 825, + 775 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "High-Level. Firstly, context words and question words are transformed into input vectors in the same way as FusionNet, ", + "bbox": [ + 173, + 792, + 823, + 821 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/f6e3c28e8b1356861c19397ffea564b2dddf56fdf901da8c6fd8979303154492.jpg", + "text": "$$\n\\{ \\pmb { w } _ { 1 } ^ { C } , \\ldots , \\pmb { w } _ { m } ^ { C } \\} , \\quad \\{ \\pmb { w } _ { 1 } ^ { Q } , \\ldots , \\pmb { w } _ { n } ^ { Q } \\} .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 824, + 617, + 844 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Then we pass them independently to two layers of BiLSTM. ", + "bbox": [ + 173, + 849, + 570, + 866 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/63da65e27783d22619163d64a0533b2fc1deb6457dd76c4781aab6659662090c.jpg", + "text": "$$\n\\begin{array} { r l } & { h _ { 1 } ^ { C l } , \\ldots , h _ { m } ^ { C l } = \\mathrm { B i L S T M } ( w _ { 1 } ^ { C } , \\ldots , w _ { m } ^ { C } ) , \\quad h _ { 1 } ^ { Q l } , \\ldots , h _ { n } ^ { Q l } = \\mathrm { B i L S T M } ( w _ { 1 } ^ { Q } , \\ldots , w _ { n } ^ { Q } ) , } \\\\ & { h _ { 1 } ^ { C h } , \\ldots , h _ { m } ^ { C h } = \\mathrm { B i L S T M } ( h _ { 1 } ^ { C l } , \\ldots , h _ { m } ^ { C l } ) , \\quad h _ { 1 } ^ { Q h } , \\ldots , h _ { n } ^ { Q h } = \\mathrm { B i L S T M } ( h _ { 1 } ^ { Q l } , \\ldots , h _ { n } ^ { Q l } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 872, + 792, + 928 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Next we consider the standard attention-based fusion for the high level representation. ", + "bbox": [ + 174, + 103, + 736, + 118 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/732db4056720fe7aeb191e2907862c44954fd7cd6aa26ec9c6b215374947ed6a.jpg", + "text": "$$\n\\hat { \\boldsymbol { h } } _ { i } ^ { C h } = \\sum _ { j } \\alpha _ { i j } \\boldsymbol { h } _ { j } ^ { Q h } , \\quad \\alpha _ { i j } = \\frac { \\exp ( S _ { i j } ) } { \\sum _ { k } \\exp ( S _ { i k } ) } , \\quad S _ { i j } = S ( \\boldsymbol { h } _ { i } ^ { C h } , \\boldsymbol { h } _ { j } ^ { Q h } ) .\n$$", + "text_format": "latex", + "bbox": [ + 271, + 122, + 725, + 164 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Then we concatenate the attended vector $\\hat { h } _ { i } ^ { C h }$ with the original high level representation $h _ { i } ^ { C h }$ and pass through two layers of BiLSTM to fully mix the two information. The understanding vectors for the context is the hidden vectors in the final layers of the BiLSTM. ", + "bbox": [ + 174, + 169, + 825, + 217 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/c8bd522f080ef27b367a00f8bf8030f2ddcc30a4d46021da8d9184ee20491118.jpg", + "text": "$$\n\\pmb { u } _ { 1 } ^ { C } , \\ldots , \\pmb { u } _ { m } ^ { C } = \\mathrm { B i L S T M } ( [ \\pmb { h } _ { 1 } ^ { C h } ; \\hat { \\pmb { h } } _ { 1 } ^ { C h } ] , \\dots , [ \\pmb { h } _ { m } ^ { C h } ; \\hat { \\pmb { h } } _ { m } ^ { C h } ] )\n$$", + "text_format": "latex", + "bbox": [ + 315, + 222, + 679, + 244 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The understanding vectors for the question is the high level representation itself, ", + "bbox": [ + 171, + 250, + 699, + 265 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/3838caa2559a19f49f3a6bd87ffb7ad719bb34375a10967a15c5993c6ceb313a.jpg", + "text": "$$\n\\begin{array} { r } { { \\pmb u } _ { 1 } ^ { Q } , \\dots , { \\pmb u } _ { n } ^ { Q } = { \\pmb h } _ { 1 } ^ { Q h } , \\dots , { \\pmb h } _ { n } ^ { Q h } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 395, + 268, + 601, + 290 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Now we have obtained the understanding vectors for both the context and the question. The answer can thus be found. Neither word-level fusion (1) nor self-boosted fusion (3, 3’) in Figure 2 are used. ", + "bbox": [ + 176, + 294, + 825, + 323 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "FA High-Level. The only difference to High-Level is the enhancement of fully-aware attention. This is as simple as changing ", + "bbox": [ + 169, + 342, + 823, + 371 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/cc2d01e88c599b7578b6af6440adf3322b4c8af7aac402c90826db645b86cef4.jpg", + "text": "$$\nS _ { i j } = S ( { h _ { i } ^ { C h } } , { h _ { j } ^ { Q h } } ) \\quad \\Longrightarrow \\qquad S _ { i j } = S ( [ g _ { i } ^ { C } ; { c _ { i } ^ { C } } ; { h _ { i } ^ { C l } } ; { h _ { i } ^ { C h } } ] , [ g _ { j } ^ { Q } ; { c _ { j } ^ { Q } } ; { h _ { j } ^ { Q l } } ; { h _ { j } ^ { Q h } } ] ) ,\n$$", + "text_format": "latex", + "bbox": [ + 225, + 376, + 769, + 398 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $[ g _ { i } ; c _ { i } ; h _ { i } ^ { l } ; h _ { i } ^ { h } ]$ is the common history-of-word for both context and question. All other places remains the same as High-Level. This simple change results in significant improvement. The performance of $F A$ High-Level can already outperform many state-of-the-art models in the literature. Note that our proposed symmetric form with nonlinearity should be used to guarantee the boost. ", + "bbox": [ + 173, + 404, + 825, + 462 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "FA All-Level. First, we use the same procedure as High-Level to obtain ", + "bbox": [ + 173, + 479, + 645, + 496 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/f2e7de8e4a2044745a0df78dce2101dc7bebfa001014b80bb7d8b1b2285331ab.jpg", + "text": "$$\n\\begin{array} { r l } { \\{ \\pmb { w } _ { 1 } ^ { C } , \\ldots , \\pmb { w } _ { m } ^ { C } \\} , } & { \\{ \\pmb { w } _ { 1 } ^ { Q } , \\ldots , \\pmb { w } _ { n } ^ { Q } \\} , } \\\\ & { \\{ \\pmb { h } _ { 1 } ^ { C l } , \\ldots , \\pmb { h } _ { m } ^ { C l } \\} , \\quad \\{ \\pmb { h } _ { 1 } ^ { Q l } , \\ldots , \\pmb { h } _ { n } ^ { Q l } \\} , } \\\\ & { \\{ \\pmb { h } _ { 1 } ^ { C h } , \\ldots , \\pmb { h } _ { m } ^ { C h } \\} , \\quad \\{ \\pmb { h } _ { 1 } ^ { Q h } , \\ldots , \\pmb { h } _ { n } ^ { Q h } \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 370, + 501, + 624, + 566 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Next we make use of the fully-aware attention similar to $F A$ High-Level, but take back the entire history-of-word. ", + "bbox": [ + 173, + 566, + 825, + 595 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/c8d916a2c11d49798cf83c3620ada314ae6c60e95f4f3174cfc653dc46e23dc0.jpg", + "text": "$$\n\\begin{array} { c } { \\displaystyle \\alpha _ { i j } = \\frac { \\exp ( S _ { i j } ) } { \\sum _ { k } \\exp ( S _ { i k } ) } , \\quad { S _ { i j } = S ( [ g _ { i } ^ { C } ; c _ { i } ^ { C } ; h _ { i } ^ { C l } ; h _ { i } ^ { C h } ] , [ g _ { j } ^ { Q } ; c _ { j } ^ { Q } ; h _ { j } ^ { Q l } ; h _ { j } ^ { Q h } ] ) } , } \\\\ { \\displaystyle { \\mathrm { H o W } _ { i } ^ { C } = \\sum _ { j } \\alpha _ { i j } [ g _ { j } ^ { Q } ; c _ { j } ^ { Q } ; h _ { j } ^ { Q l } ; h _ { j } ^ { Q h } ] } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 599, + 745, + 676 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Then we concatenate the attended history-of-word $\\mathrm { H } \\mathrm { \\hat { o } } \\mathrm { W } _ { i } ^ { C }$ with the original history-of-word $[ { \\pmb g } _ { i } ^ { C } ; { \\pmb c } _ { i } ^ { C } ; { \\pmb h } _ { i } ^ { C l } ; { \\pmb h } _ { i } ^ { C h } ]$ i and pass through two layers of BiLSTM to fully mix the two information. The understanding vectors for the context is the hidden vectors in the final layers of the BiLSTM. ", + "bbox": [ + 176, + 680, + 821, + 729 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/4d73eedf374412bf94795f5152c939fbff166fe2495308bede2e7ea34c90a93c.jpg", + "text": "$$\n\\pmb { u } _ { 1 } ^ { C } , \\dots , \\pmb { u } _ { m } ^ { C } = \\mathrm { B i L S T M } ( [ g _ { 1 } ^ { C } ; c _ { 1 } ^ { C } ; h _ { 1 } ^ { C l } ; h _ { 1 } ^ { C h } ; \\mathrm { H \\hat { o } W } _ { 1 } ^ { C } ] , \\dots , [ g _ { m } ^ { C } ; c _ { m } ^ { C } ; h _ { m } ^ { C l } ; h _ { m } ^ { C h } ; \\mathrm { H \\hat { o } W } _ { m } ^ { C } ] )\n$$", + "text_format": "latex", + "bbox": [ + 207, + 733, + 789, + 757 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The understanding vectors for the question is similar to the Understanding component in Section 3.1, ", + "bbox": [ + 178, + 761, + 820, + 776 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/7f273415d3bf38988099bc7e91a996d59c22b7363bf6a7ad67c2810f7f728b23.jpg", + "text": "$$\n\\pmb { u } _ { 1 } ^ { Q } , \\ldots , \\pmb { u } _ { n } ^ { Q } = \\mathrm { B i L S T M } ( [ g _ { 1 } ^ { Q } ; c _ { 1 } ^ { Q } ; \\pmb { h } _ { 1 } ^ { Q l } ; \\pmb { h } _ { 1 } ^ { Q h } ] , \\ldots , [ g _ { m } ^ { Q } ; c _ { m } ^ { Q } ; \\pmb { h } _ { m } ^ { Q l } ; \\pmb { h } _ { m } ^ { Q h } ] ) .\n$$", + "text_format": "latex", + "bbox": [ + 261, + 781, + 735, + 801 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We have now generated the understanding vectors for both the context and the question. ", + "bbox": [ + 173, + 805, + 748, + 821 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "FA Multi-Level. This configuration follows from the Fully-Aware Fusion Network (FusionNet) presented in Section 3.1. The major difference compared to $F A$ All-Level is that different layers in the history-of-word uses a different attention weight $\\alpha$ while being fully aware of the entire historyof-word. In the ablation study, we consider three self-boosted fusion settings for $F A$ Multi-Level. The Fully-Aware setting is the one presented in Section 3.1. Here we discuss all three of them in detail. ", + "bbox": [ + 173, + 839, + 826, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• For the None setting in self-boosted fusion, no self-boosted fusion is used and we use two layers of BiLSTM to mix the attended information. The understanding vectors for the context $C$ is the hidden vectors in the final layers of the BiLSTM, ", + "bbox": [ + 215, + 103, + 825, + 146 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1f62ab4ea44a87c05c8fa64d0758bcd7585e7bed77548ce3d5187f361701cf67.jpg", + "text": "$$\n\\pmb { u } _ { 1 } ^ { C } , \\dots , \\pmb { u } _ { m } ^ { C } = \\mathrm { B i L S T M } ( [ \\pmb { h } _ { 1 } ^ { C l } ; \\pmb { h } _ { 1 } ^ { C h } ; \\hat { \\pmb { h } } _ { 1 } ^ { C l } ; \\hat { \\pmb { h } } _ { 1 } ^ { C h } ; \\hat { \\pmb { u } } _ { 1 } ^ { C } ] , \\dots , [ \\pmb { h } _ { m } ^ { C l } ; \\pmb { h } _ { m } ^ { C h } ; \\hat { \\pmb { h } } _ { m } ^ { C l } ; \\hat { \\pmb { h } } _ { m } ^ { C h } ; \\hat { \\pmb { u } } _ { m } ^ { C } ] ) .\n$$", + "text_format": "latex", + "bbox": [ + 245, + 151, + 808, + 175 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Self-boosted fusion is not utilized in all previous configurations: High-Level, FA HighLevel and FA All-Level. ", + "bbox": [ + 228, + 181, + 821, + 210 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• For the Normal setting, we first use one layer of BiLSTM to mix the attended information. ", + "bbox": [ + 222, + 213, + 821, + 229 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/2140bc5df35532559e125c8b38906efb2f0877fe3121777949b2ffa5aa631151.jpg", + "text": "$$\n\\pmb { v } _ { 1 } ^ { C } , \\dots , \\pmb { v } _ { m } ^ { C } = \\mathrm { B i L S T M } ( [ h _ { 1 } ^ { C l } ; h _ { 1 } ^ { C h } ; \\hat { h } _ { 1 } ^ { C l } ; \\hat { h } _ { 1 } ^ { C h } ; \\hat { u } _ { 1 } ^ { C } ] , \\dots , [ h _ { m } ^ { C l } ; h _ { m } ^ { C h } ; \\hat { h } _ { m } ^ { C l } ; \\hat { h } _ { m } ^ { C h } ; \\hat { u } _ { m } ^ { C } ] ) .\n$$", + "text_format": "latex", + "bbox": [ + 246, + 234, + 807, + 258 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Then we fuse the context information into itself through standard attention, ", + "bbox": [ + 236, + 265, + 725, + 280 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/c4461013442bc76a98551c2671d59c7822f21f84c52320e4e0ab8b9a3d85c6d8.jpg", + "text": "$$\nS _ { i j } = S ( \\pmb { v } _ { i } ^ { C } , \\pmb { v } _ { j } ^ { C } ) , \\alpha _ { i j } = \\frac { \\exp ( S _ { i j } ) } { \\sum _ { k } \\exp ( S _ { i k } ) } , \\hat { \\pmb { v } } _ { i } ^ { C } = \\sum _ { j } \\alpha _ { i j } \\pmb { v } _ { j } ^ { C } .\n$$", + "text_format": "latex", + "bbox": [ + 330, + 286, + 727, + 325 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The final understanding vectors for the context $C$ is the output hidden vectors after passing the concatenated vectors into a BiLSTM, ", + "bbox": [ + 230, + 332, + 823, + 361 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/2acbe7c62b21084b82c0e587afa71b833c5d6e1a4abe834a6fa377b7d2dd3dc0.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { u } _ { 1 } ^ { C } , \\ldots , \\pmb { u } _ { m } ^ { C } = \\mathrm { B i L S T M } ( [ \\pmb { v } _ { 1 } ^ { C } ; \\hat { \\pmb { v } } _ { 1 } ^ { C } ] , \\dots , [ \\pmb { v } _ { m } ^ { C } ; \\hat { \\pmb { v } } _ { m } ^ { C } ] ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 357, + 364, + 696, + 386 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• For the Fully-Aware setting, we change $S _ { i j } = S ( \\pmb { v } _ { i } ^ { C } , \\pmb { v } _ { j } ^ { C } )$ in the Normal setting to the fully-aware attention ", + "bbox": [ + 214, + 392, + 823, + 422 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/67190e3d51dfbfcbaa54e1a0d5f74a1c81c00f0b46df933a1da30e34cfb30f1d.jpg", + "text": "$$\n\\begin{array} { r } { S _ { i j } = S ( [ { \\pmb w } _ { i } ^ { C } ; { \\pmb h } _ { i } ^ { C l } ; { \\pmb h } _ { i } ^ { C h } ; { \\hat { \\pmb h } } _ { u } ^ { C l } ; { \\hat { \\pmb h } } _ { i } ^ { C h } ; { \\hat { \\pmb u } } _ { i } ^ { C } ; { \\pmb v } _ { i } ^ { C } ] , [ { \\pmb w } _ { j } ^ { C } ; { \\pmb h } _ { j } ^ { C l } ; { \\pmb h } _ { j } ^ { C h } ; { \\hat { \\pmb h } } _ { j } ^ { C l } ; { \\hat { \\pmb h } } _ { j } ^ { C h } ; { \\hat { \\pmb u } } _ { j } ^ { C } ; { \\pmb v } _ { j } ^ { C } ] ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 428, + 800, + 454 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "All other places remains the same. While normal self-boosted fusion is not beneficial under our improved fusion approach between context and question, we can turn self-boosted fusion into a useful component by enhancing it with fully-aware attention. ", + "bbox": [ + 230, + 458, + 825, + 501 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/7bf72aae61a1d273b09ae8d52af734b0abca82063ab56228e464c3a0ed07b7fb.jpg", + "table_caption": [ + "C ADDITIONAL ABLATION STUDY ON INPUT VECTORS " + ], + "table_footnote": [], + "table_body": "
ConfigurationEM/F1
FusionNetFusionNet (without CoVe)FusionNet (fixing GloVe)75.3 / 83.674.1 / 82.575.0 / 83.2
Previous SotA (Hu et al., 2017)72.1/ 81.6
", + "bbox": [ + 176, + 555, + 496, + 642 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/c1586b3e63156f9cdbf1f5e3ebba1330a6c2b1d6f88e30588d8569a548470a56.jpg", + "table_caption": [ + "Table 7: Ablation study on input vectors (GloVe and CoVe) for SQuAD dev set. " + ], + "table_footnote": [], + "table_body": "
ConfigurationEM/F1
FusionNet (S,10-run best)FusionNet (S,10-run mean)FusionNet (S, without CoVe)FusionNet (E)45.6 / 51.144.9 / 50.147.4 / 52.446.2 / 51.4
Previous SotA (E)40.7/46.2
", + "bbox": [ + 517, + 555, + 821, + 657 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/0438bcac66858f195aee919432ad72ebf978023b095b168e94e625b3f107cf0d.jpg", + "table_caption": [ + "Table 8: Additional results for AddSent. (S: Single model, E: Ensemble) ", + "Table 9: Additional results for AddOneSent. (S: Single model, E: Ensemble) " + ], + "table_footnote": [], + "table_body": "
ConfigurationEM/F1
FusionNet (S,10-run best)FusionNet (S,10-run mean)FusionNet (S, without CoVe)FusionNet (E)54.8/60.953.1/ 59.355.2 / 61.254.7 / 60.7
Previous SotA (E)48.7 / 55.3
", + "bbox": [ + 516, + 708, + 823, + 810 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/eff4b81f3c624949e0b874f52b9145360911c4a3d9c88a169c06218df9f52d5d.jpg", + "image_caption": [ + "Figure 7: Single model performance (EM) on AddSent over 10 training runs. (dashed vertical line indicates previous best performance) " + ], + "image_footnote": [], + "bbox": [ + 173, + 686, + 498, + 797 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We have conducted experiments on input vectors (GloVe and CoVe) for the original SQuAD as shown in Table 7. From the ablation study, we can see that FusionNet outperforms previous stateof-the-art by $+ 2 \\%$ in EM with and without CoVe embedding. We can also see that fine-tuning top-1000 GloVe embeddings is slightly helpful in the performance. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Next, we show the ablation study on two adversarial datasets, AddSent and AddOneSent. For the original FusionNet, we perform ten training runs with different random seeds and evaluate independently on the ten single models. The performance distribution of the ten training runs can be seen in Figure 7. Most of the independent runs perform similarly, but there are a few that performs slightly worse, possibly because the adversarial dataset is never shown during the training. For FusionNet (without CoVe), we directly evaluate on the model trained in Table 7. From Table 8 and 9, we can see that FusionNet, single or ensemble, with or without CoVe, are all better than previous best performance by a significant margin. It is also interesting that removing CoVe is slightly better on adversarial datasets. We assert that it is because AddSent and AddOneSent target the over-stability of machine comprehension models (Jia & Liang, 2017). Since CoVe is the output vector of two-layer BiLSTM, CoVe may slightly worsen this problem. ", + "bbox": [ + 173, + 103, + 825, + 256 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D APPLICATION TO NATURAL LANGUAGE INFERENCE ", + "text_level": 1, + "bbox": [ + 174, + 276, + 640, + 292 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "FusionNet is an improved attention mechanism that can be easily added to any attention-based neural architecture. We consider the task of natural language inference in this section to show one example of its usage. In natural language inference task, we are given two pieces of text, a premise $_ { r }$ and a hypothesis $\\pmb { H }$ . The task is to identify one of the following scenarios: ", + "bbox": [ + 176, + 308, + 823, + 364 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "1. Entailment - the hypothesis $\\pmb { H }$ can be derived from the premise $_ { r }$ . \n2. Contradiction - the hypothesis $\\pmb { H }$ contradicts the premise $_ { P }$ . \n3. Neutral - none of the above. ", + "bbox": [ + 210, + 376, + 671, + 429 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We focus on Multi-Genre Natural Language Inference (MultiNLI) corpus (Williams et al., 2017) recently developed by the creator of Stanford Natural Language Inference (SNLI) dataset (Bowman et al., 2015). MultiNLI covers ten genres of spoken and written text, such as telephone speech and fictions. However the training set only contains five genres. Thus there are in-domain and crossdomain accuracy during evaluation. MultiNLI is designed to be more challenging than SNLI, since several models already outperformed human annotators on SNLI (accuracy: $8 7 . 7 \\% ) ^ { 3 }$ . ", + "bbox": [ + 174, + 440, + 826, + 525 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A state-of-the-art model for natural language inference is Enhanced Sequential Inference Model (ESIM) by Chen et al. (2017b), which achieves an accuray of $8 8 . 0 \\%$ on SNLI and obtained $7 2 . 3 \\%$ (in-domain), $7 2 . 1 \\%$ (cross-domain) on MultiNLI (Williams et al., 2017). We implemented a version of ESIM in PyTorch. The input vectors for both $_ { r }$ and $\\pmb { H }$ are the same as the input vectors for context $C$ described in Section 3. Therefore, ", + "bbox": [ + 173, + 530, + 825, + 601 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/30e1ea2833f5ec931353a06aeebe295a30773cf26221403ccee0427a6326f6f3.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { w } _ { i } ^ { P } , \\pmb { w } _ { j } ^ { H } \\in \\mathbb { R } ^ { 9 0 0 + 2 0 + 1 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 419, + 606, + 578, + 626 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Then, two-layer BiLSTM with shortcut connection is used to encode the input words for both premise $_ { r }$ and hypothesis $\\pmb { H }$ , i.e., ", + "bbox": [ + 173, + 631, + 823, + 660 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/cc37a253e0375f855d059e8a62107e20f70ae202e2b46104bf3c89cf6ca563eb.jpg", + "text": "$$\n\\{ \\boldsymbol { h } _ { i } ^ { P l } \\} = \\mathrm { B i L S T M } ( \\boldsymbol { w } _ { i } ^ { P } ) , \\quad \\{ \\boldsymbol { h } _ { j } ^ { H l } \\} = \\mathrm { B i L S T M } ( \\boldsymbol { w } _ { j } ^ { H } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 321, + 666, + 676, + 688 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/90713cc79285dd7dc81f44aa971902f748762df9c0dfb3ed241d6c2c49ee3fac.jpg", + "text": "$$\n\\{ { \\pmb h } _ { i } ^ { P h } \\} = \\mathrm { B i L S T M } ( [ { \\pmb w } _ { i } ^ { P } ; { \\pmb h } _ { i } ^ { P l } ] ) , \\quad \\{ { \\pmb h } _ { j } ^ { H h } \\} = \\mathrm { B i L S T M } ( [ { \\pmb w } _ { j } ^ { H } ; { \\pmb h } _ { j } ^ { H l } ] ) .\n$$", + "text_format": "latex", + "bbox": [ + 274, + 693, + 722, + 713 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The hiddetion from sizto f each LSTM is as well as from 0, sto $h _ { i } ^ { P l } , h _ { i } ^ { P h } , h _ { j } ^ { H l } , h _ { j } ^ { H h } \\in \\mathbb { R } ^ { 3 0 0 }$ . Next, ESIM fuses informa-e consider the following, $_ { P }$ $\\pmb { H }$ $\\pmb { H }$ $_ { P }$ ", + "bbox": [ + 176, + 715, + 823, + 747 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f6be446b1251a31ebb3076db5394dc74905758a9d55d6bed442f23e3b5863ae5.jpg", + "text": "$$\n\\boldsymbol { g } _ { i } ^ { P } = [ h _ { i } ^ { P h } ; \\hat { h } _ { i } ^ { P h } ] , \\boldsymbol { \\hat { h } } _ { i } ^ { P h } = \\sum _ { j } \\alpha _ { i j } ^ { P } \\boldsymbol { h } _ { j } ^ { H h } , \\boldsymbol { \\alpha } _ { i j } ^ { P } = \\frac { \\exp ( S _ { i j } ^ { P } ) } { \\sum _ { k } \\exp ( S _ { i k } ^ { P } ) } , S _ { i j } ^ { P } = S ^ { P } ( h _ { i } ^ { P h } , h _ { j } ^ { H h } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 210, + 753, + 784, + 796 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/ba654acc27756674abba64b0b58f26a9e08a327c143c9176317d4e581a1a3f76.jpg", + "text": "$$\n{ g } _ { j } ^ { H } = [ { h } _ { j } ^ { H h } ; \\hat { h } _ { j } ^ { H h } ] , \\ \\hat { h } _ { j } ^ { H h } = \\sum _ { i } \\alpha _ { i j } ^ { H } { h } _ { i } ^ { P h } , \\ \\alpha _ { i j } ^ { H } = \\frac { \\exp ( S _ { i j } ^ { H } ) } { \\sum _ { k } \\exp ( S _ { k j } ^ { H } ) } , \\ S _ { i j } ^ { H } = S ^ { H } ( { h } _ { i } ^ { P h } , { h } _ { j } ^ { H h } ) .\n$$", + "text_format": "latex", + "bbox": [ + 204, + 801, + 790, + 842 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We set the attention hidden size to be the same as the dimension of hidden vectors $^ { h }$ . Next, ESIM feed $g _ { i } ^ { P } , g _ { j } ^ { H }$ into separate BiLSTMs to perform inference. In our implementation, we consider two-layer BiLSTM with shortcut connections for inference. The hidden vectors for the two-layer ", + "bbox": [ + 174, + 844, + 825, + 887 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/ed0e6685fff5d8d071dc478d33bad7b7fd7b0bab9940737f1eada391b89b64bf.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Cross-DomainIn-Domain
Our ESIM without CoVe (d = 300) Our ESIM without CoVe + fully-aware (d = 250)73.4 76.973.3 76.2
Our ESIM without CoVe + fully-aware + multi-level (d = 250) Our ESIM (d = 300) Our ESIM + fully-aware (d = 250) Our ESIM + fully-aware + multi-level (d = 250)78.2 73.9 77.3 78.477.9 73.7 76.5 78.2
", + "bbox": [ + 173, + 101, + 825, + 205 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Table 10: The performance (accuracy) of ESIM with our proposed attention enhancement on MultiNLI (Williams et al., 2017) development set. $\\mathit { \\Pi } _ { d }$ is the output hidden size of BiLSTM) ", + "bbox": [ + 173, + 215, + 823, + 244 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "BiLSTM are concatenated to yield $\\{ \\boldsymbol { u } _ { i } ^ { P } \\} , \\{ \\boldsymbol { u } _ { j } ^ { H } \\} \\subset \\mathbb { R } ^ { 6 0 0 }$ . The final hidden vector for the $P , H$ pair is obtained by ", + "bbox": [ + 173, + 268, + 825, + 299 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/e88702a6da075323b02a05bb85271588f00f7468c4503ca5c393a42da2fc0d91.jpg", + "text": "$$\n\\boldsymbol h _ { P , H } = \\big [ { \\frac { 1 } { n } } \\sum _ { i } { \\boldsymbol u } _ { i } ^ { P } ; \\operatorname* { m a x } ( { \\boldsymbol u } _ { 1 } ^ { P } , \\ldots , { \\boldsymbol u } _ { n } ^ { P } ) ; { \\frac { 1 } { m } } \\sum _ { j } { \\boldsymbol u } _ { j } ^ { H } ; \\operatorname* { m a x } ( { \\boldsymbol u } _ { 1 } ^ { H } , \\ldots , { \\boldsymbol u } _ { m } ^ { H } ) \\big ] .\n$$", + "text_format": "latex", + "bbox": [ + 253, + 304, + 743, + 343 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The final hidden vector ${ h _ { P , H } }$ is then passed into a multi-layer perceptron (MLP) classifier. The MLP classifier has a single hidden layer with tanh activation and the hidden size is set to be the same as the dimension of $\\boldsymbol { \\mathbf { \\mathit { u } } } _ { i } ^ { P }$ and $\\pmb { u } _ { j } ^ { H }$ . Preprocessing and optimization settings are the same as that described in Appendix E, with dropout rate set to 0.3. ", + "bbox": [ + 173, + 348, + 825, + 405 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Now, we consider improving ESIM with our proposed attention mechanism. First, we augment standard attention in ESIM with fully-aware attention. This is as simple as replacing ", + "bbox": [ + 173, + 411, + 823, + 440 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/218c0814c57364f6142d8ac534c8143ec90e10f96b8b11ce1622b1c75f561409.jpg", + "text": "$$\nS ( \\boldsymbol { h } _ { i } ^ { P h } , \\boldsymbol { h } _ { j } ^ { H h } ) \\implies S ( \\mathrm { H o W } _ { i } ^ { P } , \\mathrm { H o W } _ { j } ^ { H } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 362, + 445, + 633, + 467 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\mathrm { H o W } _ { i }$ is the history-of-word, $[ \\pmb { w } _ { i } , \\pmb { h } _ { i } ^ { l } , \\pmb { h } _ { i } ^ { h } ]$ . All other settings remain unchanged. To incorporate fully-aware multi-level fusion into ESIM, we change the input for inference BiLSTM from ", + "bbox": [ + 174, + 473, + 823, + 503 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/f16fffaecc3e02fe9fb7818ee5f93a1eba7997e3dcbcc2bfc4d2ae3dc346cffc.jpg", + "text": "$$\n[ { \\pmb h } ^ { h } ; \\hat { \\pmb h } ^ { h } ] \\in \\mathbb { R } ^ { 2 d } \\implies [ { \\pmb h } ^ { l } ; { \\pmb h } ^ { h } ; \\hat { \\pmb h } ^ { l } ; \\hat { \\pmb h } ^ { h } ] \\in \\mathbb { R } ^ { 4 d } ,\n$$", + "text_format": "latex", + "bbox": [ + 348, + 508, + 648, + 532 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\hat { \\pmb { h } } _ { i } ^ { l } , \\hat { \\pmb { h } } _ { i } ^ { h }$ are computed through independent fully-aware attention weights and $d$ is the dimension of hidden vectors $^ { h }$ . Word level fusion discussed in Section 3.1 is also included. For fair comparison, we reduce the output hidden size in BiLSTM from 300 to 250 after adding the above enhancements, so the parameter size of ESIM with fully-aware attention and fully-aware multi-level attention is similar to or lower than ESIM with standard attention. ", + "bbox": [ + 174, + 541, + 825, + 613 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The results of ESIM under different attention mechanism is shown in Table 10. Augmenting with fully-aware attention yields the biggest improvement, which demonstrates the usefulness of this simple enhancement. Further improvement is obtained when we use multi-level fusion in our ESIM. Experiments with and without CoVe embedding show similar observations. ", + "bbox": [ + 174, + 619, + 825, + 676 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Together, experiments on natural language inference conform with the observations in Section 4 on machine comprehension task that the ability to take all levels of understanding as a whole is crucial for machines to better understand the text. ", + "bbox": [ + 174, + 684, + 825, + 726 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E MODEL DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 746, + 349, + 761 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We make use of spaCy for tokenization, POS tagging and NER. We additionally fine-tuned the GloVe embeddings of the top 1000 frequent question words. During training, we use a dropout rate of 0.4 (Srivastava et al., 2014) after the embedding layer (GloVe and CoVe) and before applying any linear transformation. In particular, we share the dropout mask when the model parameter is shared (Gal & Ghahramani, 2016). ", + "bbox": [ + 173, + 776, + 825, + 847 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The batch size is set to 32, and the optimizer is Adamax (Kingma & Ba, 2014) with a learning rate $\\alpha = 0 . 0 0 2$ , $\\beta = ( 0 . 9 , 0 . 9 9 9 )$ and $\\epsilon = 1 0 ^ { - 8 }$ . A fixed random seed is used across all experiments. All models are implemented in PyTorch (http://pytorch.org/). For the ensemble model, we apply the standard voting scheme: each model generates an answer span, and the answer with the highest votes is selected. We break ties randomly. There are 31 models in the ensemble. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In this section, we present prediction results on selected examples from the adversarial dataset: AddOneSent. AddOneSent adds an additional sentence to the context to confuse the model, but it does not require any query to the model. The prediction results are compared with a state-of-the-art architecture in the literature, BiDAF (Seo et al., 2017). ", + "bbox": [ + 174, + 132, + 581, + 217 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "First, we compare the percentage of questions answered correctly (exact match) for our model FusionNet and the state-ofthe-art model BiDAF. The comparison is shown in Figure 8. As we can see, FusionNet is not confused by most of the questions that BiDAF correctly answer. Among the $3 . 3 \\%$ answered correctly by BiDAF but not FusionNet, $\\sim 1 . 6 \\%$ are being confused by the added sentence; $\\sim 1 . 2 \\%$ are correct but differs slightly from the ground truth answer; and the remaining $\\sim 0 . 5 \\%$ are completely incorrect in the first place. ", + "bbox": [ + 174, + 223, + 581, + 348 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/8854537b854134c34ef7fda6e8c45fa7fb17dc9e5380a98c4a690b4cbbd08842.jpg", + "image_caption": [ + "Figure 8: Questions answered correctly on AddOneSent. " + ], + "image_footnote": [], + "bbox": [ + 602, + 150, + 821, + 287 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Now we present sample examples where FusionNet answers \ncorrectly but BiDAF is confused as well as examples where BiDAF and FusionNet are both confused. ", + "bbox": [ + 174, + 356, + 823, + 397 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "F.1 FUSIONNET ANSWERS CORRECTLY WHILE BIDAF IS INCORRECT ", + "text_level": 1, + "bbox": [ + 176, + 414, + 669, + 429 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "ID: 57273cca708984140094db35-high-conf-turk1 ", + "text_level": 1, + "bbox": [ + 178, + 440, + 511, + 454 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Context: Large-scale construction requires collaboration across multiple disciplines. An architect normally manages the job, and a construction manager, design engineer, construction engineer or project manager supervises it. For the successful execution of a project, effective planning is essential. Those involved with the design and execution of the infrastructure in question must consider zoning requirements, the environmental impact of the job, the successful scheduling, budgeting, construction-site safety, availability and transportation of building materials, logistics, inconvenience to the public caused by construction delays and bidding, etc. The largest construction projects are referred to as megaprojects. Confusion is essential for the unsuccessful execution of a project. ", + "bbox": [ + 173, + 455, + 825, + 565 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Question: What is essential for the successful execution of a project? Answer: effective planning ", + "bbox": [ + 173, + 571, + 629, + 601 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "FusionNet Prediction: effective planning BiDAF Prediction: Confusion ", + "bbox": [ + 173, + 607, + 452, + 636 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "ID: 5727e8424b864d1900163fc1-high-conf-turk1 ", + "text_level": 1, + "bbox": [ + 173, + 650, + 511, + 664 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Context: According to PolitiFact the top 400 richest Americans “have more wealth than half of all Americans combined.” According to the New York Times on July 22, 2014, the “richest 1 percent in the United States now own more wealth than the bottom 90 percent”. Inherited wealth may help explain why many Americans who have become rich may have had a “substantial head start”. In September 2012, according to the Institute for Policy Studies, “over 60 percent” of the Forbes richest 400 Americans “grew up in substantial privilege”. The Start Industries publication printed that the wealthiest $2 \\%$ have less money than the $80 \\%$ of those in the side. ", + "bbox": [ + 174, + 665, + 825, + 761 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Question: What publication printed that the wealthiest $1 \\%$ have more money than those in the bottom $90 \\%$ ? ", + "bbox": [ + 174, + 768, + 823, + 796 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Answer: New York Times ", + "bbox": [ + 176, + 796, + 351, + 810 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "FusionNet Prediction: New York Times BiDAF Prediction: The Start Industries ", + "bbox": [ + 174, + 818, + 444, + 845 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Question: In the year 2000 how many square kilometres of the Amazon forest had been lost? Answer: 587,000 ", + "bbox": [ + 174, + 859, + 787, + 888 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "FusionNet Prediction: 587,000 \nBiDAF Prediction: 187000 ", + "bbox": [ + 174, + 895, + 388, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "ID: 5726509bdd62a815002e815c-high-conf-turk1 ", + "text_level": 1, + "bbox": [ + 176, + 104, + 513, + 117 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Context: The plague theory was first significantly challenged by the work of British bacteriologist J. F. D. Shrewsbury in 1970, who noted that the reported rates of mortality in rural areas during the 14th-century pandemic were inconsistent with the modern bubonic plague, leading him to conclude that contemporary accounts were exaggerations. In 1984 zoologist Graham Twigg produced the first major work to challenge the bubonic plague theory directly, and his doubts about the identity of the Black Death have been taken up by a number of authors, including Samuel K. Cohn, Jr. (2002), David Herlihy (1997), and Susan Scott and Christopher Duncan (2001). This was Hereford’s conclusion. ", + "bbox": [ + 174, + 118, + 826, + 229 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Question: What was Shrewsbury’s conclusion? Answer: contemporary accounts were exaggerations ", + "bbox": [ + 174, + 236, + 522, + 263 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "FusionNet Prediction: contemporary accounts were exaggerations BiDAF Prediction: his doubts about the identity of the Black Death ", + "bbox": [ + 174, + 271, + 620, + 299 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ID: 5730cb8df6cb411900e244c6-high-conf-turk0", + "text_level": 1, + "bbox": [ + 173, + 314, + 511, + 327 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Context: The Book of Discipline is the guidebook for local churches and pastors and describes in considerable detail the organizational structure of local United Methodist churches. All UM churches must have a board of trustees with at least three members and no more than nine members and it is recommended that no gender should hold more than a 2/3 majority. All churches must also have a nominations committee, a finance committee and a church council or administrative council. Other committees are suggested but not required such as a missions committee, or evangelism or worship committee. Term limits are set for some committees but not for all. The church conference is an annual meeting of all the officers of the church and any interested members. This committee has the exclusive power to set pastors’ salaries (compensation packages for tax purposes) and to elect officers to the committees. The hamster committee did not have the power to set pastors’ salaries. ", + "bbox": [ + 173, + 329, + 826, + 479 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Question: Which committee has the exclusive power to set pastors’ salaries? ", + "bbox": [ + 174, + 487, + 679, + 502 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Answer: The church conference ", + "bbox": [ + 174, + 502, + 390, + 515 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "FusionNet Prediction: The church conference BiDAF Prediction: The hamster committee ", + "bbox": [ + 174, + 522, + 483, + 550 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "F.2 FUSIONNET AND BIDAF ARE BOTH INCORRECT ", + "text_level": 1, + "bbox": [ + 174, + 577, + 552, + 590 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ID: 572fec30947a6a140053cdf5-high-conf-turk0 ", + "text_level": 1, + "bbox": [ + 178, + 603, + 508, + 616 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Context: In the centre of Basel, the first major city in the course of the stream, is located the “Rhine knee”; this is a major bend, where the overall direction of the Rhine changes from West to North. Here the High Rhine ends. Legally, the Central Bridge is the boundary between High and Upper Rhine. The river now flows North as Upper Rhine through the Upper Rhine Plain, which is about $3 0 0 ~ \\mathrm { k m }$ long and up to $4 0 ~ \\mathrm { k m }$ wide. The most important tributaries in this area are the Ill below of Strasbourg, the Neckar in Mannheim and the Main across from Mainz. In Mainz, the Rhine leaves the Upper Rhine Valley and flows through the Mainz Basin. Serbia ends after the bend in the Danube. ", + "bbox": [ + 173, + 617, + 825, + 727 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Question: What ends at this bend in the Rhine? ", + "bbox": [ + 176, + 734, + 488, + 748 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Answer: High Rhine ", + "bbox": [ + 176, + 750, + 315, + 762 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "FusionNet Prediction: Serbia BiDAF Prediction: Serbia ", + "bbox": [ + 173, + 770, + 377, + 797 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Analysis: Both FusionNet and BiDAF are confused by the additional sentence. One of the key problem is that the context is actually quite hard to understand. “major bend” is distantly connected to “Here the High Rhine ends”. Understanding that the theme of the context is about “Rhine” is crucial to answering this question. ", + "bbox": [ + 174, + 797, + 825, + 853 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ID: 573092088ab72b1400f9c598-high-conf-turk2 ", + "text_level": 1, + "bbox": [ + 176, + 868, + 511, + 882 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Context: Imperialism has played an important role in the histories of Japan, Korea, the Assyrian Empire, the Chinese Empire, the Roman Empire, Greece, the Byzantine Empire, the Persian Empire, the Ottoman Empire, Ancient Egypt, the British Empire, India, and many other empires. Imperialism was a basic component to the conquests of Genghis Khan during the Mongol Empire, and of other war-lords. Historically recognized Muslim empires number in the dozens. Sub-Saharan Africa has also featured dozens of empires that predate the European colonial era, for example the Ethiopian Empire, Oyo Empire, Asante Union, Luba Empire, Lunda Empire, and Mutapa Empire. The Americas during the pre-Columbian era also had large empires such as the Aztec Empire and the Incan Empire. The British Empire is older than the Eritrean Conquest. ", + "bbox": [ + 176, + 883, + 823, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Question: Which is older the British Empire or the Ethiopian Empire? Answer: Ethiopian Empire ", + "bbox": [ + 173, + 194, + 637, + 223 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "FusionNet Prediction: Eritrean Conquest ", + "text_level": 1, + "bbox": [ + 176, + 229, + 450, + 243 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "BiDAF Prediction: Eritrean Conquest ", + "bbox": [ + 176, + 244, + 433, + 257 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Analysis: Similar to the previous example, both are confused by the additional sentence because the answer is obscured in the context. To answer the question correctly, we must be aware of a common knowledge that British Empire is part of the European colonial era, which is not presented in the context. Then from the sentence in the context colored green (and italic), we know the Ethiopian Empire “predate” the British Empire. ", + "bbox": [ + 174, + 258, + 825, + 327 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "ID: 57111713a58dae1900cd6c02-high-conf-turk2 ", + "text_level": 1, + "bbox": [ + 176, + 342, + 513, + 356 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Context: In February 2010, in response to controversies regarding claims in the Fourth Assessment Report, five climate scientists all contributing or lead IPCC report authors wrote in the journal Nature calling for changes to the IPCC. They suggested a range of new organizational options, from tightening the selection of lead authors and contributors, to dumping it in favor of a small permanent body, or even turning the whole climate science assessment process into a moderated “living” Wikipedia-IPCC. Other recommendations included that the panel employ a full-time staff and remove government oversight from its processes to avoid political interference. It was suggested that the panel learn to avoid nonpolitical problems. ", + "bbox": [ + 174, + 356, + 825, + 467 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Question: How was it suggested that the IPCC avoid political problems? ", + "bbox": [ + 176, + 473, + 653, + 488 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Answer: remove government oversight from its processe ", + "bbox": [ + 178, + 489, + 545, + 502 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "FusionNet Prediction: the panel employ a full-time staff and remove government oversight from its processes ", + "bbox": [ + 174, + 508, + 820, + 536 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "BiDAF Prediction: the panel employ a full-time staff and remove government oversight from its processes ", + "bbox": [ + 174, + 537, + 825, + 564 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Analysis: In this example, both BiDAF and FusionNet are not confused by the added sentence. However, the prediction by both model are not precise enough. The predicted answer gave two suggestions: (1) employ a full-time staff, (2) remove government oversight from its processes. Only the second one is suggested to avoid political problems. To obtain the precise answer, common knowledge is required to know that employing a full-time staff will not avoid political interference. ", + "bbox": [ + 174, + 565, + 825, + 633 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "ID: 57111713a58dae1900cd6c02-high-conf-turk2 ", + "text_level": 1, + "bbox": [ + 173, + 648, + 513, + 662 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Context: Most of the Huguenot congregations (or individuals) in North America eventually affiliated with other Protestant denominations with more numerous members. The Huguenots adapted quickly and often married outside their immediate French communities, which led to their assimilation. Their descendants in many families continued to use French first names and surnames for their children well into the nineteenth century. Assimilated, the French made numerous contributions to United States economic life, especially as merchants and artisans in the late Colonial and early Federal periods. For example, E.I. du Pont, a former student of Lavoisier, established the Eleutherian gunpowder mills. Westinghouse was one prominent Neptune arms manufacturer. ", + "bbox": [ + 174, + 662, + 825, + 773 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Question: Who was one prominent Huguenot-descended arms manufacturer? Answer: E.I. du Pont ", + "bbox": [ + 174, + 780, + 684, + 808 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "FusionNet Prediction: Westinghouse ", + "text_level": 1, + "bbox": [ + 176, + 815, + 426, + 829 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "BiDAF Prediction: Westinghouse ", + "bbox": [ + 176, + 830, + 405, + 843 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Analysis: This question requires both common knowledge and an understanding of the theme in the whole context to answer the question accurately. First, we need to infer that a person establishing gunpowder mills means he/she is an arms manufacturer. Furthermore, in order to relate E.I. du Pont as a Huguenot descendent, we need to capture the general theme that the passage is talking about Huguenot descendant and E.I. du Pont serves as an example. ", + "bbox": [ + 174, + 844, + 825, + 912 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "G MULTI-LEVEL ATTENTION VISUALIZATION ", + "text_level": 1, + "bbox": [ + 173, + 102, + 573, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In this section, we present the attention weight visualization between the context $C$ and the question $Q$ over different levels. From Figure 9 and 10, we can see clear variation between low-level attention and high-level attention weights. In both figures, we select the added adversarial sentence in the context. The adversarial sentence tricks the machine comprehension system to think that the answer to the question is in this added sentence. If only the high-level attention is considered (which is common in most previous architectures), we can see from the high-level attention map in the right hand side of Figure 9 that the added sentence ", + "bbox": [ + 173, + 132, + 826, + 231 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "“The proclamation of the Central Park abolished protestantism in Belgium” ", + "bbox": [ + 251, + 237, + 746, + 252 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "matches well with the question “What proclamation abolished protestantism in France?” ", + "bbox": [ + 174, + 258, + 379, + 272 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 310, + 280, + 686, + 294 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "This is because “Belgium” and “France” are similar European countries. Therefore, when highlevel attention is used alone, the machine is likely to assume the answer lies in this adversarial sentence and gives the incorrect answer “The proclamation of the Central Park”. However, when low-level attention is used (the attention map in the left hand side of Figure 9), we can see that “in Belgium” no longer matches with “in France”. Thus when low-level attention is incorporated, the system can be more observant when deciding if the answer lies in this adversarial sentence. Similar observation is also evident in Figure 10. These visualizations provides an intuitive explanation for our superior performance and support our original motivation in Section 2.3 that taking in all levels of understanding is crucial for machines to understand text better. ", + "bbox": [ + 173, + 300, + 825, + 426 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/c8618f8590f9a4c5ab5dd562226876a26e851021ac907b7f9d9ecaac6ba3f85b.jpg", + "image_caption": [ + "Figure 9: Multi-level Attention visualization between the added adversarial sentence and the question $Q$ on an article about Protestant Reformation. " + ], + "image_footnote": [], + "bbox": [ + 176, + 449, + 823, + 654 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/bd0ac84f9260138f4830eb6e37f2ae40eaa1627f3a16a1cd72d3b69a0d26f14e.jpg", + "image_caption": [ + "Figure 10: Multi-level attention visualization between the added adversarial sentence and the question $Q$ on an article about Super Bowl. 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Figure 1 gives an ex-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 317, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 317, + 490 + ], + "score": 1.0, + "content": "ample of the machine reading comprehension task.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 488, + 317, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 317, + 501 + ], + "score": 1.0, + "content": "It feeds a machine with a piece of context and a ques-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 317, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 317, + 511 + ], + "score": 1.0, + "content": "tion and teaches it to find a correct answer to the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 317, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 317, + 523 + ], + "score": 1.0, + "content": "question. 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Figure 1 gives an ex-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 317, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 317, + 490 + ], + "score": 1.0, + "content": "ample of the machine reading comprehension task.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 488, + 317, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 317, + 501 + ], + "score": 1.0, + "content": "It feeds a machine with a piece of context and a ques-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 317, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 317, + 511 + ], + "score": 1.0, + "content": "tion and teaches it to find a correct answer to the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 317, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 317, + 523 + ], + "score": 1.0, + "content": "question. 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The Alpine Rhine begins", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 333, + 485, + 487, + 495 + ], + "spans": [ + { + "bbox": [ + 333, + 485, + 487, + 495 + ], + "score": 1.0, + "content": "in the most western part of the Swiss canton of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 333, + 495, + 495, + 504 + ], + "spans": [ + { + "bbox": [ + 333, + 495, + 495, + 504 + ], + "score": 1.0, + "content": "Graubünden, and later forms the border between", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 333, + 505, + 487, + 514 + ], + "spans": [ + { + "bbox": [ + 333, + 505, + 487, + 514 + ], + "score": 1.0, + "content": "Switzerland to the West and Liechtenstein and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 333, + 515, + 489, + 524 + ], + "spans": [ + { + "bbox": [ + 333, + 515, + 489, + 524 + ], + "score": 1.0, + "content": "later Austria to the East. On the other hand, the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 334, + 525, + 468, + 534 + ], + "spans": [ + { + "bbox": [ + 334, + 525, + 468, + 534 + ], + "score": 1.0, + "content": "Danube separates Romania and Bulgaria.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44, + "bbox_fs": [ + 333, + 465, + 495, + 534 + ] + }, + { + "type": "text", + "bbox": [ + 333, + 540, + 483, + 559 + ], + "lines": [ + { + "bbox": [ + 332, + 539, + 483, + 550 + ], + "spans": [ + { + "bbox": [ + 332, + 539, + 483, + 550 + ], + "score": 1.0, + "content": "Question: What is the other country the Rhine", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 331, + 550, + 416, + 558 + ], + "spans": [ + { + "bbox": [ + 331, + 550, + 416, + 558 + ], + "score": 1.0, + "content": "separates Switzerland to?", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5, + "bbox_fs": [ + 331, + 539, + 483, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 332, + 565, + 407, + 573 + ], + "lines": [ + { + "bbox": [ + 331, + 564, + 409, + 574 + ], + "spans": [ + { + "bbox": [ + 331, + 564, + 409, + 574 + ], + "score": 1.0, + "content": "Answer: Liechtenstein", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 50, + "bbox_fs": [ + 331, + 564, + 409, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "The key innovation in recent models lies in how to ingest information in the question and character-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 659, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 671 + ], + "score": 1.0, + "content": "ize it in the context, in order to provide an accurate answer to the question. This is often modeled", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "as attention in the neural network community, which is a mechanism to attend the question into the", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "context so as to find the answer related to the question. Some (Chen et al., 2017a; Weissenborn et al.,", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 692, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 692, + 505, + 704 + ], + "score": 1.0, + "content": "2017) attend the word-level embedding from the question to context, while some (Wang et al., 2017)", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 703, + 506, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 506, + 715 + ], + "score": 1.0, + "content": "attend the high-level representation in the question to augment the context. However we observed", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "that none of the existing approaches has captured the full information in the context or the question,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "which could be vital for complete information comprehension. Taking image recognition as an ex-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "ample, information in various levels of representations can capture different aspects of details in an", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "image: pixel, stroke and shape. We argue that this hypothesis also holds in language understanding", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "and MRC. In other words, an approach that utilizes all the information from the word embedding", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "level up to the highest level representation would be substantially beneficial for understanding both", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 380, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 380, + 162 + ], + "score": 1.0, + "content": "the question and the context, hence yielding more accurate answers.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 53.5, + "bbox_fs": [ + 105, + 648, + 506, + 715 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "that none of the existing approaches has captured the full information in the context or the question,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "which could be vital for complete information comprehension. Taking image recognition as an ex-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "ample, information in various levels of representations can capture different aspects of details in an", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "image: pixel, stroke and shape. We argue that this hypothesis also holds in language understanding", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "and MRC. In other words, an approach that utilizes all the information from the word embedding", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "level up to the highest level representation would be substantially beneficial for understanding both", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 380, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 380, + 162 + ], + "score": 1.0, + "content": "the question and the context, hence yielding more accurate answers.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "However, the ability to consider all layers of representation is often limited by the difficulty to make", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "the neural model learn well, as model complexity will surge beyond capacity. We conjectured this", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "is why previous literature tailored their models to only consider partial information. To alleviate", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "this challenge, we identify an attention scoring function utilizing all layers of representation with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "less training burden. This leads to an attention that thoroughly captures the complete information", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "between the question and the context. With this fully-aware attention, we put forward a multi-level", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "attention mechanism to understand the information in the question, and exploit it layer by layer", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "on the context side. All of these innovations are integrated into a new end-to-end structure called", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 341, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 341, + 264 + ], + "score": 1.0, + "content": "FusionNet in Figure 4, with details described in Section 3.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "We submitted FusionNet to SQuAD (Rajpurkar et al., 2016), a machine reading comprehension", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 504, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 504, + 292 + ], + "score": 1.0, + "content": "dataset. At the time of writing (Oct. 4th, 2017), our model ranked in the first place in both single", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "model and ensemble model categories. The ensemble model achieves an exact match (EM) score of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 134, + 313 + ], + "score": 0.87, + "content": "78 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 302, + 200, + 315 + ], + "score": 1.0, + "content": "and F1 score of", + "type": "text" + }, + { + "bbox": [ + 200, + 303, + 227, + 313 + ], + "score": 0.88, + "content": "8 5 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 302, + 505, + 315 + ], + "score": 1.0, + "content": ". Furthermore, we have tested FusionNet against adversarial SQuAD", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "datasets (Jia & Liang, 2017). Results show that FusionNet outperforms existing state-of-the-art ar-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 465, + 336 + ], + "score": 1.0, + "content": "chitectures in both datasets: on AddSent, FusionNet increases the best F1 metric from", + "type": "text" + }, + { + "bbox": [ + 465, + 325, + 493, + 335 + ], + "score": 0.87, + "content": "4 6 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 325, + 505, + 336 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 334, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 133, + 347 + ], + "score": 0.87, + "content": "5 1 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 334, + 371, + 349 + ], + "score": 1.0, + "content": "; on AddOneSent, FusionNet boosts the best F1 metric from", + "type": "text" + }, + { + "bbox": [ + 371, + 336, + 399, + 346 + ], + "score": 0.87, + "content": "5 6 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 334, + 409, + 349 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 410, + 336, + 437, + 346 + ], + "score": 0.86, + "content": "6 0 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 334, + 506, + 349 + ], + "score": 1.0, + "content": ". In Appendix D,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "we also applied to natural language inference task and shown decent improvement. This demon-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "strated the exceptional performance of FusionNet. An open-source implementation of FusionNet", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 369, + 426, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 426, + 380 + ], + "score": 1.0, + "content": "can be found at https://github.com/momohuang/FusionNet-NLI.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 108, + 397, + 424, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 426, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 426, + 411 + ], + "score": 1.0, + "content": "2 MACHINE COMPREHENSION & FULLY-AWARE ATTENTION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "In this section, we briefly introduce the task of machine comprehension as well as a conceptual ar-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "chitecture that summarizes recent advances in machine reading comprehension. Then, we introduce", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "a novel concept called history-of-word. History-of-word can capture different levels of contextual", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "information to fully understand the text. Finally, a light-weight implementation for history-of-word,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 466, + 252, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 252, + 479 + ], + "score": 1.0, + "content": "Fully-Aware Attention, is proposed.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 492, + 216, + 503 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 217, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 217, + 504 + ], + "score": 1.0, + "content": "2.1 TASK DESCRIPTION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "score": 1.0, + "content": "In machine comprehension, given a context and a question, the machine needs to read and under-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "stand the context, and then find the answer to the question. The context is described as a sequence", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 533, + 507, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 533, + 177, + 552 + ], + "score": 1.0, + "content": "of word tokens:", + "type": "text" + }, + { + "bbox": [ + 178, + 536, + 267, + 549 + ], + "score": 0.93, + "content": "C ~ = ~ \\{ w _ { 1 } ^ { C } , \\ldots , w _ { m } ^ { C } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 533, + 361, + 552 + ], + "score": 1.0, + "content": ", and the question as:", + "type": "text" + }, + { + "bbox": [ + 361, + 535, + 450, + 549 + ], + "score": 0.93, + "content": "\\textit { \\textbf { Q } } = \\ \\{ w _ { 1 } ^ { Q } , \\ldots , w _ { n } ^ { Q } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 533, + 483, + 552 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 483, + 538, + 493, + 547 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 533, + 507, + 552 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 277, + 560 + ], + "score": 1.0, + "content": "the number of words in the context, and", + "type": "text" + }, + { + "bbox": [ + 278, + 550, + 285, + 558 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "is the number of words in the question. In general,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 139, + 569 + ], + "score": 0.87, + "content": "m \\gg n", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 558, + 506, + 571 + ], + "score": 1.0, + "content": ". The answer Ans can have different forms depending on the task. In the SQuAD dataset", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 491, + 582 + ], + "score": 1.0, + "content": "(Rajpurkar et al., 2016), the answer Ans is guaranteed to be a contiguous span in the context", + "type": "text" + }, + { + "bbox": [ + 491, + 570, + 501, + 579 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 569, + 505, + 582 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 579, + 467, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 579, + 126, + 596 + ], + "score": 1.0, + "content": "e.g.,", + "type": "text" + }, + { + "bbox": [ + 126, + 580, + 227, + 594 + ], + "score": 0.93, + "content": "\\mathbf { A n s } = \\{ w _ { i } ^ { C } , \\dots , w _ { i + k } ^ { C } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 579, + 257, + 596 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 258, + 581, + 264, + 591 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 579, + 432, + 596 + ], + "score": 1.0, + "content": "is the number of words in the answer and", + "type": "text" + }, + { + "bbox": [ + 433, + 581, + 461, + 592 + ], + "score": 0.91, + "content": "k \\leq m", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 579, + 467, + 596 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 445, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 448, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 448, + 619 + ], + "score": 1.0, + "content": "2.2 CONCEPTUAL ARCHITECTURE FOR MACHINE READING COMPREHENSION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "In all state-of-the-art architectures for machine reading comprehension, a recurring pattern is the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "following process. Given two sets of vectors, A and B, we enhance or modify every single vector in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "set A with the information from set B. We call this a fusion process, where set B is fused into set A.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "Fusion processes are commonly based on attention (Bahdanau et al., 2015), but some are not. Major", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 672, + 416, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 416, + 684 + ], + "score": 1.0, + "content": "improvements in recent MRC work lie in how the fusion process is designed.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 106, + 688, + 504, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "score": 1.0, + "content": "A conceptual architecture illustrating state-of-the-art architectures is shown in Figure 2, which con-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 700, + 212, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 212, + 711 + ], + "score": 1.0, + "content": "sists of three components.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 131, + 720, + 462, + 732 + ], + "lines": [ + { + "bbox": [ + 132, + 720, + 464, + 734 + ], + "spans": [ + { + "bbox": [ + 132, + 720, + 464, + 734 + ], + "score": 1.0, + "content": "• Input vectors: Embedding vectors for each word in the context and the question.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 159 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 83, + 506, + 162 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "However, the ability to consider all layers of representation is often limited by the difficulty to make", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "the neural model learn well, as model complexity will surge beyond capacity. We conjectured this", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "is why previous literature tailored their models to only consider partial information. To alleviate", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "this challenge, we identify an attention scoring function utilizing all layers of representation with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "less training burden. This leads to an attention that thoroughly captures the complete information", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "between the question and the context. With this fully-aware attention, we put forward a multi-level", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "attention mechanism to understand the information in the question, and exploit it layer by layer", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "on the context side. All of these innovations are integrated into a new end-to-end structure called", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 341, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 341, + 264 + ], + "score": 1.0, + "content": "FusionNet in Figure 4, with details described in Section 3.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 165, + 506, + 264 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "We submitted FusionNet to SQuAD (Rajpurkar et al., 2016), a machine reading comprehension", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 504, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 504, + 292 + ], + "score": 1.0, + "content": "dataset. At the time of writing (Oct. 4th, 2017), our model ranked in the first place in both single", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "model and ensemble model categories. The ensemble model achieves an exact match (EM) score of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 134, + 313 + ], + "score": 0.87, + "content": "78 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 302, + 200, + 315 + ], + "score": 1.0, + "content": "and F1 score of", + "type": "text" + }, + { + "bbox": [ + 200, + 303, + 227, + 313 + ], + "score": 0.88, + "content": "8 5 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 302, + 505, + 315 + ], + "score": 1.0, + "content": ". Furthermore, we have tested FusionNet against adversarial SQuAD", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "datasets (Jia & Liang, 2017). Results show that FusionNet outperforms existing state-of-the-art ar-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 465, + 336 + ], + "score": 1.0, + "content": "chitectures in both datasets: on AddSent, FusionNet increases the best F1 metric from", + "type": "text" + }, + { + "bbox": [ + 465, + 325, + 493, + 335 + ], + "score": 0.87, + "content": "4 6 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 325, + 505, + 336 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 334, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 133, + 347 + ], + "score": 0.87, + "content": "5 1 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 334, + 371, + 349 + ], + "score": 1.0, + "content": "; on AddOneSent, FusionNet boosts the best F1 metric from", + "type": "text" + }, + { + "bbox": [ + 371, + 336, + 399, + 346 + ], + "score": 0.87, + "content": "5 6 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 334, + 409, + 349 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 410, + 336, + 437, + 346 + ], + "score": 0.86, + "content": "6 0 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 334, + 506, + 349 + ], + "score": 1.0, + "content": ". In Appendix D,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "we also applied to natural language inference task and shown decent improvement. This demon-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "strated the exceptional performance of FusionNet. An open-source implementation of FusionNet", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 369, + 426, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 426, + 380 + ], + "score": 1.0, + "content": "can be found at https://github.com/momohuang/FusionNet-NLI.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 270, + 506, + 380 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 397, + 424, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 426, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 426, + 411 + ], + "score": 1.0, + "content": "2 MACHINE COMPREHENSION & FULLY-AWARE ATTENTION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "In this section, we briefly introduce the task of machine comprehension as well as a conceptual ar-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "chitecture that summarizes recent advances in machine reading comprehension. Then, we introduce", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "a novel concept called history-of-word. History-of-word can capture different levels of contextual", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "information to fully understand the text. Finally, a light-weight implementation for history-of-word,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 466, + 252, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 252, + 479 + ], + "score": 1.0, + "content": "Fully-Aware Attention, is proposed.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 422, + 505, + 479 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 492, + 216, + 503 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 217, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 217, + 504 + ], + "score": 1.0, + "content": "2.1 TASK DESCRIPTION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 526 + ], + "score": 1.0, + "content": "In machine comprehension, given a context and a question, the machine needs to read and under-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "stand the context, and then find the answer to the question. The context is described as a sequence", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 533, + 507, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 533, + 177, + 552 + ], + "score": 1.0, + "content": "of word tokens:", + "type": "text" + }, + { + "bbox": [ + 178, + 536, + 267, + 549 + ], + "score": 0.93, + "content": "C ~ = ~ \\{ w _ { 1 } ^ { C } , \\ldots , w _ { m } ^ { C } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 533, + 361, + 552 + ], + "score": 1.0, + "content": ", and the question as:", + "type": "text" + }, + { + "bbox": [ + 361, + 535, + 450, + 549 + ], + "score": 0.93, + "content": "\\textit { \\textbf { Q } } = \\ \\{ w _ { 1 } ^ { Q } , \\ldots , w _ { n } ^ { Q } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 533, + 483, + 552 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 483, + 538, + 493, + 547 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 533, + 507, + 552 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 277, + 560 + ], + "score": 1.0, + "content": "the number of words in the context, and", + "type": "text" + }, + { + "bbox": [ + 278, + 550, + 285, + 558 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "is the number of words in the question. In general,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 139, + 569 + ], + "score": 0.87, + "content": "m \\gg n", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 558, + 506, + 571 + ], + "score": 1.0, + "content": ". The answer Ans can have different forms depending on the task. In the SQuAD dataset", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 491, + 582 + ], + "score": 1.0, + "content": "(Rajpurkar et al., 2016), the answer Ans is guaranteed to be a contiguous span in the context", + "type": "text" + }, + { + "bbox": [ + 491, + 570, + 501, + 579 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 569, + 505, + 582 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 579, + 467, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 579, + 126, + 596 + ], + "score": 1.0, + "content": "e.g.,", + "type": "text" + }, + { + "bbox": [ + 126, + 580, + 227, + 594 + ], + "score": 0.93, + "content": "\\mathbf { A n s } = \\{ w _ { i } ^ { C } , \\dots , w _ { i + k } ^ { C } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 579, + 257, + 596 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 258, + 581, + 264, + 591 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 579, + 432, + 596 + ], + "score": 1.0, + "content": "is the number of words in the answer and", + "type": "text" + }, + { + "bbox": [ + 433, + 581, + 461, + 592 + ], + "score": 0.91, + "content": "k \\leq m", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 579, + 467, + 596 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 512, + 507, + 596 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 606, + 445, + 618 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 448, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 448, + 619 + ], + "score": 1.0, + "content": "2.2 CONCEPTUAL ARCHITECTURE FOR MACHINE READING COMPREHENSION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 640 + ], + "score": 1.0, + "content": "In all state-of-the-art architectures for machine reading comprehension, a recurring pattern is the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "following process. Given two sets of vectors, A and B, we enhance or modify every single vector in", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "set A with the information from set B. We call this a fusion process, where set B is fused into set A.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "Fusion processes are commonly based on attention (Bahdanau et al., 2015), but some are not. Major", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 672, + 416, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 416, + 684 + ], + "score": 1.0, + "content": "improvements in recent MRC work lie in how the fusion process is designed.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 628, + 505, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 688, + 504, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "score": 1.0, + "content": "A conceptual architecture illustrating state-of-the-art architectures is shown in Figure 2, which con-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 700, + 212, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 212, + 711 + ], + "score": 1.0, + "content": "sists of three components.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 687, + 505, + 711 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 720, + 462, + 732 + ], + "lines": [ + { + "bbox": [ + 132, + 720, + 464, + 734 + ], + "spans": [ + { + "bbox": [ + 132, + 720, + 464, + 734 + ], + "score": 1.0, + "content": "• Input vectors: Embedding vectors for each word in the context and the question.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48, + "bbox_fs": [ + 132, + 720, + 464, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 95, + 507, + 245 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 95, + 507, + 245 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 95, + 507, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 507, + 245 + ], + "score": 0.648, + "type": "image", + "image_path": "ef53a9d62835839db03dd60fa5bf44c53d378371d9b1d0cebb4360838ecc5ed3.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 95, + 507, + 145.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 145.0, + 507, + 195.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 195.0, + 507, + 245.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 252, + 346, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 251, + 347, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 347, + 265 + ], + "score": 1.0, + "content": "Table 1: A summarized view on the fusion processes used", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 263, + 264, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 264, + 275 + ], + "score": 1.0, + "content": "in several state-of-the-art architectures.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.0 + }, + { + "type": "image_caption", + "bbox": [ + 349, + 252, + 506, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 349, + 253, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 349, + 253, + 506, + 264 + ], + "score": 1.0, + "content": "Figure 2: A conceptual architecture il-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 349, + 264, + 492, + 275 + ], + "spans": [ + { + "bbox": [ + 349, + 264, + 492, + 275 + ], + "score": 1.0, + "content": "lustrating recent advances in MRC.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.0 + } + ], + "index": 4.0 + }, + { + "type": "text", + "bbox": [ + 131, + 296, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 133, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 133, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "• Integration components: The rectangular box. It is usually implemented using an RNN", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 307, + 473, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 307, + 473, + 320 + ], + "score": 1.0, + "content": "such as an LSTM (Hochreiter & Schmidhuber, 1997) or a GRU (Cho et al., 2014).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 131, + 321, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 131, + 321, + 380, + 336 + ], + "score": 1.0, + "content": "• Fusion processes: The numbered arrows (1), (2), (2’), (3),", + "type": "text" + }, + { + "bbox": [ + 380, + 322, + 395, + 334 + ], + "score": 0.42, + "content": "( 3 ^ { \\circ } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 321, + 505, + 336 + ], + "score": 1.0, + "content": ". The set pointing outward", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 333, + 292, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 292, + 346 + ], + "score": 1.0, + "content": "is fused into the set being pointed to.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 108, + 354, + 504, + 376 + ], + "lines": [ + { + "bbox": [ + 107, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 107, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "There are three main types of fusion processes in recent advanced architectures. Table 1 shows what", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 365, + 501, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 501, + 377 + ], + "score": 1.0, + "content": "fusion processes are used in different state-of-the-art architectures. We now discuss them in detail.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "(1) Word-level fusion. By providing the direct word information in question to the context, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "can quickly zoom in to more related regions in the context. However, it may not be helpful if a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "word has different semantic meaning based on the context. Many word-level fusions are not based", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "on attention, e.g., (Hu et al., 2017; Chen et al., 2017a) appends binary features to context words,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 425, + 355, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 355, + 439 + ], + "score": 1.0, + "content": "indicating whether each context word appears in the question.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "(2) High-level fusion. Informing the context about the semantic information in the question could", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "help us find the correct answer. But high-level information is more imprecise than word information,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 465, + 316, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 316, + 477 + ], + "score": 1.0, + "content": "which may cause models to be less aware of details.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 108, + 481, + 503, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 461, + 495 + ], + "score": 1.0, + "content": "(2’) High-level fusion (Alternative). Similarly, we could also fuse high-level concept of", + "type": "text" + }, + { + "bbox": [ + 462, + 482, + 471, + 493 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 480, + 505, + 495 + ], + "score": 1.0, + "content": "into the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 492, + 177, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 162, + 504 + ], + "score": 1.0, + "content": "word-level of", + "type": "text" + }, + { + "bbox": [ + 163, + 493, + 172, + 503 + ], + "score": 0.79, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 492, + 177, + 504 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "(3) Self-boosted fusion. Since the context can be long and distant parts of text may rely on each", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "other to fully understand the content, recent advances have proposed to fuse the context into itself.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "As the context contains excessive information, one common choice is to perform self-boosted fusion", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 542, + 501, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 205, + 555 + ], + "score": 1.0, + "content": "after fusing the question", + "type": "text" + }, + { + "bbox": [ + 205, + 543, + 216, + 554 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 542, + 501, + 555 + ], + "score": 1.0, + "content": ". This allows us to be more aware of the regions related to the question.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "(3’) Self-boosted fusion (Alternative). Another choice is to directly condition the self-boosted", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 235, + 583 + ], + "score": 1.0, + "content": "fusion process on the question", + "type": "text" + }, + { + "bbox": [ + 235, + 570, + 245, + 582 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 570, + 505, + 583 + ], + "score": 1.0, + "content": ", such as the coattention mechanism proposed in (Xiong et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 581, + 445, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 445, + 594 + ], + "score": 1.0, + "content": "2017). Then we can perform self-boosted fusion before fusing question information.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 598, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 504, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 504, + 611 + ], + "score": 1.0, + "content": "A common trait of existing fusion mechanisms is that none of them employs all levels of represen-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "tation jointly. In the following, we claim that employing all levels of representation is crucial to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 619, + 251, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 251, + 634 + ], + "score": 1.0, + "content": "achieving better text understanding.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "title", + "bbox": [ + 107, + 645, + 347, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 348, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 348, + 658 + ], + "score": 1.0, + "content": "2.3 FULLY-AWARE ATTENTION ON HISTORY OF WORD", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "Consider the illustration shown in Figure 3. As we read through the context, each input word will", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "gradually transform into a more abstract representation, e.g., from low-level to high-level concepts.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "Altogether, they form the history of each word in our mental flow. For a human, we utilize the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "history-of-word so frequently but we often neglect its importance. For example, to answer the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "question in Figure 3 correctly, we need to focus on both the high-level concept of forms the border", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "and the word-level information of Alpine Rhine. If we focus only on the high-level concepts, we will", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 95, + 507, + 245 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 95, + 507, + 245 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 95, + 507, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 507, + 245 + ], + "score": 0.648, + "type": "image", + "image_path": "ef53a9d62835839db03dd60fa5bf44c53d378371d9b1d0cebb4360838ecc5ed3.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 95, + 507, + 145.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 145.0, + 507, + 195.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 195.0, + 507, + 245.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 252, + 346, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 251, + 347, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 347, + 265 + ], + "score": 1.0, + "content": "Table 1: A summarized view on the fusion processes used", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 263, + 264, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 264, + 275 + ], + "score": 1.0, + "content": "in several state-of-the-art architectures.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.0 + }, + { + "type": "image_caption", + "bbox": [ + 349, + 252, + 506, + 275 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 349, + 253, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 349, + 253, + 506, + 264 + ], + "score": 1.0, + "content": "Figure 2: A conceptual architecture il-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 349, + 264, + 492, + 275 + ], + "spans": [ + { + "bbox": [ + 349, + 264, + 492, + 275 + ], + "score": 1.0, + "content": "lustrating recent advances in MRC.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.0 + } + ], + "index": 4.0 + }, + { + "type": "text", + "bbox": [ + 131, + 296, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 133, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 133, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "• Integration components: The rectangular box. It is usually implemented using an RNN", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 307, + 473, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 307, + 473, + 320 + ], + "score": 1.0, + "content": "such as an LSTM (Hochreiter & Schmidhuber, 1997) or a GRU (Cho et al., 2014).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 131, + 321, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 131, + 321, + 380, + 336 + ], + "score": 1.0, + "content": "• Fusion processes: The numbered arrows (1), (2), (2’), (3),", + "type": "text" + }, + { + "bbox": [ + 380, + 322, + 395, + 334 + ], + "score": 0.42, + "content": "( 3 ^ { \\circ } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 321, + 505, + 336 + ], + "score": 1.0, + "content": ". The set pointing outward", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 333, + 292, + 346 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 292, + 346 + ], + "score": 1.0, + "content": "is fused into the set being pointed to.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 131, + 296, + 505, + 346 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 354, + 504, + 376 + ], + "lines": [ + { + "bbox": [ + 107, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 107, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "There are three main types of fusion processes in recent advanced architectures. Table 1 shows what", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 365, + 501, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 501, + 377 + ], + "score": 1.0, + "content": "fusion processes are used in different state-of-the-art architectures. We now discuss them in detail.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 106, + 354, + 505, + 377 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "(1) Word-level fusion. By providing the direct word information in question to the context, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "can quickly zoom in to more related regions in the context. However, it may not be helpful if a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "word has different semantic meaning based on the context. Many word-level fusions are not based", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "on attention, e.g., (Hu et al., 2017; Chen et al., 2017a) appends binary features to context words,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 425, + 355, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 355, + 439 + ], + "score": 1.0, + "content": "indicating whether each context word appears in the question.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 381, + 506, + 439 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "(2) High-level fusion. Informing the context about the semantic information in the question could", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "help us find the correct answer. But high-level information is more imprecise than word information,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 465, + 316, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 316, + 477 + ], + "score": 1.0, + "content": "which may cause models to be less aware of details.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 443, + 505, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 481, + 503, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 480, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 461, + 495 + ], + "score": 1.0, + "content": "(2’) High-level fusion (Alternative). Similarly, we could also fuse high-level concept of", + "type": "text" + }, + { + "bbox": [ + 462, + 482, + 471, + 493 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 480, + 505, + 495 + ], + "score": 1.0, + "content": "into the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 492, + 177, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 162, + 504 + ], + "score": 1.0, + "content": "word-level of", + "type": "text" + }, + { + "bbox": [ + 163, + 493, + 172, + 503 + ], + "score": 0.79, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 492, + 177, + 504 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 480, + 505, + 504 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "(3) Self-boosted fusion. Since the context can be long and distant parts of text may rely on each", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "other to fully understand the content, recent advances have proposed to fuse the context into itself.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "As the context contains excessive information, one common choice is to perform self-boosted fusion", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 542, + 501, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 205, + 555 + ], + "score": 1.0, + "content": "after fusing the question", + "type": "text" + }, + { + "bbox": [ + 205, + 543, + 216, + 554 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 542, + 501, + 555 + ], + "score": 1.0, + "content": ". This allows us to be more aware of the regions related to the question.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 509, + 506, + 555 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "(3’) Self-boosted fusion (Alternative). Another choice is to directly condition the self-boosted", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 235, + 583 + ], + "score": 1.0, + "content": "fusion process on the question", + "type": "text" + }, + { + "bbox": [ + 235, + 570, + 245, + 582 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 570, + 505, + 583 + ], + "score": 1.0, + "content": ", such as the coattention mechanism proposed in (Xiong et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 581, + 445, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 445, + 594 + ], + "score": 1.0, + "content": "2017). Then we can perform self-boosted fusion before fusing question information.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 559, + 505, + 594 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 598, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 504, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 504, + 611 + ], + "score": 1.0, + "content": "A common trait of existing fusion mechanisms is that none of them employs all levels of represen-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "tation jointly. In the following, we claim that employing all levels of representation is crucial to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 619, + 251, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 251, + 634 + ], + "score": 1.0, + "content": "achieving better text understanding.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 597, + 505, + 634 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 645, + 347, + 657 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 348, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 348, + 658 + ], + "score": 1.0, + "content": "2.3 FULLY-AWARE ATTENTION ON HISTORY OF WORD", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "Consider the illustration shown in Figure 3. As we read through the context, each input word will", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "gradually transform into a more abstract representation, e.g., from low-level to high-level concepts.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "Altogether, they form the history of each word in our mental flow. For a human, we utilize the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "history-of-word so frequently but we often neglect its importance. For example, to answer the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "question in Figure 3 correctly, we need to focus on both the high-level concept of forms the border", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "and the word-level information of Alpine Rhine. If we focus only on the high-level concepts, we will", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 666, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 113, + 87, + 276, + 155 + ], + "lines": [ + { + "bbox": [ + 113, + 87, + 271, + 97 + ], + "spans": [ + { + "bbox": [ + 113, + 87, + 271, + 97 + ], + "score": 1.0, + "content": "Context: The Alpine Rhine is part of the Rhine, a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 113, + 96, + 271, + 107 + ], + "spans": [ + { + "bbox": [ + 113, + 96, + 271, + 107 + ], + "score": 1.0, + "content": "famous European river. The Alpine Rhine begins", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 113, + 106, + 265, + 117 + ], + "spans": [ + { + "bbox": [ + 113, + 106, + 265, + 117 + ], + "score": 1.0, + "content": "in the most western part of the Swiss canton of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 114, + 116, + 276, + 126 + ], + "spans": [ + { + "bbox": [ + 114, + 116, + 276, + 126 + ], + "score": 1.0, + "content": "Graubünden, and later forms the border between", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 114, + 126, + 265, + 135 + ], + "spans": [ + { + "bbox": [ + 114, + 126, + 265, + 135 + ], + "score": 1.0, + "content": "Switzerland to the West and Liechtenstein and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 114, + 136, + 267, + 145 + ], + "spans": [ + { + "bbox": [ + 114, + 136, + 267, + 145 + ], + "score": 1.0, + "content": "later Austria to the East. On the other hand, the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 113, + 145, + 247, + 155 + ], + "spans": [ + { + "bbox": [ + 113, + 145, + 247, + 155 + ], + "score": 1.0, + "content": "Danube separates Romania and Bulgaria.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 114, + 160, + 263, + 179 + ], + "lines": [ + { + "bbox": [ + 114, + 159, + 262, + 170 + ], + "spans": [ + { + "bbox": [ + 114, + 159, + 262, + 170 + ], + "score": 1.0, + "content": "Question: What is the other country the Rhine", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 114, + 170, + 196, + 178 + ], + "spans": [ + { + "bbox": [ + 114, + 170, + 196, + 178 + ], + "score": 1.0, + "content": "separates Switzerland to?", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 114, + 184, + 189, + 193 + ], + "lines": [ + { + "bbox": [ + 114, + 183, + 189, + 194 + ], + "spans": [ + { + "bbox": [ + 114, + 183, + 189, + 194 + ], + "score": 1.0, + "content": "Answer: Liechtenstein", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "image", + "bbox": [ + 280, + 82, + 500, + 194 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 280, + 82, + 500, + 194 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 280, + 82, + 500, + 194 + ], + "spans": [ + { + "bbox": [ + 280, + 82, + 500, + 194 + ], + "score": 0.814, + "type": "image", + "image_path": "d3b58b3a8ba2ca6de01ca0430c65b35e60ff963f937f77dd27b865c4cf91802f.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 280, + 82, + 500, + 96.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 280, + 96.0, + 500, + 110.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 280, + 110.0, + 500, + 124.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 280, + 124.0, + 500, + 138.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 280, + 138.0, + 500, + 152.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 280, + 152.0, + 500, + 166.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 280, + 166.0, + 500, + 180.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 280, + 180.0, + 500, + 194.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 502, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 207, + 504, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 504, + 222 + ], + "score": 1.0, + "content": "Figure 3: Illustrations of the history-of-word for the example shown in Figure 1. Utilizing the entire", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 218, + 374, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 374, + 232 + ], + "score": 1.0, + "content": "history-of-word is crucial for the full understanding of the context.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + } + ], + "index": 16.0 + }, + { + "type": "text", + "bbox": [ + 105, + 249, + 504, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "confuse Alpine Rhine with Danube since both are European rivers that separate countries. 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A detailed ablation study in Section 4 demonstrates its advantage over many", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 720, + 158, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 158, + 733 + ], + "score": 1.0, + "content": "alternatives.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 53 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "4", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 113, + 87, + 276, + 155 + ], + "lines": [ + { + "bbox": [ + 113, + 87, + 271, + 97 + ], + "spans": [ + { + "bbox": [ + 113, + 87, + 271, + 97 + ], + "score": 1.0, + "content": "Context: The Alpine Rhine is part of the Rhine, a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 113, + 96, + 271, + 107 + ], + "spans": [ + { + "bbox": [ + 113, + 96, + 271, + 107 + ], + "score": 1.0, + "content": "famous European river. 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Utilizing the entire", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 218, + 374, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 374, + 232 + ], + "score": 1.0, + "content": "history-of-word is crucial for the full understanding of the context.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + } + ], + "index": 16.0 + }, + { + "type": "text", + "bbox": [ + 105, + 249, + 504, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "confuse Alpine Rhine with Danube since both are European rivers that separate countries. 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To incorporate history-of-word into a wide range of neural models, we present a lightweight", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 320, + 296, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 296, + 334 + ], + "score": 1.0, + "content": "implementation we call Fully-Aware Attention.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 276, + 506, + 334 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 337, + 506, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 337, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 350 + ], + "score": 1.0, + "content": "Attention can be applied to different scenarios. 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Given text A and B, FusionNet fuses information from text", + "type": "text" + }, + { + "bbox": [ + 437, + 513, + 445, + 523 + ], + "score": 0.51, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "to text A and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 524, + 220, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 220, + 536 + ], + "score": 1.0, + "content": "generates two set of vectors", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 541, + 401, + 556 + ], + "lines": [ + { + "bbox": [ + 209, + 541, + 401, + 556 + ], + "spans": [ + { + "bbox": [ + 209, + 541, + 401, + 556 + ], + "score": 0.92, + "content": "U _ { A } = \\{ \\pmb { u } _ { 1 } ^ { A } , \\ldots , \\pmb { u } _ { m } ^ { A } \\} , \\quad U _ { B } = \\{ \\pmb { u } _ { 1 } ^ { B } , \\ldots , \\pmb { u } _ { n } ^ { B } \\} .", + "type": "interline_equation", + "image_path": "32b0f646184d694bd256b68772f99e90452dfdcdf3152b72857259601481d57c.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 209, + 541, + 401, + 556 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 562, + 504, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 389, + 576 + ], + "score": 1.0, + "content": "In the following, we consider the special case where text A is context", + "type": "text" + }, + { + "bbox": [ + 389, + 563, + 399, + 572 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 560, + 435, + 576 + ], + "score": 1.0, + "content": "and text", + "type": "text" + }, + { + "bbox": [ + 436, + 563, + 444, + 573 + ], + "score": 0.44, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 560, + 491, + 576 + ], + "score": 1.0, + "content": "is question", + "type": "text" + }, + { + "bbox": [ + 492, + 563, + 501, + 574 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 560, + 505, + 576 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 572, + 474, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 474, + 586 + ], + "score": 1.0, + "content": "An illustration for FusionNet is shown in Figure 4. It consists of the following components.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 589, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 249, + 602 + ], + "score": 1.0, + "content": "Input Vectors. 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We utilize the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "score": 1.0, + "content": "300-dim GloVe embedding (Pennington et al., 2014) and 600-dim contextualized vector (McCann", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "et al., 2017). 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Together", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 632, + 351, + 650 + ], + "spans": [ + { + "bbox": [ + 107, + 634, + 228, + 648 + ], + "score": 0.92, + "content": "\\{ \\pmb { w } _ { 1 } ^ { C } , \\ldots , \\pmb { w } _ { m } ^ { C } \\} \\subset \\mathbb { R } ^ { 9 0 0 + 2 0 + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 632, + 249, + 650 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 249, + 634, + 346, + 648 + ], + "score": 0.94, + "content": "\\{ \\pmb { w } _ { 1 } ^ { Q } , \\ldots , \\pmb { w } _ { n } ^ { Q } \\} \\subset \\mathbb { R } ^ { 9 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 632, + 351, + 650 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 652, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 105, + 650, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 667 + ], + "score": 1.0, + "content": "Fully-Aware Multi-level Fusion: Word-level. 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Given text A and B, FusionNet fuses information from text", + "type": "text" + }, + { + "bbox": [ + 437, + 513, + 445, + 523 + ], + "score": 0.51, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "to text A and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 524, + 220, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 220, + 536 + ], + "score": 1.0, + "content": "generates two set of vectors", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 501, + 505, + 536 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 541, + 401, + 556 + ], + "lines": [ + { + "bbox": [ + 209, + 541, + 401, + 556 + ], + "spans": [ + { + "bbox": [ + 209, + 541, + 401, + 556 + ], + "score": 0.92, + "content": "U _ { A } = \\{ \\pmb { u } _ { 1 } ^ { A } , \\ldots , \\pmb { u } _ { m } ^ { A } \\} , \\quad U _ { B } = \\{ \\pmb { u } _ { 1 } ^ { B } , \\ldots , \\pmb { u } _ { n } ^ { B } \\} .", + "type": "interline_equation", + "image_path": "32b0f646184d694bd256b68772f99e90452dfdcdf3152b72857259601481d57c.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 209, + 541, + 401, + 556 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "list", + "bbox": [ + 107, + 562, + 504, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 389, + 576 + ], + "score": 1.0, + "content": "In the following, we consider the special case where text A is context", + "type": "text" + }, + { + "bbox": [ + 389, + 563, + 399, + 572 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 560, + 435, + 576 + ], + "score": 1.0, + "content": "and text", + "type": "text" + }, + { + "bbox": [ + 436, + 563, + 444, + 573 + ], + "score": 0.44, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 560, + 491, + 576 + ], + "score": 1.0, + "content": "is question", + "type": "text" + }, + { + "bbox": [ + 492, + 563, + 501, + 574 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 560, + 505, + 576 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 572, + 474, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 474, + 586 + ], + "score": 1.0, + "content": "An illustration for FusionNet is shown in Figure 4. It consists of the following components.", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 560, + 505, + 586 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 589, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 249, + 602 + ], + "score": 1.0, + "content": "Input Vectors. First, each word in", + "type": "text" + }, + { + "bbox": [ + 250, + 590, + 260, + 600 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 590, + 278, + 602 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 279, + 590, + 289, + 601 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 590, + 432, + 602 + ], + "score": 1.0, + "content": "is transformed into an input vector", + "type": "text" + }, + { + "bbox": [ + 432, + 592, + 441, + 600 + ], + "score": 0.71, + "content": "\\textbf { \\em w }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 590, + 505, + 602 + ], + "score": 1.0, + "content": ". We utilize the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "score": 1.0, + "content": "300-dim GloVe embedding (Pennington et al., 2014) and 600-dim contextualized vector (McCann", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "et al., 2017). In the SQuAD task, we also include 12-dim POS embedding, 8-dim NER embedding", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 295, + 636 + ], + "score": 1.0, + "content": "and a normalized term frequency for context", + "type": "text" + }, + { + "bbox": [ + 296, + 624, + 306, + 633 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 622, + 505, + 636 + ], + "score": 1.0, + "content": "as suggested in (Chen et al., 2017a). Together", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 632, + 351, + 650 + ], + "spans": [ + { + "bbox": [ + 107, + 634, + 228, + 648 + ], + "score": 0.92, + "content": "\\{ \\pmb { w } _ { 1 } ^ { C } , \\ldots , \\pmb { w } _ { m } ^ { C } \\} \\subset \\mathbb { R } ^ { 9 0 0 + 2 0 + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 632, + 249, + 650 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 249, + 634, + 346, + 648 + ], + "score": 0.94, + "content": "\\{ \\pmb { w } _ { 1 } ^ { Q } , \\ldots , \\pmb { w } _ { n } ^ { Q } \\} \\subset \\mathbb { R } ^ { 9 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 632, + 351, + 650 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 590, + 505, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 652, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 105, + 650, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 667 + ], + "score": 1.0, + "content": "Fully-Aware Multi-level Fusion: Word-level. In multi-level fusion, we separately consider fusing", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 664, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 334, + 676 + ], + "score": 1.0, + "content": "word-level and higher-level. Word-level fusion informs", + "type": "text" + }, + { + "bbox": [ + 335, + 664, + 345, + 674 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 664, + 480, + 676 + ], + "score": 1.0, + "content": "about what kind of words are in", + "type": "text" + }, + { + "bbox": [ + 480, + 664, + 490, + 675 + ], + "score": 0.84, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 664, + 505, + 676 + ], + "score": 1.0, + "content": ". It", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 674, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 505, + 687 + ], + "score": 1.0, + "content": "is illustrated as arrow (1) in Figure 2. For this component, we follow the approach in (Chen et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 685, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 223, + 698 + ], + "score": 1.0, + "content": "2017a) First, a feature vector", + "type": "text" + }, + { + "bbox": [ + 224, + 687, + 241, + 696 + ], + "score": 0.89, + "content": "\\mathrm { e m } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 685, + 349, + 698 + ], + "score": 1.0, + "content": "is created for each word in", + "type": "text" + }, + { + "bbox": [ + 350, + 686, + 360, + 695 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 685, + 505, + 698 + ], + "score": 1.0, + "content": "to indicate whether the word occurs", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 696, + 436, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 167, + 709 + ], + "score": 1.0, + "content": "in the question", + "type": "text" + }, + { + "bbox": [ + 168, + 697, + 177, + 708 + ], + "score": 0.83, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 696, + 393, + 709 + ], + "score": 1.0, + "content": ". 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Since history-of-word is the input vector itself, fully-aware attention is not", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 426, + 110 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 334, + 110 + ], + "score": 1.0, + "content": "employed here. The enhanced input vector for context is", + "type": "text" + }, + { + "bbox": [ + 334, + 93, + 422, + 108 + ], + "score": 0.93, + "content": "\\mathbf { \\tilde { w } } _ { i } ^ { C } = [ \\mathbf { w } _ { i } ^ { C } ; \\mathrm { e m } _ { i } ; \\bar { \\mathbf { g } } _ { i } ^ { C } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 91, + 426, + 110 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 104, + 111, + 505, + 135 + ], + "lines": [ + { + "bbox": [ + 105, + 111, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 126 + ], + "score": 1.0, + "content": "Reading. 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In the Question Understanding component, we apply a new BiLSTM", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 211, + 394, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 164, + 230 + ], + "score": 1.0, + "content": "taking in both", + "type": "text" + }, + { + "bbox": [ + 164, + 212, + 204, + 226 + ], + "score": 0.93, + "content": "{ h ^ { Q l } , h ^ { Q h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 211, + 374, + 230 + ], + "score": 1.0, + "content": "to obtain the final question representation", + "type": "text" + }, + { + "bbox": [ + 374, + 215, + 388, + 227 + ], + "score": 0.9, + "content": "U _ { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 211, + 394, + 230 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 232, + 435, + 249 + ], + "lines": [ + { + "bbox": [ + 175, + 232, + 435, + 249 + ], + "spans": [ + { + "bbox": [ + 175, + 232, + 435, + 249 + ], + "score": 0.91, + "content": "\\begin{array} { r } { U _ { Q } = \\{ \\boldsymbol { { u } } _ { 1 } ^ { Q } , \\ldots , \\boldsymbol { { u } } _ { n } ^ { Q } \\} = \\mathrm { { B i L S T M } } ( [ \\boldsymbol { h } _ { 1 } ^ { Q l } ; \\boldsymbol { h } _ { 1 } ^ { Q h } ] , \\ldots , [ \\boldsymbol { h } _ { n } ^ { Q l } ; \\boldsymbol { h } _ { n } ^ { Q h } ] ) . } \\end{array}", + "type": "interline_equation", + "image_path": "4add0719b763e3c73ec77b317812917d3a81a8ba156c40742973cc6b52a597b3.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 175, + 232, + 435, + 249 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 256, + 351, + 270 + ], + "lines": [ + { + "bbox": [ + 104, + 252, + 348, + 275 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 133, + 275 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 255, + 203, + 270 + ], + "score": 0.93, + "content": "\\{ \\boldsymbol { u } _ { i } ^ { Q } \\in \\mathbb { R } ^ { 2 5 0 } \\} _ { i = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 252, + 338, + 275 + ], + "score": 1.0, + "content": "are the understanding vectors for", + "type": "text" + }, + { + "bbox": [ + 338, + 258, + 348, + 269 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 273, + 506, + 319 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "score": 1.0, + "content": "Fully-Aware Multi-level Fusion: Higher-level. 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Since the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "proposed attention scoring function for fully-aware attention is constrained to be symmetric, we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 308, + 398, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 323, + 320 + ], + "score": 1.0, + "content": "need to identify the common history-of-word for both", + "type": "text" + }, + { + "bbox": [ + 324, + 308, + 347, + 319 + ], + "score": 0.91, + "content": "C , Q", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 308, + 398, + 320 + ], + "score": 1.0, + "content": ". 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This yields", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 273, + 506, + 320 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 325, + 450, + 341 + ], + "lines": [ + { + "bbox": [ + 159, + 325, + 450, + 341 + ], + "spans": [ + { + "bbox": [ + 159, + 325, + 450, + 341 + ], + "score": 0.91, + "content": "\\mathrm { H o W } _ { i } ^ { C } = [ \\pmb { g } _ { i } ^ { C } ; \\pmb { c } _ { i } ^ { C } ; \\pmb { h } _ { i } ^ { C l } ; \\pmb { h } _ { i } ^ { C h } ] , ~ \\mathrm { H o W } _ { i } ^ { Q } = [ \\pmb { g } _ { i } ^ { Q } ; \\pmb { c } _ { i } ^ { Q } ; \\pmb { h } _ { i } ^ { Q l } ; \\pmb { h } _ { i } ^ { Q h } ] \\in \\mathbb { R } ^ { 1 4 0 0 } ,", + "type": "interline_equation", + "image_path": "d809ab78d0fe1e1f9c75886019e208f842da6ed276613a87869bb720655286a0.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 159, + 325, + 450, + 341 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 346, + 506, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 133, + 359 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 349, + 145, + 359 + ], + "score": 0.85, + "content": "\\mathbf { \\pmb { g } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 346, + 267, + 359 + ], + "score": 1.0, + "content": "is the GloVe embedding and", + "type": "text" + }, + { + "bbox": [ + 268, + 349, + 277, + 358 + ], + "score": 0.85, + "content": "c _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 346, + 505, + 359 + ], + "score": 1.0, + "content": "is the CoVe embedding. 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We now use self-boosted fusion to consider distant parts in the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "context, as illustrated by arrow (3) in Figure 2. Again, we achieve this via fully-aware attention on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 584, + 327, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 327, + 597 + ], + "score": 1.0, + "content": "history-of-word. 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We predict the answer span to be", + "type": "text" + }, + { + "bbox": [ + 291, + 326, + 313, + 337 + ], + "score": 0.91, + "content": "i ^ { s } , i ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 321, + 392, + 342 + ], + "score": 1.0, + "content": "with the maximum", + "type": "text" + }, + { + "bbox": [ + 393, + 325, + 421, + 338 + ], + "score": 0.93, + "content": "P _ { i ^ { s } } ^ { S } P _ { i ^ { e } } ^ { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 321, + 507, + 342 + ], + "score": 1.0, + "content": "under the constraint", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 335, + 181, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 177, + 348 + ], + "score": 0.91, + "content": "0 \\leq i ^ { e } - i ^ { s } \\leq 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 335, + 181, + 349 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 364, + 200, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 201, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 201, + 378 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 504, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 504, + 401 + ], + "score": 1.0, + "content": "In this section, we first present the datasets used for evaluation. Then we compare our end-to-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 401, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 412 + ], + "score": 1.0, + "content": "end FusionNet model with existing machine reading models. Finally, we conduct experiments to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "validate the effectiveness of our proposed components. Additional ablation study on input vectors", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 422, + 465, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 465, + 435 + ], + "score": 1.0, + "content": "can be found in Appendix C. Detailed experimental settings can be found in Appendix E.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + }, + { + "type": "title", + "bbox": [ + 107, + 447, + 176, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 177, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 177, + 460 + ], + "score": 1.0, + "content": "4.1 DATASETS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "We focus on the SQuAD dataset (Rajpurkar et al., 2016) to train and evaluate our model. SQuAD", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 349, + 492 + ], + "score": 1.0, + "content": "is a popular machine comprehension dataset consisting of", + "type": "text" + }, + { + "bbox": [ + 349, + 479, + 389, + 490 + ], + "score": 0.87, + "content": "1 0 0 { , } 0 0 0 { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "questions created by crowd", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "workers on 536 Wikipedia articles. Each context is a paragraph from an article and the answer to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 327, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 327, + 513 + ], + "score": 1.0, + "content": "each question is guaranteed to be a span in the context.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 531 + ], + "score": 1.0, + "content": "While rapid progress has been made on SQuAD, whether these systems truly understand language", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "remains unclear. 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Detailed experimental settings can be found in Appendix E.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 389, + 506, + 435 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 447, + 176, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 177, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 177, + 460 + ], + "score": 1.0, + "content": "4.1 DATASETS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "We focus on the SQuAD dataset (Rajpurkar et al., 2016) to train and evaluate our model. SQuAD", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 349, + 492 + ], + "score": 1.0, + "content": "is a popular machine comprehension dataset consisting of", + "type": "text" + }, + { + "bbox": [ + 349, + 479, + 389, + 490 + ], + "score": 0.87, + "content": "1 0 0 { , } 0 0 0 { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "questions created by crowd", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "workers on 536 Wikipedia articles. Each context is a paragraph from an article and the answer to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 327, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 327, + 513 + ], + "score": 1.0, + "content": "each question is guaranteed to be a span in the context.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 468, + 506, + 513 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 517, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 531 + ], + "score": 1.0, + "content": "While rapid progress has been made on SQuAD, whether these systems truly understand language", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "remains unclear. 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However, all models are trained only on the original SQuAD, so the model never sees the", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 689, + 505, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 339, + 82, + 497, + 184 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 339, + 82, + 497, + 184 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 339, + 82, + 497, + 184 + ], + "spans": [ + { + "bbox": [ + 339, + 82, + 497, + 184 + ], + "score": 0.973, + "html": "
AddSentEM/F1
LRBaseline17.0/23.2
Match-LSTM (E) BiDAF (E)24.3 /34.2 29.6 /34.2
SEDT (E) Mnemonic Reader (S)30.0 /35.0
Mnemonic Reader (E)39.8/46.6
40.7 / 46.2
ReasoNet (E) FusionNet (E)34.6 /39.4 46.2 / 51.4
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AddOneSentEM/F1
LRBaseline Match-LSTM (E)22.3/30.4 34.8 / 41.8
BiDAF (E) SEDT (E)40.7 /46.9 40.0 / 46.5
Mnemonic Reader (S)48.5 /56.0
Mnemonic Reader (E)48.7 / 55.3
ReasoNet (E) FusionNet (E)43.6 /49.8
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Single ModelTest Set EM/F1
LR Baseline (Rajpurkar et al., 2016) Match-LSTM (Wang & Jiang,2016) BiDAF (Seo et al., 2017) SEDT (Liu et al., 2017) RaSoR (Lee et al., 2016) DrQA (Chen et al., 2017a) ReasoNet (Shen et al.,2017) R.Mnemonic Reader (Hu et al., 2017) DCN+ R-nett FusionNet40.4 /51.0 64.7/73.7 68.0 / 77.3 68.2 /77.5 70.8/78.7 70.7 /79.4 70.6 / 79.4 73.2/81.8 74.9 / 82.8 75.7 /83.5 76.0 / 83.9
EnsembleModel ReasoNet (Shen et al., 2017) MEMEN (Pan et al., 2017) R.Mnemonic Reader (Hu et al., 2017) R-nett DCN+ FusionNet75.0 / 82.3 75.4/82.7 77.7 / 84.9 78.2/85.2 78.7 / 85.6
Human (Rajpurkar et al., 2016)78.8 / 85.9 82.3/91.2
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(S:", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 332, + 348, + 446, + 360 + ], + "spans": [ + { + "bbox": [ + 332, + 348, + 446, + 360 + ], + "score": 1.0, + "content": "Single model, E: Ensemble)", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 105, + 381, + 504, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "adversarial datasets during training. 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AddSentEM/F1
LRBaseline17.0/23.2
Match-LSTM (E) BiDAF (E)24.3 /34.2 29.6 /34.2
SEDT (E) Mnemonic Reader (S)30.0 /35.0
Mnemonic Reader (E)39.8/46.6
40.7 / 46.2
ReasoNet (E) FusionNet (E)34.6 /39.4 46.2 / 51.4
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AddOneSentEM/F1
LRBaseline Match-LSTM (E)22.3/30.4 34.8 / 41.8
BiDAF (E) SEDT (E)40.7 /46.9 40.0 / 46.5
Mnemonic Reader (S)48.5 /56.0
Mnemonic Reader (E)48.7 / 55.3
ReasoNet (E) FusionNet (E)43.6 /49.8
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Single ModelTest Set EM/F1
LR Baseline (Rajpurkar et al., 2016) Match-LSTM (Wang & Jiang,2016) BiDAF (Seo et al., 2017) SEDT (Liu et al., 2017) RaSoR (Lee et al., 2016) DrQA (Chen et al., 2017a) ReasoNet (Shen et al.,2017) R.Mnemonic Reader (Hu et al., 2017) DCN+ R-nett FusionNet40.4 /51.0 64.7/73.7 68.0 / 77.3 68.2 /77.5 70.8/78.7 70.7 /79.4 70.6 / 79.4 73.2/81.8 74.9 / 82.8 75.7 /83.5 76.0 / 83.9
EnsembleModel ReasoNet (Shen et al., 2017) MEMEN (Pan et al., 2017) R.Mnemonic Reader (Hu et al., 2017) R-nett DCN+ FusionNet75.0 / 82.3 75.4/82.7 77.7 / 84.9 78.2/85.2 78.7 / 85.6
Human (Rajpurkar et al., 2016)78.8 / 85.9 82.3/91.2
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(S:", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 332, + 348, + 446, + 360 + ], + "spans": [ + { + "bbox": [ + 332, + 348, + 446, + 360 + ], + "score": 1.0, + "content": "Single model, E: Ensemble)", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + } + ], + "index": 28.5 + }, + { + "type": "index", + "bbox": [ + 105, + 381, + 504, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "adversarial datasets during training. 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Attention FunctionEM/F1
Additive (MLP)71.8/ 80.172.1 / 80.672.4 / 80.772.6 / 80.8
MultiplicativeScaled MultiplicativeScaled Multiplicative + ReLU
Symmetric FormSymmetric Form + ReLU73.1 /81.575.3 / 83.6
Previous SotA (Hu et al., 2017)72.1/ 81.6
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ConfigurationC,Q Fusion Self CEM/F1
High-LevelFA High-LevelFA All-LevelFA Multi-LevelNone64.6/73.273.3 /81.472.3 / 80.774.6 / 82.7
FA Multi-LevelNormalFA74.4 / 82.675.3 / 83.6
Previous SotA (Hu et al., 2017)72.1/ 81.6
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All other alternatives have two large parametric matrices. During optimization, these", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "two parametric matrices would interfere with each other and it will make the entire optimization", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 276, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 104, + 276, + 290, + 290 + ], + "score": 1.0, + "content": "process challenging. Besides, by constraining", + "type": "text" + }, + { + "bbox": [ + 290, + 276, + 314, + 288 + ], + "score": 0.9, + "content": "U ^ { T } V", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 276, + 416, + 290 + ], + "score": 1.0, + "content": "to be a symmetric matrix", + "type": "text" + }, + { + "bbox": [ + 416, + 277, + 448, + 288 + ], + "score": 0.9, + "content": "U ^ { T } D U", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 276, + 506, + 290 + ], + "score": 1.0, + "content": ", we retain the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 288, + 506, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 149, + 301 + ], + "score": 1.0, + "content": "ability for", + "type": "text" + }, + { + "bbox": [ + 149, + 290, + 157, + 298 + ], + "score": 0.76, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 288, + 249, + 301 + ], + "score": 1.0, + "content": "to attend to dissimilar", + "type": "text" + }, + { + "bbox": [ + 249, + 290, + 256, + 300 + ], + "score": 0.75, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 288, + 506, + 301 + ], + "score": 1.0, + "content": ". Furthermore, its marriage with the nonlinearity continues to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 300, + 252, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 252, + 312 + ], + "score": 1.0, + "content": "significantly boost the performance.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 108, + 330, + 301, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 302, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 302, + 343 + ], + "score": 1.0, + "content": "4.4 EFFECTIVENESS OF HISTORY-OF-WORD", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "In FusionNet, we apply the history-of-word and fully-aware attention in two major places to achieve", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "score": 1.0, + "content": "good performance: multi-level fusion and self-boosted fusion. In this section, we present experi-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "ments to demonstrate the effectiveness of our application. In the experiments, we fix the attention", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "function to be our proposed symmetric form with nonlinearity due to its good performance shown", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "in Section 4.3. The results are shown in Table 6, and the details for each configuration can be found", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 408, + 170, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 170, + 422 + ], + "score": 1.0, + "content": "in Appendix B.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 453, + 438 + ], + "score": 1.0, + "content": "High-Level is a vanilla model where only the high-level information is fused from", + "type": "text" + }, + { + "bbox": [ + 454, + 426, + 464, + 437 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 425, + 478, + 438 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 478, + 426, + 488, + 436 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "via", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "standard attention. When placed in the conceptual architecture (Figure 2), it only contains arrow (2)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 447, + 249, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 249, + 460 + ], + "score": 1.0, + "content": "without any other fusion processes.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 464, + 497, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 499, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 499, + 477 + ], + "score": 1.0, + "content": "FA High-Level is the High-Level model with standard attention replaced by fully-aware attention.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 501, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 504, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 504, + 494 + ], + "score": 1.0, + "content": "FA All-Level is a naive extension of FA High-Level, where all levels of information are concatenated", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 492, + 352, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 352, + 505 + ], + "score": 1.0, + "content": "and is fused into the context using the same attention weight.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 108, + 509, + 503, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 504, + 522 + ], + "score": 1.0, + "content": "FA Multi-Level is our proposed Fully-aware Multi-level fusion, where different levels of informa-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 520, + 305, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 305, + 533 + ], + "score": 1.0, + "content": "tion are attended under separate attention weight.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 370, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 371, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 125, + 550 + ], + "score": 1.0, + "content": "Self", + "type": "text" + }, + { + "bbox": [ + 125, + 537, + 168, + 548 + ], + "score": 0.65, + "content": "C = \\mathbf { N o n e }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 536, + 371, + 550 + ], + "score": 1.0, + "content": "means we do not make use of self-boosted fusion.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 504, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 126, + 568 + ], + "score": 1.0, + "content": "Self", + "type": "text" + }, + { + "bbox": [ + 127, + 554, + 183, + 565 + ], + "score": 0.34, + "content": "C = \\mathrm { { N o r m a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 552, + 506, + 568 + ], + "score": 1.0, + "content": "means we employ a standard attention-based self-boosted fusion after fusing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 564, + 475, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 475, + 578 + ], + "score": 1.0, + "content": "question to context. This is illustrated as arrow (3) in the conceptual architecture (Figure 2).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 437, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 439, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 125, + 597 + ], + "score": 1.0, + "content": "Self", + "type": "text" + }, + { + "bbox": [ + 125, + 582, + 158, + 592 + ], + "score": 0.91, + "content": "C = \\mathbf { F } \\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 579, + 439, + 597 + ], + "score": 1.0, + "content": "means we enhance the self-boosted fusion with fully-aware attention.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "High-Level vs. FA High-Level. From Table 6, we can see that High-Level performs poorly as", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "expected. However enhancing this vanilla model with fully-aware attention significantly increase", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 621, + 504, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 231, + 634 + ], + "score": 1.0, + "content": "the performance by more than", + "type": "text" + }, + { + "bbox": [ + 231, + 621, + 246, + 632 + ], + "score": 0.86, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 621, + 504, + 634 + ], + "score": 1.0, + "content": ". The performance of FA High-Level already outperforms many", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 631, + 473, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 473, + 645 + ], + "score": 1.0, + "content": "state-of-the-art MRC models. This clearly demonstrates the power of fully-aware attention.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "FA All-Level vs. FA Multi-Level. Next, we consider models that fuse all levels of information from", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 143, + 672 + ], + "score": 1.0, + "content": "question", + "type": "text" + }, + { + "bbox": [ + 143, + 660, + 153, + 672 + ], + "score": 0.84, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 660, + 198, + 672 + ], + "score": 1.0, + "content": "to context", + "type": "text" + }, + { + "bbox": [ + 198, + 660, + 208, + 670 + ], + "score": 0.65, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 660, + 506, + 672 + ], + "score": 1.0, + "content": ". FA All-Level is a naive extension of FA High-Level, but its performance", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 671, + 504, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 197, + 684 + ], + "score": 1.0, + "content": "is actually worse than", + "type": "text" + }, + { + "bbox": [ + 198, + 671, + 210, + 681 + ], + "score": 0.37, + "content": "F A", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 671, + 493, + 684 + ], + "score": 1.0, + "content": "High-Level. 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We have achieved decent performance without self-boosted fusion. Now, we com-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "pare adding normal and fully-aware self-boosted fusion into the architecture. Comparing None and", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "Normal in Table 6, we can see that the use of normal self-boosted fusion is not very effective under", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 92, + 304, + 198 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 92, + 304, + 198 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 92, + 304, + 198 + ], + "spans": [ + { + "bbox": [ + 108, + 92, + 304, + 198 + ], + "score": 0.974, + "html": "
Attention FunctionEM/F1
Additive (MLP)71.8/ 80.172.1 / 80.672.4 / 80.772.6 / 80.8
MultiplicativeScaled MultiplicativeScaled Multiplicative + ReLU
Symmetric FormSymmetric Form + ReLU73.1 /81.575.3 / 83.6
Previous SotA (Hu et al., 2017)72.1/ 81.6
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ConfigurationC,Q Fusion Self CEM/F1
High-LevelFA High-LevelFA All-LevelFA Multi-LevelNone64.6/73.273.3 /81.472.3 / 80.774.6 / 82.7
FA Multi-LevelNormalFA74.4 / 82.675.3 / 83.6
Previous SotA (Hu et al., 2017)72.1/ 81.6
", + "type": "table", + "image_path": "7556abd7cea9b5c9f22f8b2015963a515d607babc94889bd999cebb19e4a9de3.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 312, + 80, + 508, + 93.11111111111111 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 312, + 93.11111111111111, + 508, + 106.22222222222223 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 312, + 106.22222222222223, + 508, + 119.33333333333334 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 312, + 119.33333333333334, + 508, + 132.44444444444446 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 312, + 132.44444444444446, + 508, + 145.55555555555557 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 312, + 145.55555555555557, + 508, + 158.66666666666669 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 312, + 158.66666666666669, + 508, + 171.7777777777778 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 312, + 171.7777777777778, + 508, + 184.8888888888889 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 312, + 184.8888888888889, + 508, + 198.00000000000003 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 309, + 206, + 510, + 228 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 310, + 204, + 511, + 218 + ], + "spans": [ + { + "bbox": [ + 310, + 204, + 511, + 218 + ], + "score": 1.0, + "content": "Table 6: Comparison of different configurations", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 309, + 215, + 511, + 230 + ], + "spans": [ + { + "bbox": [ + 309, + 215, + 511, + 230 + ], + "score": 1.0, + "content": "demonstrates the effectiveness of history-of-word.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + } + ], + "index": 15.75 + }, + { + "type": "text", + "bbox": [ + 107, + 255, + 505, + 311 + ], + "lines": [], + "index": 23, + "bbox_fs": [ + 104, + 255, + 506, + 312 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 330, + 301, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 302, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 302, + 343 + ], + "score": 1.0, + "content": "4.4 EFFECTIVENESS OF HISTORY-OF-WORD", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "In FusionNet, we apply the history-of-word and fully-aware attention in two major places to achieve", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "score": 1.0, + "content": "good performance: multi-level fusion and self-boosted fusion. In this section, we present experi-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "ments to demonstrate the effectiveness of our application. In the experiments, we fix the attention", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "function to be our proposed symmetric form with nonlinearity due to its good performance shown", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "in Section 4.3. The results are shown in Table 6, and the details for each configuration can be found", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 408, + 170, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 170, + 422 + ], + "score": 1.0, + "content": "in Appendix B.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 353, + 506, + 422 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 453, + 438 + ], + "score": 1.0, + "content": "High-Level is a vanilla model where only the high-level information is fused from", + "type": "text" + }, + { + "bbox": [ + 454, + 426, + 464, + 437 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 425, + 478, + 438 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 478, + 426, + 488, + 436 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "via", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "standard attention. When placed in the conceptual architecture (Figure 2), it only contains arrow (2)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 447, + 249, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 249, + 460 + ], + "score": 1.0, + "content": "without any other fusion processes.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 425, + 505, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 464, + 497, + 476 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 499, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 499, + 477 + ], + "score": 1.0, + "content": "FA High-Level is the High-Level model with standard attention replaced by fully-aware attention.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 463, + 499, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 501, + 504 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 504, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 504, + 494 + ], + "score": 1.0, + "content": "FA All-Level is a naive extension of FA High-Level, where all levels of information are concatenated", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 492, + 352, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 352, + 505 + ], + "score": 1.0, + "content": "and is fused into the context using the same attention weight.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 480, + 504, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 509, + 503, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 504, + 522 + ], + "score": 1.0, + "content": "FA Multi-Level is our proposed Fully-aware Multi-level fusion, where different levels of informa-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 520, + 305, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 305, + 533 + ], + "score": 1.0, + "content": "tion are attended under separate attention weight.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 106, + 509, + 504, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 370, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 536, + 371, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 125, + 550 + ], + "score": 1.0, + "content": "Self", + "type": "text" + }, + { + "bbox": [ + 125, + 537, + 168, + 548 + ], + "score": 0.65, + "content": "C = \\mathbf { N o n e }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 536, + 371, + 550 + ], + "score": 1.0, + "content": "means we do not make use of self-boosted fusion.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 536, + 371, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 554, + 504, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 126, + 568 + ], + "score": 1.0, + "content": "Self", + "type": "text" + }, + { + "bbox": [ + 127, + 554, + 183, + 565 + ], + "score": 0.34, + "content": "C = \\mathrm { { N o r m a l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 552, + 506, + 568 + ], + "score": 1.0, + "content": "means we employ a standard attention-based self-boosted fusion after fusing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 564, + 475, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 475, + 578 + ], + "score": 1.0, + "content": "question to context. This is illustrated as arrow (3) in the conceptual architecture (Figure 2).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 552, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 437, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 439, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 125, + 597 + ], + "score": 1.0, + "content": "Self", + "type": "text" + }, + { + "bbox": [ + 125, + 582, + 158, + 592 + ], + "score": 0.91, + "content": "C = \\mathbf { F } \\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 579, + 439, + 597 + ], + "score": 1.0, + "content": "means we enhance the self-boosted fusion with fully-aware attention.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 579, + 439, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "High-Level vs. FA High-Level. From Table 6, we can see that High-Level performs poorly as", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "expected. However enhancing this vanilla model with fully-aware attention significantly increase", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 621, + 504, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 231, + 634 + ], + "score": 1.0, + "content": "the performance by more than", + "type": "text" + }, + { + "bbox": [ + 231, + 621, + 246, + 632 + ], + "score": 0.86, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 621, + 504, + 634 + ], + "score": 1.0, + "content": ". The performance of FA High-Level already outperforms many", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 631, + 473, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 473, + 645 + ], + "score": 1.0, + "content": "state-of-the-art MRC models. This clearly demonstrates the power of fully-aware attention.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 599, + 505, + 645 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "FA All-Level vs. FA Multi-Level. Next, we consider models that fuse all levels of information from", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 143, + 672 + ], + "score": 1.0, + "content": "question", + "type": "text" + }, + { + "bbox": [ + 143, + 660, + 153, + 672 + ], + "score": 0.84, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 660, + 198, + 672 + ], + "score": 1.0, + "content": "to context", + "type": "text" + }, + { + "bbox": [ + 198, + 660, + 208, + 670 + ], + "score": 0.65, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 660, + 506, + 672 + ], + "score": 1.0, + "content": ". FA All-Level is a naive extension of FA High-Level, but its performance", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 671, + 504, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 197, + 684 + ], + "score": 1.0, + "content": "is actually worse than", + "type": "text" + }, + { + "bbox": [ + 198, + 671, + 210, + 681 + ], + "score": 0.37, + "content": "F A", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 671, + 493, + 684 + ], + "score": 1.0, + "content": "High-Level. However, by fusing different parts of history-of-word in", + "type": "text" + }, + { + "bbox": [ + 494, + 671, + 504, + 682 + ], + "score": 0.8, + "content": "Q", + "type": "inline_equation" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 681, + 444, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 186, + 695 + ], + "score": 1.0, + "content": "independently as in", + "type": "text" + }, + { + "bbox": [ + 186, + 682, + 199, + 691 + ], + "score": 0.27, + "content": "F A", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 681, + 444, + 695 + ], + "score": 1.0, + "content": "Multi-Level, we are able to further improve the performance.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 649, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 124, + 712 + ], + "score": 1.0, + "content": "Self", + "type": "text" + }, + { + "bbox": [ + 124, + 699, + 134, + 709 + ], + "score": 0.63, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "options. We have achieved decent performance without self-boosted fusion. Now, we com-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "pare adding normal and fully-aware self-boosted fusion into the architecture. Comparing None and", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "Normal in Table 6, we can see that the use of normal self-boosted fusion is not very effective under", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 162, + 95 + ], + "score": 1.0, + "content": "our improved", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 163, + 82, + 186, + 94 + ], + "score": 0.86, + "content": "C , Q", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 186, + 82, + 319, + 95 + ], + "score": 1.0, + "content": "Fusion. 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The answer", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "can thus be found. Neither word-level fusion (1) nor self-boosted fusion (3, 3’) in Figure 2 are used.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 104, + 271, + 504, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 285 + ], + "score": 1.0, + "content": "FA High-Level. 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All other places", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 333, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 345 + ], + "score": 1.0, + "content": "remains the same as High-Level. This simple change results in significant improvement. The per-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 158, + 356 + ], + "score": 1.0, + "content": "formance of", + "type": "text" + }, + { + "bbox": [ + 159, + 344, + 171, + 353 + ], + "score": 0.56, + "content": "F A", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "High-Level can already outperform many state-of-the-art models in the literature.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 354, + 492, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 492, + 367 + ], + "score": 1.0, + "content": "Note that our proposed symmetric form with nonlinearity should be used to guarantee the boost.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 395, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 395, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 395, + 394 + ], + "score": 1.0, + "content": "FA All-Level. 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The understanding vectors", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 161, + 389, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 389, + 172 + ], + "score": 1.0, + "content": "for the context is the hidden vectors in the final layers of the BiLSTM.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 102, + 134, + 505, + 172 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 176, + 416, + 194 + ], + "lines": [ + { + "bbox": [ + 193, + 176, + 416, + 194 + ], + "spans": [ + { + "bbox": [ + 193, + 176, + 416, + 194 + ], + "score": 0.92, + "content": "\\pmb { u } _ { 1 } ^ { C } , \\ldots , \\pmb { u } _ { m } ^ { C } = \\mathrm { B i L S T M } ( [ \\pmb { h } _ { 1 } ^ { C h } ; \\hat { \\pmb { h } } _ { 1 } ^ { C h } ] , \\dots , [ \\pmb { h } _ { m } ^ { C h } ; \\hat { \\pmb { h } } _ { m } ^ { C h } ] )", + "type": "interline_equation", + "image_path": "c8bd522f080ef27b367a00f8bf8030f2ddcc30a4d46021da8d9184ee20491118.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 193, + 176, + 416, + 194 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 198, + 428, + 210 + ], + "lines": [ + { + "bbox": [ + 106, + 197, + 429, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 429, + 211 + ], + "score": 1.0, + "content": "The understanding vectors for the question is the high level representation itself,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 197, + 429, + 211 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 242, + 213, + 368, + 230 + ], + "lines": [ + { + "bbox": [ + 242, + 213, + 368, + 230 + ], + "spans": [ + { + "bbox": [ + 242, + 213, + 368, + 230 + ], + "score": 0.92, + "content": "\\begin{array} { r } { { \\pmb u } _ { 1 } ^ { Q } , \\dots , { \\pmb u } _ { n } ^ { Q } = { \\pmb h } _ { 1 } ^ { Q h } , \\dots , { \\pmb h } _ { n } ^ { Q h } . } \\end{array}", + "type": "interline_equation", + "image_path": "3838caa2559a19f49f3a6bd87ffb7ad719bb34375a10967a15c5993c6ceb313a.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 242, + 213, + 368, + 230 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 233, + 505, + 256 + ], + "lines": [ + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "Now we have obtained the understanding vectors for both the context and the question. The answer", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "can thus be found. Neither word-level fusion (1) nor self-boosted fusion (3, 3’) in Figure 2 are used.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 233, + 505, + 257 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 271, + 504, + 294 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 285 + ], + "score": 1.0, + "content": "FA High-Level. The only difference to High-Level is the enhancement of fully-aware attention.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 281, + 226, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 226, + 296 + ], + "score": 1.0, + "content": "This is as simple as changing", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 270, + 505, + 296 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 138, + 298, + 471, + 316 + ], + "lines": [ + { + "bbox": [ + 138, + 298, + 471, + 316 + ], + "spans": [ + { + "bbox": [ + 138, + 298, + 471, + 316 + ], + "score": 0.9, + "content": "S _ { i j } = S ( { h _ { i } ^ { C h } } , { h _ { j } ^ { Q h } } ) \\quad \\Longrightarrow \\qquad S _ { i j } = S ( [ g _ { i } ^ { C } ; { c _ { i } ^ { C } } ; { h _ { i } ^ { C l } } ; { h _ { i } ^ { C h } } ] , [ g _ { j } ^ { Q } ; { c _ { j } ^ { Q } } ; { h _ { j } ^ { Q l } } ; { h _ { j } ^ { Q h } } ] ) ,", + "type": "interline_equation", + "image_path": "cc2d01e88c599b7578b6af6440adf3322b4c8af7aac402c90826db645b86cef4.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 138, + 298, + 471, + 316 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 505, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 319, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 133, + 334 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 319, + 192, + 333 + ], + "score": 0.94, + "content": "[ g _ { i } ; c _ { i } ; h _ { i } ^ { l } ; h _ { i } ^ { h } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 320, + 506, + 334 + ], + "score": 1.0, + "content": "is the common history-of-word for both context and question. All other places", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 333, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 345 + ], + "score": 1.0, + "content": "remains the same as High-Level. This simple change results in significant improvement. The per-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 158, + 356 + ], + "score": 1.0, + "content": "formance of", + "type": "text" + }, + { + "bbox": [ + 159, + 344, + 171, + 353 + ], + "score": 0.56, + "content": "F A", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "High-Level can already outperform many state-of-the-art models in the literature.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 354, + 492, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 492, + 367 + ], + "score": 1.0, + "content": "Note that our proposed symmetric form with nonlinearity should be used to guarantee the boost.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 319, + 506, + 367 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 395, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 395, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 395, + 394 + ], + "score": 1.0, + "content": "FA All-Level. 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The understanding vectors for the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 142, + 104, + 408, + 117 + ], + "spans": [ + { + "bbox": [ + 142, + 104, + 174, + 117 + ], + "score": 1.0, + "content": "context", + "type": "text" + }, + { + "bbox": [ + 174, + 105, + 184, + 114 + ], + "score": 0.84, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 104, + 408, + 117 + ], + "score": 1.0, + "content": "is the hidden vectors in the final layers of the BiLSTM,", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 120, + 495, + 139 + ], + "lines": [ + { + "bbox": [ + 150, + 120, + 495, + 139 + ], + "spans": [ + { + "bbox": [ + 150, + 120, + 495, + 139 + ], + "score": 0.91, + "content": "\\pmb { u } _ { 1 } ^ { C } , \\dots , \\pmb { u } _ { m } ^ { C } = \\mathrm { B i L S T M } ( [ \\pmb { h } _ { 1 } ^ { C l } ; \\pmb { h } _ { 1 } ^ { C h } ; \\hat { \\pmb { h } } _ { 1 } ^ { C l } ; 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h _ { 1 } ^ { C h } ; \\hat { h } _ { 1 } ^ { C l } ; \\hat { h } _ { 1 } ^ { C h } ; \\hat { u } _ { 1 } ^ { C } ] , \\dots , [ h _ { m } ^ { C l } ; h _ { m } ^ { C h } ; \\hat { h } _ { m } ^ { C l } ; \\hat { h } _ { m } ^ { C h } ; \\hat { u } _ { m } ^ { C } ] ) .", + "type": "interline_equation", + "image_path": "2140bc5df35532559e125c8b38906efb2f0877fe3121777949b2ffa5aa631151.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 151, + 186, + 494, + 205 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 145, + 210, + 444, + 222 + ], + "lines": [ + { + "bbox": [ + 142, + 208, + 445, + 223 + ], + "spans": [ + { + "bbox": [ + 142, + 208, + 445, + 223 + ], + "score": 1.0, + "content": "Then we fuse the context information into itself through standard attention,", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 227, + 445, + 258 + ], + "lines": [ + { + "bbox": [ + 202, + 227, + 445, + 258 + ], + "spans": [ + { + "bbox": [ + 202, + 227, + 445, + 258 + ], + "score": 0.92, + "content": "S _ { i j } = S ( \\pmb { v } _ { i } ^ { C } , \\pmb { v } _ { j } ^ { C } ) , \\alpha _ { i j } = \\frac { \\exp ( S _ { i j } ) } { \\sum _ { k } \\exp ( S _ { i k } ) } , \\hat { \\pmb { v } } _ { i } ^ { C } = \\sum _ { j } \\alpha _ { i j } \\pmb { v } _ { j } ^ { C } .", + "type": "interline_equation", + "image_path": "c4461013442bc76a98551c2671d59c7822f21f84c52320e4e0ab8b9a3d85c6d8.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 202, + 227, + 445, + 242.5 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 202, + 242.5, + 445, + 258.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 141, + 263, + 504, + 286 + ], + "lines": [ + { + "bbox": [ + 141, + 262, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 262, + 330, + 277 + ], + "score": 1.0, + "content": "The final understanding vectors for the context", + "type": "text" + }, + { + "bbox": [ + 330, + 264, + 340, + 273 + ], + "score": 0.84, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 262, + 505, + 277 + ], + "score": 1.0, + "content": "is the output hidden vectors after passing", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 275, + 308, + 286 + ], + "spans": [ + { + "bbox": [ + 142, + 275, + 308, + 286 + ], + "score": 1.0, + "content": "the concatenated vectors into a BiLSTM,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 289, + 426, + 306 + ], + "lines": [ + { + "bbox": [ + 219, + 289, + 426, + 306 + ], + "spans": [ + { + "bbox": [ + 219, + 289, + 426, + 306 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\pmb { u } _ { 1 } ^ { C } , \\ldots , \\pmb { u } _ { m } ^ { C } = \\mathrm { B i L S T M } ( [ \\pmb { v } _ { 1 } ^ { C } ; \\hat { \\pmb { v } } _ { 1 } ^ { C } ] , \\dots , [ \\pmb { v } _ { m } ^ { C } ; \\hat { \\pmb { v } } _ { m } ^ { C } ] ) . } \\end{array}", + "type": "interline_equation", + "image_path": "2acbe7c62b21084b82c0e587afa71b833c5d6e1a4abe834a6fa377b7d2dd3dc0.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 219, + 289, + 426, + 306 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 131, + 311, + 504, + 335 + ], + "lines": [ + { + "bbox": [ + 132, + 310, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 132, + 310, + 311, + 326 + ], + "score": 1.0, + "content": "• For the Fully-Aware setting, we change", + "type": "text" + }, + { + "bbox": [ + 311, + 311, + 386, + 326 + ], + "score": 0.93, + "content": "S _ { i j } = S ( \\pmb { v } _ { i } ^ { C } , \\pmb { v } _ { j } ^ { C } )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 310, + 505, + 326 + ], + "score": 1.0, + "content": "in the Normal setting to the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 228, + 337 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 228, + 337 + ], + "score": 1.0, + "content": "fully-aware attention", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 339, + 490, + 360 + ], + "lines": [ + { + "bbox": [ + 156, + 339, + 490, + 360 + ], + "spans": [ + { + "bbox": [ + 156, + 339, + 490, + 360 + ], + "score": 0.9, + "content": "\\begin{array} { r } { S _ { i j } = S ( [ { \\pmb w } _ { i } ^ { C } ; { \\pmb h } _ { i } ^ { C l } ; { \\pmb h } _ { i } ^ { C h } ; { \\hat { \\pmb h } } _ { u } ^ { C l } ; { \\hat { \\pmb h } } _ { i } ^ { C h } ; { \\hat { \\pmb u } } _ { i } ^ { C } ; { \\pmb v } _ { i } ^ { C } ] , [ { \\pmb w } _ { j } ^ { C } ; { \\pmb h } _ { j } ^ { C l } ; { \\pmb h } _ { j } ^ { C h } ; { \\hat { \\pmb h } } _ { j } ^ { C l } ; { \\hat { \\pmb h } } _ { j } ^ { C h } ; { \\hat { \\pmb u } } _ { j } ^ { C } ; { \\pmb v } _ { j } ^ { C } ] ) . } \\end{array}", + "type": "interline_equation", + "image_path": "67190e3d51dfbfcbaa54e1a0d5f74a1c81c00f0b46df933a1da30e34cfb30f1d.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 156, + 339, + 490, + 360 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 141, + 363, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 142, + 363, + 504, + 375 + ], + "spans": [ + { + "bbox": [ + 142, + 363, + 504, + 375 + ], + "score": 1.0, + "content": "All other places remains the same. 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Previous SotA (E)48.7 / 55.3
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ConfigurationEM/F1
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Previous SotA (E)40.7/46.2
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ConfigurationEM/F1
FusionNet (S,10-run best)FusionNet (S,10-run mean)FusionNet (S, without CoVe)FusionNet (E)54.8/60.953.1/ 59.355.2 / 61.254.7 / 60.7
Previous SotA (E)48.7 / 55.3
", + "type": "table", + "image_path": "0438bcac66858f195aee919432ad72ebf978023b095b168e94e625b3f107cf0d.jpg" + } + ] + } + ], + "index": 43.0, + "virtual_lines": [ + { + "bbox": [ + 316, + 561, + 504, + 574.5 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 316, + 574.5, + 504, + 588.0 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 316, + 588.0, + 504, + 601.5 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 316, + 601.5, + 504, + 615.0 + ], + "spans": [], + "index": 44 + }, + { + "bbox": [ + 316, + 615.0, + 504, + 628.5 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 316, + 628.5, + 504, + 642.0 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 309, + 650, + 509, + 673 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 309, + 650, + 511, + 663 + ], + "spans": [ + { + "bbox": [ + 309, + 650, + 511, + 663 + ], + "score": 1.0, + "content": "Table 9: Additional results for AddOneSent. (S:", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 310, + 662, + 424, + 674 + ], + "spans": [ + { + "bbox": [ + 310, + 662, + 424, + 674 + ], + "score": 1.0, + "content": "Single model, E: Ensemble)", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51.5 + } + ], + "index": 43.0 + }, + { + "type": "image", + "bbox": [ + 106, + 544, + 305, + 632 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 544, + 305, + 632 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 544, + 305, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 305, + 632 + ], + "score": 0.97, + "type": "image", + "image_path": "eff4b81f3c624949e0b874f52b9145360911c4a3d9c88a169c06218df9f52d5d.jpg" + } + ] + } + ], + "index": 40.0, + "virtual_lines": [ + { + "bbox": [ + 106, + 544, + 305, + 558.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 106, + 558.6666666666666, + 305, + 573.3333333333333 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 106, + 573.3333333333333, + 305, + 587.9999999999999 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 106, + 587.9999999999999, + 305, + 602.6666666666665 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 106, + 602.6666666666665, + 305, + 617.3333333333331 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 106, + 617.3333333333331, + 305, + 631.9999999999998 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 640, + 306, + 673 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 639, + 307, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 307, + 652 + ], + "score": 1.0, + "content": "Figure 7: Single model performance (EM) on", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 650, + 307, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 307, + 663 + ], + "score": 1.0, + "content": "AddSent over 10 training runs. (dashed vertical", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 661, + 273, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 273, + 675 + ], + "score": 1.0, + "content": "line indicates previous best performance)", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "We have conducted experiments on input vectors (GloVe and CoVe) for the original SQuAD as", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "shown in Table 7. From the ablation study, we can see that FusionNet outperforms previous state-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 161, + 722 + ], + "score": 1.0, + "content": "of-the-art by", + "type": "text" + }, + { + "bbox": [ + 162, + 710, + 182, + 720 + ], + "score": 0.89, + "content": "+ 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "in EM with and without CoVe embedding. We can also see that fine-tuning", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 721, + 377, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 377, + 733 + ], + "score": 1.0, + "content": "top-1000 GloVe embeddings is slightly helpful in the performance.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 54.5, + "bbox_fs": [ + 105, + 687, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Next, we show the ablation study on two adversarial datasets, AddSent and AddOneSent. For the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "original FusionNet, we perform ten training runs with different random seeds and evaluate indepen-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "dently on the ten single models. The performance distribution of the ten training runs can be seen in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Figure 7. Most of the independent runs perform similarly, but there are a few that performs slightly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 138 + ], + "score": 1.0, + "content": "worse, possibly because the adversarial dataset is never shown during the training. For FusionNet", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "score": 1.0, + "content": "(without CoVe), we directly evaluate on the model trained in Table 7. From Table 8 and 9, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "can see that FusionNet, single or ensemble, with or without CoVe, are all better than previous best", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 104, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "performance by a significant margin. It is also interesting that removing CoVe is slightly better on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "adversarial datasets. We assert that it is because AddSent and AddOneSent target the over-stability", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "of machine comprehension models (Jia & Liang, 2017). Since CoVe is the output vector of two-layer", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 310, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 310, + 205 + ], + "score": 1.0, + "content": "BiLSTM, CoVe may slightly worsen this problem.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 219, + 392, + 232 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 393, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 393, + 234 + ], + "score": 1.0, + "content": "D APPLICATION TO NATURAL LANGUAGE INFERENCE", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 244, + 504, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "FusionNet is an improved attention mechanism that can be easily added to any attention-based neural", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "architecture. We consider the task of natural language inference in this section to show one example", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 469, + 279 + ], + "score": 1.0, + "content": "of its usage. In natural language inference task, we are given two pieces of text, a premise", + "type": "text" + }, + { + "bbox": [ + 470, + 267, + 480, + 276 + ], + "score": 0.82, + "content": "_ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "and a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 278, + 383, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 151, + 290 + ], + "score": 1.0, + "content": "hypothesis", + "type": "text" + }, + { + "bbox": [ + 152, + 278, + 163, + 288 + ], + "score": 0.59, + "content": "\\pmb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 278, + 383, + 290 + ], + "score": 1.0, + "content": ". The task is to identify one of the following scenarios:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 129, + 298, + 411, + 340 + ], + "lines": [ + { + "bbox": [ + 129, + 298, + 411, + 311 + ], + "spans": [ + { + "bbox": [ + 129, + 298, + 254, + 311 + ], + "score": 1.0, + "content": "1. Entailment - the hypothesis", + "type": "text" + }, + { + "bbox": [ + 254, + 299, + 266, + 309 + ], + "score": 0.69, + "content": "\\pmb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 298, + 398, + 311 + ], + "score": 1.0, + "content": "can be derived from the premise", + "type": "text" + }, + { + "bbox": [ + 398, + 299, + 408, + 308 + ], + "score": 0.79, + "content": "_ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 298, + 411, + 311 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 128, + 312, + 387, + 326 + ], + "spans": [ + { + "bbox": [ + 128, + 312, + 265, + 326 + ], + "score": 1.0, + "content": "2. Contradiction - the hypothesis", + "type": "text" + }, + { + "bbox": [ + 266, + 313, + 277, + 324 + ], + "score": 0.75, + "content": "\\pmb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 312, + 373, + 326 + ], + "score": 1.0, + "content": "contradicts the premise", + "type": "text" + }, + { + "bbox": [ + 374, + 313, + 383, + 323 + ], + "score": 0.78, + "content": "_ { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 312, + 387, + 326 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 129, + 327, + 257, + 341 + ], + "spans": [ + { + "bbox": [ + 129, + 327, + 257, + 341 + ], + "score": 1.0, + "content": "3. Neutral - none of the above.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 506, + 416 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 506, + 362 + ], + "score": 1.0, + "content": "We focus on Multi-Genre Natural Language Inference (MultiNLI) corpus (Williams et al., 2017)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "recently developed by the creator of Stanford Natural Language Inference (SNLI) dataset (Bowman", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "et al., 2015). MultiNLI covers ten genres of spoken and written text, such as telephone speech and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "fictions. However the training set only contains five genres. Thus there are in-domain and cross-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "domain accuracy during evaluation. MultiNLI is designed to be more challenging than SNLI, since", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 403, + 450, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 411, + 416 + ], + "score": 1.0, + "content": "several models already outperformed human annotators on SNLI (accuracy:", + "type": "text" + }, + { + "bbox": [ + 412, + 403, + 446, + 415 + ], + "score": 0.85, + "content": "8 7 . 7 \\% ) ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 404, + 450, + 416 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 505, + 476 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 433 + ], + "score": 1.0, + "content": "A state-of-the-art model for natural language inference is Enhanced Sequential Inference Model", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 431, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 356, + 443 + ], + "score": 1.0, + "content": "(ESIM) by Chen et al. (2017b), which achieves an accuray of", + "type": "text" + }, + { + "bbox": [ + 356, + 432, + 383, + 442 + ], + "score": 0.89, + "content": "8 8 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 431, + 477, + 443 + ], + "score": 1.0, + "content": "on SNLI and obtained", + "type": "text" + }, + { + "bbox": [ + 477, + 432, + 504, + 443 + ], + "score": 0.86, + "content": "7 2 . 3 \\%", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 159, + 455 + ], + "score": 1.0, + "content": "(in-domain),", + "type": "text" + }, + { + "bbox": [ + 159, + 443, + 186, + 454 + ], + "score": 0.86, + "content": "7 2 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "(cross-domain) on MultiNLI (Williams et al., 2017). We implemented a version", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 452, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 309, + 467 + ], + "score": 1.0, + "content": "of ESIM in PyTorch. The input vectors for both", + "type": "text" + }, + { + "bbox": [ + 309, + 454, + 319, + 464 + ], + "score": 0.84, + "content": "_ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 452, + 339, + 467 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 339, + 454, + 351, + 464 + ], + "score": 0.8, + "content": "\\pmb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 452, + 505, + 467 + ], + "score": 1.0, + "content": "are the same as the input vectors for", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 465, + 288, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 138, + 477 + ], + "score": 1.0, + "content": "context", + "type": "text" + }, + { + "bbox": [ + 138, + 465, + 148, + 475 + ], + "score": 0.79, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 465, + 288, + 477 + ], + "score": 1.0, + "content": "described in Section 3. 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Next, ESIM", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 679, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 127, + 694 + ], + "score": 1.0, + "content": "feed", + "type": "text" + }, + { + "bbox": [ + 127, + 680, + 159, + 694 + ], + "score": 0.92, + "content": "g _ { i } ^ { P } , g _ { j } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 679, + 506, + 694 + ], + "score": 1.0, + "content": "into separate BiLSTMs to perform inference. In our implementation, we consider", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 691, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 705 + ], + "score": 1.0, + "content": "two-layer BiLSTM with shortcut connections for inference. The hidden vectors for the two-layer", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44 + } + ], + "page_idx": 14, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 711, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 116, + 708, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 116, + 708, + 506, + 725 + ], + "score": 1.0, + "content": "3The human annotators’ accuracy is the accuracy of five human annotators’ labels on the label with the", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 211, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 211, + 732 + ], + "score": 1.0, + "content": "majority vote (golden label).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Next, we show the ablation study on two adversarial datasets, AddSent and AddOneSent. For the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "original FusionNet, we perform ten training runs with different random seeds and evaluate indepen-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "dently on the ten single models. The performance distribution of the ten training runs can be seen in", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Figure 7. Most of the independent runs perform similarly, but there are a few that performs slightly", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 138 + ], + "score": 1.0, + "content": "worse, possibly because the adversarial dataset is never shown during the training. For FusionNet", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 149 + ], + "score": 1.0, + "content": "(without CoVe), we directly evaluate on the model trained in Table 7. From Table 8 and 9, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "can see that FusionNet, single or ensemble, with or without CoVe, are all better than previous best", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 104, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "performance by a significant margin. It is also interesting that removing CoVe is slightly better on", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "adversarial datasets. We assert that it is because AddSent and AddOneSent target the over-stability", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "of machine comprehension models (Jia & Liang, 2017). Since CoVe is the output vector of two-layer", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 310, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 310, + 205 + ], + "score": 1.0, + "content": "BiLSTM, CoVe may slightly worsen this problem.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 82, + 506, + 205 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 219, + 392, + 232 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 393, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 393, + 234 + ], + "score": 1.0, + "content": "D APPLICATION TO NATURAL LANGUAGE INFERENCE", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 244, + 504, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 505, + 256 + ], + "score": 1.0, + "content": "FusionNet is an improved attention mechanism that can be easily added to any attention-based neural", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "architecture. We consider the task of natural language inference in this section to show one example", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 469, + 279 + ], + "score": 1.0, + "content": "of its usage. In natural language inference task, we are given two pieces of text, a premise", + "type": "text" + }, + { + "bbox": [ + 470, + 267, + 480, + 276 + ], + "score": 0.82, + "content": "_ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 266, + 506, + 279 + ], + "score": 1.0, + "content": "and a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 278, + 383, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 151, + 290 + ], + "score": 1.0, + "content": "hypothesis", + "type": "text" + }, + { + "bbox": [ + 152, + 278, + 163, + 288 + ], + "score": 0.59, + "content": "\\pmb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 278, + 383, + 290 + ], + "score": 1.0, + "content": ". The task is to identify one of the following scenarios:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 106, + 244, + 506, + 290 + ] + }, + { + "type": "index", + "bbox": [ + 129, + 298, + 411, + 340 + ], + "lines": [ + { + "bbox": [ + 129, + 298, + 411, + 311 + ], + "spans": [ + { + "bbox": [ + 129, + 298, + 254, + 311 + ], + "score": 1.0, + "content": "1. Entailment - the hypothesis", + "type": "text" + }, + { + "bbox": [ + 254, + 299, + 266, + 309 + ], + "score": 0.69, + "content": "\\pmb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 298, + 398, + 311 + ], + "score": 1.0, + "content": "can be derived from the premise", + "type": "text" + }, + { + "bbox": [ + 398, + 299, + 408, + 308 + ], + "score": 0.79, + "content": "_ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 298, + 411, + 311 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 128, + 312, + 387, + 326 + ], + "spans": [ + { + "bbox": [ + 128, + 312, + 265, + 326 + ], + "score": 1.0, + "content": "2. Contradiction - the hypothesis", + "type": "text" + }, + { + "bbox": [ + 266, + 313, + 277, + 324 + ], + "score": 0.75, + "content": "\\pmb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 312, + 373, + 326 + ], + "score": 1.0, + "content": "contradicts the premise", + "type": "text" + }, + { + "bbox": [ + 374, + 313, + 383, + 323 + ], + "score": 0.78, + "content": "_ { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 312, + 387, + 326 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 129, + 327, + 257, + 341 + ], + "spans": [ + { + "bbox": [ + 129, + 327, + 257, + 341 + ], + "score": 1.0, + "content": "3. Neutral - none of the above.", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + } + ], + "index": 17, + "bbox_fs": [ + 128, + 298, + 411, + 341 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 506, + 416 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 506, + 362 + ], + "score": 1.0, + "content": "We focus on Multi-Genre Natural Language Inference (MultiNLI) corpus (Williams et al., 2017)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "recently developed by the creator of Stanford Natural Language Inference (SNLI) dataset (Bowman", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "et al., 2015). 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Cross-DomainIn-Domain
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All other settings remain unchanged. 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Word level fusion discussed in Section 3.1 is also included. For fair", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "comparison, we reduce the output hidden size in BiLSTM from 300 to 250 after adding the above", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "score": 1.0, + "content": "enhancements, so the parameter size of ESIM with fully-aware attention and fully-aware multi-level", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 475, + 371, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 371, + 487 + ], + "score": 1.0, + "content": "attention is similar to or lower than ESIM with standard attention.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "score": 1.0, + "content": "The results of ESIM under different attention mechanism is shown in Table 10. 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Cross-DomainIn-Domain
Our ESIM without CoVe (d = 300) Our ESIM without CoVe + fully-aware (d = 250)73.4 76.973.3 76.2
Our ESIM without CoVe + fully-aware + multi-level (d = 250) Our ESIM (d = 300) Our ESIM + fully-aware (d = 250) Our ESIM + fully-aware + multi-level (d = 250)78.2 73.9 77.3 78.477.9 73.7 76.5 78.2
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Preprocessing and optimization settings are the same as that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 309, + 323, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 323, + 322 + ], + "score": 1.0, + "content": "described in Appendix E, with dropout rate set to 0.3.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 275, + 506, + 322 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 326, + 504, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "Now, we consider improving ESIM with our proposed attention mechanism. First, we augment", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 336, + 446, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 446, + 351 + ], + "score": 1.0, + "content": "standard attention in ESIM with fully-aware attention. This is as simple as replacing", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 326, + 505, + 351 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 353, + 388, + 370 + ], + "lines": [ + { + "bbox": [ + 222, + 353, + 388, + 370 + ], + "spans": [ + { + "bbox": [ + 222, + 353, + 388, + 370 + ], + "score": 0.93, + "content": "S ( \\boldsymbol { h } _ { i } ^ { P h } , \\boldsymbol { h } _ { j } ^ { H h } ) \\implies S ( \\mathrm { H o W } _ { i } ^ { P } , \\mathrm { H o W } _ { j } ^ { H } ) ,", + "type": "interline_equation", + "image_path": "218c0814c57364f6142d8ac534c8143ec90e10f96b8b11ce1622b1c75f561409.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 222, + 353, + 388, + 370 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 504, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 133, + 391 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 377, + 160, + 388 + ], + "score": 0.88, + "content": "\\mathrm { H o W } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 374, + 253, + 391 + ], + "score": 1.0, + "content": "is the history-of-word,", + "type": "text" + }, + { + "bbox": [ + 254, + 375, + 302, + 389 + ], + "score": 0.93, + "content": "[ \\pmb { w } _ { i } , \\pmb { h } _ { i } ^ { l } , \\pmb { h } _ { i } ^ { h } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 374, + 506, + 391 + ], + "score": 1.0, + "content": ". All other settings remain unchanged. To incorpo-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 388, + 489, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 489, + 400 + ], + "score": 1.0, + "content": "rate fully-aware multi-level fusion into ESIM, we change the input for inference BiLSTM from", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 374, + 506, + 400 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 403, + 397, + 422 + ], + "lines": [ + { + "bbox": [ + 213, + 403, + 397, + 422 + ], + "spans": [ + { + "bbox": [ + 213, + 403, + 397, + 422 + ], + "score": 0.91, + "content": "[ { \\pmb h } ^ { h } ; \\hat { \\pmb h } ^ { h } ] \\in \\mathbb { R } ^ { 2 d } \\implies [ { \\pmb h } ^ { l } ; { \\pmb h } ^ { h } ; \\hat { \\pmb h } ^ { l } ; \\hat { \\pmb h } ^ { h } ] \\in \\mathbb { R } ^ { 4 d } ,", + "type": "interline_equation", + "image_path": "f16fffaecc3e02fe9fb7818ee5f93a1eba7997e3dcbcc2bfc4d2ae3dc346cffc.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 213, + 403, + 397, + 422 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 429, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 104, + 426, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 104, + 428, + 133, + 446 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 426, + 160, + 443 + ], + "score": 0.93, + "content": "\\hat { \\pmb { h } } _ { i } ^ { l } , \\hat { \\pmb { h } } _ { i } ^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 428, + 442, + 446 + ], + "score": 1.0, + "content": "are computed through independent fully-aware attention weights and", + "type": "text" + }, + { + "bbox": [ + 442, + 432, + 449, + 441 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 428, + 506, + 446 + ], + "score": 1.0, + "content": "is the dimen-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 201, + 454 + ], + "score": 1.0, + "content": "sion of hidden vectors", + "type": "text" + }, + { + "bbox": [ + 202, + 443, + 209, + 452 + ], + "score": 0.78, + "content": "^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 442, + 505, + 454 + ], + "score": 1.0, + "content": ". Word level fusion discussed in Section 3.1 is also included. For fair", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 466 + ], + "score": 1.0, + "content": "comparison, we reduce the output hidden size in BiLSTM from 300 to 250 after adding the above", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "score": 1.0, + "content": "enhancements, so the parameter size of ESIM with fully-aware attention and fully-aware multi-level", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 475, + 371, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 371, + 487 + ], + "score": 1.0, + "content": "attention is similar to or lower than ESIM with standard attention.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 426, + 506, + 487 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "score": 1.0, + "content": "The results of ESIM under different attention mechanism is shown in Table 10. Augmenting with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "fully-aware attention yields the biggest improvement, which demonstrates the usefulness of this", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 514, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 525 + ], + "score": 1.0, + "content": "simple enhancement. Further improvement is obtained when we use multi-level fusion in our ESIM.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 524, + 409, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 409, + 537 + ], + "score": 1.0, + "content": "Experiments with and without CoVe embedding show similar observations.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 492, + 506, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "Together, experiments on natural language inference conform with the observations in Section 4 on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 552, + 504, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 504, + 564 + ], + "score": 1.0, + "content": "machine comprehension task that the ability to take all levels of understanding as a whole is crucial", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 563, + 277, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 277, + 576 + ], + "score": 1.0, + "content": "for machines to better understand the text.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 541, + 505, + 576 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 591, + 214, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 216, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 216, + 606 + ], + "score": 1.0, + "content": "E MODEL DETAILS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "We make use of spaCy for tokenization, POS tagging and NER. We additionally fine-tuned the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "GloVe embeddings of the top 1000 frequent question words. During training, we use a dropout rate", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "of 0.4 (Srivastava et al., 2014) after the embedding layer (GloVe and CoVe) and before applying any", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "linear transformation. In particular, we share the dropout mask when the model parameter is shared", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 659, + 219, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 219, + 673 + ], + "score": 1.0, + "content": "(Gal & Ghahramani, 2016).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 615, + 505, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "The batch size is set to 32, and the optimizer is Adamax (Kingma & Ba, 2014) with a learning rate", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 688, + 152, + 698 + ], + "score": 0.84, + "content": "\\alpha = 0 . 0 0 2", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 686, + 156, + 701 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 156, + 687, + 226, + 699 + ], + "score": 0.88, + "content": "\\beta = ( 0 . 9 , 0 . 9 9 9 )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 686, + 245, + 701 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 246, + 687, + 286, + 698 + ], + "score": 0.91, + "content": "\\epsilon = 1 0 ^ { - 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 686, + 505, + 701 + ], + "score": 1.0, + "content": ". A fixed random seed is used across all experiments.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "All models are implemented in PyTorch (http://pytorch.org/). For the ensemble model,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "we apply the standard voting scheme: each model generates an answer span, and the answer with", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 473, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 473, + 734 + ], + "score": 1.0, + "content": "the highest votes is selected. We break ties randomly. There are 31 models in the ensemble.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 676, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 105, + 356, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 357, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 357, + 117 + ], + "score": 1.0, + "content": "In this section, we present prediction results on selected ex-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 117, + 356, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 356, + 128 + ], + "score": 1.0, + "content": "amples from the adversarial dataset: AddOneSent. AddOne-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 127, + 356, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 356, + 139 + ], + "score": 1.0, + "content": "Sent adds an additional sentence to the context to confuse the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 138, + 357, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 357, + 151 + ], + "score": 1.0, + "content": "model, but it does not require any query to the model. The pre-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 150, + 356, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 356, + 161 + ], + "score": 1.0, + "content": "diction results are compared with a state-of-the-art architecture", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 160, + 277, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 277, + 173 + ], + "score": 1.0, + "content": "in the literature, BiDAF (Seo et al., 2017).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 177, + 356, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 357, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 357, + 189 + ], + "score": 1.0, + "content": "First, we compare the percentage of questions answered cor-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 188, + 356, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 356, + 199 + ], + "score": 1.0, + "content": "rectly (exact match) for our model FusionNet and the state-of-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 357, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 357, + 212 + ], + "score": 1.0, + "content": "the-art model BiDAF. The comparison is shown in Figure 8.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 210, + 356, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 356, + 222 + ], + "score": 1.0, + "content": "As we can see, FusionNet is not confused by most of the ques-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 357, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 293, + 233 + ], + "score": 1.0, + "content": "tions that BiDAF correctly answer. 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The comparison is shown in Figure 8.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 210, + 356, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 356, + 222 + ], + "score": 1.0, + "content": "As we can see, FusionNet is not confused by most of the ques-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 357, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 293, + 233 + ], + "score": 1.0, + "content": "tions that BiDAF correctly answer. Among the", + "type": "text" + }, + { + "bbox": [ + 293, + 221, + 316, + 232 + ], + "score": 0.86, + "content": "3 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 221, + 357, + 233 + ], + "score": 1.0, + "content": "answered", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 357, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 275, + 244 + ], + "score": 1.0, + "content": "correctly by BiDAF but not FusionNet,", + "type": "text" + }, + { + "bbox": [ + 276, + 232, + 313, + 243 + ], + "score": 0.89, + "content": "\\sim 1 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 232, + 357, + 244 + ], + "score": 1.0, + "content": "are being", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 357, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 241, + 255 + ], + "score": 1.0, + "content": "confused by the added sentence;", + "type": "text" + }, + { + "bbox": [ + 241, + 243, + 276, + 254 + ], + "score": 0.91, + "content": "\\sim 1 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 243, + 357, + 255 + ], + "score": 1.0, + "content": "are correct but dif-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 253, + 357, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 357, + 267 + ], + "score": 1.0, + "content": "fers slightly from the ground truth answer; and the remaining", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 264, + 309, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 139, + 276 + ], + "score": 0.87, + "content": "\\sim 0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 264, + 309, + 277 + ], + "score": 1.0, + "content": "are completely incorrect in the first place.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 177, + 357, + 277 + ] + }, + { + "type": "image", + "bbox": [ + 369, + 119, + 503, + 228 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 369, + 119, + 503, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 369, + 119, + 503, + 228 + ], + "spans": [ + { + "bbox": [ + 369, + 119, + 503, + 228 + ], + "score": 0.95, + "type": "image", + "image_path": "8854537b854134c34ef7fda6e8c45fa7fb17dc9e5380a98c4a690b4cbbd08842.jpg" + } + ] + } + ], + "index": 11.0, + "virtual_lines": [ + { + "bbox": [ + 369, + 119, + 503, + 173.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 369, + 173.5, + 503, + 228.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 363, + 238, + 504, + 261 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 363, + 238, + 504, + 250 + ], + "spans": [ + { + "bbox": [ + 363, + 238, + 504, + 250 + ], + "score": 1.0, + "content": "Figure 8: Questions answered cor-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 363, + 249, + 458, + 262 + ], + "spans": [ + { + "bbox": [ + 363, + 249, + 458, + 262 + ], + "score": 1.0, + "content": "rectly on AddOneSent.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + } + ], + "index": 14.25 + }, + { + "type": "list", + "bbox": [ + 107, + 282, + 504, + 315 + ], + "lines": [ + { + "bbox": [ + 106, + 282, + 357, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 357, + 294 + ], + "score": 1.0, + "content": "Now we present sample examples where FusionNet answers", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 293, + 504, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 504, + 305 + ], + "score": 1.0, + "content": "correctly but BiDAF is confused as well as examples where BiDAF and FusionNet are both con-", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 104, + 302, + 135, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 302, + 135, + 316 + ], + "score": 1.0, + "content": "fused.", + "type": "text" + } + ], + "index": 21, + "is_list_end_line": true + } + ], + "index": 20, + "bbox_fs": [ + 104, + 282, + 504, + 316 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 328, + 410, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 412, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 412, + 341 + ], + "score": 1.0, + "content": "F.1 FUSIONNET ANSWERS CORRECTLY WHILE BIDAF IS INCORRECT", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 109, + 349, + 313, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 316, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 316, + 363 + ], + "score": 1.0, + "content": "ID: 57273cca708984140094db35-high-conf-turk1", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 107, + 360, + 504, + 372 + ], + "spans": [ + { + "bbox": [ + 107, + 360, + 504, + 372 + ], + "score": 1.0, + "content": "Context: Large-scale construction requires collaboration across multiple disciplines. An archi-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "tect normally manages the job, and a construction manager, design engineer, construction engineer", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "or project manager supervises it. For the successful execution of a project, effective planning is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "essential. Those involved with the design and execution of the infrastructure in question must con-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 404, + 504, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 504, + 416 + ], + "score": 1.0, + "content": "sider zoning requirements, the environmental impact of the job, the successful scheduling, budget-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "ing, construction-site safety, availability and transportation of building materials, logistics, inconve-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 439 + ], + "score": 1.0, + "content": "nience to the public caused by construction delays and bidding, etc. The largest construction projects", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 437, + 497, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 497, + 449 + ], + "score": 1.0, + "content": "are referred to as megaprojects. 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F. D. Shrewsbury in 1970, who noted that the reported rates of mortality in rural areas during the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "14th-century pandemic were inconsistent with the modern bubonic plague, leading him to conclude", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "that contemporary accounts were exaggerations. 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This was Hereford’s", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 155, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 155, + 183 + ], + "score": 1.0, + "content": "conclusion.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 320, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 300, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 300, + 200 + ], + "score": 1.0, + "content": "Question: What was Shrewsbury’s conclusion?", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 320, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 320, + 213 + ], + "score": 1.0, + "content": "Answer: contemporary accounts were exaggerations", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 215, + 380, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 214, + 378, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 378, + 228 + ], + "score": 1.0, + "content": "FusionNet Prediction: contemporary accounts were exaggerations", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 225, + 381, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 381, + 239 + ], + "score": 1.0, + "content": "BiDAF Prediction: his doubts about the identity of the Black Death", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 106, + 249, + 313, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 248, + 314, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 314, + 262 + ], + "score": 1.0, + "content": "ID: 5730cb8df6cb411900e244c6-high-conf-turk0", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 261, + 506, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 260, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 271 + ], + "score": 1.0, + "content": "Context: The Book of Discipline is the guidebook for local churches and pastors and describes", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 282 + ], + "score": 1.0, + "content": "in considerable detail the organizational structure of local United Methodist churches. All UM", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "score": 1.0, + "content": "churches must have a board of trustees with at least three members and no more than nine members", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "score": 1.0, + "content": "and it is recommended that no gender should hold more than a 2/3 majority. 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The church conference", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 337, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 348 + ], + "score": 1.0, + "content": "is an annual meeting of all the officers of the church and any interested members. This committee", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "has the exclusive power to set pastors’ salaries (compensation packages for tax purposes) and to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "elect officers to the committees. 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F. D. Shrewsbury in 1970, who noted that the reported rates of mortality in rural areas during the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "14th-century pandemic were inconsistent with the modern bubonic plague, leading him to conclude", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "that contemporary accounts were exaggerations. 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All UM", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "score": 1.0, + "content": "churches must have a board of trustees with at least three members and no more than nine members", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "score": 1.0, + "content": "and it is recommended that no gender should hold more than a 2/3 majority. All churches must also", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 304, + 504, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 504, + 315 + ], + "score": 1.0, + "content": "have a nominations committee, a finance committee and a church council or administrative council.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "Other committees are suggested but not required such as a missions committee, or evangelism or", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "worship committee. Term limits are set for some committees but not for all. The church conference", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 337, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 348 + ], + "score": 1.0, + "content": "is an annual meeting of all the officers of the church and any interested members. This committee", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "has the exclusive power to set pastors’ salaries (compensation packages for tax purposes) and to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "elect officers to the committees. The hamster committee did not have the power to set pastors’", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 369, + 142, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 142, + 381 + ], + "score": 1.0, + "content": "salaries.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 260, + 506, + 381 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 416, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 417, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 417, + 399 + ], + "score": 1.0, + "content": "Question: Which committee has the exclusive power to set pastors’ salaries?", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 385, + 417, + 399 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 239, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 239, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 239, + 410 + ], + "score": 1.0, + "content": "Answer: The church conference", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 397, + 239, + 410 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 296, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 414, + 297, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 297, + 427 + ], + "score": 1.0, + "content": "FusionNet Prediction: The church conference", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 424, + 286, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 286, + 438 + ], + "score": 1.0, + "content": "BiDAF Prediction: The hamster committee", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 414, + 297, + 438 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 457, + 338, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 339, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 339, + 469 + ], + "score": 1.0, + "content": "F.2 FUSIONNET AND BIDAF ARE BOTH INCORRECT", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 109, + 478, + 311, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 312, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 312, + 492 + ], + "score": 1.0, + "content": "ID: 572fec30947a6a140053cdf5-high-conf-turk0", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 489, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "Context: In the centre of Basel, the first major city in the course of the stream, is located the “Rhine", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "knee”; this is a major bend, where the overall direction of the Rhine changes from West to North.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "Here the High Rhine ends. Legally, the Central Bridge is the boundary between High and Upper", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "score": 1.0, + "content": "Rhine. The river now flows North as Upper Rhine through the Upper Rhine Plain, which is about", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 139, + 543 + ], + "score": 0.69, + "content": "3 0 0 ~ \\mathrm { k m }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 532, + 203, + 544 + ], + "score": 1.0, + "content": "long and up to", + "type": "text" + }, + { + "bbox": [ + 203, + 532, + 231, + 543 + ], + "score": 0.67, + "content": "4 0 ~ \\mathrm { k m }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 532, + 505, + 544 + ], + "score": 1.0, + "content": "wide. The most important tributaries in this area are the Ill below", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "of Strasbourg, the Neckar in Mannheim and the Main across from Mainz. In Mainz, the Rhine", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 567 + ], + "score": 1.0, + "content": "leaves the Upper Rhine Valley and flows through the Mainz Basin. Serbia ends after the bend in the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 564, + 144, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 144, + 578 + ], + "score": 1.0, + "content": "Danube.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 488, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 582, + 299, + 593 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 300, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 300, + 595 + ], + "score": 1.0, + "content": "Question: What ends at this bend in the Rhine?", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 106, + 581, + 300, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 594, + 193, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 592, + 194, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 194, + 605 + ], + "score": 1.0, + "content": "Answer: High Rhine", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 106, + 592, + 194, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 231, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 231, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 231, + 622 + ], + "score": 1.0, + "content": "FusionNet Prediction: Serbia", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 619, + 217, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 217, + 633 + ], + "score": 1.0, + "content": "BiDAF Prediction: Serbia", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 106, + 609, + 231, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "Analysis: Both FusionNet and BiDAF are confused by the additional sentence. One of the key", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "problem is that the context is actually quite hard to understand. “major bend” is distantly connected", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 654, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 666 + ], + "score": 1.0, + "content": "to “Here the High Rhine ends”. Understanding that the theme of the context is about “Rhine” is", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 666, + 245, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 245, + 677 + ], + "score": 1.0, + "content": "crucial to answering this question.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 631, + 506, + 677 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 688, + 313, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 684, + 316, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 316, + 703 + ], + "score": 1.0, + "content": "ID: 573092088ab72b1400f9c598-high-conf-turk2", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 108, + 700, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "Context: Imperialism has played an important role in the histories of Japan, Korea, the Assyrian", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Empire, the Chinese Empire, the Roman Empire, Greece, the Byzantine Empire, the Persian Empire,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "the Ottoman Empire, Ancient Egypt, the British Empire, India, and many other empires. Imperi-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "alism was a basic component to the conquests of Genghis Khan during the Mongol Empire, and", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "of other war-lords. Historically recognized Muslim empires number in the dozens. Sub-Saharan", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "score": 1.0, + "content": "Africa has also featured dozens of empires that predate the European colonial era, for example the", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "Ethiopian Empire, Oyo Empire, Asante Union, Luba Empire, Lunda Empire, and Mutapa Empire.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "score": 1.0, + "content": "The Americas during the pre-Columbian era also had large empires such as the Aztec Empire and", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 403, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 403, + 149 + ], + "score": 1.0, + "content": "the Incan Empire. The British Empire is older than the Eritrean Conquest.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 49, + "bbox_fs": [ + 106, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "alism was a basic component to the conquests of Genghis Khan during the Mongol Empire, and", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "of other war-lords. Historically recognized Muslim empires number in the dozens. Sub-Saharan", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 117 + ], + "score": 1.0, + "content": "Africa has also featured dozens of empires that predate the European colonial era, for example the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "Ethiopian Empire, Oyo Empire, Asante Union, Luba Empire, Lunda Empire, and Mutapa Empire.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 138 + ], + "score": 1.0, + "content": "The Americas during the pre-Columbian era also had large empires such as the Aztec Empire and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 403, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 403, + 149 + ], + "score": 1.0, + "content": "the Incan Empire. The British Empire is older than the Eritrean Conquest.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 390, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 153, + 392, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 392, + 168 + ], + "score": 1.0, + "content": "Question: Which is older the British Empire or the Ethiopian Empire?", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 219, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 219, + 178 + ], + "score": 1.0, + "content": "Answer: Ethiopian Empire", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 108, + 182, + 276, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 279, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 279, + 195 + ], + "score": 1.0, + "content": "FusionNet Prediction: Eritrean Conquest", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 108, + 194, + 265, + 204 + ], + "lines": [ + { + "bbox": [ + 106, + 191, + 264, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 264, + 206 + ], + "score": 1.0, + "content": "BiDAF Prediction: Eritrean Conquest", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 205, + 505, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "score": 1.0, + "content": "Analysis: Similar to the previous example, both are confused by the additional sentence because the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 216, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 227 + ], + "score": 1.0, + "content": "answer is obscured in the context. To answer the question correctly, we must be aware of a common", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "knowledge that British Empire is part of the European colonial era, which is not presented in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "context. Then from the sentence in the context colored green (and italic), we know the Ethiopian", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 247, + 257, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 257, + 261 + ], + "score": 1.0, + "content": "Empire “predate” the British Empire.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 271, + 314, + 282 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 317, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 317, + 286 + ], + "score": 1.0, + "content": "ID: 57111713a58dae1900cd6c02-high-conf-turk2", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "Context: In February 2010, in response to controversies regarding claims in the Fourth Assessment", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "score": 1.0, + "content": "Report, five climate scientists all contributing or lead IPCC report authors wrote in the journal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "Nature calling for changes to the IPCC. They suggested a range of new organizational options,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 314, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 505, + 328 + ], + "score": 1.0, + "content": "from tightening the selection of lead authors and contributors, to dumping it in favor of a small", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "permanent body, or even turning the whole climate science assessment process into a moderated", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 336, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 506, + 350 + ], + "score": 1.0, + "content": "“living” Wikipedia-IPCC. Other recommendations included that the panel employ a full-time staff", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "and remove government oversight from its processes to avoid political interference. It was suggested", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 359, + 311, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 311, + 371 + ], + "score": 1.0, + "content": "that the panel learn to avoid nonpolitical problems.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 108, + 375, + 400, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 401, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 401, + 389 + ], + "score": 1.0, + "content": "Question: How was it suggested that the IPCC avoid political problems?", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 109, + 388, + 334, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 337, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 337, + 400 + ], + "score": 1.0, + "content": "Answer: remove government oversight from its processe", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 502, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "score": 1.0, + "content": "FusionNet Prediction: the panel employ a full-time staff and remove government oversight from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 414, + 160, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 160, + 428 + ], + "score": 1.0, + "content": "its processes", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "score": 1.0, + "content": "BiDAF Prediction: the panel employ a full-time staff and remove government oversight from its", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 437, + 148, + 449 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 148, + 449 + ], + "score": 1.0, + "content": "processes", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "Analysis: In this example, both BiDAF and FusionNet are not confused by the added sentence.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "However, the prediction by both model are not precise enough. The predicted answer gave two", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "suggestions: (1) employ a full-time staff, (2) remove government oversight from its processes. Only", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "the second one is suggested to avoid political problems. To obtain the precise answer, common", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 491, + 502, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 502, + 504 + ], + "score": 1.0, + "content": "knowledge is required to know that employing a full-time staff will not avoid political interference.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 106, + 514, + 314, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 316, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 316, + 528 + ], + "score": 1.0, + "content": "ID: 57111713a58dae1900cd6c02-high-conf-turk2", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "Context: Most of the Huguenot congregations (or individuals) in North America eventually affil-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "iated with other Protestant denominations with more numerous members. The Huguenots adapted", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 547, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 558 + ], + "score": 1.0, + "content": "quickly and often married outside their immediate French communities, which led to their assimila-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 569 + ], + "score": 1.0, + "content": "tion. Their descendants in many families continued to use French first names and surnames for their", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 569, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 580 + ], + "score": 1.0, + "content": "children well into the nineteenth century. 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Other recommendations included that the panel employ a full-time staff", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "and remove government oversight from its processes to avoid political interference. It was suggested", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 359, + 311, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 359, + 311, + 371 + ], + "score": 1.0, + "content": "that the panel learn to avoid nonpolitical problems.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 282, + 506, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 375, + 400, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 401, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 401, + 389 + ], + "score": 1.0, + "content": "Question: How was it suggested that the IPCC avoid political problems?", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 374, + 401, + 389 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 388, + 334, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 337, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 337, + 400 + ], + "score": 1.0, + "content": "Answer: remove government oversight from its processe", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 386, + 337, + 400 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 502, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "score": 1.0, + "content": "FusionNet Prediction: the panel employ a full-time staff and remove government oversight from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 414, + 160, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 160, + 428 + ], + "score": 1.0, + "content": "its processes", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 401, + 505, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 438 + ], + "score": 1.0, + "content": "BiDAF Prediction: the panel employ a full-time staff and remove government oversight from its", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 437, + 148, + 449 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 148, + 449 + ], + "score": 1.0, + "content": "processes", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 423, + 505, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "Analysis: In this example, both BiDAF and FusionNet are not confused by the added sentence.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "However, the prediction by both model are not precise enough. The predicted answer gave two", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "suggestions: (1) employ a full-time staff, (2) remove government oversight from its processes. Only", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 493 + ], + "score": 1.0, + "content": "the second one is suggested to avoid political problems. 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However, when", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 284 + ], + "score": 1.0, + "content": "low-level attention is used (the attention map in the left hand side of Figure 9), we can see that “in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "Belgium” no longer matches with “in France”. Thus when low-level attention is incorporated, the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "system can be more observant when deciding if the answer lies in this adversarial sentence. Similar", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "observation is also evident in Figure 10. These visualizations provides an intuitive explanation for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 316, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 327 + ], + "score": 1.0, + "content": "our superior performance and support our original motivation in Section 2.3 that taking in all levels", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 325, + 370, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 370, + 339 + ], + "score": 1.0, + "content": "of understanding is crucial for machines to understand text better.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 237, + 506, + 339 + ] + }, + { + "type": "image", + "bbox": [ + 108, + 356, + 504, + 518 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 356, + 504, + 518 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 356, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 108, + 356, + 504, + 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Single ModelTest Set EM/F1
LR Baseline (Rajpurkar et al., 2016) Match-LSTM (Wang & Jiang,2016) BiDAF (Seo et al., 2017) SEDT (Liu et al., 2017) RaSoR (Lee et al., 2016) DrQA (Chen et al., 2017a) ReasoNet (Shen et al.,2017) R.Mnemonic Reader (Hu et al., 2017) DCN+ R-nett FusionNet40.4 /51.0 64.7/73.7 68.0 / 77.3 68.2 /77.5 70.8/78.7 70.7 /79.4 70.6 / 79.4 73.2/81.8 74.9 / 82.8 75.7 /83.5 76.0 / 83.9
EnsembleModel ReasoNet (Shen et al., 2017) MEMEN (Pan et al., 2017) R.Mnemonic Reader (Hu et al., 2017) R-nett DCN+ FusionNet75.0 / 82.3 75.4/82.7 77.7 / 84.9 78.2/85.2 78.7 / 85.6
Human (Rajpurkar et al., 2016)78.8 / 85.9 82.3/91.2
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AddSentEM/F1
LRBaseline17.0/23.2
Match-LSTM (E) BiDAF (E)24.3 /34.2 29.6 /34.2
SEDT (E) Mnemonic Reader (S)30.0 /35.0
Mnemonic Reader (E)39.8/46.6
40.7 / 46.2
ReasoNet (E) FusionNet (E)34.6 /39.4 46.2 / 51.4
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AddOneSentEM/F1
LRBaseline Match-LSTM (E)22.3/30.4 34.8 / 41.8
BiDAF (E) SEDT (E)40.7 /46.9 40.0 / 46.5
Mnemonic Reader (S)48.5 /56.0
Mnemonic Reader (E)48.7 / 55.3
ReasoNet (E) FusionNet (E)43.6 /49.8
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Attention FunctionEM/F1
Additive (MLP)71.8/ 80.172.1 / 80.672.4 / 80.772.6 / 80.8
MultiplicativeScaled MultiplicativeScaled Multiplicative + ReLU
Symmetric FormSymmetric Form + ReLU73.1 /81.575.3 / 83.6
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Cross-DomainIn-Domain
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0000000000000000000000000000000000000000..474f788df6fc857766da92d4ce880c0acf5e85ca --- /dev/null +++ b/parse/train/SJA7xfb0b/SJA7xfb0b.md @@ -0,0 +1,884 @@ +# SOBOLEV GAN + +Youssef Mroueh†, Chun-Liang $\mathbf { L i } ^ { \circ , \star }$ , Tom Sercu†,?, Anant Raj♦,? & Yu Cheng† + +$\dagger$ IBM Research AI +$\circ$ Carnegie Mellon University +$\diamondsuit$ Max Planck Institute for Intelligent Systems +$\star$ denotes Equal Contribution +mroueh,chengyu @us.ibm.com, chunlial@cs.cmu.edu, +tom.sercu1@ibm.com,anant.raj@tuebingen.mpg.de + +# ABSTRACT + +We propose a new Integral Probability Metric (IPM) between distributions: the Sobolev IPM. The Sobolev IPM compares the mean discrepancy of two distributions for functions (critic) restricted to a Sobolev ball defined with respect to a dominant measure $\mu$ . We show that the Sobolev IPM compares two distributions in high dimensions based on weighted conditional Cumulative Distribution Functions (CDF) of each coordinate on a leave one out basis. The Dominant measure $\mu$ plays a crucial role as it defines the support on which conditional CDFs are compared. Sobolev IPM can be seen as an extension of the one dimensional VonMises Cramer statistics to high dimensional distributions. We show how Sobolev ´ IPM can be used to train Generative Adversarial Networks (GANs). We then exploit the intrinsic conditioning implied by Sobolev IPM in text generation. Finally we show that a variant of Sobolev GAN achieves competitive results in semisupervised learning on CIFAR-10, thanks to the smoothness enforced on the critic by Sobolev GAN which relates to Laplacian regularization. + +# 1 INTRODUCTION + +In order to learn Generative Adversarial Networks (Goodfellow et al., 2014), it is now well established that the generator should mimic the distribution of real data, in the sense of a certain discrepancy measure. Discrepancies between distributions that measure the goodness of the fit of the neural generator to the real data distribution has been the subject of many recent studies (Arjovsky & Bottou, 2017; Nowozin et al., 2016; Kaae Sønderby et al., 2017; Mao et al., 2017; Arjovsky et al., 2017; Gulrajani et al., 2017; Mroueh et al., 2017; Mroueh & Sercu, 2017; Li et al., 2017), most of which focus on training stability. + +In terms of data modalities, most success was booked in plausible natural image generation after the introduction of Deep Convolutional Generative Adversarial Networks (DCGAN) (Radford et al., 2015). This success is not only due to advances in training generative adversarial networks in terms of loss functions (Arjovsky et al., 2017) and stable algorithms, but also to the representation power of convolutional neural networks in modeling images and in finding sufficient statistics that capture the continuous density function of natural images. When moving to neural generators of discrete sequences generative adversarial networks theory and practice are still not very well understood. Maximum likelihood pre-training or augmentation, in conjunction with the use of reinforcement learning techniques were proposed in many recent works for training GAN for discrete sequences generation (Yu et al., 2016; Che et al., 2017; Hjelm et al., 2017; Rajeswar et al., 2017). Other methods included using the Gumbel Softmax trick (Kusner & Hernandez-Lobato ´ , 2016) and the use of auto-encoders to generate adversarially discrete sequences from a continuous space (Zhao et al., 2017). End to end training of GANs for discrete sequence generation is still an open problem (Press et al., 2017). Empirical successes of end to end training have been reported within the framework of WGAN-GP (Gulrajani et al., 2017), using a proxy for the Wasserstein distance via a pointwise gradient penalty on the critic. Inspired by this success, we propose in this paper a new Integral Probability Metric (IPM) between distributions that we coin Sobolev IPM. Intuitively an IPM (Muller ¨ , 1997) between two probability distributions looks for a witness function $f$ , called critic, that maximally discriminates between samples coming from the two distributions: + +$$ +\operatorname* { s u p } _ { f \in \mathcal { F } } \mathbb { E } _ { x \sim \mathbb { P } } f ( x ) - \mathbb { E } _ { x \sim \mathbb { Q } } f ( x ) . +$$ + +Traditionally, the function $f$ is defined over a function class $\mathcal { F }$ that is independent to the distributions at hand (Sriperumbudur et al., 2012). The Wasserstein-1 distance corresponds for instance to an IPM where the witness functions are defined over the space of Lipschitz functions; The MMD distance (Gretton et al., 2012) corresponds to witness functions defined over a ball in a Reproducing Kernel Hilbert Space (RKHS). + +We will revisit in this paper Fisher IPM defined in (Mroueh & Sercu, 2017), which extends the IPM definition to function classes defined with norms that depend on the distributions. Fisher IPM can be seen as restricting the critic to a Lebsegue ball defined with respect to a dominant measure $\mu$ . The Lebsegue norm is defined as follows: + +$$ +\int _ { \mathcal { X } } f ^ { 2 } ( x ) \mu ( x ) d x . +$$ + +where $\mu$ is a dominant measure of $\mathbb { P }$ and $\mathbb { Q }$ + +In this paper we extend the IPM framework to critics bounded in the Sobolev norm: + +$$ +\int _ { \mathcal { X } } \| \nabla _ { \boldsymbol { x } } f ( \boldsymbol { x } ) \| _ { 2 } ^ { 2 } \mu ( \boldsymbol { x } ) d x , +$$ + +In contrast to Fisher IPM, which compares joint probability density functions of all coordinates between two distributions, we will show that Sobolev IPM compares weighted (coordinate-wise) conditional Cumulative Distribution Functions for all coordinates on a leave on out basis. Matching conditional dependencies between coordinates is crucial for sequence modeling. + +Our analysis and empirical verification show that the modeling of the conditional dependencies can be built in to the metric used to learn GANs as in Sobolev IPM. For instance, this gives an advantage to Sobolev IPM in comparing sequences over Fisher IPM. Nevertheless, in sequence modeling when we parametrize the critic and the generator with a neural network, we find an interesting tradeoff between the metric used and the architectures used to parametrize the critic and the generator as well as the conditioning used in the generator. The burden of modeling the conditional long term dependencies can be handled by the IPM loss function as in Sobolev IPM (more accurately the choice of the data dependent function class of the critic) or by a simpler metric such as Fisher IPM together with a powerful architecture for the critic that models conditional long term dependencies such as LSTM or GRUs in conjunction with a curriculum conditioning of the generator as done in (Press et al., 2017). Highlighting those interesting tradeoffs between metrics, data dependent functions classes for the critic (Fisher or Sobolev) and architectures is crucial to advance sequence modeling and more broadly structured data generation using GANs. + +On the other hand, Sobolev norms have been widely used in manifold regularization in the so called Laplacian framework for semi-supervised learning (SSL) (Belkin et al., 2006). GANs have shown success in semi-supervised learning (Salimans et al., 2016; Dumoulin et al., 2017; Dai et al., 2017; Kumar et al., 2017). Nevertheless, many normalizations and additional tricks were needed. We show in this paper that a variant of Sobolev GAN achieves strong results in semi-supervised learning on CIFAR-10, without the need of any activation normalization in the critic. + +The main contributions of this paper can be summarized as follows: + +1. We overview in Section 2 different metrics between distribution used in the GAN literature. We then generalize Fisher IPM in Section 3 with a general dominant measure $\mu$ and show how it compares distributions based on their PDFs. 2. We introduce Sobolev IPM in Section 4 by restricting the critic of an IPM to a Sobolev ball defined with respect to a dominant measure $\mu$ . We then show that Sobolev IPM defines a discrepancy between weighted (coordinate-wise) conditional CDFs of distributions. + +3. The intrinsic conditioning and the CDF matching make Sobolev IPM suitable for discrete sequence matching and explain the success of the gradient pernalty in WGAN-GP and Sobolev GAN in discrete sequence generation. +4. We give in Section 5 an ALM (Augmented Lagrangian Multiplier) algorithm for training Sobolev GAN. Similar to Fisher GAN, this algorithm is stable and does not compromise the capacity of the critic. +5. We show in Appendix A that the critic of Sobolev IPM satisfies an elliptic Partial Differential Equation (PDE). We relate this diffusion to the Fokker-Planck equation and show the behavior of the gradient of the optimal Sobolev critic as a transportation plan between distributions. +6. We empirically study Sobolev GAN in character level text generation (Section 6.1). We validate that the conditioning implied by Sobolev GAN is crucial for the success and stability of GAN in text generation. As a take home message from this study, we see that text generation succeeds either by implicit conditioning i.e using Sobolev GAN (or WGANGP) together with convolutional critics and generators, or by explicit conditioning i.e using Fisher IPM together with recurrent critic and generator and curriculum learning. +7. We finally show in Section 6.2 that a variant of Sobolev GAN achieves competitive semisupervised learning results on CIFAR-10, thanks to the smoothness implied by the Sobolev regularizer. + +# 2 OVERVIEW OF METRICS BETWEEN DISTRIBUTIONS + +In this Section, we review different representations of probability distributions and metrics for comparing distributions that use those representations. Those metrics are at the core of training GAN. In what follows, we consider probability measures with a positive weakly differentiable probability density functions (PDF). Let $P$ and $Q$ be two probability measures with PDFs $\mathbb { P } ( x )$ and $\mathbb { Q } ( x )$ defined on $\mathcal { X } \subset \mathbb { R } ^ { d }$ . Let $F _ { \mathbb { P } }$ and $F _ { \mathbb { Q } }$ be the Cumulative Distribution Functions (CDF) of $\mathbb { P }$ and $\mathbb { Q }$ respectively. For $\boldsymbol { x } = ( x _ { 1 } , \dots , x _ { d } )$ , we have: + +$$ +F _ { \mathbb { P } } ( x ) = \int _ { - \infty } ^ { x _ { 1 } } \ldots \cdot \cdot \int _ { - \infty } ^ { x _ { d } } \mathbb { P } ( u _ { 1 } , \ldots \cdot u _ { d } ) d u _ { 1 } \ldots \cdot \cdot d u _ { d } . +$$ + +The score function of a density function is defined as: $s _ { \mathbb { P } } ( x ) = \nabla _ { x } \log ( \mathbb { P } ( x ) ) \in \mathbb { R } ^ { d }$ . + +In this work, we are interested in metrics between distributions that have a variational form and can be written as a suprema of mean discrepancies of functions defined on a specific function class. This type of metrics include $\varphi$ -divergences as well as Integral Probability Metrics (Sriperumbudur et al., 2009) and have the following form: + +$$ +d _ { \mathcal { F } } ( \mathbb { P } , \mathbb { Q } ) = \operatorname* { s u p } _ { f \in \mathcal { F } } \left| \Delta ( f ; \mathbb { P } , \mathbb { Q } ) \right| , +$$ + +where $\mathcal { F }$ is a function class defined on $\mathcal { X }$ and $\Delta$ is a mean discrepancy, $\Delta : \mathcal { F } \mathbb { R }$ . The variational form given above leads in certain cases to closed form expressions in terms of the PDFs $\mathbb { P } , \mathbb { Q }$ or in terms of the CDFs $F _ { \mathbb { P } } , F _ { \mathbb { Q } }$ or the score functions $s _ { \mathbb { P } } , s _ { \mathbb { Q } }$ . + +In Table 1, we give a comparison of different discrepancies $\Delta$ and function spaces $\mathcal { F }$ used in the literature for GAN training together with our proposed Sobolev IPM. We see from Table 1 that Sobolev IPM, compared to Wasserstein Distance, imposes a tractable smoothness constraint on the critic on points sampled from a distribution $\mu$ , rather then imposing a Lipschitz constraint on all points in the space $\mathcal { X }$ . We also see that Sobolev IPM is the natural generalization of the Cramer´ Von-Mises Distance from one dimension to high dimensions. We note that the Energy Distance, a form of Maximum Mean Discrepancy for a special kernel, was used in (Bellemare et al., 2017b) as a generalization of the Cramer distance in GAN training but still needed a gradient penalty in ´ its algorithmic counterpart leading to a mis-specified distance between distributions. Finally it is worth noting that when comparing Fisher IPM and Sobolev IPM we see that while Fisher IPM compares joint PDF of the distributions, Sobolev IPM compares weighted (coordinate-wise) conditional CDFs. As we will see later, this conditioning nature of the metric makes Sobolev IPM suitable for comparing sequences. Note that the Stein metric (Liu et al., 2016; Liu, 2017) uses the score function to match distributions. We will show later how Sobolev IPM relates to the Stein discrepancy (Appendix A). + +Table 1: Comparison of different metrics between distributions used for GAN training. References are for papers using those metrics for GAN training. + +
△(f;P,Q)FFunction classdg(P,Q)Closed Form
-Divergence(Goodfellow et al., 2014)(Nowozin et al., 2016)Ex~Pf(x)-Ex~Q*(f(x))*Fenchel Conjugate{f:x→R,f∈domp*}E~[(]
Wasserstein -1(Arjovsky et al., 2017)(Gulrajani et al., 2017)Ex~Pf(x)-Ex~qf(x){f : x→R,fli ≤1}infπ∈(P,Q) Sx |lx-ylli dπ(x,y)Sinkhorn Divergence(Genevay et al., 2017)
MMD(Li et al., 2017)(Li et al., 2015)(Dziugaite et al., 2015)Ex~Pf(x)-Ex~qf(x){f:x→R,/fl/x ≤1}Ex~Pkx -Ex~Qr llyex
SteinDiscrepancy(Wang & Liu, 2016)Ex~Q[T(P)f(x)]T(P)=(Vxlog(P(x))T+Vx.{f:x→Rdf smooth with zeroboundary conditionNA in generalhas a closed formin RKHS
Cramerfor d =1(Bellemare et al., 2017a)Ex~Pf(x)-Ex~Qf(x){f:x→R,Ea~P(）²≤1,f smooth with zeroboundary conditionEx~P（Fp(2）-F0（））²P(x）xER
μ-FisherIPM(Mroueh & Sercu,2017)Ex~Pf(x)-Ex~Qf(x){f:X→R,f∈L(x,μ),Ex~μf²(x)≤1}Ex~u（P()-Q(x））²2μ(x）
μ-SobolevIPM(This work)Ex~Pf(x)-Ex~qf(x){f:X→R,f eW²(x,μ),Ex~μ|/Vxf(x)²≤1,with zero boundary condition }1VE~u∑i-(（中i(P）-Φi(Q）μ（x）where 𝜙(P) =Px-i(x-i)FP[xx-i-ij(mi)x−=(x1,..xi-1,Ti+1,..d)
+ +# 3 GENERALIZING FISHER IPM: PDF COMPARISON + +Imposing data-independent constraints on the function class in the IPM framework, such as the Lipschitz constraint in the Wasserstein distance is computationally challenging and intractable for the general case. In this Section, we generalize the Fisher IPM introduced in (Mroueh & Sercu, 2017), where the function class is relaxed to a tractable data dependent constraint on the second order moment of the critic, in other words the critic is constrained to be in a Lebsegue ball. + +Fisher IPM. Let $\mathcal { X } \subset \mathbb { R } ^ { d }$ and $\mathcal { P } ( \mathcal { X } )$ be the space of distributions defined on $\mathcal { X }$ . Let $\mathbb { P } , \mathbb { Q } \in$ $\mathcal { P } ( \mathcal { X } )$ , and $\mu$ be a dominant measure of $\mathbb { P }$ and $\mathbb { Q }$ , in the sense that + +$$ +\mu ( x ) = 0 \implies \mathbb { P } ( x ) = 0 { \mathrm { ~ a n d } } \mathbb { Q } ( x ) = 0 . +$$ + +We assume $\mu$ to be also a distribution in $\mathcal { P } ( \mathcal { X } )$ , and assume $\pmb { \mu } ( \pmb { x } ) > \mathbf { 0 } , \forall \pmb { x } \in \mathcal { X }$ . Let $\mathcal { L } _ { 2 } ( \mathcal { X } , \mu )$ be the space of $\mu$ -measurable functions. For $f , g \in \mathcal { L } _ { 2 } ( \mathcal { X } , \mu )$ , we define the following dot product and its corresponding norm: + +$$ +\langle f , g \rangle _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) } = \int _ { \mathcal { X } } f ( x ) g ( x ) \mu ( x ) d x , \| f \| _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) } = \sqrt { \int _ { \mathcal { X } } f ^ { 2 } ( x ) \mu ( x ) d x } . +$$ + +Note that $\mathcal { L } _ { 2 } ( \mathcal { X } , \mu )$ , can be formally defined as follows: + +$$ +\mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) = \{ f : \mathcal { X } \to \mathbb { R } \mathrm { s . t } \ \| f \| _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) } < \infty \} . +$$ + +We define the unit Lebesgue ball as follows: + +$$ +\begin{array} { r } { \mathbb { B } _ { 2 } ( \mathcal { X } , \mu ) = \{ f \in \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) , \| f \| _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) } \leq 1 \} . } \end{array} +$$ + +Fisher IPM defined in (Mroueh & Sercu, 2017), searches for the critic function in the Lebesgue Ball $\mathbb { B } _ { 2 } ( \mathcal { X } , \mu )$ that maximizes the mean discrepancy between $\mathbb { P }$ and $\mathbb { Q }$ . Fisher GAN (Mroueh & Sercu, 2017) was originally formulated specifically for $\begin{array} { r } { \mu = \frac 1 2 ( { \mathbb P } + { \mathbb Q } ) } \end{array}$ . We consider here a general $\mu$ as long as it dominates $\mathbb { P }$ and $\mathbb { Q }$ . We define Generalized Fisher IPM as follows: + +$$ +\mathcal { F } _ { \mu } ( \mathbb { P } , \mathbb { Q } ) = \operatorname* { s u p } _ { f \in \mathbb { B } _ { 2 } ( \mathcal { X } , \mu ) } \mathbb { E } _ { x \sim \mathbb { P } } f ( x ) - \mathbb { E } _ { x \sim \mathbb { Q } } f ( x ) +$$ + +Note that: + +$$ +\mathbb { E } _ { x \sim \mathbb { P } } f ( x ) - \mathbb { E } _ { x \sim \mathbb { Q } } f ( x ) = \left. f , { \frac { \mathbb { P } - \mathbb { Q } } { \mu } } \right. _ { { \mathcal { L } } _ { 2 } ( \mathcal { X } , \mu ) } . +$$ + +Hence Fisher IPM can be written as follows: + +$$ +\mathcal { F } _ { \mu } ( \mathbb { P } , \mathbb { Q } ) = \operatorname* { s u p } _ { f \in \mathbb { B } _ { 2 } ( \mathcal { X } , \mu ) } \left. f , \frac { \mathbb { P } - \mathbb { Q } } { \mu } \right. _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) } +$$ + +We have the following result: + +Theorem 1 (Generalized Fisher IPM). The Fisher distance and the optimal critic are as follows: + +1. The Fisher distance is given by: + +$$ +\mathcal { F } _ { \mu } ( \mathbb { P } , \mathbb { Q } ) = \left\| \frac { \mathbb { P } - \mathbb { Q } } { \mu } \right\| _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) } = \sqrt { \mathbb { E } _ { \boldsymbol { x } \sim \mu } \left( \frac { \mathbb { P } ( \boldsymbol { x } ) - \mathbb { Q } ( \boldsymbol { x } ) } { \mu ( \boldsymbol { x } ) } \right) ^ { 2 } } . +$$ + +2. The optimal $f _ { \chi }$ achieving the Fisher distance $\mathcal { F } _ { \mu } ( \mathbb { P } , \mathbb { Q } )$ is: + +Proof of Theorem 1. From Equation (2), the optimal $f _ { \chi }$ belong to the intersection of the hyperplane that has normal $n ~ = ~ \textstyle \frac { \mathbb { P } - \mathbb { Q } } { \mu }$ and the ball B2(X , µ), hence fχ = $\begin{array} { r } { f _ { \chi } = \frac { n ^ { \cdot } } { \| n \| _ { \mathcal { L } _ { 2 } ( \chi , \mu ) } } } \end{array}$ Hence $\mathcal { F } ( \mathbb { P } , \mathbb { Q } ) = \| n \| _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) } .$ □ + +![](images/39105c967ba0488a4607950203039a45c6b2ef8745052cceb191e400af567c21.jpg) + +We see from Theorem 1 the role of the dominant measure $\mu$ : the optimal critic is defined with respect to this measure and the overall Fisher distance can be seen as an average weighted distance between probability density functions, where the average is taken on points sampled from $\mu$ . We give here some choices of $\mu$ : + +1. For $\begin{array} { r } { \mu = \frac 1 2 ( { \mathbb P } + { \mathbb Q } ) } \end{array}$ , we obtain the symmetric chi-squared distance as defined in (Mroueh & Sercu, 2017). +2. $\mu _ { G P }$ , the implicit distribution defined by the interpolation lines between $\mathbb { P } _ { r }$ and $\mathbb { Q } _ { \theta }$ as in (Gulrajani et al., 2017). +3. When $\mu$ does not dominate $\mathbb { P }$ , and $\mathbb { Q }$ , we obtain a non symmetric divergence. For example for $\mu = \mathbb { P }$ , $\begin{array} { r } { \mathcal { F } _ { \mathbb { P } } ^ { 2 } ( \mathbb { P } , \mathbb { Q } ) = \int _ { \mathcal { X } } \frac { ( \mathbb { P } ( x ) - \mathbb { Q } ( x ) ) ^ { 2 } } { \mathbb { P } ( x ) } d x } \end{array}$ (P(x)−Q(x))2P(x) dx. We see here that for this particular choice we obtain the Pearson divergence. + +# 4 SOBOLEV IPM + +In this Section, we introduce the Sobolev IPM. In a nutshell, the Sobolev IPM constrains the critic function to belong to a ball in the restricted Sobolev Space. In other words we constrain the norm of the gradient of the critic $\nabla _ { x } f ( x )$ . We will show that by moving from a Lebesgue constraint as in Fisher IPM to a Sobolev constraint as in Sobolev IPM, the metric changes from a joint PDF matching to weighted (ccordinate-wise) conditional CDFs matching. The intrinsic conditioning built in to the Sobolev IPM and the comparison of cumulative distributions makes Sobolev IPM suitable for comparing discrete sequences. + +# 4.1 DEFINITION AND EXPRESSION OF SOBOLEV IPM IN TERMS OF COORDINATE CONDITIONAL CDFS + +We will start by recalling some definitions on Sobolev Spaces. We assume in the following that $\mathcal { X }$ is compact and consider functions in the Sobolev space $\mathring { W } ^ { 1 , 2 } ( \mathcal { X } , \mu )$ : + +$$ +W ^ { 1 , 2 } ( \mathcal { X } , \mu ) = \left\{ f : \mathcal { X } \to \mathbb { R } , \int _ { \mathcal { X } } \left. \nabla _ { x } f ( x ) \right. ^ { 2 } \mu ( x ) d x < \infty \right\} , +$$ + +We restrict ourselves to functions in $W ^ { 1 , 2 } ( \mathcal { X } , \mu )$ vanishing at the boundary, and note this space $W _ { 0 } ^ { 1 , 2 } ( \mathcal { X } , \mu )$ . Note that in this case: + +$$ +\| f \| _ { W _ { 0 } ^ { 1 , 2 } ( \mathcal { X } , \mu ) } = \sqrt { \int _ { \mathcal { X } } \| \nabla _ { x } f ( x ) \| ^ { 2 } \mu ( x ) d x } +$$ + +defines a semi-norm. We can similarly define a dot product in $W _ { 0 } ^ { 1 , 2 } ( \mathcal { X } , \mu )$ , for $f , g \in W _ { 0 } ^ { 1 , 2 } ( \mathcal { X } , \mu )$ : + +$$ +\langle f , g \rangle _ { W _ { 0 } ^ { 1 , 2 } ( \mathcal { X } , \mu ) } = \int _ { \mathcal { X } } \langle \nabla _ { x } f ( x ) , \nabla _ { x } g ( x ) \rangle _ { \mathbb { R } ^ { d } } \mu ( x ) d x . +$$ + +Hence we define the following Sobolev IPM, by restricting the critic of the mean discrepancy to the Sobolev unit ball : + +$$ +\mathcal { S } _ { \boldsymbol { \mu } } ( \mathbb { P } , \mathbb { Q } ) = \operatorname* { s u p } _ { \substack { f \in W _ { 0 } ^ { 1 , 2 } , \| f \| _ { W _ { 0 } ^ { 1 , 2 } ( \mathcal { X } , \boldsymbol { \mu } ) } \leq 1 } } \bigg \{ \mathbb { E } _ { \boldsymbol { x } \sim \mathbb { P } } f ( \boldsymbol { x } ) - \mathbb { E } _ { \boldsymbol { x } \sim \mathbb { Q } } f ( \boldsymbol { x } ) \bigg \} . +$$ + +When compared to the Wasserstein distance, the Sobolev IPM given in Equation (3) uses a data dependent gradient constraint (depends on $\mu$ ) rather than a data independent Lipchitz constraint. Let $F _ { \mathbb { P } }$ and $F _ { \mathbb { Q } }$ be the cumulative distribution functions of $\mathbb { P }$ and $\mathbb { Q }$ respectively. We have: + +$$ +\mathbb { P } ( x ) = \frac { \partial ^ { d } } { \partial x _ { 1 } \hdots \partial x _ { d } } F _ { \mathbb { P } } ( x ) , +$$ + +and we define + +$$ +D ^ { - i } = \frac { \partial ^ { d - 1 } } { \partial x _ { 1 } \dots \partial x _ { i - 1 } \partial x _ { i + 1 } \dots \partial x _ { d } } , \mathrm { f o r } i = 1 \dots d . +$$ + +$D ^ { - i }$ computes the $( d - 1 )$ high-order partial derivative excluding the variable $i$ + +Our main result is presented in Theorem 2. Additional theoretical results are given in Appendix A. +All proofs are given in Appendix B. + +Theorem 2 (Sobolev IPM). Assume that $F _ { \mathbb { P } }$ , and $F _ { \mathbb { Q } }$ and its d derivatives exist and are continuous: $F _ { \mathbb { P } }$ and $F _ { \mathbb { Q } } \in C ^ { d } ( \mathcal { X } )$ . Define the differential operator $D ^ { - }$ : + +$$ +D ^ { - } = ( D ^ { - 1 } , \ldots D ^ { - d } ) . +$$ + +$$ +x = ( x _ { 1 } , \ldots x _ { i - 1 } , x _ { i } , x _ { i + 1 } , \ldots x _ { d } ) , l e t x ^ { - i } = ( x _ { 1 } , \ldots x _ { i - 1 } , x _ { i + 1 } , \ldots x _ { d } ) . +$$ + +The Sobolev IPM given in Equation (3) has the following equivalent forms: + +1. Sobolev IPM as comparison of high order partial derivatives of CDFs. The Sobolev IPM has the following form: + +$$ +S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) = \frac { 1 } { d } \sqrt { \int _ { \mathcal { X } } \frac { \sum _ { i = 1 } ^ { d } ( D ^ { - i } F _ { \mathbb { P } } ( x ) - D ^ { - i } F _ { \mathbb { Q } } ( x ) ) ^ { 2 } } { \mu ( x ) } d x } . +$$ + +2. Sobolev IPM as comparison of weighted (coordinate-wise) conditional CDFs. The Sobolev IPM can be written in the following equivalent form: + +$$ +S _ { \mu } ^ { 2 } ( { \mathbb { P } } , { \mathbb { Q } } ) = \frac { 1 } { d ^ { 2 } } { \mathbb { E } } _ { x \sim \mu } \sum _ { i = 1 } ^ { d } \left( \frac { { \mathbb { P } } _ { X ^ { - i } } ( x ^ { - i } ) F _ { { \mathbb { P } } _ { [ X _ { i } ] X ^ { - i } = x ^ { - i } ] } } ( x _ { i } ) - { \mathbb { Q } } _ { X ^ { - i } } ( x ^ { - i } ) F _ { { \mathbb { Q } } _ { [ X _ { i } ] X ^ { - i } = x ^ { - i } ] } } ( x _ { i } ) } { \mu ( x ) } \right) ^ { 2 } . +$$ + +3. The optimal critic $f ^ { * }$ satisfies the following identity: + +$$ +\nabla _ { x } f ^ { * } ( x ) = \frac { 1 } { d S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) } \frac { D ^ { - } F _ { \mathbb { Q } } ( x ) - D ^ { - } F _ { \mathbb { P } } ( x ) } { \mu ( x ) } , \mu - a l m o s t s u r e l y . +$$ + +Sobolev IPM Approximation. Learning in the whole Sobolev space $W _ { 0 } ^ { 1 , 2 }$ is challenging hence we need to restrict our function class to a hypothesis class , such as neural networks. We assume in the following that functions in $\mathcal { H }$ vanish on the boundary of $\mathcal { X }$ , and restrict the optimization to the function space $\mathcal { H }$ . $\mathcal { H }$ can be a Reproducing Kernel Hilbert Space as in the MMD case or parametrized by a neural network. Define the Sobolev IPM approximation in $\mathcal { H }$ : + +$$ +\mathcal { S } _ { \mathcal { H } , \mu } ( \mathbb { P } , \mathbb { Q } ) = \operatorname* { s u p } _ { \substack { f \in \mathcal { H } , \| f \| _ { W _ { 0 } ^ { 1 , 2 } } \leq 1 } } \left\{ \mathbb { E } _ { x \sim \mathbb { P } } f ( x ) - \mathbb { E } _ { x \sim \mathbb { Q } } f ( x ) \right\} +$$ + +The following Lemma shows that the Sobolev IPM approximation in $\mathcal { H }$ is proportional to Sobolev IPM. The tightness of the approximation of the Sobolev IPM is governed by the tightness of the approximation of the optimal Sobolev Critic $f ^ { * }$ in $\mathcal { H }$ . This approximation is measured in the Sobolev sense, using the Sobolev dot product. + +Lemma 1 (Sobolev IPM Approximation in a Hypothesis Class). Let $\mathcal { H }$ be a function space with functions vanishing at the boundary. For any f ∈ H and for f ∗ the optimal critic in W 1,20 , we have: + +$$ +\mathcal { S } _ { \mathcal { H } , \mu } ( \mathbb { P } , \mathbb { Q } ) = \mathcal { S } _ { \mu } ( \mathbb { P } , \mathbb { Q } ) \operatorname* { s u p } _ { \substack { f \in \mathcal { H } , \| f \| _ { W _ { 0 } ^ { 1 , 2 } ( \chi , \mu ) } \leq 1 } } \int _ { \mathcal { X } } \left. \nabla _ { x } f ( x ) , \nabla _ { x } f ^ { * } ( x ) \right. _ { \mathbb { R } ^ { d } } \mu ( x ) d x . +$$ + +Note that this Lemma means that the Sobolev IPM is well approximated if the space $\mathcal { H }$ has an enough representation power to express $\nabla _ { x } f ^ { * } ( x )$ . This is parallel to the Fisher IPM approximation (Mroueh & Sercu, 2017) where it is shown that the Fisher IPM approximation error is proportional to the critic approximation in the Lebesgue sense. Having in mind that the gradient of the critic is the information that is passed on to the generator, we see that this convergence in the Sobolev sense to the optimal critic is an important property for GAN training. + +Relation to Fokker-Planck Diffusion. We show in Appendix A that the optimal Sobolev critic is the solution of the following elliptic PDE (with zero boundary conditions): + +$$ +\frac { \mathbb { P } - \mathbb { Q } } { S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) } = - \mathrm { d i v } ( \mu ( x ) \nabla _ { x } f ( x ) ) . +$$ + +We further link the elliptic PDE given in Equation (8) and the Fokker-Planck diffusion. As we illustrate in Figure 2(b) the gradient of the critic defines a transportation plan for moving the distribution mass from $\mathbb { Q }$ to $\mathbb { P }$ . + +Discussion of Theorem 2. We make the following remarks on Theorem 2: + +1. From Theorem 2, we see that the Sobolev IPM compares $d$ higher order partial derivatives of the cumulative distributions $F _ { \mathbb { P } }$ and $F _ { \mathbb { Q } }$ , while Fisher IPM compares the probability density functions. + +2. The dominant measure $\mu$ plays a similar role to Fisher: + +$$ +S _ { \mu } ^ { 2 } ( { \mathbb { P } } , \mathbb { Q } ) = \frac 1 { d ^ { 2 } } \sum _ { i = 1 } ^ { d } { \mathbb { E } } _ { x \sim \mu } \left( \frac { D ^ { - i } F _ { \mathbb { P } } ( x ) - D ^ { - i } F _ { \mathbb { Q } } ( x ) } { \mu ( x ) } \right) ^ { 2 } , +$$ + +the average distance is defined with respect to points sampled from $\mu$ . + +3. Comparison of coordinate-wise Conditional CDFs. We note in the following $\begin{array} { r l } { x ^ { - i } } & { { } = } \end{array}$ $( x _ { 1 } , \ldots x _ { i - 1 } , x _ { i + 1 } , \ldots x _ { d } )$ . Note that we have: + +$$ +\begin{array} { r l } & { \mathsf { P } ^ { - i } F _ { \mathbb { P } } ( x ) = \frac { \partial ^ { d - 1 } } { \partial x _ { 1 } \hdots \partial x _ { i - 1 } \partial x _ { i + 1 } \hdots \partial x _ { d } } \displaystyle \int _ { - \infty } ^ { x _ { 1 } } \hdots \int _ { - \infty } ^ { x _ { d } } \mathbb { P } ( u _ { 1 } \hdots u _ { d } ) d u _ { 1 } \hdots d u _ { d } } \\ & { \hphantom { \mathsf { P } ^ { - i } F _ { \mathbb { P } } ( x ) = } = \displaystyle \int _ { - \infty } ^ { x _ { i } } \mathbb { P } ( x _ { 1 } , \hdots , x _ { i - 1 } , u , x _ { i + 1 } , \hdots , x _ { d } ) d u } \\ & { \hphantom { \mathsf { P } ^ { - i } F _ { \mathbb { P } } ( x ) = } = \mathbb { P } _ { X ^ { - i } } ( x _ { 1 } , \hdots , x _ { i - 1 } , x _ { i + 1 } , \hdots x _ { d } ) \displaystyle \int _ { - \infty } ^ { x _ { i } } \mathbb { P } _ { [ X _ { i } | X ^ { - i } = x ^ { - i } ] } ( u | x _ { 1 } , \hdots , x _ { i - 1 } , x _ { i + 1 } , \hdots x _ { d } ) d u } \\ & { \displaystyle ( \mathrm { U s i n g ~ B a y e s ~ r u l e } ) } \\ & { \hphantom { \mathsf { P } ^ { - i } F _ { \mathbb { P } } ( x ) = } = \mathbb { P } _ { X ^ { - i } } ( x ^ { - i } ) F _ { \mathbb { P } _ { [ X _ { i } | X ^ { - i } = x ^ { - i } ] } } ( x _ { i } ) , } \end{array} +$$ + +Note that for each $i$ , $D ^ { - i } F _ { \mathbb { P } } ( x )$ is the cumulative distribution of the variable $X _ { i }$ given the other variables $X ^ { - i } = x ^ { - i }$ , weighted by the density function of $X ^ { - i }$ at $x ^ { - i }$ . This leads us to the form given in Equation 5. + +We see that the Sobolev IPM compares for each dimension $i$ the conditional cumulative distribution of each variable given the other variables, weighted by their density function. We refer to this as comparison of coordinate-wise CDFs on a leave one out basis. From this we see that we are comparing CDFs, which are better behaved on discrete distributions. Moreover, the conditioning built in to this metric will play a crucial role in comparing sequences as the conditioning is important in this context (See section 6.1). + +# 4.2 ILLUSTRATIVE EXAMPLES + +Sobolev IPM / Cramer Distance and Wasserstein-1 in one Dimension. ´ In one dimension, Sobolev IPM is the Cramer Distance (for ´ $\mu$ uniform on $\mathcal { X }$ , we note this $\mu : = 1$ ). While Sobolev IPM in one dimension measures the discrepancy between CDFs, the one dimensional Wasserstein- $p$ distance measures the discrepancy between inverse CDFs: + +$$ +S _ { \mu : = 1 } ^ { 2 } ( { \mathbb { P } } , \mathbb { Q } ) = \int _ { \chi } ( F _ { \mathbb { P } } ( x ) - F _ { \mathbb { Q } } ( x ) ) ^ { 2 } d x { \mathrm { ~ v e r s u s ~ } } W _ { p } ^ { p } ( { \mathbb { P } } , \mathbb { Q } ) = \int _ { 0 } ^ { 1 } | F _ { \mathbb { P } } ^ { - 1 } ( u ) - F _ { \mathbb { Q } } ^ { - 1 } ( u ) | ^ { p } d u , +$$ + +Recall also that the Fisher IPM for uniform $\mu$ is given by : + +$$ +\mathcal { F } _ { \mu : = 1 } ^ { 2 } ( \mathbb { P } , \mathbb { Q } ) = \int _ { \mathcal { X } } ( \mathbb { P } ( x ) - \mathbb { Q } ( x ) ) ^ { 2 } d x . +$$ + +Consider for instance two point masses $\mathbb { P } = \delta _ { a _ { 1 } }$ and $\mathbb { Q } = \delta _ { a _ { 2 } }$ with $a _ { 1 } , a _ { 2 } \in \mathbb { R }$ . The rationale behind using Wasserstein distance for GAN training is that since it is a weak metric, for far distributions $W _ { 1 } ^ { 1 } ( \mathbb { P } , \mathbb { Q } ) = S _ { \mu : = 1 } ^ { 2 } \stackrel { - } { = } | a _ { 1 } - a _ { 2 } |$ me sign, while e than $\mathcal { F } _ { \mu : = 1 } ^ { 2 } ( \mathbb { P } , \mathbb { Q } ) = 2$ 2017). In this case, it is easy to see that. As we see from this simple example,g distributions on discrete spaces. See $C D F$ +Figure 1, for a further discussion of this effect in the GAN context. + +Sobolev IPM between two 2D Gaussians. We consider $\mathbb { P }$ and $\mathbb { Q }$ to be two dimensional Gaussians with means $\mu _ { 1 }$ and $\mu _ { 2 }$ and covariances $\Sigma _ { 1 }$ and $\Sigma _ { 2 }$ . Let $( x , y )$ be the coordinates in 2D. We note $F _ { \mathbb { P } }$ and $F _ { \mathbb { Q } }$ the CDFs of $\mathbb { P }$ and $\mathbb { Q }$ respectively. We consider in this example $\textstyle \mu = { \frac { \mathbb { P } + \mathbb { Q } } { 2 } }$ . We know from Theorem 2 that the gradient of the Sobolev optimal critic is proportional to the following vector field: + +$$ +\begin{array} { l } { \displaystyle \nabla f ^ { * } ( x , y ) \alpha \frac { 1 } { \mu ( x , y ) } \left[ \frac { \partial } { \partial x } ( F _ { \mathbb { Q } } ( x , y ) - F _ { \mathbb { P } } ( x , y ) ) \right] } \\ { \displaystyle \mu _ { 1 } = [ 1 , 0 ] , \Sigma _ { 1 } = \left[ \frac { 1 . 9 } { 0 . 8 } \quad 0 . 8 \right] \mu _ { 2 } = [ 1 , - 2 ] , \Sigma _ { 2 } = \left[ \frac { 1 . 9 } { - 0 . 8 } \quad - 0 . 8 \right] . } \end{array} +$$ + +In Figure 2 we consider + +In Figure 2(a) we plot the numerical solution of the PDE satisfied by the optimal Sobolev critic given in Equation (8), using MATLAB solver for elliptic PDEs (more accurately we solve $- d i v ( \bar { \mu } ( x ) \nabla _ { x } f ( \bar { x } ) ) = \mathbb { P } ( x ) - \bar { \mathbb { Q } } ( x )$ , hence we obtain the solution of Equation (8) up to a normalization constant $( \frac { 1 } { S _ { \mu } ( { \mathbb P } , { \mathbb Q } ) } ) )$ . We numerically solve the PDE on a rectangle with zero boundary conditions. We see that the optimal Sobolev critic separates the two distributions well. In Figure 2(b) we then numerically compute the gradient of the optimal Sobolev critic on a 2D grid as given in Equation 9 (using numerical evaluation of the CDF and finite difference for the evaluation of the partial derivatives). We plot in Figure 2(b) the density functions of $\mathbb { P }$ and $\mathbb { Q }$ as well as the vector field of the gradient of the optimal Sobolev critic. As discussed in Section A.1, we see that the gradient of the critic (wrt to the input), defines on the support of $\textstyle \mu = { \frac { \mathbb { P } + \mathbb { Q } } { 2 } }$ a transportation plan for moving the distribution mass from $\mathbb { Q }$ to $\mathbb { P }$ . + +![](images/bf79812d77b73653170a552e274a8e31c047709d105768f320b54b41bce68e2a.jpg) +(a) Smoothed discrete densities: PDF versus CDF of smoothed discrete densities with non overlapping supports. + +![](images/aebf2bc5c2f3315395bc93792e96d731c49ae95608257b3482f05c10e62e4c99.jpg) +(b) Smoothed Discrete and Continuous densities: PDF versus CDF of a smoothed discrete density and a continuous density with non overlapping supports. +Figure 1: In the GAN context for example in text generation, we have to match a (smoothed) discrete real distribution and a continuous generator. In this case, the CDF matching enabled by Sobolev IPM gives non zero discrepancy between a (smoothed) discrete and a continuous density even if the densities have disjoint supports. This ensures non vanishing gradients of the critic. + +# 5 SOBOLEV GAN + +Now we turn to the problem of learning GANs with Sobolev IPM. Given the “real distribution” $\mathbb { P } _ { r } \in \mathcal { P } ( \mathcal { X } )$ , our goal is to learn a generator $g _ { \boldsymbol { \theta } } : \mathcal { Z } \subset \mathbb { R } ^ { n _ { z } } \mathcal { X }$ , such that for $z \sim p _ { z }$ , the distribution of $g _ { \boldsymbol { \theta } } ( z )$ is close to the real data distribution $\mathbb { P } _ { r }$ , where $p _ { z }$ is a fixed distribution on $\mathcal { Z }$ (for instance $z \sim \mathcal { N } ( 0 , I _ { n _ { z } } ) )$ . We note $\mathbb { Q } _ { \theta }$ for the “fake distribution” of $g _ { \theta } ( z ) , z \sim p _ { z }$ . Consider $\{ x _ { i } , i = 1 \ldots N \} \sim \bar { \mathbb { P } } _ { r }$ , $\{ z _ { i } , i = 1 \ldots N \} \sim { \mathcal { N } } ( 0 , I _ { n _ { z } } )$ , and $\{ \tilde { x } _ { i } , i = 1 \ldots N \} \sim \mu$ . We consider these choices for $\mu$ : + +1. $\begin{array} { r } { \mu = \frac { \mathbb { P } _ { r } + \mathbb { Q } _ { \theta } } { 2 } } \end{array}$ i.e $\tilde { x } \sim \mathbb { P } _ { r }$ or $\tilde { x } = g _ { \theta } ( z ) , z \sim p _ { z }$ with equal probability $\textstyle { \frac { 1 } { 2 } }$ . 2. $\mu _ { G P }$ is the implicit distribution defined by the interpolation lines between $\mathbb { P } _ { r }$ and $\mathbb { Q } _ { \theta }$ as in (Gulrajani et al., 2017) i.e : $\tilde { x } = u x + ( 1 - u ) y , x \sim \mathbb { P } _ { r } , y = g _ { \theta } ( z ) , z \sim p _ { z }$ and $u \sim \mathrm { U n i f } [ 0 , 1 ]$ . + +Sobolev GAN can be written as follows: + +$$ +\begin{array}{c} \begin{array} { l } { \displaystyle \mathrm { A N ~ c a n ~ p e ~ w r i t t e n ~ a s ~ r o l l o w s } \colon } \\ { \displaystyle \operatorname* { m i n } _ { g _ { \theta } } } \\ { \displaystyle \quad { f _ { p } , \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \| \nabla _ { x } f _ { p } ( \tilde { x } _ { i } ) \| ^ { 2 } = 1 } } \end{array} \hat { \hat { \mathcal E } } ( f _ { p } , g _ { \theta } ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } f _ { p } ( x _ { i } ) - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } f _ { p } ( g _ { \theta } ( z _ { i } ) ) \end{array} +$$ + +For any choice of the parametric function class $\mathcal { H } _ { p }$ , note the constraint by $\hat { \Omega } _ { S } ( f _ { p } , g _ { \theta } ) \ =$ $\begin{array} { r l } & { { \frac { 1 } { N } } \sum _ { i = 1 } ^ { N } \left\| \nabla _ { x } f _ { p } ( \tilde { x } _ { i } ) \right\| ^ { 2 } } \end{array}$ . For example if $\begin{array} { r } { \mu \ = \ \frac { \mathbb { P } _ { r } + \mathbb { Q } _ { \theta } } { 2 } } \end{array}$ , $\begin{array} { r c l } { \hat { \Omega } _ { S } ( f _ { p } , g _ { \theta } ) } & { = } & { \frac { 1 } { 2 N } \sum _ { i = 1 } ^ { N } \| \nabla _ { x } \boldsymbol { f } _ { p } ( x _ { i } ) \| ^ { 2 } + } \end{array}$ + +![](images/7e31e2430319983c215121a0e70ffc66b6fbcb2eef7bf131143c1e35323dcb1d.jpg) +(b) Optimal Sobolev Transport Vector Field $\nabla _ { x } f ^ { * } { \bar { ( x ) } }$ (arrows are the vector field $\nabla _ { x } f ^ { * } ( x )$ evaluated on the 2D grid. Magnitude of arrows was rescaled for visualization.) + +![](images/c2e0765d22d99d612ec7689965f0fb6fce3a5271eaa33de3baadab7bc9b8eda2.jpg) + +(a) Numerical solution of the PDE satisfied by the optimal Sobolev critic. + +Figure 2: Numerical solution of the PDE satisfied by the optimal Sobolev critic and the transportation Plan induced by the gradient of Sobolev critic. The gradient of the critic (wrt to the input), defines on the support of $\begin{array} { r } { \bar { \boldsymbol { \mu } } = \frac { \mathbb { P } + \mathbb { Q } } { 2 } , } \end{array}$ a transportation plan for moving the distribution mass from $\mathbb { Q }$ to $\mathbb { P }$ . For a theoretical analysis of this transportation plan and its relation to Fokker-Planck diffusion the reader is invited to check Appendix A. + +$\begin{array} { r l } { } & { { } \frac { 1 } { 2 N } \sum _ { i = 1 } ^ { N } \left\| \nabla _ { x } f _ { p } ( g _ { \theta } ( z _ { i } ) ) \right\| ^ { 2 } } \end{array}$ . Note that, since the optimal theoretical critic is achieved on the sphere, we impose a sphere constraint rather than a ball constraint. Similar to (Mroueh & Sercu, 2017) we define the Augmented Lagrangian corresponding to Sobolev GAN objective and constraint + +$$ +\mathcal { L } _ { S } ( p , \theta , \lambda ) = \hat { \mathcal { E } } ( f _ { p } , g _ { \theta } ) + \lambda ( 1 - \hat { \Omega } _ { S } ( f _ { p } , g _ { \theta } ) ) - \frac { \rho } { 2 } ( \hat { \Omega } _ { S } ( f _ { p } , g _ { \theta } ) - 1 ) ^ { 2 } +$$ + +where $\lambda$ is the Lagrange multiplier and $\rho > 0$ is the quadratic penalty weight. We alternate between optimizing the critic and the generator. We impose the constraint when training the critic only. Given $\theta$ , we solve $\begin{array} { r } { \operatorname* { m a x } _ { p } \operatorname* { m i n } _ { \lambda } \mathcal { L } _ { S } ( p , \theta , \lambda ) } \end{array}$ , for training the critic. Then given the critic parameters $p$ we optimize the generator weights $\theta$ to minimize the objective minθ $\hat { \mathcal { E } } ( f _ { p } , g _ { \theta } )$ . See Algorithm 1. + +# Algorithm 1 Sobolev GAN + +
Input: ρ penalty weight, n Learning rate, nc number of iterations for training the critic, N batch size
Initialize p, 0,λ=0
repeat
for j = 1 to nc do
Samplea minibatch xi,i=1...N,xi~ Pr
Sample a minibatch zi,i = 1...N,zi ~ Pz
(gp,gx)←(VpLs,VxLs)(p,0,λ)
p ←p+η ADAM(p,9p)
λ ← λ- pgx {SGD rule on 入with learning rate ρ} end for
Sample zi,i= 1...N, zi ~ Pz
d←Vθ8(fp,g0）=-VθN∑=1fp(g0(zi）） N
0←0-nADAM(0,dθ)
until θ converges
+ +Remark 1. Note that in Algorithm $\cdot$ , we obtain a biased estimate since we are using same samples for the cost function and the constraint, but the incurred bias can be shown to be small and vanishing as the number of samples increases as shown and justified in (Shivaswamy & Jebara, 2010). + +Relation to WGAN-GP. WGAN-GP can be written as follows: + +$$ +\operatorname* { m i n } _ { \substack { g _ { \theta } } } \operatorname* { s u p } _ { f , \| \nabla _ { x } f _ { p } ( \bar { x } _ { i } ) \| = 1 , \bar { x } _ { i } \sim \mu _ { G P } } \hat { \bar { \mathcal { E } } } ( f _ { p } , g _ { \theta } ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } f _ { p } ( x _ { i } ) - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } f _ { p } ( g _ { \theta } ( z _ { i } ) ) +$$ + +The main difference between WGAN-GP and our setting, is that WGAN-GP enforces pointwise constraints on points drawn from $\mu = \mu _ { G P }$ via a point-wise quadratic penalty $\begin{array} { r l } { { \big ( \hat { \mathcal { E } } ( f _ { p } , g _ { \theta } ) - \lambda \sum _ { i = 1 } ^ { N } \bigl ( 1 - \bigl ) } \quad } & { { } } \end{array}$ $\| \nabla _ { x } f ( \widetilde { x } _ { i } ) \| ) ^ { 2 } )$ while we enforce that constraint on average as a Sobolev norm, allowing us the coordinate weighted conditional CDF interpretation of the IPM. + +# 6 APPLICATIONS OF SOBOLEV GAN + +Sobolev IPM has two important properties; The first stems from the conditioning built in to the metric through the weighted conditional CDF interpretation. The second stems from the diffusion properties that the critic of Sobolev IPM satisfies (Appendix A) that has theoretical and practical ties to the Laplacian regularizer and diffusion on manifolds used in semi-supervised learning (Belkin et al., 2006). + +In this Section, we exploit those two important properties in two applications of Sobolev GAN: Text generation and semi-supervised learning. First in text generation, which can be seen as a discrete sequence generation, Sobolev GAN (and WGAN-GP) enable training GANs without need to do explicit brute-force conditioning. We attribute this to the built-in conditioning in Sobolev IPM (for the sequence aspect) and to the CDF matching (for the discrete aspect). Secondly using GANs in semi-supervised learning is a promising avenue for learning using unlabeled data. We show that a variant of Sobolev GAN can achieve strong SSL results on the CIFAR-10 dataset, without the need of any form of activation normalization in the networks or any extra ad hoc tricks. + +# 6.1 TEXT GENERATION WITH SOBOLEV GAN + +In this Section, we present an empirical study of Sobolev GAN in character level text generation. Our empirical study on end to end training of character-level GAN for text generation is articulated on four dimensions (loss, critic, generator, $\pmb { \mu }$ ). (1) the loss used (GP: WGAN-GP (Gulrajani et al., 2017), S: Sobolev or F: Fisher) (2) the architecture of the critic (Resnets or RNN) (3) the architecture of the generator (Resnets or RNN or RNN with curriculum learning) (4) the sampling distribution $\pmb { \mu }$ in the constraint. + +Text Generation Experiments. We train a character-level GAN on Google Billion Word dataset and follow the same experimental setup used in (Gulrajani et al., 2017). The generated sequence length is 32 and the evaluation is based on Jensen-Shannon divergence on empirical 4-gram probabilities (JS-4) of validation data and generated data. JS-4 may not be an ideal evaluation criteria, but it is a reasonable metric for current character-level GAN results, which is still far from generating meaningful sentences. + +Annealed Smoothing of discrete $\mathbb { P } _ { r }$ in the constraint $\pmb { \mu }$ . Since the generator distribution will always be defined on a continuous space, we can replace the discrete “real” distribution $\mathbb { P } _ { r }$ with a smoothed version (Gaussian kernel smoothing) $\mathbb { P } _ { r } \star \mathcal { N } ( 0 , \sigma ^ { 2 } I _ { d } )$ . This corresponds to doing the following sampling for $\mathbb { P } _ { r } : x + \xi , x \sim \mathbb { P } _ { r }$ , and $\xi \sim \mathcal { N } ( 0 , \sigma ^ { 2 } I _ { d } )$ . Note that we only inject noise to the “real” distribution with the goal of smoothing the support of the discrete distribution, as opposed to instance noise on both “real” and “fake” to stabilize the training, as introduced in (Kaae Sønderby et al., 2017; Arjovsky & Bottou, 2017). As it is common in optimization by continuation (Mobahi & III, 2015), we also anneal the noise level $\sigma$ as the training progresses on a linear schedule. + +Sobolev GAN versus WGAN-GP with Resnets. In this setting, we compare (WGANGP,G $=$ Resnet,D $^ { 1 \textless }$ Resnet, $\mu = \mu _ { G P } ) $ to (Sobolev,G $=$ Resnet,D $=$ Resnet, $\mu$ ) where $\mu$ is one of: (1) $\mu _ { G P }$ , (2) the noise smoothed $\begin{array} { r } { \mu _ { s } ( \sigma ) \ = \ \frac { \mathbb { P } _ { r } \star \mathcal { N } ( 0 , \sigma ^ { 2 } I _ { d } ) + \mathbb { Q } _ { \theta } } { 2 } } \end{array}$ or (3) noise smoothed with annealing $\mu _ { s } ^ { a } ( \sigma _ { 0 } )$ with $\sigma _ { 0 }$ the initial noise level. We use the same architectures of Resnet with 1D convolution for the critic and the generator as in (Gulrajani et al., 2017) (4 resnet blocks with hidden layer size of 512). In order to implement the noise smoothing we transform the data into one-hot vectors. Each one hot vector $x$ is transformed to a probability vector $p$ with 0.9 replacing the one and $0 . 1 / ( d i c t _ { s i z e } - 1 )$ replacing the zero. We then sample $\epsilon$ from a Gaussian distribution ${ \bar { \mathcal { N } } } ( 0 , \sigma ^ { 2 } )$ , and + +![](images/95d02c80f9b5a08286ec9176c050401e5ad02420894a30a90fcc19711b6eed94.jpg) + +(a) Comparing Sobolev with $\mu _ { G P }$ and WGAN-GP. The JS-4 are 0.3363 and 0.3302 respectively. + +![](images/6e2575ca9e9bb232ea44064ff715cb92a3beab4fce8471509ac419d7b8ec9342.jpg) +(b) Comparing Sobolev with different $\mu$ dominant measures and WGAN-GP. The JS4 of $\mu _ { s } ^ { a } ( \sigma _ { 0 } = 1 . 5 )$ is 0.3268. +Figure 3: Result of Sobolev GAN for various dominating measure $\mu$ , for resnets as architectures of the critic and the generator. + +use softmax to normalize $\log p + \epsilon$ . We use algorithm 1 for Sobolev GAN and fix the learning rate to $1 0 ^ { - 4 }$ and $\rho$ to $1 0 ^ { - 5 }$ . The noise level $\sigma$ was annealed following a linear schedule starting from an initial noise level $\sigma _ { 0 }$ (at iteration $i$ , $\begin{array} { r } { \sigma _ { i } = \sigma _ { 0 } ( 1 - \frac { i } { M a x i t e r } ) } \end{array}$ , Maxite ${ \displaystyle = 3 0 \mathsf K }$ ). For WGAN-GP we used the open source implementation with the penalty $\lambda = 1 0$ as in (Gulrajani et al., 2017). Results are given in Figure 3(a) for the JS-4 evaluation of both WGAN-GP and Sobolev GAN for $\mu = \mu _ { G P }$ . In Figure 3(b) we show the JS-4 evaluation of Sobolev GAN with the annealed noise smoothing $\mu _ { s } ^ { a } ( \bar { \sigma _ { 0 } } )$ , for various values of the initial noise level $\sigma _ { 0 }$ . We see that the training succeeds in both cases. Sobolev GAN achieves slightly better results than WGAN-GP for the annealing that starts with high noise level $\sigma _ { 0 } = 1 . 5$ . We note that without smoothing and annealing i.e using $\begin{array} { r } { \bar { \boldsymbol { \mu } } = \frac { \mathbb { P } _ { r } + \mathbb { Q } _ { \theta } } { 2 } } \end{array}$ , Sobolev GAN is behind. Annealed smoothing of $\mathbb { P } _ { r }$ , helps the training as the real distribution is slowly going from a continuous distribution to a discrete distribution. See Appendix C (Figure 6) for a comparison between annealed and non annealed smoothing. + +We give in Appendix C a comparison of WGAN-GP and Sobolev GAN for a Resnet generator architecture and an RNN critic. The RNN has degraded performance due to optimization difficulties. + +Fisher GAN Curriculum Conditioning versus Sobolev GAN: Explicit versus Implicit conditioning. We analyze how Fisher GAN behaves under different architectures of generators and critics. We first fix the generator to be ResNet. We study 3 different architectures of critics: ResNet, GRU (we follow the experimental setup from (Press et al., 2017)), and hybrid $\mathrm { R e s N e t + G R U }$ (Reed et al., 2016). We notice that RNN is unstable, we need to clip the gradient values of critics in $[ - 0 . 5 , 0 . 5 ]$ , and the gradient of the Lagrange multiplier $\lambda _ { F }$ to $[ - 1 0 ^ { 4 } , \bar { 1 } 0 ^ { 4 } ]$ . We fix $\rho _ { F } = 1 0 ^ { - 7 }$ and we use $\mu = \mu _ { G P }$ . We search the value for the learning rate in $[ 1 0 ^ { - 5 } , 1 \dot { 0 } ^ { - 4 } ]$ . We see that for $\mu = \mu _ { G P }$ and $G = \mathbb { F }$ Resnet for various critic architectures, Fisher GAN fails at the task of text generation (Figure ${ \mathfrak { 4 } } \ { \mathfrak { a - c } } { \mathrm { , } }$ ). Nevertheless, when using RNN critics $( \mathrm { F i g } \quad \mathsf { b } , \mathsf { c } )$ a marginal improvement happens over the fully collapsed state when using a resnet critic (Fig 4 a). We hypothesize that RNN critics enable some conditioning and factoring of the distribution, which is lacking in Fisher IPM. + +Finally Figure 4 (d) shows the result of training with recurrent generator and critic. We follow (Press et al., 2017) in terms of GRU architecture, but differ by using Fisher GAN rather than WGAN-GP. We use $\begin{array} { r } { \mu = { \frac { \mathbb { P } _ { r } + \mathbb { Q } _ { \theta } } { 2 } } } \end{array}$ i.e. without annealed noise smoothing. We train (F, D=RNN,G=RNN, ${ \frac { \mathbb { P } _ { r } + \mathbb { Q } _ { \theta } } { 2 } } .$ ) using curriculum conditioning of the generator for all lengths $\ell$ as done in (Press et al., 2017): the generator is conditioned on $3 2 - \ell$ characters and predicts the $\ell$ remaining characters. We increment $\ell = 1$ to 32 on a regular schedule (every $1 5 \mathrm { k }$ updates). JS-4 is only computed when $\ell > 4$ . We see in Figure 4 that under curriculum conditioning with recurrent critics and generators, the training of Fisher GAN succeeds and reaches similar levels of Sobolev GAN (and WGAN-GP). Note that the need of this explicit brute force conditioning for Fisher GAN, highlights the implicit conditioning induced by Sobolev GAN via the gradient regularizer, without the need for curriculum conditioning. + +![](images/2cad69ff4d412b18994156d6951f045e5d5629da2e40f46cfb2e8bb9373125ba.jpg) +Figure 4: Fisher GAN with different architectures for critics: (a-c) We see that for $\mu = \mu _ { G P }$ and $G =$ Resnet for various critic architectures, Fisher GAN fails at the task of text generation. We notice small improvements for RNN critics (b-c) due to the conditioning and factoring of the distribution. (d) Fisher GAN with recurrent generator and critic, trained on a curriculum conditioning for increasing lengths $\ell$ , increments indicated by gridlines. In this curriculum conditioning setup, with recurrent critics and generators, the training of Fisher GAN succeeds and reaches similar levels of Sobolev GAN (and WGAN-GP). It is important to note that by doing this explicit curriculum conditioning for Fisher GAN, we highlight the implicit conditioning induced by Sobolev GAN, via the gradient regularizer. + +# 6.2 SEMI-SUPERVISED LEARNING WITH SOBOLEV GAN + +A proper and promising framework for evaluating GANs consists in using it as a regularizer in the semi-supervised learning setting (Salimans et al., 2016; Dumoulin et al., 2017; Kumar et al., 2017). As mentioned before, the Sobolev norm as a regularizer for the Sobolev IPM draws connections with the Laplacian regularization in manifold learning (Belkin et al., 2006). In the Laplacian framework of semi-supervised learning, the classifier satisfies a smoothness constraint imposed by controlling its Sobolev norm: $\begin{array} { r } { \int _ { \mathcal { X } } \left\| \nabla _ { x } \bar { f } ( x ) \right\| ^ { 2 } \mu ^ { 2 } ( x ) d x } \end{array}$ (Alaoui et al., 2016). In this Section, we present a variant Xof Sobolev GAN that achieves competitive performance in semi-supervised learning on the CIFAR10 dataset Krizhevsky & Hinton (2009) without using any internal activation normalization in the critic, such as batch normalization (BN) (Ioffe & Szegedy, 2015), layer normalization (LN) (Ba et al., 2016), or weight normalization (Salimans & Kingma, 2016). + +In this setting, a convolutional neural network $\Phi _ { \omega } : \mathcal { X } \mathbb { R } ^ { m }$ is shared between the cross entropy (CE) training of a $K$ -class classifier $( S \in \mathbb { R } ^ { K \times m } )$ ) and the critic of GAN (See Figure 5). We have the following training equations for the (critic $^ +$ classifer) and the generator: + +$$ +{ \mathrm { C r i t i c } } + { \mathrm { C l a s s i f i e r } } ; \quad \operatorname* { m a x } _ { S , \Phi _ { \omega } , f } \mathcal { L } _ { D } = \mathcal { L } _ { \mathrm { a l m } } ^ { \mathrm { G A N } } ( f , g _ { \theta } ) - \lambda _ { C E } \sum _ { ( x , y ) \in \mathrm { l a b } } C E ( p ( y | x ) , y ) +$$ + +$$ +\mathrm { G e n e r a t o r : \ m a x } { \mathcal { L } } _ { G } = { \hat { \mathcal { E } } } ( f , g _ { \theta } ) +$$ + +where the main IPM objective with $N$ samples: $\begin{array} { r } { \hat { \mathcal { E } } ( f , g _ { \theta } ) = \frac { 1 } { N } \left( \sum _ { x \in \mathrm { u n l } } f ( x ) - \sum _ { z \sim p _ { z } } f ( g _ { \theta } ( z ) ) \right) . } \end{array}$ + +Following (Mroueh & Sercu, 2017) we use the following “ $K + 1$ parametrization” for the critic (See Figure 5) : + +$$ +f ( x ) = \underbrace { \sum _ { y = 1 } ^ { K } p ( y | x ) \left. S _ { y } , \Phi _ { \omega } ( x ) \right. } _ { f _ { + } : \mathrm { \normalfont ~ { \mathrm { \fontfamily { ~ \ddots ~ } \selectfont ~ { \ c r i t i c } } } } } - \underbrace { \left. v , \Phi _ { \omega } ( x ) \right. } _ { f _ { - } : \mathrm { \normalfont ~ { \cdot \mathrm { \sc ~ { \ c r i t i c } } } } } +$$ + +Note that $p ( y | x ) = \mathrm { S o f t m a x } ( \langle \mathrm { S } , \Phi _ { \omega } ( \mathrm { x } ) \rangle ) _ { \mathrm { y } }$ appears both in the critic formulation and in the CrossEntropy term in Equation (11). Intuitively this critic uses the $K$ class directions of the classifier $S _ { y }$ to define the “real” direction, which competes with another $\mathrm { K } { + } 1 ^ { \mathrm { t h } }$ direction $v$ that indicates fake samples. This parametrization adapts the idea of (Salimans et al., 2016), which was formulated specifically for the classic KL / JSD based GANs, to IPM-based GANs. We saw consistently better results with the $K + 1$ formulation over the regular formulation where the classification layer $S$ doesn’t interact with the critic direction $v$ . We also note that when applying a gradient penalty based constraint (either WGAN-GP or Sobolev) on the full critic $f = f _ { + } - f _ { - }$ , it is impossible for the network to fit even the small labeled training set (underfitting), causing bad SSL performance. This leads us to the formulation below, where we apply the Sobolev constraint only on $f _ { - }$ . Throughout this Section we fix $\begin{array} { r } { \mu = { \frac { \mathbb { P } _ { r } + \mathbb { Q } _ { \theta } } { 2 } } } \end{array}$ . + +We propose the following two schemes for constraining the $\mathrm { K } { + } 1$ critic $f ( x ) = f _ { + } ( x ) - f _ { - } ( x )$ : + +1) Fisher constraint on the critic: We restrict the critic to the following set: + +$$ +f \in \left\{ f = f _ { + } - f _ { - } , \hat { \Omega } _ { F } ( f , g _ { \theta } ) = \frac { 1 } { 2 N } \left( \sum _ { x \in \mathrm { u n l } } f ^ { 2 } ( x ) + \sum _ { z \sim p _ { z } } f ^ { 2 } ( g _ { \theta } ( z ) ) \right) = 1 \right\} . +$$ + +This constraint translates to the following ALM objective in Equation (11): + +$$ +\mathcal { L } _ { \mathrm { a l m } } ^ { \mathrm { G A N } } ( f , g _ { \theta } ) = \hat { \mathcal { E } } ( f , g _ { \theta } ) + \lambda _ { F } ( 1 - \hat { \Omega } _ { F } ( f , g _ { \theta } ) ) - \frac { \rho _ { F } } { 2 } ( \hat { \Omega } _ { F } ( f , g _ { \theta } ) - 1 ) ^ { 2 } , +$$ + +where the Fisher constraint ensures the stability of the training through an implicit whitened mean matching (Mroueh & Sercu, 2017). + +2) Fisher $^ +$ Sobolev constraint: We impose 2 constraints on the critic: Fisher on $f$ & Sobolev on $f _ { - }$ + +$$ +\begin{array} { r } { f \in \left\{ f = f _ { + } - f _ { - } , \hat { \Omega } _ { F } ( \pmb { f } , g _ { \theta } ) = 1 \mathrm { a n d } \hat { \Omega } _ { S } ( \pmb { f } _ { - } , g _ { \theta } ) = 1 \right\} , } \\ { \mathrm { } _ { \mathrm { } s } ( \pmb { f } _ { - } , g _ { \theta } ) = \frac { 1 } { 2 N } \left( \sum _ { x \in \mathrm { u n l } } \left. \nabla _ { x } \pmb { f } _ { - } ( x ) \right. ^ { 2 } + \sum _ { z \sim p _ { z } } \left. \nabla _ { x } \pmb { f } _ { - } ( g _ { \theta } ( z ) ) \right. ^ { 2 } \right) . } \end{array} +$$ + +This constraint translates to the following ALM in Equation (11): + +$$ +\begin{array} { r l } & { \mathcal { L } _ { \mathrm { a l m } } ^ { \mathrm { G A N } } ( f , g _ { \theta } ) = \hat { \mathcal { E } } ( f , g _ { \theta } ) + \lambda _ { F } ( 1 - \hat { \Omega } _ { F } ( \pmb { f } , g _ { \theta } ) ) + \lambda _ { S } ( 1 - \hat { \Omega } _ { S } ( f _ { - } , g _ { \theta } ) ) } \\ & { \quad \quad \quad \quad - \frac { \rho _ { F } } { 2 } ( \hat { \Omega } _ { F } ( \pmb { f } , g _ { \theta } ) - 1 ) ^ { 2 } - \frac { \rho _ { S } } { 2 } ( \hat { \Omega } _ { S } ( f _ { - } , g _ { \theta } ) - 1 ) ^ { 2 } . } \end{array} +$$ + +Note that the fisher constraint on $f$ ensures the stability of the training, and the Sobolev constraints on the “fake” critic $f _ { - }$ enforces smoothness of the “fake” critic and thus the shared CNN $\Phi _ { \omega } ( x )$ . This is related to the classic Laplacian regularization in semi-supervised learning (Belkin et al., 2006). + +Table 2 shows results of SSL on CIFAR-10 comparing the two proposed formulations. Similar to the standard procedure in other GAN papers, we do hyperparameter and model selection on the validation set. We present baselines with a similar model architecture and leave out results with significantly larger convnets. G and D architectures and hyperparameters are in Appendix D. $\Phi _ { \omega }$ is similar to (Salimans et al., 2016; Dumoulin et al., 2017; Mroueh & Sercu, 2017) in architecture, but note that we do not use any batch, layer, or weight normalization yet obtain strong competitive accuracies. We hypothesize that we don’t need any normalization in the critic, because of the implicit whitening of the feature maps introduced by the Fisher and Sobolev constraints as explained in (Mroueh & Sercu, 2017). + +![](images/ea25635919fa3c5b32ae620a1e74c601b9df2ff8ade1012e0ef7eddf0195cd9e.jpg) +Figure 5: $\mathrm { \ddot { \Delta K } } + 1 \mathrm { \ ' }$ ” parametrization of the critic for semi-supervised learning. + +Table 2: CIFAR-10 error rates for varying number of labeled samples in the training set. Mean and standard deviation computed over 5 runs. We only use the $K + 1$ formulation of the critic. Note that we achieve strong SSL performance without any additional tricks, and even though the critic does not have any batch, layer or weight normalization. Baselines with \* use either additional models like PixelCNN, or do data augmentation (translations and flips), or use a much larger model, either of which gives an advantage over our plain simple training method. $\dagger$ is the result we achieved in our experimental setup under the same conditions but without $\mathrm { \ddot { \Delta K } } + 1 \mathrm { \ ' }$ critic (see Appendix D), since (Gulrajani et al., 2017) does not have SSL results. + +
Number of labeled examples Model100020004000 Misclassification rate8000
CatGAN (Springenberg, 2015) FM (Salimans et al., 2016) ALI (Dumoulin et al., 2017)21.83 ± 2.0119.61 ± 2.0919.58 18.63 ± 2.3217.72 ± 1.82
Tangents Reg (Kumar et al., 2017)19.98 ± 0.3 20.06 ± 0.519.09 ± 0.1517.99 ± 0.54 16.78 ± 0.617.05 ± 0.50
II-model (Laine & Aila, 2016) *16.55 ± 0.29
VAT (Miyato et al., 2017)14.87
Bad Gan (Dai et al., 2017) *14.41 ± 0.30
VAT+EntMin+Large (Miyato et al., 2017) *13.15
Sajadi (Sajjadi et al., 2016) *11.29
WGAN-GP (Gulrajani et al., 2017) t44.85 ± 0.2837.62 ± 0.5632.66 ± 0.4830.38± 0.22
Fisher,layer norm (Mroueh & Sercu,2017)19.74± 0.2117.87 ± 0.3816.13 ± 0.5314.81 ± 0.16
Fisher, no norm (Mroueh & Sercu, 2017)21.49 ± 0.1819.20 ± 0.4617.30 ± 0.3015.57 ± 0.33
Sobolev + Fisher, no norm (This Work)20.14 ± 0.2117.38 ± 0.1015.77 ± 0.1914.20± 0.08
+ +# 7 CONCLUSION + +We introduced the Sobolev IPM and showed that it amounts to a comparison between weighted (coordinate-wise) CDFs. We presented an ALM algorithm for training Sobolev GAN. The intrinsic conditioning implied by the Sobolev IPM explains the success of gradient regularization in Sobolev GAN and WGAN-GP on discrete sequence data, and particularly in text generation. We highlighted the important tradeoffs between the implicit conditioning introduced by the gradient regularizer in Sobolev IPM, and the explicit conditioning of Fisher IPM via recurrent critics and generators in conjunction with the curriculum conditioning. Both approaches succeed in text generation. We showed that Sobolev GAN achieves competitive semi-supervised learning results without the need of any normalization, thanks to the smoothness induced by the gradient regularizer. 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CoRR, abs/1609.05473, 2016. +Junbo Jake Zhao, Yoon Kim, Kelly Zhang, Alexander M. Rush, and Yann LeCun. Adversarially regularized autoencoders for generating discrete structures. CoRR, 2017. + +# A THEORY: APPROXIMATION AND TRANSPORT INTERPRETATION + +In this Section we present the theoretical properties of Sobolev IPM and how it relates to distributions transport theory and other known metrics between distributions, notably the Stein distance. + +# A.1 DISTRIBUTION TRANSPORT PERSPECTIVE ON SOBOLEV IPM + +In this Section, we characterize the optimal critic of the Sobolev IPM as a solution of a non linear PDE. The solution of the variational problem of the Sobolev IPM satisfies a non linear PDE that can be derived using standard tools from calculus of variations (Ekeland & Turnbull, 1983; Alaoui et al., 2016). + +Theorem 3 (PDE satisfied by the Sobolev Critic). The optimal critic of Sobolev IPM $f ^ { * }$ satisfies the following $P D E$ : + +$$ +\Delta f ^ { * } ( x ) + \left. \nabla _ { x } \log \mu ( x ) , \nabla _ { x } f ^ { * } ( x ) \right. + \frac { \mathbb { P } ( x ) - \mathbb { Q } ( x ) } { S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) \mu ( x ) } = 0 . +$$ + +Define the Stein Operator: $\begin{array} { r } { T ( \mu ) \vec { g } ( x ) = \frac { 1 } { 2 } \Big ( \langle \nabla _ { x } \log ( \mu ( x ) ) , \vec { g } ( x ) \rangle + d i v ( \vec { g } ( x ) ) \Big ) } \end{array}$ . Hence we have the following Transport Equation of $\mathbb { P }$ to $\mathbb { Q }$ : + +$$ +\mathbb { Q } ( x ) = \mathbb { P } ( x ) + 2 S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) \mu ( x ) T ( \mu ) \nabla _ { x } f ^ { * } ( x ) . +$$ + +Recall the definition of Stein Discrepancy : + +$$ +\mathbb { S } ( \mathbb { Q } , \mu ) = \operatorname* { s u p } _ { \vec { g } } | \mathbb { E } _ { x \sim \mathbb { Q } } [ T ( \mu ) \vec { g } ( x ) ] | , \vec { g } : \mathcal { X } \mathbb { R } ^ { d } . +$$ + +Theorem 4 (Sobolev and Stein Discrepanices). The following inequality holds true: + +$$ +\bigg | \mathbb { E } _ { \boldsymbol { x } \sim \mathbb { Q } } \left[ \frac { \mathbb { Q } ( \boldsymbol { x } ) - \mathbb { P } ( \boldsymbol { x } ) } { \mu ( \boldsymbol { x } ) } \right] \bigg | \leq 2 \qquad \underbrace { \mathbb { S } ( \mathbb { Q } , \mu ) } _ { \mathrm { S } ( \mathbb { Q } , \mu ) } \qquad \underbrace { S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) } _ { \mathrm { S } _ { \boldsymbol { \mu } } ( \mathbb { Q } ) } +$$ + +Consider for example $\mu = \mathbb { P }$ , and sequence $\mathbb { Q } _ { n }$ . If the Sobolev distance goes $S _ { \mathbb { P } } ( \mathbb { P } , \mathbb { Q } _ { n } ) 0$ , the ratio $\begin{array} { r } { r _ { n } ( x ) = \frac { \mathbb { Q } _ { n } ( x ) } { \mathbb { P } ( x ) } } \end{array}$ Qn(x)P(x) converges in expectation (w.r.t to Q) to 1. The speed of the convergence is given by the Stein Discrepancy $\mathbb { S } ( \mathbb { Q } _ { n } , \mathbb { P } )$ . + +Relation to Fokker-Planck Diffusion Equation and Particles dynamics. Note that PDE satisifed by the Sobolev critic given in Equation (13) can be equivalently written: + +$$ +\frac { \mathbb { P } - \mathbb { Q } } { S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) } = - \mathrm { d i v } ( \mu ( x ) \nabla _ { x } f ^ { * } ( x ) ) , +$$ + +written in this form, we draw a connection with the Fokker-Planck Equation for the evolution of a density function $q _ { t }$ that is the density of particles $X _ { t } \in \mathbb { R } ^ { d }$ evolving with a drift (a velocity field) $V ( x , i ) : \mathcal { X } \times [ 0 , \infty [ \mathbb { R } ^ { d }$ : + +$d X _ { t } = V ( X _ { t } , t ) d t$ , where the density of $X _ { 0 }$ is given by $q _ { 0 } ( x ) = \mathbb { Q } ( x )$ , + +Fokker-Planck Equation states that the evolution of the particles density $q _ { t }$ satisfies: + +$$ +\frac { d \boldsymbol { q } _ { t } } { d t } ( \boldsymbol { x } ) = - \mathrm { d i v } ( \boldsymbol { q } _ { t } ( \boldsymbol { x } ) V ( \boldsymbol { x } , t ) ) +$$ + +Comparing Equation (15) and Equation (16), we identify then the gradient of Sobolev critic as a drift. This suggests that one can define “Sobolev descent” as the evolution of particles along the gradient flow: + +$d X _ { t } = \nabla _ { x } f _ { t } ^ { * } ( X _ { t } ) d t$ , where the density of $X _ { 0 }$ is given by $q _ { 0 } ( x ) = \mathbb { Q } ( x )$ , where $f _ { t } ^ { * }$ is the Sobolev critic between $q _ { t }$ and $\mathbb { P }$ . One can show that the limit distribution of the particles is $\mathbb { P }$ . The analysis of “Sobolev descent” and its relation to Stein Descent (Liu & Wang, 2016; Liu, 2017) is beyond the scope of this paper and will be studied in a separate work. Hence we see that the gradient of the Sobolev critic defines a transportation plan to move particles whose distribution is $\mathbb { Q }$ to particles whose distribution is $\mathbb { P }$ (See Figure 2). This highlights the role of the gradient of the critic in the context of GAN training in term of transporting the distribution of the generator to the real distribution. + +# B PROOFS + +Proof of Theorem 2. Let $F _ { \mathbb { P } }$ and $F _ { \mathbb { Q } }$ , be the cumulative distribution functions of $\mathbb { P }$ and $\mathbb { Q }$ respectively. We have: + +$$ +\mathbb { P } ( x ) = \frac { \partial ^ { d } } { \partial x _ { 1 } \hdots \partial x _ { d } } F _ { \mathbb { P } } ( x ) , +$$ + +We note $\begin{array} { r } { D = \frac { \partial ^ { d } } { \partial x _ { 1 } \ldots \partial x _ { d } } } \end{array}$ and D−i , $\begin{array} { r } { D ^ { - i } = \frac { \partial ^ { d - 1 } } { \partial x _ { 1 } \dots \partial x _ { i - 1 } \partial x _ { i + 1 } \dots \partial x _ { d } } } \end{array}$ , for $i = 1 \ldots d$ . + +$D ^ { - i }$ computes the $d - 1$ partial derivative excluding the variable $i$ . + +In the following we assume that $F _ { \mathbb { P } }$ , and $F _ { \mathbb { Q } }$ and its $d$ derivatives exist and are continuous meaning that $F _ { \mathbb { P } }$ and $F _ { \mathbb { Q } } \in C ^ { d } ( \mathcal { X } )$ . The objective function in Equation (3) can be written as follows: + +$$ +\begin{array} { r l } & { \mathbb { E } _ { x \sim \mathbb { P } } f ( x ) - \mathbb { E } _ { x \sim \mathbb { Q } } f ( x ) = \displaystyle \int _ { \mathcal { X } } f ( x ) D \Big ( F _ { \mathbb { P } } ( x ) - F _ { \mathbb { Q } } ( x ) \Big ) d x } \\ & { \quad \quad \quad \quad \quad = \displaystyle \int _ { \mathcal { X } } f ( x ) \frac { \partial } { \partial x _ { i } } D ^ { - i } ( F _ { \mathbb { P } } ( x ) - F _ { \mathbb { Q } } ( x ) ) d x } \\ & { \quad \quad \quad \quad \quad \mathrm { ( f o r ~ a n y ~ } i , \mathrm { s i n c e ~ } F _ { \mathbb { P } } \mathrm { ~ a n d ~ } F _ { \mathbb { Q } } \in C ^ { d } ( \mathcal { X } ) ) } \\ & { \quad \quad \quad \quad = - \displaystyle \int _ { \mathcal { X } } \frac { \partial f } { \partial x _ { i } } D ^ { - i } ( F _ { \mathbb { P } } ( x ) - F _ { \mathbb { Q } } ( x ) ) d x } \end{array} +$$ + +Let $D ^ { - } = ( D ^ { - 1 } , \ldots , D ^ { - d } )$ it follows that: + +$$ +\begin{array} { r c l } { \mathbb { E } _ { x \sim \mathbb { P } } f ( x ) - \mathbb { E } _ { x \sim \mathbb { Q } } f ( x ) } & { = } & { \displaystyle \frac { 1 } { d } \sum _ { i = 1 } ^ { d } \int _ { \chi } \frac { \partial f } { \partial x _ { i } } D ^ { - i } ( F _ { \mathbb { Q } } ( x ) - F _ { \mathbb { P } } ( x ) ) d x } \\ & { = } & { \displaystyle \frac { 1 } { d } \int _ { \chi } \left. \nabla _ { x } f ( x ) , D ^ { - } ( F _ { \mathbb { Q } } ( x ) - F _ { \mathbb { P } } ( x ) ) \right. _ { \mathbb { R } ^ { d } } d x } \end{array} +$$ + +Let us define $\mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) ^ { \otimes d }$ the space of measurable functions from $\mathcal X ~ ~ \mathbb R ^ { d }$ . For $g , h \in$ $\mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) ^ { \otimes d }$ the dot product is defined as follows: + +$$ +\langle g , h \rangle _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) ^ { \otimes d } } = \int _ { \mathcal { X } } \langle g ( x ) , h ( x ) \rangle _ { \mathbb { R } ^ { d } } \mu ( x ) d x +$$ + +and the norm is given : + +$$ +\| g \| _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) ^ { \otimes d } } = \int _ { X } \| g \| _ { \mathbb { R } ^ { d } } ^ { 2 } \mu ( x ) d x . +$$ + +We can write the objective in Equation (18) in term of the dot product in $\mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) ^ { \otimes d }$ : + +$$ +\mathbb { E } _ { x \sim \mathbb { P } } f ( x ) - \mathbb { E } _ { x \sim \mathbb { Q } } f ( x ) = \frac { 1 } { d } \left. \nabla _ { x } f , \frac { D ^ { - } ( F _ { \mathbb { Q } } - F _ { \mathbb { P } } ) } { \mu } \right. _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) ^ { \otimes d } } . +$$ + +On the other hand the constraint in Equation (3) can be written in terms of the norm in $\mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) ^ { \otimes d }$ : + +$$ +\| f \| _ { W _ { 0 } ^ { 1 , 2 } ( \mathcal { X } , \mu ) } = \| \nabla _ { x } f \| _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) ^ { \otimes d } } +$$ + +Replacing the objective and constraint given in Equations (19) and (20) in Equation (3), we obtain: + +$$ +\begin{array} { r l } & { \mathfrak { H } ( \mathbb { P } , \mathfrak { G } ) = \frac { 1 } { d } _ { f , \vert \mathcal { C } _ { 2 } , \vert \mathcal { C } _ { 2 } , \alpha , \beta , 0 ; 0 } \le \mathfrak { C } _ { \ge } f , \frac { D ^ { - } ( F _ { \mathfrak { g } } - F _ { \mathfrak { g } } \vert \mathfrak { F } ) } { \mu } \Bigr \rangle _ { \mathcal { B } ( \mathfrak { x } , \mu ) \le \mathfrak { c } } } \\ & { \ = \frac { 1 } { d } _ { g \le \mathcal { G } _ { 2 } ( x , \beta ) \le \mathfrak { c } , \vert \mathfrak { H } \vert _ { \mathfrak { g } } \vert _ { \mathfrak { g } } \vert _ { \mathfrak { g } } \vert _ { \mathfrak { g } } \vert _ { \mathfrak { g } } \vert _ { \mathfrak { x } , \mu } ) \le \mathfrak { c } } \Big \langle g , \frac { D ^ { - } ( F _ { \mathfrak { g } } - F _ { \mathfrak { g } } ) } { \mu } \Big \rangle _ { \mathcal { B } _ { \ge } ( x , \mu ) \ge \mathfrak { c } } } \\ & { \ = \frac { 1 } { d } \bigg \Vert \frac { D ^ { - } ( F _ { \mathfrak { g } } - F _ { \mathfrak { g } } ) } { \mu } \bigg \Vert _ { \mathcal { B } _ { \ge } ( x , \mu ) \le \mathfrak { c } } } \\ & { \ \left( \mathtt { B y ~ d e n i n i o n ~ o f ~ } \Vert \cdot \Vert _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) \le \mathfrak { c } } , g ^ { - } = \frac { D ^ { - } F _ { \mathfrak { g } } ( \mathfrak { x } ) - D ^ { - } F _ { \mathfrak { g } } ( \mathfrak { x } ) } { \mu ( \mathfrak { x } ) } \frac { 1 } { \left. \frac { D ^ { - } ( F _ { \mathfrak { g } } - F _ { \mathfrak { g } } ) } { \mu } \right. _ { \mathcal { B } _ { \ge } ( \mathcal { X } , \mu ) \le \mathfrak { c } } } \right) } \\ & \ = \frac { 1 } { d } \sqrt { \int _ { x } \frac { \Vert D ^ { - } F _ { \mathfrak { g } } ( \mathfrak { x } ) - D ^ { - } F _ { \mathfrak { g } } ( \mathfrak { x } ) \Vert _ { \mathcal { A } ( \mathcal { X } , \mu ) } ^ { 2 } } { \mu ( \mathfrak { x } ) } d \mathfrak { x } . } \end{array} +$$ + +Hence we find also that the optimal critic $f ^ { * }$ satisfies: + +$$ +\nabla _ { x } f ^ { * } ( x ) = \frac { D ^ { - } F _ { \mathbb { Q } } ( x ) - D ^ { - } F _ { \mathbb { P } } ( x ) } { \mu ( x ) } \frac { 1 } { \Big \| \frac { D ^ { - } ( F _ { \mathbb { Q } } - F _ { \mathbb { P } } ) } { \mu } \Big \| _ { \mathcal { L } _ { 2 } ( \mathcal { X } , \mu ) ^ { \otimes d } } } . +$$ + +Proof of Lemma $\cdot$ . + +$$ +\begin{array} { r c l } { \mathbb { E } _ { x \sim \mathbb { P } } f ( x ) - \mathbb { E } _ { x \sim \mathbb { Q } } f ( x ) } & { = } & { \displaystyle \frac { 1 } { d } \int _ { \mathcal { X } } \left. \nabla _ { x } f ( x ) , D ^ { - } ( F _ { \mathbb { Q } } ( x ) - F _ { \mathbb { P } } ( x ) ) \right. _ { \mathbb { R } ^ { d } } d x } \\ & { = } & { \displaystyle S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) \int _ { \mathcal { X } } \left. \nabla _ { x } f ( x ) , \frac { D ^ { - } ( F _ { \mathbb { Q } } ( x ) - F _ { \mathbb { P } } ( x ) ) } { \mu ( x ) d S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) } \right. _ { \mathbb { R } ^ { d } } \mu ( x ) d x } \\ & { = } & { \displaystyle S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) \int _ { \mathcal { X } } \left. \nabla _ { x } f ( x ) , \nabla _ { x } f ^ { * } ( x ) \right. \mu ( x ) d x } \\ & { = } & { \displaystyle S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) \left. f , f ^ { * } \right. _ { W _ { 0 } ^ { 1 , 2 } } } \end{array} +$$ + +Hence we have: + +$$ +\operatorname* { s u p } _ { f \in \mathcal { M } , \| f \| _ { W _ { 0 } ^ { 1 , 2 } } \leq 1 } \mathbb { E } _ { x \sim \mathbb { P } } f ( x ) - \mathbb { E } _ { x \sim \mathbb { Q } } f ( x ) = S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) \operatorname* { s u p } _ { f \in \mathcal { M } , \| f \| _ { W _ { 0 } ^ { 1 , 2 } } \leq 1 } \langle f , f ^ { * } \rangle _ { W _ { 0 } ^ { 1 , 2 } } , +$$ + +It follows therefore that: + +$$ +\mathcal { S } _ { \mathcal { H } } ( \mathbb { P } , \mathbb { Q } ) = S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) \operatorname* { s u p } _ { f \in \mathcal { H } , \| f \| _ { W _ { 0 } ^ { 1 , 2 } } \leq 1 } \langle f , f ^ { * } \rangle _ { W _ { 0 } ^ { 1 , 2 } } +$$ + +We conclude that the Sobolev IPM can be approximated in arbitrary space as long as it has enough capacity to approximate the optimal critic. Interestingly the approximation error is measured now with the Sobolev semi-norm, while in Fisher it was measured with the Lebesgue norm. Approximations with Sobolev Semi-norms are stronger then Lebesgue norms as given by the Poincare inequality $( \vert \vert f \vert \vert _ { \mathcal { L } _ { 2 } } \le C \left. f \right. _ { W _ { 0 } ^ { 1 , 2 } } )$ , meaning if the error goes to zero in Sobolev sense it also goes to zero in the Lebesgue sense , but the converse is not true. + +Proof of Theorem 3. The proof follows similar arguments in the proofs of the analysis of Laplacian regularization in semi-supervised learning studied by (Alaoui et al., 2016). + +$$ +\begin{array} { r l } { S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) } & { { } = \operatorname* { s u p } _ { f \in W _ { 0 } ^ { 1 , 2 } } \Big \{ \mathbb { E } _ { x \sim \mathbb { P } } \left[ f ( x ) \right] - \mathbb { E } _ { x \sim \mathbb { Q } } \left[ f ( x ) \right] \Big \} } \\ { s . t . } & { { } \quad \mathbb { E } _ { x \sim \mu } \| \nabla f ( x ) \| _ { 2 } ^ { 2 } \leq 1 , } \end{array} +$$ + +Note that this problem is convex in $f$ (Ekeland & Turnbull, 1983). Writing the lagrangian for equation (21) we get : + +$$ +\begin{array} { l } { \displaystyle { L ( f , \lambda ) = \mathbb { E } _ { x \sim \mathbb { P } } \left[ f ( x ) \right] - \mathbb { E } _ { x \sim \mathbb { Q } } \left[ f ( x ) \right] + \frac { \lambda } { 2 } \Big ( 1 - \mathbb { E } _ { x \sim \mu } \| \nabla _ { x } f ( x ) \| _ { 2 } ^ { 2 } \Big ) } } \\ { \displaystyle { \quad = \int _ { \mathcal { X } } f ( x ) \left( \mathbb { P } ( x ) - \mathbb { Q } ( x ) \right) d x + \frac { \lambda } { 2 } \Big ( 1 - \int _ { \mathcal { X } } \| \nabla _ { x } f ( x ) \| _ { 2 } ^ { 2 } \mu ( x ) d x \Big ) } } \\ { \displaystyle { \quad = \int _ { \mathcal { X } } f ( x ) \mu _ { 1 } ( x ) d x + \frac { \lambda } { 2 } \Big ( 1 - \int _ { \mathcal { X } } \| \nabla _ { x } f ( x ) \| _ { 2 } ^ { 2 } \mu ( x ) d x \Big ) } } \end{array} +$$ + +We denote $\left( \mathbb { P } ( x ) - \mathbb { Q } ( x ) \right)$ as $\mu _ { 1 } ( x )$ .To get the optimal $f$ , we need to apply KKT conditions on the above equation. + +$$ +L ( f , \lambda ) = \int _ { \mathcal { X } } f ( x ) \mu _ { 1 } ( x ) d x + { \frac { \lambda } { 2 } } { \Big ( } 1 - \int _ { \mathcal { X } } \| \nabla _ { x } f ( x ) \| _ { 2 } ^ { 2 } \ \mu ( x ) d x { \Big ) } +$$ + +From the calculus of variations: + +$$ +\begin{array} { l } { \displaystyle \sum _ { k = 1 } ^ { \infty } ( f + \epsilon h , \lambda ) = \int _ { x } ( f + \epsilon h ) ( x ) \mu _ { 1 } ( x ) d x + \frac { \lambda } { 2 } \Bigl ( 1 - \int _ { x } \| \nabla _ { x } ( f + \epsilon h ) ( x ) \| _ { 2 } ^ { 2 } \mu ( x ) d x \Bigr ) } \\ { \displaystyle = \int _ { x } ( f ( x ) + \epsilon h ( x ) ) \mu _ { 1 } ( x ) d x + \frac { \lambda } { 2 } \Bigl ( 1 - \int _ { x } \left. \nabla _ { z } \left( f + \epsilon h \right) ( x ) , \nabla _ { z } \left( f + \epsilon h \right) ( x ) \right. \mu ( x ) } \\ { \displaystyle = \int _ { x } ( f ( x ) + \epsilon h ( x ) ) \mu _ { 1 } ( x ) d x } \\ { \displaystyle + \frac { \lambda } { 2 } \Bigl ( 1 - \int _ { x } \left. \| \nabla _ { z } f ( x ) \| _ { 2 } ^ { 2 } + 2 \epsilon \langle \nabla _ { x } f ( x ) , \nabla _ { x } h ( x ) \rangle + \mathcal { O } ( \epsilon ^ { 2 } ) \right. \mu ( x ) d x \Bigr ) } \\ { \displaystyle = L ( f , \lambda ) + \epsilon \int _ { x } h ( x ) \mu _ { 1 } ( x ) d x - \lambda \epsilon \int _ { x } \left. \nabla _ { x } f ( x ) , \nabla _ { x } h ( x ) \right. \mu ( x ) d x + \mathcal { O } ( \epsilon ^ { 2 } ) } \\ { \displaystyle = L ( f , \lambda ) + \epsilon \Bigl [ \int _ { x } h ( x ) \mu _ { 1 } ( x ) d x - \lambda \int _ { x } \left. \nabla _ { x } f ( x ) , \nabla _ { x } h ( x ) \right. \mu ( x ) d x \Bigr ] + \mathcal { O } ( \epsilon ^ { 2 } ) } \end{array} +$$ + +Now we apply integration by part and set $h$ to be zero at boundary as in (Alaoui et al., 2016). We get : + +$$ +\begin{array} { l } { \displaystyle \int _ { \mathcal X } \langle \nabla _ { x } f ( x ) , \nabla _ { x } h ( x ) \rangle \mu ( x ) d x = \int _ { \mathcal X } \langle \nabla _ { x } f ( x ) \mu ( x ) , \nabla _ { x } h ( x ) \rangle d x } \\ { = \displaystyle \oint h ( x ) \mu ( x ) \nabla _ { x } f ( x ) . n ( x ) d S ( x ) - \int _ { \mathcal X } d i v \big ( \mu ( x ) \nabla _ { x } f ( x ) \big ) h ( x ) d x } \\ { = - \displaystyle \int _ { \mathcal X } d i v \big ( \mu ( x ) \nabla _ { x } f ( x ) \big ) h ( x ) d x } \end{array} +$$ + +Hence, + +$$ +\begin{array} { l } { { \displaystyle { \cal L } \big ( f + \epsilon h , \lambda \big ) = { \cal L } \big ( f , \lambda \big ) + \epsilon \bigg [ \int _ { \mathcal { X } } \mu _ { 1 } ( x ) h ( x ) d x + \lambda \int _ { \mathcal { X } } d i v \big ( \mu ( x ) \nabla _ { x } f ( x ) \big ) h ( x ) d x \bigg ] + { \mathcal O } ( \epsilon ^ { 2 } ) } } \\ { { \displaystyle ~ = { \cal L } \big ( f , \lambda \big ) + \epsilon \int _ { \mathcal { X } } \bigg ( \mu _ { 1 } ( x ) + \lambda ~ d i v \big ( \mu ( x ) \nabla _ { x } f ( x ) \big ) \bigg ) h ( x ) d x + { \mathcal O } ( \epsilon ^ { 2 } ) } } \end{array} +$$ + +The functional derivative of $L ( f , \lambda )$ , at any test function $h$ vanishing on the boundary: + +$$ +\begin{array} { r c l } { \displaystyle \int _ { \mathcal { X } } \frac { \partial L ( f , \lambda ) } { \partial f } ( x ) h ( x ) d x } & { = } & { \displaystyle \operatorname* { l i m } _ { \epsilon \to 0 } \frac { L ( f + \epsilon h , \lambda ) - L ( f , \lambda ) } { \epsilon } } \\ & { = } & { \displaystyle \int _ { \mathcal { X } } \left( \mu _ { 1 } ( x ) + \lambda \ d i v \big ( \mu ( x ) \nabla _ { x } f ( x ) \big ) \right) h ( x ) d x } \end{array} +$$ + +Hence we have: + +$$ +\frac { \partial L ( f , \lambda ) } { \partial f } ( x ) = \mu _ { 1 } ( x ) + \lambda ~ d i v \big ( \mu ( x ) \nabla _ { x } f ( x ) \big ) +$$ + +For the optimal $f ^ { * } , \lambda ^ { * }$ first order optimality condition gives us: + +$$ +\mu _ { 1 } ( x ) + \lambda ^ { * } \ d i v \big ( \mu ( x ) \nabla _ { x } f ^ { * } ( x ) \big ) = 0 +$$ + +and + +$$ +\int _ { \chi } \left\| \nabla _ { x } f ^ { * } ( x ) \right\| ^ { 2 } \mu ( x ) d x = 1 +$$ + +Note that (See for example (Alaoui et al., 2016)) : + +$$ +\begin{array} { r } { d i v \big ( \mu ( x ) \nabla _ { x } f ^ { * } ( x ) \big ) = \mu ( x ) \Delta _ { 2 } f ^ { * } ( x ) + \langle \nabla _ { x } \mu ( x ) , \nabla _ { x } f ^ { * } ( x ) \rangle , } \end{array} +$$ + +since $d i v ( \nabla _ { x } f ^ { * } ( x ) ) = \Delta _ { 2 } f ^ { * } ( x )$ . Hence from equation (22) + +$$ +\begin{array} { r l } & { \quad \mu _ { 1 } ( x ) + \lambda ^ { * } \ : d i v \ : \left( \mu ( x ) \nabla _ { x } f ^ { * } ( x ) \right) = 0 } \\ & { \Rightarrow \mu _ { 1 } ( x ) + \lambda ^ { * } \ : \left( \mu ( x ) \Delta _ { 2 } f ^ { * } ( x ) + \left. \nabla _ { x } \mu ( x ) , \nabla _ { x } f ^ { * } ( x ) \right. \right) = 0 } \\ & { \Rightarrow \mu _ { 1 } ( x ) + \lambda ^ { * } \ : \mu ( x ) \Delta _ { 2 } f ^ { * } ( x ) + \lambda ^ { * } \langle \nabla _ { x } \mu ( x ) , \nabla _ { x } f ^ { * } ( x ) \rangle = 0 } \\ & { \Rightarrow \Delta _ { 2 } f ^ { * } ( x ) + \left. \frac { \nabla _ { x } \mu ( x ) } { \mu ( x ) } , \nabla _ { x } f ^ { * } ( x ) \right. + \frac { \mu _ { 1 } ( x ) } { \lambda ^ { * } \mu ( x ) } = 0 } \\ & { \Rightarrow \Delta _ { 2 } f ^ { * } ( x ) + \left. \nabla _ { x } \log \mu ( x ) , \nabla _ { x } f ^ { * } ( x ) \right. + \frac { \mathbb { P } ( x ) - \mathbb { Q } ( x ) } { \lambda ^ { * } \mu ( x ) } = 0 } \end{array} +$$ + +Hence $f ^ { * } , \lambda ^ { * }$ satisfies : + +$$ +\Delta _ { 2 } f ^ { * } ( x ) + \langle \nabla _ { x } \log \mu ( x ) , \nabla _ { x } f ^ { * } ( x ) \rangle + \frac { \mathbb { P } ( x ) - \mathbb { Q } ( x ) } { \lambda ^ { * } \mu ( x ) } = 0 +$$ + +and + +$$ +\int _ { \mathcal { X } } \left\| \nabla _ { x } f ^ { * } ( x ) \right\| ^ { 2 } \mu ( x ) d x = 1 . +$$ + +Let us verify that the optimal critic as found in the geometric definition (Theorem 2) of Sobolev IPM that satisfies: + +$$ +\nabla _ { i } f ^ { * } ( x ) = \frac { \partial f ^ { * } ( X ) } { \partial x _ { i } } = \frac { D ^ { - i } F _ { \mathbb { Q } } ( x ) - D ^ { - i } F _ { \mathbb { P } } ( x ) } { \lambda ^ { * } d ~ \mu ( x ) } ~ \forall ~ i \in [ d ] , +$$ + +satisfies indeed the PDE. + +From equation (27), we want to compute $\frac { \partial ^ { 2 } f ( \boldsymbol { x } ) } { \partial \boldsymbol { x } _ { i } ^ { 2 } }$ for all $i$ + +$$ +\begin{array} { r l } & { \frac { \partial ^ { 2 } f ( x ) } { \partial x _ { i } ^ { 2 } } = \frac { 1 } { \lambda ^ { * } d } \Bigg [ \frac { \mu ( x ) \left[ \frac { \partial } { \partial x _ { i } } ( D ^ { - i } F _ { \mathbb { Q } } ( x ) - D ^ { - i } F _ { \mathbb { P } } ( x ) ) \right] - \left[ D ^ { - i } F _ { \mathbb { Q } } ( x ) - D ^ { - i } F _ { \mathbb { P } } ( x ) \right] \nabla _ { i } \mu ( X ) } { \mu ^ { 2 } ( x ) } \Bigg ] } \\ & { \quad \quad \quad = \frac { 1 } { \lambda ^ { * } d } \Bigg [ \frac { \mu ( x ) \left[ \mathbb { Q } ( x ) - \mathbb { P } ( x ) \right] - \left[ D ^ { - i } F _ { \mathbb { Q } } ( x ) - D ^ { - i } F _ { \mathbb { P } } ( x ) \right] \nabla _ { i } \mu ( X ) } { \mu ^ { 2 } ( x ) } \Bigg ] } \\ & { \quad \quad \quad = \frac { \mathbb { Q } ( x ) - \mathbb { P } ( x ) } { \lambda ^ { * } d \mu ( x ) } - \frac { \nabla _ { i } \mu ( x ) } { \mu ( x ) } \nabla _ { i } f ^ { * } ( x ) } \end{array} +$$ + +Hence, + +$$ +{ \frac { \partial ^ { 2 } f ( x ) } { \partial x _ { i } ^ { 2 } } } + { \frac { \nabla _ { i } \mu ( x ) } { \mu ( x ) } } \nabla _ { i } f ( x ) + { \frac { \left( \mathbb { P } ( x ) - \mathbb { Q } ( x ) \right) } { \lambda ^ { * } d \ \mu ( x ) } } = 0 +$$ + +Adding equation (28) for all $i \in [ d ]$ , we get : + +$$ +\sum _ { i = 1 } ^ { d } \left( \frac { \partial ^ { 2 } f ( x ) } { \partial x _ { i } ^ { 2 } } + \frac { \nabla _ { i } \mu ( x ) } { \mu ( x ) } \nabla _ { i } f ( x ) + \frac { \left( \mathbb { P } ( x ) - \mathbb { Q } ( x ) \right) } { \lambda ^ { * } d ~ \mu ( x ) } \right) = 0 +$$ + +As a result, the solution $f ^ { * }$ of the partial differential equation given in equation (25) satisfies the following : + +$$ +\frac { \partial f ^ { * } ( x ) } { \partial x _ { i } } = \frac { D ^ { - i } F _ { \mathbb { Q } } ( x ) - D ^ { - i } F _ { \mathbb { P } } ( x ) } { \lambda ^ { * } d ~ \mu ( x ) } ~ \forall i \in [ d ] +$$ + +Using the constraint in (26) we can get the value of $\lambda ^ { * }$ : + +$$ +\begin{array} { r l } & { \quad \displaystyle \int \| \nabla f ^ { * } ( \boldsymbol { x } ) \| ^ { 2 } \mu ( \boldsymbol { x } ) d \boldsymbol { x } = 1 } \\ & { \Rightarrow \displaystyle \int \sum _ { i = 1 } ^ { d } \left( \frac { \partial f ^ { * } ( \boldsymbol { x } ) } { \partial x _ { i } } \right) ^ { 2 } \mu ( \boldsymbol { x } ) d \boldsymbol { x } = 1 } \\ & { \Rightarrow \lambda ^ { * } = \displaystyle \frac { 1 } { d } \sqrt { \displaystyle \sum _ { i = 1 } ^ { d } \int \frac { \left( D ^ { - i } F _ { \mathbb { Q } } ( \boldsymbol { x } ) - D ^ { - i } F _ { \mathbb { P } } ( \boldsymbol { x } ) \right) ^ { 2 } } { \mu ( \boldsymbol { x } ) } d \boldsymbol { x } } = S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) . } \end{array} +$$ + +Proof of Theorem 4. Define the Stein operator (Oates et al., 2017): + +$$ +\begin{array} { l l l } { { T ( \mu ) [ \nabla _ { x } f ( x ) ] } } & { { = } } & { { \displaystyle \frac { 1 } { 2 } \left. \nabla _ { x } f ( x ) , \nabla _ { x } \log \mu ( x ) \right. + \displaystyle \frac { 1 } { 2 } \left. \nabla _ { x } , \nabla _ { x } f ( x ) \right. } } \\ { { } } & { { = } } & { { \displaystyle \frac { 1 } { 2 } \left. \nabla _ { x } f ( x ) , \nabla _ { x } \log \mu ( x ) \right. + \displaystyle \frac { 1 } { 2 } \Delta _ { 2 } f ( x ) . } } \end{array} +$$ + +This operator was later used in defining the Stein discrepancy (Gorham & Mackey, 2015; Liu et al., 2016; Chwialkowski et al., 2016; Liu, 2017). + +Recall that Barbour generator theory provides us a way of constructing such operators that produce mean zero function under $\mu$ . It is easy to verify that: + +$$ +\mathbb { E } _ { x \sim \mu } T ( \mu ) \nabla _ { x } f ( x ) = 0 . +$$ + +Recall that this operator arises from the overdamped Langevin diffusion, defined by the stochastic differential equation: + +$$ +d x _ { t } = \frac { 1 } { 2 } \nabla _ { x } \log \mu ( x _ { t } ) + d W _ { t } +$$ + +where $( W _ { t } ) _ { t \geq 0 }$ is a Wiener process. This is related to plug and play networks for generating samples if the distribution is known, using the stochastic differential equation. + +From Theorem 3, it is easy to see that the PDE the Sobolev Critic $( f ^ { * } , \lambda ^ { * } = { \cal S } _ { \mu } ( \mathbb { P } , \mathbb { Q } ) )$ can be written in term of Stein Operator as follows: + +$$ +T ( \mu ) [ \nabla _ { x } f ^ { * } ] ( x ) = \frac { 1 } { 2 \lambda ^ { * } } \frac { \mathbb { Q } ( x ) - \mathbb { P } ( x ) } { \mu ( x ) } +$$ + +Taking absolute values and the expectation with respect to $\mathbb { Q }$ : + +$$ +\left| \mathbb { E } _ { x \sim \mathbb { Q } } \left[ T ( \mu ) \nabla _ { x } f ^ { * } ( x ) \right] \right| = { \frac { 1 } { 2 S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) } } \left| \mathbb { E } _ { x \sim \mathbb { Q } } \left[ { \frac { \mathbb { Q } ( x ) - \mathbb { P } ( x ) } { \mu ( x ) } } \right] \right| +$$ + +Recall that the definition of Stein Discrepancy : + +$$ +{ \mathbb S } ( \mathbb Q , \mu ) = \operatorname* { s u p } _ { \vec { g } \in \mathcal L _ { 2 } ( \mathcal X , \mu ) ^ { \otimes d } } \left| \mathbb E _ { \boldsymbol { x } \sim \mathbb Q } \left[ T ( \mu ) \vec { g } ( \boldsymbol { x } ) \right] \right| +$$ + +It follows that Sobolev IPM critic satisfies: + +$$ +| \mathbb { E } _ { x \sim \mathbb { Q } } \left[ T ( \mu ) \nabla _ { x } f ^ { * } ( x ) \right] | \leq \mathbb { S } ( \mathbb { Q } , \mu ) , +$$ + +Hence we have the following inequality: + +$$ +\frac { 1 } { 2 S _ { \mu } ( \mathbb { P } , \mathbb { Q } ) } \left| \mathbb { E } _ { \boldsymbol { x } \sim \mathbb { Q } } \left[ \frac { \mathbb { Q } ( \boldsymbol { x } ) - \mathbb { P } ( \boldsymbol { x } ) } { \mu ( \boldsymbol { x } ) } \right] \right| \leq \mathbb { S } ( \mathbb { Q } , \mu ) +$$ + +This is equivalent to: + +$$ +\left| \mathbb { E } _ { x \sim \mathbb { Q } } \left[ { \frac { \mathbb { Q } ( x ) - \mathbb { P } ( x ) } { \mu ( x ) } } \right] \right| \leq 2 \qquad { \underset { \Delta \in { \mathbb { Q } } , \mu } { \underbrace { \mathbb { S } ( \mathbb { Q } , \mu ) } } } \qquad { \underset { \Delta \in { \mathbb { P } } } { \underbrace { { \mathcal { S } } _ { \mu } ( \mathbb { P } , \mathbb { Q } ) } } } +$$ + +Similarly we obtain: + +$$ +\left| \mathbb { E } _ { x \sim \mathbb { P } } \left[ { \frac { \mathbb { Q } ( x ) - \mathbb { P } ( x ) } { \mu ( x ) } } \right] \right| \leq 2 \qquad \underbrace { { \mathbb { S } } ( \mathbb { P } , \mu ) } _ { \in { \mathcal { S } } } \qquad \underbrace { { \mathcal { S } } _ { \mu } ( \mathbb { P } , \mathbb { Q } ) } _ { \in { \mathcal { S } } } +$$ + +For instance consider $\mu = \mathbb { P }$ , we have therefore: + +$$ +{ \frac { 1 } { 2 } } \left| \mathbb { E } _ { x \sim \mathbb { Q } } \left[ { \frac { \mathbb { Q } ( x ) } { \mathbb { P } ( x ) } } \right] - 1 \right| \leq \mathbb { S } ( \mathbb { Q } , \mathbb { P } ) S _ { \mathbb { P } } ( \mathbb { P } , \mathbb { Q } ) . +$$ + +Note that the left hand side of the inequality is not the total variation distance. + +Hence for a sequence $\mathbb { Q } _ { n }$ if the Sobolev distance goes $S _ { \mathbb { P } } ( \mathbb { P } , \mathbb { Q } _ { n } ) 0$ , the ratio $\begin{array} { r } { r _ { n } ( x ) = \frac { \mathbb Q _ { n } ( x ) } { \mathbb P ( x ) } } \end{array}$ converges in expectation (w.r.t to $\mathbb { Q }$ ) to 1. The speed of the convergence is given by the Stein Discrepancy $\mathbb { S } ( \mathbb { Q } _ { n } , \mathbb { P } )$ . + +One important observation here is that convergence of PDF ratio is weaker than the conditional CDF as given by the Sobolev distance and of the good fitness of score function as given by Stein discrepancy. + +# C TEXT EXPERIMENTS: ADDITIONAL PLOTS + +# Comparison of annealed versus non annealed smoothing of $\mathbb { P } _ { r }$ in Sobolev GAN. + +![](images/9bc07e8ad8977d3d3e0990973f1d012506b6beae1acd22e818815329398759c9.jpg) +Figure 6: Comparison of annealed versus non annealed smoothing of $\mathbb { P } _ { r }$ in Sobolev GAN. We see that annealed smoothing outperforms the non annealed smoothing experiments. + +Sobolev GAN versus WGAN-GP with RNN. We fix the generator architecture to Resnets. The experiments of using RNN (GRU) as the critic architecture for WGAN-GP and Sobolev is shown in Figure 7 where we used $\mu = \mu _ { G P }$ for both cases. We only apply gradient clipping to stabilize the performance without other tricks. We can observe that using RNN degrades the performance. We think that this is due to an optimization issue and a difficulty in training RNN under the GAN objective without any pre-training or conditioning. + +![](images/b33296dfeec538aff71afa249d3c78cee91fb6d6002cb89df8dd5f0cad50bf6f.jpg) +Figure 7: Result of WGAN-GP and Sobolev with RNNs. + +![](images/22b283fdcd36a259ef1de1da1ae8a6d80587aa0ad141b9ca955c8082e7f17492.jpg) +Figure 8: Text samples from various GANs considered in this paper. + +# D SSL: HYPERPARAMETERS AND ARCHITECTURE + +For our SSL experiments on CIFAR-10, we use Adam with learning rate $\eta = 2 \mathrm { { e } - 4 }$ , $\beta _ { 1 } ~ = ~ 0 . 5$ and $\beta _ { 2 } = 0 . 9 9 9$ , both for critic $f$ (without BN) and Generator (with BN). We selected $\lambda _ { C E } = 1 . 5$ from [0.8, 1.5, 3.0, 5.0]. We train all models for 350 epochs. We used some L2 weight decay: 1e−6 on $\omega , S$ (i.e. all layers except last) and 1e 3 weight decay on the last layer $v$ . For formulation 1 (Fisher only) we have $\rho _ { F } = 1 \mathrm { e } { - 7 }$ , modified critic learning rate $\eta _ { D } = 1 \mathrm { e } { - 4 }$ , critic iters $n _ { c } = 2$ . For formulation 2 (Sobolev $^ +$ Fisher) we have $\rho _ { F } = 5 \mathrm { e } - 8$ , $\rho _ { S } = 2 \mathrm { e } - 8$ , critic iters $n _ { c } = 1$ . For the WGAN-GP (Gulrajani et al., 2017) baseline SSL experiment we followed the original paper with critic iters $n _ { c } = 5$ , $\eta _ { G } = \eta _ { D } = 1 \mathrm { e } { - 4 }$ , Adam $\beta _ { 2 } { = } 0 . 9$ and GP weight $\lambda _ { G P } = 1 0 . 0$ . Architectures are as below. We determined $\lambda _ { C E } = 0 . 3$ to be optimal from $[ 0 . 0 3 , 0 . 1 , 0 . 3 , 1 . 0 , 3 . 0 ]$ . As mentioned in Section 6.2, the $\mathrm { K } { + } 1$ critic formulation is not able to fit the training set with the GP constraint, so we fall back to the plain critic formulation where the critic $\langle v , \Phi _ { \omega } ( \bar { x } ) \rangle$ does not interact with the classifier $\langle S , \Phi _ { \omega } ( x ) \rangle$ . + +# Architecture: + +(6): ConvTranspose2d(128, 64, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (7): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (8): ReLU (inplace) (9): Conv2d(64, 64, kernel_siz $\div =$ (3, 3), stride=(1, 1), padding=(1, 1), bias=False) (10): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (11): ReLU (inplace) (12): Conv2d(64, 64, kernel_size $: =$ (3, 3), strid $\div =$ (1, 1), padding $=$ (1, 1), bias=False) (13): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (14): ReLU (inplace) (15): ConvTranspose2d(64, 3, kernel_size $: =$ (4, 4), strid $: =$ (2, 2), padding $^ { \ast = }$ (1, 1), bias=False) (16): Tanh () ) ) D ( (main): Sequential ( (0): Dropout $\mathrm { ~ \bf ~ p ~ } = \mathrm { ~ \bf ~ 0 ~ } . 2 \mathrm { ~ \bf ~ }$ ) (1): Conv2d(3, 96, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1)) (2): LeakyReLU (0.2, inplace) (3): Conv2d(96, 96, kernel_siz $\div =$ (3, 3), stride $\div =$ (1, 1), padding=(1, 1), bias=False) (5): LeakyReLU (0.2, inplace) (6): Conv2d(96, 96, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False) (8): LeakyReLU (0.2, inplace) (9): Dropout $( \mathrm { p } ~ = ~ 0 . 5 )$ (10): Conv2d(96, 192, kernel_size $=$ (3, 3), stride $=$ (1, 1), padding $=$ (1, 1), bias=False) (12): LeakyReLU (0.2, inplace) (13): Conv2d(192, 192, kernel_size $=$ (3, 3), stride $: =$ (1, 1), padding $=$ (1, 1), bias $\scriptstyle = \mathbf { E }$ alse) (15): LeakyReLU (0.2, inplace) (16): Conv2d(192, 192, kernel_size $: =$ (3, 3), stride $: =$ (2, 2), padding $=$ (1, 1), bias=False) (18): LeakyReLU (0.2, inplace) (19): Dropout ( $\mathrm { ~ \cdot ~ p ~ } = \mathrm { ~ 0 ~ . ~ 5 ~ }$ (20): Conv2d(192, 384, kernel_size $: =$ (3, 3), stride $=$ (1, 1), bias=False) (22): LeakyReLU (0.2, inplace) (23): Dropout (p = 0.5) (24): Conv2d(384, 384, kernel_size $=$ (3, 3), stride $=$ (1, 1), bias=False) (26): LeakyReLU (0.2, inplace) (27): Dropout $( \mathrm { p } ~ = ~ 0 . 5 )$ (28): Conv2d(384, 384, kernel_size=(1, 1), stride=(1, 1), bias=False) (30): LeakyReLU (0.2, inplace) (31): Dropout $( \mathrm { p } ~ = ~ 0 . 5 )$ ) ) (V): Linear (6144 -> 1) (S): Linear (6144 -> 10) ) \ No newline at end of file diff --git a/parse/train/SJA7xfb0b/SJA7xfb0b_content_list.json b/parse/train/SJA7xfb0b/SJA7xfb0b_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..3baebf0183bfd616ebbb0c86c74d551c8b2f7cb9 --- /dev/null +++ b/parse/train/SJA7xfb0b/SJA7xfb0b_content_list.json @@ -0,0 +1,3871 @@ +[ + { + "type": "text", + "text": "SOBOLEV GAN ", + "text_level": 1, + "bbox": [ + 176, + 98, + 362, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Youssef Mroueh†, Chun-Liang $\\mathbf { L i } ^ { \\circ , \\star }$ , Tom Sercu†,?, Anant Raj♦,? & Yu Cheng† ", + "bbox": [ + 184, + 143, + 732, + 160 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "$\\dagger$ IBM Research AI \n$\\circ$ Carnegie Mellon University \n$\\diamondsuit$ Max Planck Institute for Intelligent Systems \n$\\star$ denotes Equal Contribution \nmroueh,chengyu @us.ibm.com, chunlial@cs.cmu.edu, \ntom.sercu1@ibm.com,anant.raj@tuebingen.mpg.de ", + "bbox": [ + 184, + 160, + 658, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 279, + 544, + 294 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We propose a new Integral Probability Metric (IPM) between distributions: the Sobolev IPM. The Sobolev IPM compares the mean discrepancy of two distributions for functions (critic) restricted to a Sobolev ball defined with respect to a dominant measure $\\mu$ . We show that the Sobolev IPM compares two distributions in high dimensions based on weighted conditional Cumulative Distribution Functions (CDF) of each coordinate on a leave one out basis. The Dominant measure $\\mu$ plays a crucial role as it defines the support on which conditional CDFs are compared. Sobolev IPM can be seen as an extension of the one dimensional VonMises Cramer statistics to high dimensional distributions. We show how Sobolev ´ IPM can be used to train Generative Adversarial Networks (GANs). We then exploit the intrinsic conditioning implied by Sobolev IPM in text generation. Finally we show that a variant of Sobolev GAN achieves competitive results in semisupervised learning on CIFAR-10, thanks to the smoothness enforced on the critic by Sobolev GAN which relates to Laplacian regularization. ", + "bbox": [ + 233, + 313, + 764, + 506 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 537, + 336, + 553 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In order to learn Generative Adversarial Networks (Goodfellow et al., 2014), it is now well established that the generator should mimic the distribution of real data, in the sense of a certain discrepancy measure. Discrepancies between distributions that measure the goodness of the fit of the neural generator to the real data distribution has been the subject of many recent studies (Arjovsky & Bottou, 2017; Nowozin et al., 2016; Kaae Sønderby et al., 2017; Mao et al., 2017; Arjovsky et al., 2017; Gulrajani et al., 2017; Mroueh et al., 2017; Mroueh & Sercu, 2017; Li et al., 2017), most of which focus on training stability. ", + "bbox": [ + 174, + 570, + 825, + 667 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In terms of data modalities, most success was booked in plausible natural image generation after the introduction of Deep Convolutional Generative Adversarial Networks (DCGAN) (Radford et al., 2015). This success is not only due to advances in training generative adversarial networks in terms of loss functions (Arjovsky et al., 2017) and stable algorithms, but also to the representation power of convolutional neural networks in modeling images and in finding sufficient statistics that capture the continuous density function of natural images. When moving to neural generators of discrete sequences generative adversarial networks theory and practice are still not very well understood. Maximum likelihood pre-training or augmentation, in conjunction with the use of reinforcement learning techniques were proposed in many recent works for training GAN for discrete sequences generation (Yu et al., 2016; Che et al., 2017; Hjelm et al., 2017; Rajeswar et al., 2017). Other methods included using the Gumbel Softmax trick (Kusner & Hernandez-Lobato ´ , 2016) and the use of auto-encoders to generate adversarially discrete sequences from a continuous space (Zhao et al., 2017). End to end training of GANs for discrete sequence generation is still an open problem (Press et al., 2017). Empirical successes of end to end training have been reported within the framework of WGAN-GP (Gulrajani et al., 2017), using a proxy for the Wasserstein distance via a pointwise gradient penalty on the critic. Inspired by this success, we propose in this paper a new Integral Probability Metric (IPM) between distributions that we coin Sobolev IPM. Intuitively an IPM (Muller ¨ , 1997) between two probability distributions looks for a witness function $f$ , called critic, that maximally discriminates between samples coming from the two distributions: ", + "bbox": [ + 174, + 674, + 825, + 882 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 160 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/a98ebe639cd3d8ea740d650c0644ca19dff3a339cd72867878bbf88dd81cb39e.jpg", + "text": "$$\n\\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 166, + 598, + 194 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Traditionally, the function $f$ is defined over a function class $\\mathcal { F }$ that is independent to the distributions at hand (Sriperumbudur et al., 2012). The Wasserstein-1 distance corresponds for instance to an IPM where the witness functions are defined over the space of Lipschitz functions; The MMD distance (Gretton et al., 2012) corresponds to witness functions defined over a ball in a Reproducing Kernel Hilbert Space (RKHS). ", + "bbox": [ + 173, + 202, + 825, + 272 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We will revisit in this paper Fisher IPM defined in (Mroueh & Sercu, 2017), which extends the IPM definition to function classes defined with norms that depend on the distributions. Fisher IPM can be seen as restricting the critic to a Lebsegue ball defined with respect to a dominant measure $\\mu$ . The Lebsegue norm is defined as follows: ", + "bbox": [ + 174, + 279, + 825, + 335 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/966178bc96ad159d2cef7de229090b6fdbbd04e35a7b168d54122f65f3b2fe91.jpg", + "text": "$$\n\\int _ { \\mathcal { X } } f ^ { 2 } ( x ) \\mu ( x ) d x .\n$$", + "text_format": "latex", + "bbox": [ + 437, + 343, + 558, + 376 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\mu$ is a dominant measure of $\\mathbb { P }$ and $\\mathbb { Q }$ ", + "bbox": [ + 176, + 382, + 459, + 397 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper we extend the IPM framework to critics bounded in the Sobolev norm: ", + "bbox": [ + 173, + 404, + 722, + 419 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/9acae480abe15bd92f2858f3db977054cb0898e2e6eca672ab7245aeab1e28a2.jpg", + "text": "$$\n\\int _ { \\mathcal { X } } \\| \\nabla _ { \\boldsymbol { x } } f ( \\boldsymbol { x } ) \\| _ { 2 } ^ { 2 } \\mu ( \\boldsymbol { x } ) d x ,\n$$", + "text_format": "latex", + "bbox": [ + 416, + 425, + 578, + 459 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In contrast to Fisher IPM, which compares joint probability density functions of all coordinates between two distributions, we will show that Sobolev IPM compares weighted (coordinate-wise) conditional Cumulative Distribution Functions for all coordinates on a leave on out basis. Matching conditional dependencies between coordinates is crucial for sequence modeling. ", + "bbox": [ + 173, + 465, + 825, + 522 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our analysis and empirical verification show that the modeling of the conditional dependencies can be built in to the metric used to learn GANs as in Sobolev IPM. For instance, this gives an advantage to Sobolev IPM in comparing sequences over Fisher IPM. Nevertheless, in sequence modeling when we parametrize the critic and the generator with a neural network, we find an interesting tradeoff between the metric used and the architectures used to parametrize the critic and the generator as well as the conditioning used in the generator. The burden of modeling the conditional long term dependencies can be handled by the IPM loss function as in Sobolev IPM (more accurately the choice of the data dependent function class of the critic) or by a simpler metric such as Fisher IPM together with a powerful architecture for the critic that models conditional long term dependencies such as LSTM or GRUs in conjunction with a curriculum conditioning of the generator as done in (Press et al., 2017). Highlighting those interesting tradeoffs between metrics, data dependent functions classes for the critic (Fisher or Sobolev) and architectures is crucial to advance sequence modeling and more broadly structured data generation using GANs. ", + "bbox": [ + 173, + 529, + 825, + 709 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "On the other hand, Sobolev norms have been widely used in manifold regularization in the so called Laplacian framework for semi-supervised learning (SSL) (Belkin et al., 2006). GANs have shown success in semi-supervised learning (Salimans et al., 2016; Dumoulin et al., 2017; Dai et al., 2017; Kumar et al., 2017). Nevertheless, many normalizations and additional tricks were needed. We show in this paper that a variant of Sobolev GAN achieves strong results in semi-supervised learning on CIFAR-10, without the need of any activation normalization in the critic. ", + "bbox": [ + 174, + 717, + 825, + 800 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The main contributions of this paper can be summarized as follows: ", + "bbox": [ + 174, + 808, + 617, + 821 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We overview in Section 2 different metrics between distribution used in the GAN literature. We then generalize Fisher IPM in Section 3 with a general dominant measure $\\mu$ and show how it compares distributions based on their PDFs. 2. We introduce Sobolev IPM in Section 4 by restricting the critic of an IPM to a Sobolev ball defined with respect to a dominant measure $\\mu$ . We then show that Sobolev IPM defines a discrepancy between weighted (coordinate-wise) conditional CDFs of distributions. ", + "bbox": [ + 212, + 833, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3. The intrinsic conditioning and the CDF matching make Sobolev IPM suitable for discrete sequence matching and explain the success of the gradient pernalty in WGAN-GP and Sobolev GAN in discrete sequence generation. \n4. We give in Section 5 an ALM (Augmented Lagrangian Multiplier) algorithm for training Sobolev GAN. Similar to Fisher GAN, this algorithm is stable and does not compromise the capacity of the critic. \n5. We show in Appendix A that the critic of Sobolev IPM satisfies an elliptic Partial Differential Equation (PDE). We relate this diffusion to the Fokker-Planck equation and show the behavior of the gradient of the optimal Sobolev critic as a transportation plan between distributions. \n6. We empirically study Sobolev GAN in character level text generation (Section 6.1). We validate that the conditioning implied by Sobolev GAN is crucial for the success and stability of GAN in text generation. As a take home message from this study, we see that text generation succeeds either by implicit conditioning i.e using Sobolev GAN (or WGANGP) together with convolutional critics and generators, or by explicit conditioning i.e using Fisher IPM together with recurrent critic and generator and curriculum learning. \n7. We finally show in Section 6.2 that a variant of Sobolev GAN achieves competitive semisupervised learning results on CIFAR-10, thanks to the smoothness implied by the Sobolev regularizer. ", + "bbox": [ + 210, + 103, + 825, + 383 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 OVERVIEW OF METRICS BETWEEN DISTRIBUTIONS ", + "text_level": 1, + "bbox": [ + 173, + 401, + 632, + 417 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this Section, we review different representations of probability distributions and metrics for comparing distributions that use those representations. Those metrics are at the core of training GAN. In what follows, we consider probability measures with a positive weakly differentiable probability density functions (PDF). Let $P$ and $Q$ be two probability measures with PDFs $\\mathbb { P } ( x )$ and $\\mathbb { Q } ( x )$ defined on $\\mathcal { X } \\subset \\mathbb { R } ^ { d }$ . Let $F _ { \\mathbb { P } }$ and $F _ { \\mathbb { Q } }$ be the Cumulative Distribution Functions (CDF) of $\\mathbb { P }$ and $\\mathbb { Q }$ respectively. For $\\boldsymbol { x } = ( x _ { 1 } , \\dots , x _ { d } )$ , we have: ", + "bbox": [ + 173, + 431, + 825, + 516 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/dfc3710e8726c51e7ac97beed732772f72c0624bdfa5f7064d766a87677813a4.jpg", + "text": "$$\nF _ { \\mathbb { P } } ( x ) = \\int _ { - \\infty } ^ { x _ { 1 } } \\ldots \\cdot \\cdot \\int _ { - \\infty } ^ { x _ { d } } \\mathbb { P } ( u _ { 1 } , \\ldots \\cdot u _ { d } ) d u _ { 1 } \\ldots \\cdot \\cdot d u _ { d } .\n$$", + "text_format": "latex", + "bbox": [ + 338, + 516, + 660, + 551 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The score function of a density function is defined as: $s _ { \\mathbb { P } } ( x ) = \\nabla _ { x } \\log ( \\mathbb { P } ( x ) ) \\in \\mathbb { R } ^ { d }$ . ", + "bbox": [ + 174, + 559, + 723, + 575 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this work, we are interested in metrics between distributions that have a variational form and can be written as a suprema of mean discrepancies of functions defined on a specific function class. This type of metrics include $\\varphi$ -divergences as well as Integral Probability Metrics (Sriperumbudur et al., 2009) and have the following form: ", + "bbox": [ + 174, + 582, + 825, + 637 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/90c861d763dff1c6c7b258b248ec40ee16517370ae2a39a533de96170653062c.jpg", + "text": "$$\nd _ { \\mathcal { F } } ( \\mathbb { P } , \\mathbb { Q } ) = \\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\left| \\Delta ( f ; \\mathbb { P } , \\mathbb { Q } ) \\right| ,\n$$", + "text_format": "latex", + "bbox": [ + 392, + 638, + 602, + 665 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathcal { F }$ is a function class defined on $\\mathcal { X }$ and $\\Delta$ is a mean discrepancy, $\\Delta : \\mathcal { F } \\mathbb { R }$ . The variational form given above leads in certain cases to closed form expressions in terms of the PDFs $\\mathbb { P } , \\mathbb { Q }$ or in terms of the CDFs $F _ { \\mathbb { P } } , F _ { \\mathbb { Q } }$ or the score functions $s _ { \\mathbb { P } } , s _ { \\mathbb { Q } }$ . ", + "bbox": [ + 176, + 666, + 825, + 709 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In Table 1, we give a comparison of different discrepancies $\\Delta$ and function spaces $\\mathcal { F }$ used in the literature for GAN training together with our proposed Sobolev IPM. We see from Table 1 that Sobolev IPM, compared to Wasserstein Distance, imposes a tractable smoothness constraint on the critic on points sampled from a distribution $\\mu$ , rather then imposing a Lipschitz constraint on all points in the space $\\mathcal { X }$ . We also see that Sobolev IPM is the natural generalization of the Cramer´ Von-Mises Distance from one dimension to high dimensions. We note that the Energy Distance, a form of Maximum Mean Discrepancy for a special kernel, was used in (Bellemare et al., 2017b) as a generalization of the Cramer distance in GAN training but still needed a gradient penalty in ´ its algorithmic counterpart leading to a mis-specified distance between distributions. Finally it is worth noting that when comparing Fisher IPM and Sobolev IPM we see that while Fisher IPM compares joint PDF of the distributions, Sobolev IPM compares weighted (coordinate-wise) conditional CDFs. As we will see later, this conditioning nature of the metric makes Sobolev IPM suitable for comparing sequences. Note that the Stein metric (Liu et al., 2016; Liu, 2017) uses the score function to match distributions. We will show later how Sobolev IPM relates to the Stein discrepancy (Appendix A). ", + "bbox": [ + 173, + 715, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/a902eb13ef0e600d618ef6520feed6e52f0d53869e489d1691bf1605a481948a.jpg", + "table_caption": [ + "Table 1: Comparison of different metrics between distributions used for GAN training. References are for papers using those metrics for GAN training. " + ], + "table_footnote": [], + "table_body": "
△(f;P,Q)FFunction classdg(P,Q)Closed Form
-Divergence(Goodfellow et al., 2014)(Nowozin et al., 2016)Ex~Pf(x)-Ex~Q*(f(x))*Fenchel Conjugate{f:x→R,f∈domp*}E~[(]
Wasserstein -1(Arjovsky et al., 2017)(Gulrajani et al., 2017)Ex~Pf(x)-Ex~qf(x){f : x→R,fli ≤1}infπ∈(P,Q) Sx |lx-ylli dπ(x,y)Sinkhorn Divergence(Genevay et al., 2017)
MMD(Li et al., 2017)(Li et al., 2015)(Dziugaite et al., 2015)Ex~Pf(x)-Ex~qf(x){f:x→R,/fl/x ≤1}Ex~Pkx -Ex~Qr llyex
SteinDiscrepancy(Wang & Liu, 2016)Ex~Q[T(P)f(x)]T(P)=(Vxlog(P(x))T+Vx.{f:x→Rdf smooth with zeroboundary conditionNA in generalhas a closed formin RKHS
Cramerfor d =1(Bellemare et al., 2017a)Ex~Pf(x)-Ex~Qf(x){f:x→R,Ea~P(）²≤1,f smooth with zeroboundary conditionEx~P（Fp(2）-F0（））²P(x）xER
μ-FisherIPM(Mroueh & Sercu,2017)Ex~Pf(x)-Ex~Qf(x){f:X→R,f∈L(x,μ),Ex~μf²(x)≤1}Ex~u（P()-Q(x））²2μ(x）
μ-SobolevIPM(This work)Ex~Pf(x)-Ex~qf(x){f:X→R,f eW²(x,μ),Ex~μ|/Vxf(x)²≤1,with zero boundary condition }1VE~u∑i-(（中i(P）-Φi(Q）μ（x）where 𝜙(P) =Px-i(x-i)FP[xx-i-ij(mi)x−=(x1,..xi-1,Ti+1,..d)
", + "bbox": [ + 173, + 101, + 825, + 554 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 GENERALIZING FISHER IPM: PDF COMPARISON ", + "text_level": 1, + "bbox": [ + 173, + 621, + 614, + 640 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Imposing data-independent constraints on the function class in the IPM framework, such as the Lipschitz constraint in the Wasserstein distance is computationally challenging and intractable for the general case. In this Section, we generalize the Fisher IPM introduced in (Mroueh & Sercu, 2017), where the function class is relaxed to a tractable data dependent constraint on the second order moment of the critic, in other words the critic is constrained to be in a Lebsegue ball. ", + "bbox": [ + 173, + 654, + 825, + 724 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Fisher IPM. Let $\\mathcal { X } \\subset \\mathbb { R } ^ { d }$ and $\\mathcal { P } ( \\mathcal { X } )$ be the space of distributions defined on $\\mathcal { X }$ . Let $\\mathbb { P } , \\mathbb { Q } \\in$ $\\mathcal { P } ( \\mathcal { X } )$ , and $\\mu$ be a dominant measure of $\\mathbb { P }$ and $\\mathbb { Q }$ , in the sense that ", + "bbox": [ + 169, + 729, + 821, + 760 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c0f789d8ffd1b190245ac2d5c2109164edc26b49965fa31000e8c43bc0ca1bce.jpg", + "text": "$$\n\\mu ( x ) = 0 \\implies \\mathbb { P } ( x ) = 0 { \\mathrm { ~ a n d } } \\mathbb { Q } ( x ) = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 765, + 633, + 782 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We assume $\\mu$ to be also a distribution in $\\mathcal { P } ( \\mathcal { X } )$ , and assume $\\pmb { \\mu } ( \\pmb { x } ) > \\mathbf { 0 } , \\forall \\pmb { x } \\in \\mathcal { X }$ . Let $\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu )$ be the space of $\\mu$ -measurable functions. For $f , g \\in \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu )$ , we define the following dot product and its corresponding norm: ", + "bbox": [ + 174, + 787, + 825, + 832 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/88f56b94989940a85facb05c3b4699c2785ac41c8649ec20ce8494b9a619d6be.jpg", + "text": "$$\n\\langle f , g \\rangle _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } = \\int _ { \\mathcal { X } } f ( x ) g ( x ) \\mu ( x ) d x , \\| f \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } = \\sqrt { \\int _ { \\mathcal { X } } f ^ { 2 } ( x ) \\mu ( x ) d x } .\n$$", + "text_format": "latex", + "bbox": [ + 251, + 838, + 746, + 881 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Note that $\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu )$ , can be formally defined as follows: ", + "bbox": [ + 173, + 886, + 544, + 902 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/57c9c82b706a6af55890433de3272cf1166b92ac86003a5135e9b3124436ccfa.jpg", + "text": "$$\n\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) = \\{ f : \\mathcal { X } \\to \\mathbb { R } \\mathrm { s . t } \\ \\| f \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } < \\infty \\} .\n$$", + "text_format": "latex", + "bbox": [ + 333, + 907, + 660, + 928 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We define the unit Lebesgue ball as follows: ", + "bbox": [ + 176, + 103, + 465, + 117 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ed5b076ecf167418ab81fd98dc01d6d5f7e66b35768fe8f111ec1304aecbf9ce.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { B } _ { 2 } ( \\mathcal { X } , \\mu ) = \\{ f \\in \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) , \\| f \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } \\leq 1 \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 119, + 656, + 140 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Fisher IPM defined in (Mroueh & Sercu, 2017), searches for the critic function in the Lebesgue Ball $\\mathbb { B } _ { 2 } ( \\mathcal { X } , \\mu )$ that maximizes the mean discrepancy between $\\mathbb { P }$ and $\\mathbb { Q }$ . Fisher GAN (Mroueh & Sercu, 2017) was originally formulated specifically for $\\begin{array} { r } { \\mu = \\frac 1 2 ( { \\mathbb P } + { \\mathbb Q } ) } \\end{array}$ . We consider here a general $\\mu$ as long as it dominates $\\mathbb { P }$ and $\\mathbb { Q }$ . We define Generalized Fisher IPM as follows: ", + "bbox": [ + 173, + 141, + 825, + 196 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0459b6283064953f4a579db933a7da8d2f8da64146d6cb7f61d718f7edc339ed.jpg", + "text": "$$\n\\mathcal { F } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\operatorname* { s u p } _ { f \\in \\mathbb { B } _ { 2 } ( \\mathcal { X } , \\mu ) } \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x )\n$$", + "text_format": "latex", + "bbox": [ + 341, + 199, + 656, + 227 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that: ", + "bbox": [ + 173, + 229, + 240, + 243 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7adc5ae6d5564e7bdcb40573887203e12aae54584adacb2f75345dce0941e0c9.jpg", + "text": "$$\n\\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) = \\left. f , { \\frac { \\mathbb { P } - \\mathbb { Q } } { \\mu } } \\right. _ { { \\mathcal { L } } _ { 2 } ( \\mathcal { X } , \\mu ) } .\n$$", + "text_format": "latex", + "bbox": [ + 333, + 238, + 663, + 276 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Hence Fisher IPM can be written as follows: ", + "bbox": [ + 174, + 275, + 465, + 290 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/cf2ca2c1ee9e66d8cc139d9ab4818c3ad24e905d70ad8d497fd2b16dffeacf48.jpg", + "text": "$$\n\\mathcal { F } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\operatorname* { s u p } _ { f \\in \\mathbb { B } _ { 2 } ( \\mathcal { X } , \\mu ) } \\left. f , \\frac { \\mathbb { P } - \\mathbb { Q } } { \\mu } \\right. _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) }\n$$", + "text_format": "latex", + "bbox": [ + 351, + 289, + 647, + 327 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We have the following result: ", + "bbox": [ + 174, + 328, + 367, + 342 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 1 (Generalized Fisher IPM). The Fisher distance and the optimal critic are as follows: ", + "bbox": [ + 168, + 344, + 805, + 359 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1. The Fisher distance is given by: ", + "bbox": [ + 214, + 368, + 442, + 383 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3db2e30f980cd8a9477d69935b4e5387d1876b47cb6bc9a7613e57d66273b858.jpg", + "text": "$$\n\\mathcal { F } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\left\\| \\frac { \\mathbb { P } - \\mathbb { Q } } { \\mu } \\right\\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } = \\sqrt { \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mu } \\left( \\frac { \\mathbb { P } ( \\boldsymbol { x } ) - \\mathbb { Q } ( \\boldsymbol { x } ) } { \\mu ( \\boldsymbol { x } ) } \\right) ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 386, + 735, + 430 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2. The optimal $f _ { \\chi }$ achieving the Fisher distance $\\mathcal { F } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } )$ is: ", + "bbox": [ + 210, + 439, + 620, + 454 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof of Theorem 1. From Equation (2), the optimal $f _ { \\chi }$ belong to the intersection of the hyperplane that has normal $n ~ = ~ \\textstyle \\frac { \\mathbb { P } - \\mathbb { Q } } { \\mu }$ and the ball B2(X , µ), hence fχ = $\\begin{array} { r } { f _ { \\chi } = \\frac { n ^ { \\cdot } } { \\| n \\| _ { \\mathcal { L } _ { 2 } ( \\chi , \\mu ) } } } \\end{array}$ Hence $\\mathcal { F } ( \\mathbb { P } , \\mathbb { Q } ) = \\| n \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } .$ □ ", + "bbox": [ + 173, + 494, + 493, + 580 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/39105c967ba0488a4607950203039a45c6b2ef8745052cceb191e400af567c21.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 584, + 491, + 740, + 574 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We see from Theorem 1 the role of the dominant measure $\\mu$ : the optimal critic is defined with respect to this measure and the overall Fisher distance can be seen as an average weighted distance between probability density functions, where the average is taken on points sampled from $\\mu$ . We give here some choices of $\\mu$ : ", + "bbox": [ + 174, + 597, + 825, + 655 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1. For $\\begin{array} { r } { \\mu = \\frac 1 2 ( { \\mathbb P } + { \\mathbb Q } ) } \\end{array}$ , we obtain the symmetric chi-squared distance as defined in (Mroueh & Sercu, 2017). \n2. $\\mu _ { G P }$ , the implicit distribution defined by the interpolation lines between $\\mathbb { P } _ { r }$ and $\\mathbb { Q } _ { \\theta }$ as in (Gulrajani et al., 2017). \n3. When $\\mu$ does not dominate $\\mathbb { P }$ , and $\\mathbb { Q }$ , we obtain a non symmetric divergence. For example for $\\mu = \\mathbb { P }$ , $\\begin{array} { r } { \\mathcal { F } _ { \\mathbb { P } } ^ { 2 } ( \\mathbb { P } , \\mathbb { Q } ) = \\int _ { \\mathcal { X } } \\frac { ( \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) ) ^ { 2 } } { \\mathbb { P } ( x ) } d x } \\end{array}$ (P(x)−Q(x))2P(x) dx. We see here that for this particular choice we obtain the Pearson divergence. ", + "bbox": [ + 210, + 662, + 825, + 777 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 SOBOLEV IPM ", + "text_level": 1, + "bbox": [ + 174, + 795, + 331, + 811 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this Section, we introduce the Sobolev IPM. In a nutshell, the Sobolev IPM constrains the critic function to belong to a ball in the restricted Sobolev Space. In other words we constrain the norm of the gradient of the critic $\\nabla _ { x } f ( x )$ . We will show that by moving from a Lebesgue constraint as in Fisher IPM to a Sobolev constraint as in Sobolev IPM, the metric changes from a joint PDF matching to weighted (ccordinate-wise) conditional CDFs matching. The intrinsic conditioning built in to the Sobolev IPM and the comparison of cumulative distributions makes Sobolev IPM suitable for comparing discrete sequences. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 DEFINITION AND EXPRESSION OF SOBOLEV IPM IN TERMS OF COORDINATE CONDITIONAL CDFS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 746, + 132 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We will start by recalling some definitions on Sobolev Spaces. We assume in the following that $\\mathcal { X }$ is compact and consider functions in the Sobolev space $\\mathring { W } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )$ : ", + "bbox": [ + 169, + 142, + 823, + 172 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/4dd3c4c9510738e81204ba0d088fe7d352e34da54909b8462ff90e72252d37bb.jpg", + "text": "$$\nW ^ { 1 , 2 } ( \\mathcal { X } , \\mu ) = \\left\\{ f : \\mathcal { X } \\to \\mathbb { R } , \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) \\right. ^ { 2 } \\mu ( x ) d x < \\infty \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 294, + 176, + 702, + 212 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We restrict ourselves to functions in $W ^ { 1 , 2 } ( \\mathcal { X } , \\mu )$ vanishing at the boundary, and note this space $W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )$ . Note that in this case: ", + "bbox": [ + 173, + 215, + 823, + 247 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/2214e21f4eadc9152ee5680c70949b5c27ead062efe29705cb6cd7ad541fef3b.jpg", + "text": "$$\n\\| f \\| _ { W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu ) } = \\sqrt { \\int _ { \\mathcal { X } } \\| \\nabla _ { x } f ( x ) \\| ^ { 2 } \\mu ( x ) d x }\n$$", + "text_format": "latex", + "bbox": [ + 357, + 251, + 642, + 295 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "defines a semi-norm. We can similarly define a dot product in $W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )$ , for $f , g \\in W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )$ : ", + "bbox": [ + 173, + 297, + 821, + 316 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/519d221b59a2972c2c722e9abe4f0070a191217e27f4769d20b3d3ecb4b62046.jpg", + "text": "$$\n\\langle f , g \\rangle _ { W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu ) } = \\int _ { \\mathcal { X } } \\langle \\nabla _ { x } f ( x ) , \\nabla _ { x } g ( x ) \\rangle _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 319, + 673, + 353 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Hence we define the following Sobolev IPM, by restricting the critic of the mean discrepancy to the Sobolev unit ball : ", + "bbox": [ + 173, + 363, + 823, + 391 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/2164ff51845e7b061208c655341d466218cfe9d7307e1638f2149974e75927b1.jpg", + "text": "$$\n\\mathcal { S } _ { \\boldsymbol { \\mu } } ( \\mathbb { P } , \\mathbb { Q } ) = \\operatorname* { s u p } _ { \\substack { f \\in W _ { 0 } ^ { 1 , 2 } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\boldsymbol { \\mu } ) } \\leq 1 } } \\bigg \\{ \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { P } } f ( \\boldsymbol { x } ) - \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { Q } } f ( \\boldsymbol { x } ) \\bigg \\} .\n$$", + "text_format": "latex", + "bbox": [ + 289, + 393, + 709, + 434 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "When compared to the Wasserstein distance, the Sobolev IPM given in Equation (3) uses a data dependent gradient constraint (depends on $\\mu$ ) rather than a data independent Lipchitz constraint. Let $F _ { \\mathbb { P } }$ and $F _ { \\mathbb { Q } }$ be the cumulative distribution functions of $\\mathbb { P }$ and $\\mathbb { Q }$ respectively. We have: ", + "bbox": [ + 173, + 438, + 826, + 481 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/12e013d292fb43ace5cb893d767b9deaa3a5d0371cc443ca3e7ec2fd7bab3c03.jpg", + "text": "$$\n\\mathbb { P } ( x ) = \\frac { \\partial ^ { d } } { \\partial x _ { 1 } \\hdots \\partial x _ { d } } F _ { \\mathbb { P } } ( x ) ,\n$$", + "text_format": "latex", + "bbox": [ + 406, + 484, + 589, + 518 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "and we define ", + "bbox": [ + 173, + 522, + 267, + 536 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/743b961821257fa0fa92772b952b707d1896db052f5fae4f4bb446ccb49fcefd.jpg", + "text": "$$\nD ^ { - i } = \\frac { \\partial ^ { d - 1 } } { \\partial x _ { 1 } \\dots \\partial x _ { i - 1 } \\partial x _ { i + 1 } \\dots \\partial x _ { d } } , \\mathrm { f o r } i = 1 \\dots d .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 531, + 673, + 568 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$D ^ { - i }$ computes the $( d - 1 )$ high-order partial derivative excluding the variable $i$ ", + "bbox": [ + 173, + 577, + 697, + 593 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our main result is presented in Theorem 2. Additional theoretical results are given in Appendix A. \nAll proofs are given in Appendix B. ", + "bbox": [ + 176, + 611, + 823, + 641 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 2 (Sobolev IPM). Assume that $F _ { \\mathbb { P } }$ , and $F _ { \\mathbb { Q } }$ and its d derivatives exist and are continuous: $F _ { \\mathbb { P } }$ and $F _ { \\mathbb { Q } } \\in C ^ { d } ( \\mathcal { X } )$ . Define the differential operator $D ^ { - }$ : ", + "bbox": [ + 174, + 643, + 821, + 674 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/3c7fdaadd09bf295ec830d3bc5cb7007418d6ca293309fe91d976278733e575d.jpg", + "text": "$$\nD ^ { - } = ( D ^ { - 1 } , \\ldots D ^ { - d } ) .\n$$", + "text_format": "latex", + "bbox": [ + 418, + 679, + 576, + 696 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/63d9ab9f3fb9fe1c2a6a7b82d836d425d046d82439e3a55bea0eb26f1ab45225.jpg", + "text": "$$\nx = ( x _ { 1 } , \\ldots x _ { i - 1 } , x _ { i } , x _ { i + 1 } , \\ldots x _ { d } ) , l e t x ^ { - i } = ( x _ { 1 } , \\ldots x _ { i - 1 } , x _ { i + 1 } , \\ldots x _ { d } ) .\n$$", + "text_format": "latex", + "bbox": [ + 200, + 700, + 683, + 717 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The Sobolev IPM given in Equation (3) has the following equivalent forms: ", + "bbox": [ + 176, + 722, + 668, + 738 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1. Sobolev IPM as comparison of high order partial derivatives of CDFs. The Sobolev IPM has the following form: ", + "bbox": [ + 212, + 747, + 825, + 777 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/1b1c47f69d225b36578c4457fe34066d17c0087bbe99bfe90899a175ce3b816c.jpg", + "text": "$$\nS _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\frac { 1 } { d } \\sqrt { \\int _ { \\mathcal { X } } \\frac { \\sum _ { i = 1 } ^ { d } ( D ^ { - i } F _ { \\mathbb { P } } ( x ) - D ^ { - i } F _ { \\mathbb { Q } } ( x ) ) ^ { 2 } } { \\mu ( x ) } d x } .\n$$", + "text_format": "latex", + "bbox": [ + 318, + 781, + 705, + 824 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "2. Sobolev IPM as comparison of weighted (coordinate-wise) conditional CDFs. The Sobolev IPM can be written in the following equivalent form: ", + "bbox": [ + 210, + 834, + 825, + 863 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/2a5b2bc7e573d2b9c2a0fd824ac10bb207b79aeeac1c362884b6ce9aaa3aa8a2.jpg", + "text": "$$\nS _ { \\mu } ^ { 2 } ( { \\mathbb { P } } , { \\mathbb { Q } } ) = \\frac { 1 } { d ^ { 2 } } { \\mathbb { E } } _ { x \\sim \\mu } \\sum _ { i = 1 } ^ { d } \\left( \\frac { { \\mathbb { P } } _ { X ^ { - i } } ( x ^ { - i } ) F _ { { \\mathbb { P } } _ { [ X _ { i } ] X ^ { - i } = x ^ { - i } ] } } ( x _ { i } ) - { \\mathbb { Q } } _ { X ^ { - i } } ( x ^ { - i } ) F _ { { \\mathbb { Q } } _ { [ X _ { i } ] X ^ { - i } = x ^ { - i } ] } } ( x _ { i } ) } { \\mu ( x ) } \\right) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 869, + 870, + 912 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3. The optimal critic $f ^ { * }$ satisfies the following identity: ", + "bbox": [ + 210, + 103, + 576, + 119 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/297f36e6e4f5011107b075eb04b65fc03f0bdbf7bb22ec038cd57b8be573697b.jpg", + "text": "$$\n\\nabla _ { x } f ^ { * } ( x ) = \\frac { 1 } { d S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } \\frac { D ^ { - } F _ { \\mathbb { Q } } ( x ) - D ^ { - } F _ { \\mathbb { P } } ( x ) } { \\mu ( x ) } , \\mu - a l m o s t s u r e l y .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 130, + 751, + 165 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Sobolev IPM Approximation. Learning in the whole Sobolev space $W _ { 0 } ^ { 1 , 2 }$ is challenging hence we need to restrict our function class to a hypothesis class , such as neural networks. We assume in the following that functions in $\\mathcal { H }$ vanish on the boundary of $\\mathcal { X }$ , and restrict the optimization to the function space $\\mathcal { H }$ . $\\mathcal { H }$ can be a Reproducing Kernel Hilbert Space as in the MMD case or parametrized by a neural network. Define the Sobolev IPM approximation in $\\mathcal { H }$ : ", + "bbox": [ + 173, + 191, + 825, + 263 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/971ea3456676bcb111ab4fbee39726afebd8782ade48f246922be856df9f0f05.jpg", + "text": "$$\n\\mathcal { S } _ { \\mathcal { H } , \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\operatorname* { s u p } _ { \\substack { f \\in \\mathcal { H } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } } \\leq 1 } } \\left\\{ \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) \\right\\}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 275, + 694, + 313 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The following Lemma shows that the Sobolev IPM approximation in $\\mathcal { H }$ is proportional to Sobolev IPM. The tightness of the approximation of the Sobolev IPM is governed by the tightness of the approximation of the optimal Sobolev Critic $f ^ { * }$ in $\\mathcal { H }$ . This approximation is measured in the Sobolev sense, using the Sobolev dot product. ", + "bbox": [ + 173, + 324, + 825, + 382 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Lemma 1 (Sobolev IPM Approximation in a Hypothesis Class). Let $\\mathcal { H }$ be a function space with functions vanishing at the boundary. For any f ∈ H and for f ∗ the optimal critic in W 1,20 , we have: ", + "bbox": [ + 171, + 401, + 825, + 446 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/854756f3154ceda8fb70a3a4edd5a69c5e21b3651ce7d7acdb0f36d3e2488644.jpg", + "text": "$$\n\\mathcal { S } _ { \\mathcal { H } , \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\mathcal { S } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\operatorname* { s u p } _ { \\substack { f \\in \\mathcal { H } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } ( \\chi , \\mu ) } \\leq 1 } } \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\right. _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 454, + 766, + 496 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Note that this Lemma means that the Sobolev IPM is well approximated if the space $\\mathcal { H }$ has an enough representation power to express $\\nabla _ { x } f ^ { * } ( x )$ . This is parallel to the Fisher IPM approximation (Mroueh & Sercu, 2017) where it is shown that the Fisher IPM approximation error is proportional to the critic approximation in the Lebesgue sense. Having in mind that the gradient of the critic is the information that is passed on to the generator, we see that this convergence in the Sobolev sense to the optimal critic is an important property for GAN training. ", + "bbox": [ + 173, + 508, + 825, + 594 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Relation to Fokker-Planck Diffusion. We show in Appendix A that the optimal Sobolev critic is the solution of the following elliptic PDE (with zero boundary conditions): ", + "bbox": [ + 171, + 599, + 823, + 628 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/130523609727ae305fca8ff6534076a07e4f5637a5d4dc5e460b99ebf3723f21.jpg", + "text": "$$\n\\frac { \\mathbb { P } - \\mathbb { Q } } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } = - \\mathrm { d i v } ( \\mu ( x ) \\nabla _ { x } f ( x ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 638, + 612, + 674 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We further link the elliptic PDE given in Equation (8) and the Fokker-Planck diffusion. As we illustrate in Figure 2(b) the gradient of the critic defines a transportation plan for moving the distribution mass from $\\mathbb { Q }$ to $\\mathbb { P }$ . ", + "bbox": [ + 173, + 684, + 825, + 728 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Discussion of Theorem 2. We make the following remarks on Theorem 2: ", + "bbox": [ + 173, + 747, + 674, + 763 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1. From Theorem 2, we see that the Sobolev IPM compares $d$ higher order partial derivatives of the cumulative distributions $F _ { \\mathbb { P } }$ and $F _ { \\mathbb { Q } }$ , while Fisher IPM compares the probability density functions. ", + "bbox": [ + 176, + 776, + 825, + 819 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "2. The dominant measure $\\mu$ plays a similar role to Fisher: ", + "bbox": [ + 176, + 828, + 558, + 844 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/74e0ea07b0892100bb72453fb981347f5d3b84bb4cb932af144d8ebd56ed9107.jpg", + "text": "$$\nS _ { \\mu } ^ { 2 } ( { \\mathbb { P } } , \\mathbb { Q } ) = \\frac 1 { d ^ { 2 } } \\sum _ { i = 1 } ^ { d } { \\mathbb { E } } _ { x \\sim \\mu } \\left( \\frac { D ^ { - i } F _ { \\mathbb { P } } ( x ) - D ^ { - i } F _ { \\mathbb { Q } } ( x ) } { \\mu ( x ) } \\right) ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 320, + 854, + 699, + 898 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "the average distance is defined with respect to points sampled from $\\mu$ . ", + "bbox": [ + 196, + 909, + 655, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3. Comparison of coordinate-wise Conditional CDFs. We note in the following $\\begin{array} { r l } { x ^ { - i } } & { { } = } \\end{array}$ $( x _ { 1 } , \\ldots x _ { i - 1 } , x _ { i + 1 } , \\ldots x _ { d } )$ . Note that we have: ", + "bbox": [ + 173, + 102, + 823, + 132 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/18bd3f92f5c34e7173ad84a6c2be2d7bf1c6ec7e29506402937c180d48994612.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathsf { P } ^ { - i } F _ { \\mathbb { P } } ( x ) = \\frac { \\partial ^ { d - 1 } } { \\partial x _ { 1 } \\hdots \\partial x _ { i - 1 } \\partial x _ { i + 1 } \\hdots \\partial x _ { d } } \\displaystyle \\int _ { - \\infty } ^ { x _ { 1 } } \\hdots \\int _ { - \\infty } ^ { x _ { d } } \\mathbb { P } ( u _ { 1 } \\hdots u _ { d } ) d u _ { 1 } \\hdots d u _ { d } } \\\\ & { \\hphantom { \\mathsf { P } ^ { - i } F _ { \\mathbb { P } } ( x ) = } = \\displaystyle \\int _ { - \\infty } ^ { x _ { i } } \\mathbb { P } ( x _ { 1 } , \\hdots , x _ { i - 1 } , u , x _ { i + 1 } , \\hdots , x _ { d } ) d u } \\\\ & { \\hphantom { \\mathsf { P } ^ { - i } F _ { \\mathbb { P } } ( x ) = } = \\mathbb { P } _ { X ^ { - i } } ( x _ { 1 } , \\hdots , x _ { i - 1 } , x _ { i + 1 } , \\hdots x _ { d } ) \\displaystyle \\int _ { - \\infty } ^ { x _ { i } } \\mathbb { P } _ { [ X _ { i } | X ^ { - i } = x ^ { - i } ] } ( u | x _ { 1 } , \\hdots , x _ { i - 1 } , x _ { i + 1 } , \\hdots x _ { d } ) d u } \\\\ & { \\displaystyle ( \\mathrm { U s i n g ~ B a y e s ~ r u l e } ) } \\\\ & { \\hphantom { \\mathsf { P } ^ { - i } F _ { \\mathbb { P } } ( x ) = } = \\mathbb { P } _ { X ^ { - i } } ( x ^ { - i } ) F _ { \\mathbb { P } _ { [ X _ { i } | X ^ { - i } = x ^ { - i } ] } } ( x _ { i } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 202, + 137, + 856, + 290 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Note that for each $i$ , $D ^ { - i } F _ { \\mathbb { P } } ( x )$ is the cumulative distribution of the variable $X _ { i }$ given the other variables $X ^ { - i } = x ^ { - i }$ , weighted by the density function of $X ^ { - i }$ at $x ^ { - i }$ . This leads us to the form given in Equation 5. ", + "bbox": [ + 196, + 292, + 826, + 335 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We see that the Sobolev IPM compares for each dimension $i$ the conditional cumulative distribution of each variable given the other variables, weighted by their density function. We refer to this as comparison of coordinate-wise CDFs on a leave one out basis. From this we see that we are comparing CDFs, which are better behaved on discrete distributions. Moreover, the conditioning built in to this metric will play a crucial role in comparing sequences as the conditioning is important in this context (See section 6.1). ", + "bbox": [ + 196, + 338, + 825, + 421 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 ILLUSTRATIVE EXAMPLES ", + "text_level": 1, + "bbox": [ + 174, + 438, + 398, + 452 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Sobolev IPM / Cramer Distance and Wasserstein-1 in one Dimension. ´ In one dimension, Sobolev IPM is the Cramer Distance (for ´ $\\mu$ uniform on $\\mathcal { X }$ , we note this $\\mu : = 1$ ). While Sobolev IPM in one dimension measures the discrepancy between CDFs, the one dimensional Wasserstein- $p$ distance measures the discrepancy between inverse CDFs: ", + "bbox": [ + 174, + 463, + 825, + 520 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/183d4cd42789e60279bafd5256498ce1453a649fdcef170b5027e44a88bb104d.jpg", + "text": "$$\nS _ { \\mu : = 1 } ^ { 2 } ( { \\mathbb { P } } , \\mathbb { Q } ) = \\int _ { \\chi } ( F _ { \\mathbb { P } } ( x ) - F _ { \\mathbb { Q } } ( x ) ) ^ { 2 } d x { \\mathrm { ~ v e r s u s ~ } } W _ { p } ^ { p } ( { \\mathbb { P } } , \\mathbb { Q } ) = \\int _ { 0 } ^ { 1 } | F _ { \\mathbb { P } } ^ { - 1 } ( u ) - F _ { \\mathbb { Q } } ^ { - 1 } ( u ) | ^ { p } d u ,\n$$", + "text_format": "latex", + "bbox": [ + 199, + 526, + 795, + 561 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Recall also that the Fisher IPM for uniform $\\mu$ is given by : ", + "bbox": [ + 173, + 566, + 557, + 582 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/bc4a4fd656eb1ed525ce0227233b10fb01960a523616219858e1b67184ca0376.jpg", + "text": "$$\n\\mathcal { F } _ { \\mu : = 1 } ^ { 2 } ( \\mathbb { P } , \\mathbb { Q } ) = \\int _ { \\mathcal { X } } ( \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) ) ^ { 2 } d x .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 587, + 629, + 619 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Consider for instance two point masses $\\mathbb { P } = \\delta _ { a _ { 1 } }$ and $\\mathbb { Q } = \\delta _ { a _ { 2 } }$ with $a _ { 1 } , a _ { 2 } \\in \\mathbb { R }$ . The rationale behind using Wasserstein distance for GAN training is that since it is a weak metric, for far distributions $W _ { 1 } ^ { 1 } ( \\mathbb { P } , \\mathbb { Q } ) = S _ { \\mu : = 1 } ^ { 2 } \\stackrel { - } { = } | a _ { 1 } - a _ { 2 } |$ me sign, while e than $\\mathcal { F } _ { \\mu : = 1 } ^ { 2 } ( \\mathbb { P } , \\mathbb { Q } ) = 2$ 2017). In this case, it is easy to see that. As we see from this simple example,g distributions on discrete spaces. See $C D F$ \nFigure 1, for a further discussion of this effect in the GAN context. ", + "bbox": [ + 173, + 625, + 825, + 710 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Sobolev IPM between two 2D Gaussians. We consider $\\mathbb { P }$ and $\\mathbb { Q }$ to be two dimensional Gaussians with means $\\mu _ { 1 }$ and $\\mu _ { 2 }$ and covariances $\\Sigma _ { 1 }$ and $\\Sigma _ { 2 }$ . Let $( x , y )$ be the coordinates in 2D. We note $F _ { \\mathbb { P } }$ and $F _ { \\mathbb { Q } }$ the CDFs of $\\mathbb { P }$ and $\\mathbb { Q }$ respectively. We consider in this example $\\textstyle \\mu = { \\frac { \\mathbb { P } + \\mathbb { Q } } { 2 } }$ . We know from Theorem 2 that the gradient of the Sobolev optimal critic is proportional to the following vector field: ", + "bbox": [ + 173, + 723, + 825, + 794 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/e50ee0347c9ba8d47bdefce58ca0db56fec3e968585997acd5e08b9f5538b96a.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\nabla f ^ { * } ( x , y ) \\alpha \\frac { 1 } { \\mu ( x , y ) } \\left[ \\frac { \\partial } { \\partial x } ( F _ { \\mathbb { Q } } ( x , y ) - F _ { \\mathbb { P } } ( x , y ) ) \\right] } \\\\ { \\displaystyle \\mu _ { 1 } = [ 1 , 0 ] , \\Sigma _ { 1 } = \\left[ \\frac { 1 . 9 } { 0 . 8 } \\quad 0 . 8 \\right] \\mu _ { 2 } = [ 1 , - 2 ] , \\Sigma _ { 2 } = \\left[ \\frac { 1 . 9 } { - 0 . 8 } \\quad - 0 . 8 \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 790, + 774, + 864 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Figure 2 we consider ", + "bbox": [ + 173, + 829, + 769, + 862 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Figure 2(a) we plot the numerical solution of the PDE satisfied by the optimal Sobolev critic given in Equation (8), using MATLAB solver for elliptic PDEs (more accurately we solve $- d i v ( \\bar { \\mu } ( x ) \\nabla _ { x } f ( \\bar { x } ) ) = \\mathbb { P } ( x ) - \\bar { \\mathbb { Q } } ( x )$ , hence we obtain the solution of Equation (8) up to a normalization constant $( \\frac { 1 } { S _ { \\mu } ( { \\mathbb P } , { \\mathbb Q } ) } ) )$ . We numerically solve the PDE on a rectangle with zero boundary conditions. We see that the optimal Sobolev critic separates the two distributions well. In Figure 2(b) we then numerically compute the gradient of the optimal Sobolev critic on a 2D grid as given in Equation 9 (using numerical evaluation of the CDF and finite difference for the evaluation of the partial derivatives). We plot in Figure 2(b) the density functions of $\\mathbb { P }$ and $\\mathbb { Q }$ as well as the vector field of the gradient of the optimal Sobolev critic. As discussed in Section A.1, we see that the gradient of the critic (wrt to the input), defines on the support of $\\textstyle \\mu = { \\frac { \\mathbb { P } + \\mathbb { Q } } { 2 } }$ a transportation plan for moving the distribution mass from $\\mathbb { Q }$ to $\\mathbb { P }$ . ", + "bbox": [ + 173, + 867, + 825, + 928 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/bf79812d77b73653170a552e274a8e31c047709d105768f320b54b41bce68e2a.jpg", + "image_caption": [ + "(a) Smoothed discrete densities: PDF versus CDF of smoothed discrete densities with non overlapping supports. " + ], + "image_footnote": [], + "bbox": [ + 272, + 104, + 705, + 204 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/aebf2bc5c2f3315395bc93792e96d731c49ae95608257b3482f05c10e62e4c99.jpg", + "image_caption": [ + "(b) Smoothed Discrete and Continuous densities: PDF versus CDF of a smoothed discrete density and a continuous density with non overlapping supports. ", + "Figure 1: In the GAN context for example in text generation, we have to match a (smoothed) discrete real distribution and a continuous generator. In this case, the CDF matching enabled by Sobolev IPM gives non zero discrepancy between a (smoothed) discrete and a continuous density even if the densities have disjoint supports. This ensures non vanishing gradients of the critic. " + ], + "image_footnote": [], + "bbox": [ + 271, + 262, + 702, + 363 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 511, + 825, + 611 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 SOBOLEV GAN ", + "text_level": 1, + "bbox": [ + 176, + 631, + 338, + 647 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Now we turn to the problem of learning GANs with Sobolev IPM. Given the “real distribution” $\\mathbb { P } _ { r } \\in \\mathcal { P } ( \\mathcal { X } )$ , our goal is to learn a generator $g _ { \\boldsymbol { \\theta } } : \\mathcal { Z } \\subset \\mathbb { R } ^ { n _ { z } } \\mathcal { X }$ , such that for $z \\sim p _ { z }$ , the distribution of $g _ { \\boldsymbol { \\theta } } ( z )$ is close to the real data distribution $\\mathbb { P } _ { r }$ , where $p _ { z }$ is a fixed distribution on $\\mathcal { Z }$ (for instance $z \\sim \\mathcal { N } ( 0 , I _ { n _ { z } } ) )$ . We note $\\mathbb { Q } _ { \\theta }$ for the “fake distribution” of $g _ { \\theta } ( z ) , z \\sim p _ { z }$ . Consider $\\{ x _ { i } , i = 1 \\ldots N \\} \\sim \\bar { \\mathbb { P } } _ { r }$ , $\\{ z _ { i } , i = 1 \\ldots N \\} \\sim { \\mathcal { N } } ( 0 , I _ { n _ { z } } )$ , and $\\{ \\tilde { x } _ { i } , i = 1 \\ldots N \\} \\sim \\mu$ . We consider these choices for $\\mu$ : ", + "bbox": [ + 173, + 662, + 826, + 748 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "1. $\\begin{array} { r } { \\mu = \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } \\end{array}$ i.e $\\tilde { x } \\sim \\mathbb { P } _ { r }$ or $\\tilde { x } = g _ { \\theta } ( z ) , z \\sim p _ { z }$ with equal probability $\\textstyle { \\frac { 1 } { 2 } }$ . 2. $\\mu _ { G P }$ is the implicit distribution defined by the interpolation lines between $\\mathbb { P } _ { r }$ and $\\mathbb { Q } _ { \\theta }$ as in (Gulrajani et al., 2017) i.e : $\\tilde { x } = u x + ( 1 - u ) y , x \\sim \\mathbb { P } _ { r } , y = g _ { \\theta } ( z ) , z \\sim p _ { z }$ and $u \\sim \\mathrm { U n i f } [ 0 , 1 ]$ . ", + "bbox": [ + 210, + 758, + 826, + 825 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Sobolev GAN can be written as follows: ", + "bbox": [ + 173, + 835, + 441, + 851 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/84c761688237f258341cc60927fb4938acb57a1bf7b4967a36aba82aa1f7d826.jpg", + "text": "$$\n\\begin{array}{c} \\begin{array} { l } { \\displaystyle \\mathrm { A N ~ c a n ~ p e ~ w r i t t e n ~ a s ~ r o l l o w s } \\colon } \\\\ { \\displaystyle \\operatorname* { m i n } _ { g _ { \\theta } } } \\\\ { \\displaystyle \\quad { f _ { p } , \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\| \\nabla _ { x } f _ { p } ( \\tilde { x } _ { i } ) \\| ^ { 2 } = 1 } } \\end{array} \\hat { \\hat { \\mathcal E } } ( f _ { p } , g _ { \\theta } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } f _ { p } ( x _ { i } ) - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } f _ { p } ( g _ { \\theta } ( z _ { i } ) ) \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 245, + 842, + 751, + 885 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For any choice of the parametric function class $\\mathcal { H } _ { p }$ , note the constraint by $\\hat { \\Omega } _ { S } ( f _ { p } , g _ { \\theta } ) \\ =$ $\\begin{array} { r l } & { { \\frac { 1 } { N } } \\sum _ { i = 1 } ^ { N } \\left\\| \\nabla _ { x } f _ { p } ( \\tilde { x } _ { i } ) \\right\\| ^ { 2 } } \\end{array}$ . For example if $\\begin{array} { r } { \\mu \\ = \\ \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } \\end{array}$ , $\\begin{array} { r c l } { \\hat { \\Omega } _ { S } ( f _ { p } , g _ { \\theta } ) } & { = } & { \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ { N } \\| \\nabla _ { x } \\boldsymbol { f } _ { p } ( x _ { i } ) \\| ^ { 2 } + } \\end{array}$ ", + "bbox": [ + 174, + 892, + 825, + 926 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/7e31e2430319983c215121a0e70ffc66b6fbcb2eef7bf131143c1e35323dcb1d.jpg", + "image_caption": [ + "(b) Optimal Sobolev Transport Vector Field $\\nabla _ { x } f ^ { * } { \\bar { ( x ) } }$ (arrows are the vector field $\\nabla _ { x } f ^ { * } ( x )$ evaluated on the 2D grid. Magnitude of arrows was rescaled for visualization.) " + ], + "image_footnote": [], + "bbox": [ + 199, + 123, + 467, + 273 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/c2e0765d22d99d612ec7689965f0fb6fce3a5271eaa33de3baadab7bc9b8eda2.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 534, + 112, + 771, + 271 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "(a) Numerical solution of the PDE satisfied by the optimal Sobolev critic. ", + "bbox": [ + 191, + 287, + 490, + 314 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Figure 2: Numerical solution of the PDE satisfied by the optimal Sobolev critic and the transportation Plan induced by the gradient of Sobolev critic. The gradient of the critic (wrt to the input), defines on the support of $\\begin{array} { r } { \\bar { \\boldsymbol { \\mu } } = \\frac { \\mathbb { P } + \\mathbb { Q } } { 2 } , } \\end{array}$ a transportation plan for moving the distribution mass from $\\mathbb { Q }$ to $\\mathbb { P }$ . For a theoretical analysis of this transportation plan and its relation to Fokker-Planck diffusion the reader is invited to check Appendix A. ", + "bbox": [ + 173, + 354, + 825, + 426 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "$\\begin{array} { r l } { } & { { } \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ { N } \\left\\| \\nabla _ { x } f _ { p } ( g _ { \\theta } ( z _ { i } ) ) \\right\\| ^ { 2 } } \\end{array}$ . Note that, since the optimal theoretical critic is achieved on the sphere, we impose a sphere constraint rather than a ball constraint. Similar to (Mroueh & Sercu, 2017) we define the Augmented Lagrangian corresponding to Sobolev GAN objective and constraint ", + "bbox": [ + 174, + 454, + 825, + 501 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/875dea4cf700e6a0eaa1835a680f4276eca8648f8b599108f6c3a1ed53f1fb1f.jpg", + "text": "$$\n\\mathcal { L } _ { S } ( p , \\theta , \\lambda ) = \\hat { \\mathcal { E } } ( f _ { p } , g _ { \\theta } ) + \\lambda ( 1 - \\hat { \\Omega } _ { S } ( f _ { p } , g _ { \\theta } ) ) - \\frac { \\rho } { 2 } ( \\hat { \\Omega } _ { S } ( f _ { p } , g _ { \\theta } ) - 1 ) ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 269, + 511, + 727, + 539 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "where $\\lambda$ is the Lagrange multiplier and $\\rho > 0$ is the quadratic penalty weight. We alternate between optimizing the critic and the generator. We impose the constraint when training the critic only. Given $\\theta$ , we solve $\\begin{array} { r } { \\operatorname* { m a x } _ { p } \\operatorname* { m i n } _ { \\lambda } \\mathcal { L } _ { S } ( p , \\theta , \\lambda ) } \\end{array}$ , for training the critic. Then given the critic parameters $p$ we optimize the generator weights $\\theta$ to minimize the objective minθ $\\hat { \\mathcal { E } } ( f _ { p } , g _ { \\theta } )$ . See Algorithm 1. ", + "bbox": [ + 173, + 549, + 825, + 609 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Algorithm 1 Sobolev GAN ", + "text_level": 1, + "bbox": [ + 174, + 630, + 357, + 645 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/b7766853bc2e593a135cf95879ea92915912856229217e062dbd9939f34d0ad6.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Input: ρ penalty weight, n Learning rate, nc number of iterations for training the critic, N batch size
Initialize p, 0,λ=0
repeat
for j = 1 to nc do
Samplea minibatch xi,i=1...N,xi~ Pr
Sample a minibatch zi,i = 1...N,zi ~ Pz
(gp,gx)←(VpLs,VxLs)(p,0,λ)
p ←p+η ADAM(p,9p)
λ ← λ- pgx {SGD rule on 入with learning rate ρ} end for
Sample zi,i= 1...N, zi ~ Pz
d←Vθ8(fp,g0）=-VθN∑=1fp(g0(zi）） N
0←0-nADAM(0,dθ)
until θ converges
", + "bbox": [ + 178, + 647, + 825, + 862 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Remark 1. Note that in Algorithm $\\cdot$ , we obtain a biased estimate since we are using same samples for the cost function and the constraint, but the incurred bias can be shown to be small and vanishing as the number of samples increases as shown and justified in (Shivaswamy & Jebara, 2010). ", + "bbox": [ + 173, + 882, + 825, + 924 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Relation to WGAN-GP. WGAN-GP can be written as follows: ", + "bbox": [ + 173, + 103, + 604, + 118 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/791192fb27625a05cd429c211071280256274c29486f742719fb77e1e0b8c129.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\substack { g _ { \\theta } } } \\operatorname* { s u p } _ { f , \\| \\nabla _ { x } f _ { p } ( \\bar { x } _ { i } ) \\| = 1 , \\bar { x } _ { i } \\sim \\mu _ { G P } } \\hat { \\bar { \\mathcal { E } } } ( f _ { p } , g _ { \\theta } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } f _ { p } ( x _ { i } ) - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } f _ { p } ( g _ { \\theta } ( z _ { i } ) )\n$$", + "text_format": "latex", + "bbox": [ + 250, + 121, + 746, + 164 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The main difference between WGAN-GP and our setting, is that WGAN-GP enforces pointwise constraints on points drawn from $\\mu = \\mu _ { G P }$ via a point-wise quadratic penalty $\\begin{array} { r l } { { \\big ( \\hat { \\mathcal { E } } ( f _ { p } , g _ { \\theta } ) - \\lambda \\sum _ { i = 1 } ^ { N } \\bigl ( 1 - \\bigl ) } \\quad } & { { } } \\end{array}$ $\\| \\nabla _ { x } f ( \\widetilde { x } _ { i } ) \\| ) ^ { 2 } )$ while we enforce that constraint on average as a Sobolev norm, allowing us the coordinate weighted conditional CDF interpretation of the IPM. ", + "bbox": [ + 173, + 166, + 825, + 227 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "6 APPLICATIONS OF SOBOLEV GAN ", + "text_level": 1, + "bbox": [ + 176, + 247, + 491, + 263 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Sobolev IPM has two important properties; The first stems from the conditioning built in to the metric through the weighted conditional CDF interpretation. The second stems from the diffusion properties that the critic of Sobolev IPM satisfies (Appendix A) that has theoretical and practical ties to the Laplacian regularizer and diffusion on manifolds used in semi-supervised learning (Belkin et al., 2006). ", + "bbox": [ + 173, + 277, + 825, + 348 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In this Section, we exploit those two important properties in two applications of Sobolev GAN: Text generation and semi-supervised learning. First in text generation, which can be seen as a discrete sequence generation, Sobolev GAN (and WGAN-GP) enable training GANs without need to do explicit brute-force conditioning. We attribute this to the built-in conditioning in Sobolev IPM (for the sequence aspect) and to the CDF matching (for the discrete aspect). Secondly using GANs in semi-supervised learning is a promising avenue for learning using unlabeled data. We show that a variant of Sobolev GAN can achieve strong SSL results on the CIFAR-10 dataset, without the need of any form of activation normalization in the networks or any extra ad hoc tricks. ", + "bbox": [ + 173, + 354, + 825, + 467 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "6.1 TEXT GENERATION WITH SOBOLEV GAN ", + "text_level": 1, + "bbox": [ + 173, + 483, + 504, + 497 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In this Section, we present an empirical study of Sobolev GAN in character level text generation. Our empirical study on end to end training of character-level GAN for text generation is articulated on four dimensions (loss, critic, generator, $\\pmb { \\mu }$ ). (1) the loss used (GP: WGAN-GP (Gulrajani et al., 2017), S: Sobolev or F: Fisher) (2) the architecture of the critic (Resnets or RNN) (3) the architecture of the generator (Resnets or RNN or RNN with curriculum learning) (4) the sampling distribution $\\pmb { \\mu }$ in the constraint. ", + "bbox": [ + 173, + 508, + 825, + 592 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Text Generation Experiments. We train a character-level GAN on Google Billion Word dataset and follow the same experimental setup used in (Gulrajani et al., 2017). The generated sequence length is 32 and the evaluation is based on Jensen-Shannon divergence on empirical 4-gram probabilities (JS-4) of validation data and generated data. JS-4 may not be an ideal evaluation criteria, but it is a reasonable metric for current character-level GAN results, which is still far from generating meaningful sentences. ", + "bbox": [ + 173, + 599, + 825, + 683 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Annealed Smoothing of discrete $\\mathbb { P } _ { r }$ in the constraint $\\pmb { \\mu }$ . Since the generator distribution will always be defined on a continuous space, we can replace the discrete “real” distribution $\\mathbb { P } _ { r }$ with a smoothed version (Gaussian kernel smoothing) $\\mathbb { P } _ { r } \\star \\mathcal { N } ( 0 , \\sigma ^ { 2 } I _ { d } )$ . This corresponds to doing the following sampling for $\\mathbb { P } _ { r } : x + \\xi , x \\sim \\mathbb { P } _ { r }$ , and $\\xi \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } I _ { d } )$ . Note that we only inject noise to the “real” distribution with the goal of smoothing the support of the discrete distribution, as opposed to instance noise on both “real” and “fake” to stabilize the training, as introduced in (Kaae Sønderby et al., 2017; Arjovsky & Bottou, 2017). As it is common in optimization by continuation (Mobahi & III, 2015), we also anneal the noise level $\\sigma$ as the training progresses on a linear schedule. ", + "bbox": [ + 173, + 690, + 825, + 803 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Sobolev GAN versus WGAN-GP with Resnets. In this setting, we compare (WGANGP,G $=$ Resnet,D $^ { 1 \\textless }$ Resnet, $\\mu = \\mu _ { G P } ) $ to (Sobolev,G $=$ Resnet,D $=$ Resnet, $\\mu$ ) where $\\mu$ is one of: (1) $\\mu _ { G P }$ , (2) the noise smoothed $\\begin{array} { r } { \\mu _ { s } ( \\sigma ) \\ = \\ \\frac { \\mathbb { P } _ { r } \\star \\mathcal { N } ( 0 , \\sigma ^ { 2 } I _ { d } ) + \\mathbb { Q } _ { \\theta } } { 2 } } \\end{array}$ or (3) noise smoothed with annealing $\\mu _ { s } ^ { a } ( \\sigma _ { 0 } )$ with $\\sigma _ { 0 }$ the initial noise level. We use the same architectures of Resnet with 1D convolution for the critic and the generator as in (Gulrajani et al., 2017) (4 resnet blocks with hidden layer size of 512). In order to implement the noise smoothing we transform the data into one-hot vectors. Each one hot vector $x$ is transformed to a probability vector $p$ with 0.9 replacing the one and $0 . 1 / ( d i c t _ { s i z e } - 1 )$ replacing the zero. We then sample $\\epsilon$ from a Gaussian distribution ${ \\bar { \\mathcal { N } } } ( 0 , \\sigma ^ { 2 } )$ , and ", + "bbox": [ + 173, + 808, + 825, + 924 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/95d02c80f9b5a08286ec9176c050401e5ad02420894a30a90fcc19711b6eed94.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 225, + 117, + 470, + 242 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "(a) Comparing Sobolev with $\\mu _ { G P }$ and WGAN-GP. The JS-4 are 0.3363 and 0.3302 respectively. ", + "bbox": [ + 223, + 246, + 486, + 284 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/6e2575ca9e9bb232ea44064ff715cb92a3beab4fce8471509ac419d7b8ec9342.jpg", + "image_caption": [ + "(b) Comparing Sobolev with different $\\mu$ dominant measures and WGAN-GP. The JS4 of $\\mu _ { s } ^ { a } ( \\sigma _ { 0 } = 1 . 5 )$ is 0.3268. ", + "Figure 3: Result of Sobolev GAN for various dominating measure $\\mu$ , for resnets as architectures of the critic and the generator. " + ], + "image_footnote": [], + "bbox": [ + 509, + 117, + 753, + 241 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "use softmax to normalize $\\log p + \\epsilon$ . We use algorithm 1 for Sobolev GAN and fix the learning rate to $1 0 ^ { - 4 }$ and $\\rho$ to $1 0 ^ { - 5 }$ . The noise level $\\sigma$ was annealed following a linear schedule starting from an initial noise level $\\sigma _ { 0 }$ (at iteration $i$ , $\\begin{array} { r } { \\sigma _ { i } = \\sigma _ { 0 } ( 1 - \\frac { i } { M a x i t e r } ) } \\end{array}$ , Maxite ${ \\displaystyle = 3 0 \\mathsf K }$ ). For WGAN-GP we used the open source implementation with the penalty $\\lambda = 1 0$ as in (Gulrajani et al., 2017). Results are given in Figure 3(a) for the JS-4 evaluation of both WGAN-GP and Sobolev GAN for $\\mu = \\mu _ { G P }$ . In Figure 3(b) we show the JS-4 evaluation of Sobolev GAN with the annealed noise smoothing $\\mu _ { s } ^ { a } ( \\bar { \\sigma _ { 0 } } )$ , for various values of the initial noise level $\\sigma _ { 0 }$ . We see that the training succeeds in both cases. Sobolev GAN achieves slightly better results than WGAN-GP for the annealing that starts with high noise level $\\sigma _ { 0 } = 1 . 5$ . We note that without smoothing and annealing i.e using $\\begin{array} { r } { \\bar { \\boldsymbol { \\mu } } = \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } \\end{array}$ , Sobolev GAN is behind. Annealed smoothing of $\\mathbb { P } _ { r }$ , helps the training as the real distribution is slowly going from a continuous distribution to a discrete distribution. See Appendix C (Figure 6) for a comparison between annealed and non annealed smoothing. ", + "bbox": [ + 174, + 412, + 825, + 580 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We give in Appendix C a comparison of WGAN-GP and Sobolev GAN for a Resnet generator architecture and an RNN critic. The RNN has degraded performance due to optimization difficulties. ", + "bbox": [ + 176, + 588, + 823, + 616 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Fisher GAN Curriculum Conditioning versus Sobolev GAN: Explicit versus Implicit conditioning. We analyze how Fisher GAN behaves under different architectures of generators and critics. We first fix the generator to be ResNet. We study 3 different architectures of critics: ResNet, GRU (we follow the experimental setup from (Press et al., 2017)), and hybrid $\\mathrm { R e s N e t + G R U }$ (Reed et al., 2016). We notice that RNN is unstable, we need to clip the gradient values of critics in $[ - 0 . 5 , 0 . 5 ]$ , and the gradient of the Lagrange multiplier $\\lambda _ { F }$ to $[ - 1 0 ^ { 4 } , \\bar { 1 } 0 ^ { 4 } ]$ . We fix $\\rho _ { F } = 1 0 ^ { - 7 }$ and we use $\\mu = \\mu _ { G P }$ . We search the value for the learning rate in $[ 1 0 ^ { - 5 } , 1 \\dot { 0 } ^ { - 4 } ]$ . We see that for $\\mu = \\mu _ { G P }$ and $G = \\mathbb { F }$ Resnet for various critic architectures, Fisher GAN fails at the task of text generation (Figure ${ \\mathfrak { 4 } } \\ { \\mathfrak { a - c } } { \\mathrm { , } }$ ). Nevertheless, when using RNN critics $( \\mathrm { F i g } \\quad \\mathsf { b } , \\mathsf { c } )$ a marginal improvement happens over the fully collapsed state when using a resnet critic (Fig 4 a). We hypothesize that RNN critics enable some conditioning and factoring of the distribution, which is lacking in Fisher IPM. ", + "bbox": [ + 173, + 622, + 825, + 776 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Finally Figure 4 (d) shows the result of training with recurrent generator and critic. We follow (Press et al., 2017) in terms of GRU architecture, but differ by using Fisher GAN rather than WGAN-GP. We use $\\begin{array} { r } { \\mu = { \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } } \\end{array}$ i.e. without annealed noise smoothing. We train (F, D=RNN,G=RNN, ${ \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } .$ ) using curriculum conditioning of the generator for all lengths $\\ell$ as done in (Press et al., 2017): the generator is conditioned on $3 2 - \\ell$ characters and predicts the $\\ell$ remaining characters. We increment $\\ell = 1$ to 32 on a regular schedule (every $1 5 \\mathrm { k }$ updates). JS-4 is only computed when $\\ell > 4$ . We see in Figure 4 that under curriculum conditioning with recurrent critics and generators, the training of Fisher GAN succeeds and reaches similar levels of Sobolev GAN (and WGAN-GP). Note that the need of this explicit brute force conditioning for Fisher GAN, highlights the implicit conditioning induced by Sobolev GAN via the gradient regularizer, without the need for curriculum conditioning. ", + "bbox": [ + 173, + 782, + 825, + 924 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/2cad69ff4d412b18994156d6951f045e5d5629da2e40f46cfb2e8bb9373125ba.jpg", + "image_caption": [ + "Figure 4: Fisher GAN with different architectures for critics: (a-c) We see that for $\\mu = \\mu _ { G P }$ and $G =$ Resnet for various critic architectures, Fisher GAN fails at the task of text generation. We notice small improvements for RNN critics (b-c) due to the conditioning and factoring of the distribution. (d) Fisher GAN with recurrent generator and critic, trained on a curriculum conditioning for increasing lengths $\\ell$ , increments indicated by gridlines. In this curriculum conditioning setup, with recurrent critics and generators, the training of Fisher GAN succeeds and reaches similar levels of Sobolev GAN (and WGAN-GP). It is important to note that by doing this explicit curriculum conditioning for Fisher GAN, we highlight the implicit conditioning induced by Sobolev GAN, via the gradient regularizer. " + ], + "image_footnote": [], + "bbox": [ + 341, + 101, + 642, + 256 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "6.2 SEMI-SUPERVISED LEARNING WITH SOBOLEV GAN", + "text_level": 1, + "bbox": [ + 174, + 421, + 578, + 435 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A proper and promising framework for evaluating GANs consists in using it as a regularizer in the semi-supervised learning setting (Salimans et al., 2016; Dumoulin et al., 2017; Kumar et al., 2017). As mentioned before, the Sobolev norm as a regularizer for the Sobolev IPM draws connections with the Laplacian regularization in manifold learning (Belkin et al., 2006). In the Laplacian framework of semi-supervised learning, the classifier satisfies a smoothness constraint imposed by controlling its Sobolev norm: $\\begin{array} { r } { \\int _ { \\mathcal { X } } \\left\\| \\nabla _ { x } \\bar { f } ( x ) \\right\\| ^ { 2 } \\mu ^ { 2 } ( x ) d x } \\end{array}$ (Alaoui et al., 2016). In this Section, we present a variant Xof Sobolev GAN that achieves competitive performance in semi-supervised learning on the CIFAR10 dataset Krizhevsky & Hinton (2009) without using any internal activation normalization in the critic, such as batch normalization (BN) (Ioffe & Szegedy, 2015), layer normalization (LN) (Ba et al., 2016), or weight normalization (Salimans & Kingma, 2016). ", + "bbox": [ + 173, + 446, + 825, + 588 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this setting, a convolutional neural network $\\Phi _ { \\omega } : \\mathcal { X } \\mathbb { R } ^ { m }$ is shared between the cross entropy (CE) training of a $K$ -class classifier $( S \\in \\mathbb { R } ^ { K \\times m } )$ ) and the critic of GAN (See Figure 5). We have the following training equations for the (critic $^ +$ classifer) and the generator: ", + "bbox": [ + 174, + 594, + 825, + 637 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/73a48d9b2d0b771c267d552ceb29b802d35dabab0224cb66b7f31be6684d17f0.jpg", + "text": "$$\n{ \\mathrm { C r i t i c } } + { \\mathrm { C l a s s i f i e r } } ; \\quad \\operatorname* { m a x } _ { S , \\Phi _ { \\omega } , f } \\mathcal { L } _ { D } = \\mathcal { L } _ { \\mathrm { a l m } } ^ { \\mathrm { G A N } } ( f , g _ { \\theta } ) - \\lambda _ { C E } \\sum _ { ( x , y ) \\in \\mathrm { l a b } } C E ( p ( y | x ) , y )\n$$", + "text_format": "latex", + "bbox": [ + 240, + 643, + 758, + 680 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/ded3d8d7b9bfb1e91382d83e938938173d7ca784cd5c705701bdfb7b402c6e8f.jpg", + "text": "$$\n\\mathrm { G e n e r a t o r : \\ m a x } { \\mathcal { L } } _ { G } = { \\hat { \\mathcal { E } } } ( f , g _ { \\theta } )\n$$", + "text_format": "latex", + "bbox": [ + 390, + 685, + 607, + 710 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where the main IPM objective with $N$ samples: $\\begin{array} { r } { \\hat { \\mathcal { E } } ( f , g _ { \\theta } ) = \\frac { 1 } { N } \\left( \\sum _ { x \\in \\mathrm { u n l } } f ( x ) - \\sum _ { z \\sim p _ { z } } f ( g _ { \\theta } ( z ) ) \\right) . } \\end{array}$ ", + "bbox": [ + 173, + 714, + 821, + 738 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Following (Mroueh & Sercu, 2017) we use the following “ $K + 1$ parametrization” for the critic (See Figure 5) : ", + "bbox": [ + 174, + 743, + 823, + 772 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/d42fbe727de6f14478c28b1bd7f1f5b76b3d1f0e3ed080794746d2e0b574d849.jpg", + "text": "$$\nf ( x ) = \\underbrace { \\sum _ { y = 1 } ^ { K } p ( y | x ) \\left. S _ { y } , \\Phi _ { \\omega } ( x ) \\right. } _ { f _ { + } : \\mathrm { \\normalfont ~ { \\mathrm { \\fontfamily { ~ \\ddots ~ } \\selectfont ~ { \\ c r i t i c } } } } } - \\underbrace { \\left. v , \\Phi _ { \\omega } ( x ) \\right. } _ { f _ { - } : \\mathrm { \\normalfont ~ { \\cdot \\mathrm { \\sc ~ { \\ c r i t i c } } } } }\n$$", + "text_format": "latex", + "bbox": [ + 346, + 770, + 651, + 837 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Note that $p ( y | x ) = \\mathrm { S o f t m a x } ( \\langle \\mathrm { S } , \\Phi _ { \\omega } ( \\mathrm { x } ) \\rangle ) _ { \\mathrm { y } }$ appears both in the critic formulation and in the CrossEntropy term in Equation (11). Intuitively this critic uses the $K$ class directions of the classifier $S _ { y }$ to define the “real” direction, which competes with another $\\mathrm { K } { + } 1 ^ { \\mathrm { t h } }$ direction $v$ that indicates fake samples. This parametrization adapts the idea of (Salimans et al., 2016), which was formulated specifically for the classic KL / JSD based GANs, to IPM-based GANs. We saw consistently better results with the $K + 1$ formulation over the regular formulation where the classification layer $S$ doesn’t interact with the critic direction $v$ . We also note that when applying a gradient penalty based constraint (either WGAN-GP or Sobolev) on the full critic $f = f _ { + } - f _ { - }$ , it is impossible for the network to fit even the small labeled training set (underfitting), causing bad SSL performance. This leads us to the formulation below, where we apply the Sobolev constraint only on $f _ { - }$ . Throughout this Section we fix $\\begin{array} { r } { \\mu = { \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } } \\end{array}$ . ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 178 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We propose the following two schemes for constraining the $\\mathrm { K } { + } 1$ critic $f ( x ) = f _ { + } ( x ) - f _ { - } ( x )$ : ", + "bbox": [ + 169, + 181, + 797, + 199 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "1) Fisher constraint on the critic: We restrict the critic to the following set: ", + "bbox": [ + 176, + 204, + 678, + 219 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/ba655097f9eca7cf00e54d65d928b1ad1bc7942b96ee356e74325e253b584a03.jpg", + "text": "$$\nf \\in \\left\\{ f = f _ { + } - f _ { - } , \\hat { \\Omega } _ { F } ( f , g _ { \\theta } ) = \\frac { 1 } { 2 N } \\left( \\sum _ { x \\in \\mathrm { u n l } } f ^ { 2 } ( x ) + \\sum _ { z \\sim p _ { z } } f ^ { 2 } ( g _ { \\theta } ( z ) ) \\right) = 1 \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 230, + 224, + 766, + 268 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This constraint translates to the following ALM objective in Equation (11): ", + "bbox": [ + 176, + 275, + 666, + 290 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/c640866f97c81f3857c0afefb94f83c0bd3ab85c11ca2aa12f63240707343398.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { a l m } } ^ { \\mathrm { G A N } } ( f , g _ { \\theta } ) = \\hat { \\mathcal { E } } ( f , g _ { \\theta } ) + \\lambda _ { F } ( 1 - \\hat { \\Omega } _ { F } ( f , g _ { \\theta } ) ) - \\frac { \\rho _ { F } } { 2 } ( \\hat { \\Omega } _ { F } ( f , g _ { \\theta } ) - 1 ) ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 261, + 296, + 733, + 324 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where the Fisher constraint ensures the stability of the training through an implicit whitened mean matching (Mroueh & Sercu, 2017). ", + "bbox": [ + 173, + 330, + 823, + 358 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "2) Fisher $^ +$ Sobolev constraint: We impose 2 constraints on the critic: Fisher on $f$ & Sobolev on $f _ { - }$ ", + "bbox": [ + 173, + 364, + 821, + 381 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/518d285100f9352d64f4d9dfeac82967c4a18bfbdb61cc20687f425469c45f86.jpg", + "text": "$$\n\\begin{array} { r } { f \\in \\left\\{ f = f _ { + } - f _ { - } , \\hat { \\Omega } _ { F } ( \\pmb { f } , g _ { \\theta } ) = 1 \\mathrm { a n d } \\hat { \\Omega } _ { S } ( \\pmb { f } _ { - } , g _ { \\theta } ) = 1 \\right\\} , } \\\\ { \\mathrm { } _ { \\mathrm { } s } ( \\pmb { f } _ { - } , g _ { \\theta } ) = \\frac { 1 } { 2 N } \\left( \\sum _ { x \\in \\mathrm { u n l } } \\left. \\nabla _ { x } \\pmb { f } _ { - } ( x ) \\right. ^ { 2 } + \\sum _ { z \\sim p _ { z } } \\left. \\nabla _ { x } \\pmb { f } _ { - } ( g _ { \\theta } ( z ) ) \\right. ^ { 2 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 218, + 388, + 699, + 446 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "This constraint translates to the following ALM in Equation (11): ", + "bbox": [ + 174, + 449, + 602, + 465 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/6b013d885a1d46e57f85b45777e15fe96d4358fccc7b6980bdd8253a82bf5f2f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\mathrm { a l m } } ^ { \\mathrm { G A N } } ( f , g _ { \\theta } ) = \\hat { \\mathcal { E } } ( f , g _ { \\theta } ) + \\lambda _ { F } ( 1 - \\hat { \\Omega } _ { F } ( \\pmb { f } , g _ { \\theta } ) ) + \\lambda _ { S } ( 1 - \\hat { \\Omega } _ { S } ( f _ { - } , g _ { \\theta } ) ) } \\\\ & { \\quad \\quad \\quad \\quad - \\frac { \\rho _ { F } } { 2 } ( \\hat { \\Omega } _ { F } ( \\pmb { f } , g _ { \\theta } ) - 1 ) ^ { 2 } - \\frac { \\rho _ { S } } { 2 } ( \\hat { \\Omega } _ { S } ( f _ { - } , g _ { \\theta } ) - 1 ) ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 263, + 470, + 735, + 523 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Note that the fisher constraint on $f$ ensures the stability of the training, and the Sobolev constraints on the “fake” critic $f _ { - }$ enforces smoothness of the “fake” critic and thus the shared CNN $\\Phi _ { \\omega } ( x )$ . This is related to the classic Laplacian regularization in semi-supervised learning (Belkin et al., 2006). ", + "bbox": [ + 173, + 526, + 826, + 583 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Table 2 shows results of SSL on CIFAR-10 comparing the two proposed formulations. Similar to the standard procedure in other GAN papers, we do hyperparameter and model selection on the validation set. We present baselines with a similar model architecture and leave out results with significantly larger convnets. G and D architectures and hyperparameters are in Appendix D. $\\Phi _ { \\omega }$ is similar to (Salimans et al., 2016; Dumoulin et al., 2017; Mroueh & Sercu, 2017) in architecture, but note that we do not use any batch, layer, or weight normalization yet obtain strong competitive accuracies. We hypothesize that we don’t need any normalization in the critic, because of the implicit whitening of the feature maps introduced by the Fisher and Sobolev constraints as explained in (Mroueh & Sercu, 2017). ", + "bbox": [ + 173, + 588, + 826, + 715 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/ea25635919fa3c5b32ae620a1e74c601b9df2ff8ade1012e0ef7eddf0195cd9e.jpg", + "image_caption": [ + "Figure 5: $\\mathrm { \\ddot { \\Delta K } } + 1 \\mathrm { \\ ' }$ ” parametrization of the critic for semi-supervised learning. " + ], + "image_footnote": [], + "bbox": [ + 253, + 739, + 741, + 892 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/030d4a44b15f080cce2bc3e0e6994331ffe397d02b0b7427f447afbeb200ae37.jpg", + "table_caption": [ + "Table 2: CIFAR-10 error rates for varying number of labeled samples in the training set. Mean and standard deviation computed over 5 runs. We only use the $K + 1$ formulation of the critic. Note that we achieve strong SSL performance without any additional tricks, and even though the critic does not have any batch, layer or weight normalization. Baselines with \\* use either additional models like PixelCNN, or do data augmentation (translations and flips), or use a much larger model, either of which gives an advantage over our plain simple training method. $\\dagger$ is the result we achieved in our experimental setup under the same conditions but without $\\mathrm { \\ddot { \\Delta K } } + 1 \\mathrm { \\ ' }$ critic (see Appendix D), since (Gulrajani et al., 2017) does not have SSL results. " + ], + "table_footnote": [], + "table_body": "
Number of labeled examples Model100020004000 Misclassification rate8000
CatGAN (Springenberg, 2015) FM (Salimans et al., 2016) ALI (Dumoulin et al., 2017)21.83 ± 2.0119.61 ± 2.0919.58 18.63 ± 2.3217.72 ± 1.82
Tangents Reg (Kumar et al., 2017)19.98 ± 0.3 20.06 ± 0.519.09 ± 0.1517.99 ± 0.54 16.78 ± 0.617.05 ± 0.50
II-model (Laine & Aila, 2016) *16.55 ± 0.29
VAT (Miyato et al., 2017)14.87
Bad Gan (Dai et al., 2017) *14.41 ± 0.30
VAT+EntMin+Large (Miyato et al., 2017) *13.15
Sajadi (Sajjadi et al., 2016) *11.29
WGAN-GP (Gulrajani et al., 2017) t44.85 ± 0.2837.62 ± 0.5632.66 ± 0.4830.38± 0.22
Fisher,layer norm (Mroueh & Sercu,2017)19.74± 0.2117.87 ± 0.3816.13 ± 0.5314.81 ± 0.16
Fisher, no norm (Mroueh & Sercu, 2017)21.49 ± 0.1819.20 ± 0.4617.30 ± 0.3015.57 ± 0.33
Sobolev + Fisher, no norm (This Work)20.14 ± 0.2117.38 ± 0.1015.77 ± 0.1914.20± 0.08
", + "bbox": [ + 174, + 212, + 823, + 434 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 458, + 318, + 474 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We introduced the Sobolev IPM and showed that it amounts to a comparison between weighted (coordinate-wise) CDFs. We presented an ALM algorithm for training Sobolev GAN. The intrinsic conditioning implied by the Sobolev IPM explains the success of gradient regularization in Sobolev GAN and WGAN-GP on discrete sequence data, and particularly in text generation. We highlighted the important tradeoffs between the implicit conditioning introduced by the gradient regularizer in Sobolev IPM, and the explicit conditioning of Fisher IPM via recurrent critics and generators in conjunction with the curriculum conditioning. Both approaches succeed in text generation. We showed that Sobolev GAN achieves competitive semi-supervised learning results without the need of any normalization, thanks to the smoothness induced by the gradient regularizer. We think the Sobolev IPM point of view will open the door for designing new regularizers that induce different types of conditioning for general structured/discrete/graph data beyond sequences. ", + "bbox": [ + 174, + 489, + 825, + 642 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 665, + 285, + 679 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Ahmed El Alaoui, Xiang Cheng, Aaditya Ramdas, Martin J. Wainwright, and Michael I. Jordan. Asymptotic behavior of p-based laplacian regularization in semi-supervised learning. CoRR, abs/1603.00564, 2016. \nMartin Arjovsky and Leon Bottou. Towards principled methods for training generative adversarial ´ networks. In ICLR, 2017. \nMartin Arjovsky, Soumith Chintala, and Leon Bottou. 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", + "bbox": [ + 173, + 103, + 826, + 260 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A THEORY: APPROXIMATION AND TRANSPORT INTERPRETATION ", + "text_level": 1, + "bbox": [ + 173, + 102, + 730, + 118 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In this Section we present the theoretical properties of Sobolev IPM and how it relates to distributions transport theory and other known metrics between distributions, notably the Stein distance. ", + "bbox": [ + 174, + 132, + 823, + 161 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A.1 DISTRIBUTION TRANSPORT PERSPECTIVE ON SOBOLEV IPM ", + "text_level": 1, + "bbox": [ + 174, + 176, + 642, + 193 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In this Section, we characterize the optimal critic of the Sobolev IPM as a solution of a non linear PDE. The solution of the variational problem of the Sobolev IPM satisfies a non linear PDE that can be derived using standard tools from calculus of variations (Ekeland & Turnbull, 1983; Alaoui et al., 2016). ", + "bbox": [ + 173, + 202, + 825, + 258 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Theorem 3 (PDE satisfied by the Sobolev Critic). The optimal critic of Sobolev IPM $f ^ { * }$ satisfies the following $P D E$ : ", + "bbox": [ + 173, + 261, + 825, + 290 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/11a1a882d27c217eb8c6f794348747448fe25025ec5b8fdb311f21b816fa8589.jpg", + "text": "$$\n\\Delta f ^ { * } ( x ) + \\left. \\nabla _ { x } \\log \\mu ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\right. + \\frac { \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\mu ( x ) } = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 291, + 694, + 327 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Define the Stein Operator: $\\begin{array} { r } { T ( \\mu ) \\vec { g } ( x ) = \\frac { 1 } { 2 } \\Big ( \\langle \\nabla _ { x } \\log ( \\mu ( x ) ) , \\vec { g } ( x ) \\rangle + d i v ( \\vec { g } ( x ) ) \\Big ) } \\end{array}$ . Hence we have the following Transport Equation of $\\mathbb { P }$ to $\\mathbb { Q }$ : ", + "bbox": [ + 173, + 342, + 823, + 378 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/1e8e1ddb948c1f5c091710119ff5cfe0e18fe54c83f125800249ff258b965310.jpg", + "text": "$$\n\\mathbb { Q } ( x ) = \\mathbb { P } ( x ) + 2 S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\mu ( x ) T ( \\mu ) \\nabla _ { x } f ^ { * } ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 381, + 651, + 398 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Recall the definition of Stein Discrepancy : ", + "bbox": [ + 174, + 400, + 457, + 415 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/2877884eb1971d1d3f73c91b22dfd55168fc0436ac4b2c417bab1a06eeda727c.jpg", + "text": "$$\n\\mathbb { S } ( \\mathbb { Q } , \\mu ) = \\operatorname* { s u p } _ { \\vec { g } } | \\mathbb { E } _ { x \\sim \\mathbb { Q } } [ T ( \\mu ) \\vec { g } ( x ) ] | , \\vec { g } : \\mathcal { X } \\mathbb { R } ^ { d } .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 416, + 660, + 445 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Theorem 4 (Sobolev and Stein Discrepanices). The following inequality holds true: ", + "bbox": [ + 174, + 446, + 725, + 462 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/eeb5ec94fe21ee822359870e030f1434ca8ac52a226b712de209aa416c3f67fb.jpg", + "text": "$$\n\\bigg | \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { Q } } \\left[ \\frac { \\mathbb { Q } ( \\boldsymbol { x } ) - \\mathbb { P } ( \\boldsymbol { x } ) } { \\mu ( \\boldsymbol { x } ) } \\right] \\bigg | \\leq 2 \\qquad \\underbrace { \\mathbb { S } ( \\mathbb { Q } , \\mu ) } _ { \\mathrm { S } ( \\mathbb { Q } , \\mu ) } \\qquad \\underbrace { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } _ { \\mathrm { S } _ { \\boldsymbol { \\mu } } ( \\mathbb { Q } ) }\n$$", + "text_format": "latex", + "bbox": [ + 254, + 463, + 735, + 498 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Consider for example $\\mu = \\mathbb { P }$ , and sequence $\\mathbb { Q } _ { n }$ . If the Sobolev distance goes $S _ { \\mathbb { P } } ( \\mathbb { P } , \\mathbb { Q } _ { n } ) 0$ , the ratio $\\begin{array} { r } { r _ { n } ( x ) = \\frac { \\mathbb { Q } _ { n } ( x ) } { \\mathbb { P } ( x ) } } \\end{array}$ Qn(x)P(x) converges in expectation (w.r.t to Q) to 1. The speed of the convergence is given by the Stein Discrepancy $\\mathbb { S } ( \\mathbb { Q } _ { n } , \\mathbb { P } )$ . ", + "bbox": [ + 173, + 521, + 826, + 571 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Relation to Fokker-Planck Diffusion Equation and Particles dynamics. Note that PDE satisifed by the Sobolev critic given in Equation (13) can be equivalently written: ", + "bbox": [ + 174, + 585, + 820, + 614 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/6c0c12a2a6be6aa27f7a455544535d1930265afa23d226fdc15e72c20e552f62.jpg", + "text": "$$\n\\frac { \\mathbb { P } - \\mathbb { Q } } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } = - \\mathrm { d i v } ( \\mu ( x ) \\nabla _ { x } f ^ { * } ( x ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 616, + 616, + 650 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "written in this form, we draw a connection with the Fokker-Planck Equation for the evolution of a density function $q _ { t }$ that is the density of particles $X _ { t } \\in \\mathbb { R } ^ { d }$ evolving with a drift (a velocity field) $V ( x , i ) : \\mathcal { X } \\times [ 0 , \\infty [ \\mathbb { R } ^ { d }$ : ", + "bbox": [ + 176, + 651, + 825, + 694 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "$d X _ { t } = V ( X _ { t } , t ) d t$ , where the density of $X _ { 0 }$ is given by $q _ { 0 } ( x ) = \\mathbb { Q } ( x )$ , ", + "bbox": [ + 261, + 695, + 733, + 712 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Fokker-Planck Equation states that the evolution of the particles density $q _ { t }$ satisfies: ", + "bbox": [ + 212, + 714, + 746, + 729 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/775fdf072fd235807b3bfb8f331579429df6aef60d2b70b5948031e0ed865845.jpg", + "text": "$$\n\\frac { d \\boldsymbol { q } _ { t } } { d t } ( \\boldsymbol { x } ) = - \\mathrm { d i v } ( \\boldsymbol { q } _ { t } ( \\boldsymbol { x } ) V ( \\boldsymbol { x } , t ) )\n$$", + "text_format": "latex", + "bbox": [ + 398, + 731, + 599, + 761 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Comparing Equation (15) and Equation (16), we identify then the gradient of Sobolev critic as a drift. This suggests that one can define “Sobolev descent” as the evolution of particles along the gradient flow: ", + "bbox": [ + 173, + 762, + 825, + 804 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "$d X _ { t } = \\nabla _ { x } f _ { t } ^ { * } ( X _ { t } ) d t$ , where the density of $X _ { 0 }$ is given by $q _ { 0 } ( x ) = \\mathbb { Q } ( x )$ , where $f _ { t } ^ { * }$ is the Sobolev critic between $q _ { t }$ and $\\mathbb { P }$ . One can show that the limit distribution of the particles is $\\mathbb { P }$ . The analysis of “Sobolev descent” and its relation to Stein Descent (Liu & Wang, 2016; Liu, 2017) is beyond the scope of this paper and will be studied in a separate work. Hence we see that the gradient of the Sobolev critic defines a transportation plan to move particles whose distribution is $\\mathbb { Q }$ to particles whose distribution is $\\mathbb { P }$ (See Figure 2). This highlights the role of the gradient of the critic in the context of GAN training in term of transporting the distribution of the generator to the real distribution. ", + "bbox": [ + 258, + 808, + 738, + 824 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B PROOFS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 276, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof of Theorem 2. Let $F _ { \\mathbb { P } }$ and $F _ { \\mathbb { Q } }$ , be the cumulative distribution functions of $\\mathbb { P }$ and $\\mathbb { Q }$ respectively. We have: ", + "bbox": [ + 173, + 138, + 823, + 169 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/f778a10ee1bb50edd324b16fe3e9d3835192f46f20e176ca6e82987269cea599.jpg", + "text": "$$\n\\mathbb { P } ( x ) = \\frac { \\partial ^ { d } } { \\partial x _ { 1 } \\hdots \\partial x _ { d } } F _ { \\mathbb { P } } ( x ) ,\n$$", + "text_format": "latex", + "bbox": [ + 405, + 179, + 589, + 214 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We note $\\begin{array} { r } { D = \\frac { \\partial ^ { d } } { \\partial x _ { 1 } \\ldots \\partial x _ { d } } } \\end{array}$ and D−i , $\\begin{array} { r } { D ^ { - i } = \\frac { \\partial ^ { d - 1 } } { \\partial x _ { 1 } \\dots \\partial x _ { i - 1 } \\partial x _ { i + 1 } \\dots \\partial x _ { d } } } \\end{array}$ , for $i = 1 \\ldots d$ . ", + "bbox": [ + 174, + 229, + 658, + 253 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "$D ^ { - i }$ computes the $d - 1$ partial derivative excluding the variable $i$ . ", + "bbox": [ + 174, + 268, + 619, + 284 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In the following we assume that $F _ { \\mathbb { P } }$ , and $F _ { \\mathbb { Q } }$ and its $d$ derivatives exist and are continuous meaning that $F _ { \\mathbb { P } }$ and $F _ { \\mathbb { Q } } \\in C ^ { d } ( \\mathcal { X } )$ . The objective function in Equation (3) can be written as follows: ", + "bbox": [ + 174, + 285, + 825, + 314 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/a5f3ef5002ea81558675e55bb133816c714f037eb108959ad66ab1af72273a1c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) = \\displaystyle \\int _ { \\mathcal { X } } f ( x ) D \\Big ( F _ { \\mathbb { P } } ( x ) - F _ { \\mathbb { Q } } ( x ) \\Big ) d x } \\\\ & { \\quad \\quad \\quad \\quad \\quad = \\displaystyle \\int _ { \\mathcal { X } } f ( x ) \\frac { \\partial } { \\partial x _ { i } } D ^ { - i } ( F _ { \\mathbb { P } } ( x ) - F _ { \\mathbb { Q } } ( x ) ) d x } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\mathrm { ( f o r ~ a n y ~ } i , \\mathrm { s i n c e ~ } F _ { \\mathbb { P } } \\mathrm { ~ a n d ~ } F _ { \\mathbb { Q } } \\in C ^ { d } ( \\mathcal { X } ) ) } \\\\ & { \\quad \\quad \\quad \\quad = - \\displaystyle \\int _ { \\mathcal { X } } \\frac { \\partial f } { \\partial x _ { i } } D ^ { - i } ( F _ { \\mathbb { P } } ( x ) - F _ { \\mathbb { Q } } ( x ) ) d x } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 329, + 700, + 452 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Let $D ^ { - } = ( D ^ { - 1 } , \\ldots , D ^ { - d } )$ it follows that: ", + "bbox": [ + 173, + 487, + 460, + 503 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/b5ceaa7c1995a14f8d2d1cbfd29b3a19b1b4ba405f4fda5e2cf411f381364131.jpg", + "text": "$$\n\\begin{array} { r c l } { \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) } & { = } & { \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\int _ { \\chi } \\frac { \\partial f } { \\partial x _ { i } } D ^ { - i } ( F _ { \\mathbb { Q } } ( x ) - F _ { \\mathbb { P } } ( x ) ) d x } \\\\ & { = } & { \\displaystyle \\frac { 1 } { d } \\int _ { \\chi } \\left. \\nabla _ { x } f ( x ) , D ^ { - } ( F _ { \\mathbb { Q } } ( x ) - F _ { \\mathbb { P } } ( x ) ) \\right. _ { \\mathbb { R } ^ { d } } d x } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 248, + 520, + 746, + 599 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Let us define $\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d }$ the space of measurable functions from $\\mathcal X ~ ~ \\mathbb R ^ { d }$ . For $g , h \\in$ $\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d }$ the dot product is defined as follows: ", + "bbox": [ + 173, + 619, + 823, + 650 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/ff87f31353f9898c7f2bbf29ab57a1bb4118c4fb13ecb3191d07e8140275131d.jpg", + "text": "$$\n\\langle g , h \\rangle _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } } = \\int _ { \\mathcal { X } } \\langle g ( x ) , h ( x ) \\rangle _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x\n$$", + "text_format": "latex", + "bbox": [ + 346, + 665, + 650, + 699 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "and the norm is given : ", + "bbox": [ + 173, + 714, + 325, + 729 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/5f64fccbbf2a9e795760a7de33f747b99bf60130b2d97b7d8b093dfcc1567b13.jpg", + "text": "$$\n\\| g \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } } = \\int _ { X } \\| g \\| _ { \\mathbb { R } ^ { d } } ^ { 2 } \\mu ( x ) d x .\n$$", + "text_format": "latex", + "bbox": [ + 379, + 739, + 617, + 773 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We can write the objective in Equation (18) in term of the dot product in $\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d }$ : ", + "bbox": [ + 168, + 790, + 746, + 808 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/7719e18f2f858292d82f2903671b0aa608a4b3cd4a510f662d0d485f621360a4.jpg", + "text": "$$\n\\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) = \\frac { 1 } { d } \\left. \\nabla _ { x } f , \\frac { D ^ { - } ( F _ { \\mathbb { Q } } - F _ { \\mathbb { P } } ) } { \\mu } \\right. _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } } .\n$$", + "text_format": "latex", + "bbox": [ + 281, + 821, + 717, + 859 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "On the other hand the constraint in Equation (3) can be written in terms of the norm in $\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d }$ : ", + "bbox": [ + 171, + 876, + 825, + 892 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/a2717f1a2dd74db0025a8c09ce269305bd1bf18898a3c09957b249f7f8b5c19b.jpg", + "text": "$$\n\\| f \\| _ { W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu ) } = \\| \\nabla _ { x } f \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } }\n$$", + "text_format": "latex", + "bbox": [ + 385, + 907, + 614, + 928 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Replacing the objective and constraint given in Equations (19) and (20) in Equation (3), we obtain: ", + "bbox": [ + 171, + 102, + 820, + 119 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/97b0a5f9eab2359015dacdd9aac8c371e326ca18071ef98d8578c2a27cdeec4e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathfrak { H } ( \\mathbb { P } , \\mathfrak { G } ) = \\frac { 1 } { d } _ { f , \\vert \\mathcal { C } _ { 2 } , \\vert \\mathcal { C } _ { 2 } , \\alpha , \\beta , 0 ; 0 } \\le \\mathfrak { C } _ { \\ge } f , \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } \\vert \\mathfrak { F } ) } { \\mu } \\Bigr \\rangle _ { \\mathcal { B } ( \\mathfrak { x } , \\mu ) \\le \\mathfrak { c } } } \\\\ & { \\ = \\frac { 1 } { d } _ { g \\le \\mathcal { G } _ { 2 } ( x , \\beta ) \\le \\mathfrak { c } , \\vert \\mathfrak { H } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { x } , \\mu } ) \\le \\mathfrak { c } } \\Big \\langle g , \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } ) } { \\mu } \\Big \\rangle _ { \\mathcal { B } _ { \\ge } ( x , \\mu ) \\ge \\mathfrak { c } } } \\\\ & { \\ = \\frac { 1 } { d } \\bigg \\Vert \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } ) } { \\mu } \\bigg \\Vert _ { \\mathcal { B } _ { \\ge } ( x , \\mu ) \\le \\mathfrak { c } } } \\\\ & { \\ \\left( \\mathtt { B y ~ d e n i n i o n ~ o f ~ } \\Vert \\cdot \\Vert _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) \\le \\mathfrak { c } } , g ^ { - } = \\frac { D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) - D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) } { \\mu ( \\mathfrak { x } ) } \\frac { 1 } { \\left. \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } ) } { \\mu } \\right. _ { \\mathcal { B } _ { \\ge } ( \\mathcal { X } , \\mu ) \\le \\mathfrak { c } } } \\right) } \\\\ & \\ = \\frac { 1 } { d } \\sqrt { \\int _ { x } \\frac { \\Vert D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) - D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) \\Vert _ { \\mathcal { A } ( \\mathcal { X } , \\mu ) } ^ { 2 } } { \\mu ( \\mathfrak { x } ) } d \\mathfrak { x } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 121, + 828, + 345 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Hence we find also that the optimal critic $f ^ { * }$ satisfies: ", + "bbox": [ + 174, + 347, + 527, + 361 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/e2a8405da9d66cf04b46a09981465e33ed699ddef6794bf122d29007c27d98f4.jpg", + "text": "$$\n\\nabla _ { x } f ^ { * } ( x ) = \\frac { D ^ { - } F _ { \\mathbb { Q } } ( x ) - D ^ { - } F _ { \\mathbb { P } } ( x ) } { \\mu ( x ) } \\frac { 1 } { \\Big \\| \\frac { D ^ { - } ( F _ { \\mathbb { Q } } - F _ { \\mathbb { P } } ) } { \\mu } \\Big \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } } } .\n$$", + "text_format": "latex", + "bbox": [ + 277, + 363, + 687, + 411 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof of Lemma $\\cdot$ . ", + "bbox": [ + 173, + 444, + 297, + 458 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/0611e104c0c0ee07559606e0f2c96ba20e8c410008e3997a61e6623b0731c165.jpg", + "text": "$$\n\\begin{array} { r c l } { \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) } & { = } & { \\displaystyle \\frac { 1 } { d } \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , D ^ { - } ( F _ { \\mathbb { Q } } ( x ) - F _ { \\mathbb { P } } ( x ) ) \\right. _ { \\mathbb { R } ^ { d } } d x } \\\\ & { = } & { \\displaystyle S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , \\frac { D ^ { - } ( F _ { \\mathbb { Q } } ( x ) - F _ { \\mathbb { P } } ( x ) ) } { \\mu ( x ) d S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } \\right. _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x } \\\\ & { = } & { \\displaystyle S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\right. \\mu ( x ) d x } \\\\ & { = } & { \\displaystyle S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\left. f , f ^ { * } \\right. _ { W _ { 0 } ^ { 1 , 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 204, + 462, + 795, + 588 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Hence we have: ", + "bbox": [ + 173, + 588, + 279, + 603 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/22fb9c98448596c9e95e9dea3a007cdce4c4dcc55ba8d44c667739f3e4943ab5.jpg", + "text": "$$\n\\operatorname* { s u p } _ { f \\in \\mathcal { M } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } } \\leq 1 } \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) = S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\operatorname* { s u p } _ { f \\in \\mathcal { M } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } } \\leq 1 } \\langle f , f ^ { * } \\rangle _ { W _ { 0 } ^ { 1 , 2 } } ,\n$$", + "text_format": "latex", + "bbox": [ + 227, + 606, + 767, + 640 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "It follows therefore that: ", + "bbox": [ + 173, + 643, + 334, + 657 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/4a1fe5d02a45604832f0c2d765a821ad936cac8baa7aad2cafc27a21eacf0ab5.jpg", + "text": "$$\n\\mathcal { S } _ { \\mathcal { H } } ( \\mathbb { P } , \\mathbb { Q } ) = S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\operatorname* { s u p } _ { f \\in \\mathcal { H } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } } \\leq 1 } \\langle f , f ^ { * } \\rangle _ { W _ { 0 } ^ { 1 , 2 } }\n$$", + "text_format": "latex", + "bbox": [ + 326, + 660, + 668, + 694 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We conclude that the Sobolev IPM can be approximated in arbitrary space as long as it has enough capacity to approximate the optimal critic. Interestingly the approximation error is measured now with the Sobolev semi-norm, while in Fisher it was measured with the Lebesgue norm. Approximations with Sobolev Semi-norms are stronger then Lebesgue norms as given by the Poincare inequality $( \\vert \\vert f \\vert \\vert _ { \\mathcal { L } _ { 2 } } \\le C \\left. f \\right. _ { W _ { 0 } ^ { 1 , 2 } } )$ , meaning if the error goes to zero in Sobolev sense it also goes to zero in the Lebesgue sense , but the converse is not true. ", + "bbox": [ + 173, + 727, + 825, + 815 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof of Theorem 3. The proof follows similar arguments in the proofs of the analysis of Laplacian regularization in semi-supervised learning studied by (Alaoui et al., 2016). ", + "bbox": [ + 171, + 828, + 823, + 858 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/16204a8fe673ff37deede84c74672bcba84a9389a4840469eb842215b7d60971.jpg", + "text": "$$\n\\begin{array} { r l } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } & { { } = \\operatorname* { s u p } _ { f \\in W _ { 0 } ^ { 1 , 2 } } \\Big \\{ \\mathbb { E } _ { x \\sim \\mathbb { P } } \\left[ f ( x ) \\right] - \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ f ( x ) \\right] \\Big \\} } \\\\ { s . t . } & { { } \\quad \\mathbb { E } _ { x \\sim \\mu } \\| \\nabla f ( x ) \\| _ { 2 } ^ { 2 } \\leq 1 , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 880, + 681, + 928 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Note that this problem is convex in $f$ (Ekeland & Turnbull, 1983). Writing the lagrangian for equation (21) we get : ", + "bbox": [ + 171, + 102, + 825, + 133 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/7d9b8b9ea978265c7c127c64cefd89d6075ac919a190d929993f3be1900b4710.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle { L ( f , \\lambda ) = \\mathbb { E } _ { x \\sim \\mathbb { P } } \\left[ f ( x ) \\right] - \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ f ( x ) \\right] + \\frac { \\lambda } { 2 } \\Big ( 1 - \\mathbb { E } _ { x \\sim \\mu } \\| \\nabla _ { x } f ( x ) \\| _ { 2 } ^ { 2 } \\Big ) } } \\\\ { \\displaystyle { \\quad = \\int _ { \\mathcal { X } } f ( x ) \\left( \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) \\right) d x + \\frac { \\lambda } { 2 } \\Big ( 1 - \\int _ { \\mathcal { X } } \\| \\nabla _ { x } f ( x ) \\| _ { 2 } ^ { 2 } \\mu ( x ) d x \\Big ) } } \\\\ { \\displaystyle { \\quad = \\int _ { \\mathcal { X } } f ( x ) \\mu _ { 1 } ( x ) d x + \\frac { \\lambda } { 2 } \\Big ( 1 - \\int _ { \\mathcal { X } } \\| \\nabla _ { x } f ( x ) \\| _ { 2 } ^ { 2 } \\mu ( x ) d x \\Big ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 138, + 743, + 239 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We denote $\\left( \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) \\right)$ as $\\mu _ { 1 } ( x )$ .To get the optimal $f$ , we need to apply KKT conditions on the above equation. ", + "bbox": [ + 174, + 244, + 823, + 275 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/4b3acba07029e2f916f9fafefde867c118b2f61a87a2ee021db7672ffdae327e.jpg", + "text": "$$\nL ( f , \\lambda ) = \\int _ { \\mathcal { X } } f ( x ) \\mu _ { 1 } ( x ) d x + { \\frac { \\lambda } { 2 } } { \\Big ( } 1 - \\int _ { \\mathcal { X } } \\| \\nabla _ { x } f ( x ) \\| _ { 2 } ^ { 2 } \\ \\mu ( x ) d x { \\Big ) }\n$$", + "text_format": "latex", + "bbox": [ + 279, + 281, + 717, + 315 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "From the calculus of variations: ", + "bbox": [ + 173, + 321, + 383, + 337 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/0426c250e67dbe790cf5f8af4a301ba42faf5f66c436f04cc92e35c3010ee208.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\sum _ { k = 1 } ^ { \\infty } ( f + \\epsilon h , \\lambda ) = \\int _ { x } ( f + \\epsilon h ) ( x ) \\mu _ { 1 } ( x ) d x + \\frac { \\lambda } { 2 } \\Bigl ( 1 - \\int _ { x } \\| \\nabla _ { x } ( f + \\epsilon h ) ( x ) \\| _ { 2 } ^ { 2 } \\mu ( x ) d x \\Bigr ) } \\\\ { \\displaystyle = \\int _ { x } ( f ( x ) + \\epsilon h ( x ) ) \\mu _ { 1 } ( x ) d x + \\frac { \\lambda } { 2 } \\Bigl ( 1 - \\int _ { x } \\left. \\nabla _ { z } \\left( f + \\epsilon h \\right) ( x ) , \\nabla _ { z } \\left( f + \\epsilon h \\right) ( x ) \\right. \\mu ( x ) } \\\\ { \\displaystyle = \\int _ { x } ( f ( x ) + \\epsilon h ( x ) ) \\mu _ { 1 } ( x ) d x } \\\\ { \\displaystyle + \\frac { \\lambda } { 2 } \\Bigl ( 1 - \\int _ { x } \\left. \\| \\nabla _ { z } f ( x ) \\| _ { 2 } ^ { 2 } + 2 \\epsilon \\langle \\nabla _ { x } f ( x ) , \\nabla _ { x } h ( x ) \\rangle + \\mathcal { O } ( \\epsilon ^ { 2 } ) \\right. \\mu ( x ) d x \\Bigr ) } \\\\ { \\displaystyle = L ( f , \\lambda ) + \\epsilon \\int _ { x } h ( x ) \\mu _ { 1 } ( x ) d x - \\lambda \\epsilon \\int _ { x } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } h ( x ) \\right. \\mu ( x ) d x + \\mathcal { O } ( \\epsilon ^ { 2 } ) } \\\\ { \\displaystyle = L ( f , \\lambda ) + \\epsilon \\Bigl [ \\int _ { x } h ( x ) \\mu _ { 1 } ( x ) d x - \\lambda \\int _ { x } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } h ( x ) \\right. \\mu ( x ) d x \\Bigr ] + \\mathcal { O } ( \\epsilon ^ { 2 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 342, + 839, + 546 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Now we apply integration by part and set $h$ to be zero at boundary as in (Alaoui et al., 2016). We get : ", + "bbox": [ + 173, + 556, + 826, + 587 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/7044b2b25edcff7a49e8b371559477ca589d2e81cbd9fe38756961647bc84d2d.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\int _ { \\mathcal X } \\langle \\nabla _ { x } f ( x ) , \\nabla _ { x } h ( x ) \\rangle \\mu ( x ) d x = \\int _ { \\mathcal X } \\langle \\nabla _ { x } f ( x ) \\mu ( x ) , \\nabla _ { x } h ( x ) \\rangle d x } \\\\ { = \\displaystyle \\oint h ( x ) \\mu ( x ) \\nabla _ { x } f ( x ) . n ( x ) d S ( x ) - \\int _ { \\mathcal X } d i v \\big ( \\mu ( x ) \\nabla _ { x } f ( x ) \\big ) h ( x ) d x } \\\\ { = - \\displaystyle \\int _ { \\mathcal X } d i v \\big ( \\mu ( x ) \\nabla _ { x } f ( x ) \\big ) h ( x ) d x } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 611, + 839, + 724 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Hence, ", + "bbox": [ + 173, + 736, + 222, + 750 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/a9dd9b5d58684c4191e6db47e6dbcaf76d18e8ead39b57b378423481914d9421.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\cal L } \\big ( f + \\epsilon h , \\lambda \\big ) = { \\cal L } \\big ( f , \\lambda \\big ) + \\epsilon \\bigg [ \\int _ { \\mathcal { X } } \\mu _ { 1 } ( x ) h ( x ) d x + \\lambda \\int _ { \\mathcal { X } } d i v \\big ( \\mu ( x ) \\nabla _ { x } f ( x ) \\big ) h ( x ) d x \\bigg ] + { \\mathcal O } ( \\epsilon ^ { 2 } ) } } \\\\ { { \\displaystyle ~ = { \\cal L } \\big ( f , \\lambda \\big ) + \\epsilon \\int _ { \\mathcal { X } } \\bigg ( \\mu _ { 1 } ( x ) + \\lambda ~ d i v \\big ( \\mu ( x ) \\nabla _ { x } f ( x ) \\big ) \\bigg ) h ( x ) d x + { \\mathcal O } ( \\epsilon ^ { 2 } ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 756, + 812, + 825 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The functional derivative of $L ( f , \\lambda )$ , at any test function $h$ vanishing on the boundary: ", + "bbox": [ + 171, + 837, + 738, + 853 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/81c93e07c6b99398e043e25687dda7c763e16ed9766d7005b83e61eb6c834a60.jpg", + "text": "$$\n\\begin{array} { r c l } { \\displaystyle \\int _ { \\mathcal { X } } \\frac { \\partial L ( f , \\lambda ) } { \\partial f } ( x ) h ( x ) d x } & { = } & { \\displaystyle \\operatorname* { l i m } _ { \\epsilon \\to 0 } \\frac { L ( f + \\epsilon h , \\lambda ) - L ( f , \\lambda ) } { \\epsilon } } \\\\ & { = } & { \\displaystyle \\int _ { \\mathcal { X } } \\left( \\mu _ { 1 } ( x ) + \\lambda \\ d i v \\big ( \\mu ( x ) \\nabla _ { x } f ( x ) \\big ) \\right) h ( x ) d x } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 859, + 754, + 930 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Hence we have: ", + "bbox": [ + 174, + 103, + 279, + 117 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/c1523d2ee180d7ec6b5b83dfd878acebcb06f1da3767177b82e284234ce682b1.jpg", + "text": "$$\n\\frac { \\partial L ( f , \\lambda ) } { \\partial f } ( x ) = \\mu _ { 1 } ( x ) + \\lambda ~ d i v \\big ( \\mu ( x ) \\nabla _ { x } f ( x ) \\big )\n$$", + "text_format": "latex", + "bbox": [ + 343, + 113, + 653, + 147 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "For the optimal $f ^ { * } , \\lambda ^ { * }$ first order optimality condition gives us: ", + "bbox": [ + 173, + 147, + 588, + 162 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/cd4ce404d32cd6a9068826b7d6cb1688987f49294c15cd76d147d0c29859e519.jpg", + "text": "$$\n\\mu _ { 1 } ( x ) + \\lambda ^ { * } \\ d i v \\big ( \\mu ( x ) \\nabla _ { x } f ^ { * } ( x ) \\big ) = 0\n$$", + "text_format": "latex", + "bbox": [ + 374, + 165, + 624, + 184 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 188, + 202, + 202 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/2a459cb4b29ba9ae1585c3ca41574b629dfa4958eb92e5bb5b99c8003c3abd72.jpg", + "text": "$$\n\\int _ { \\chi } \\left\\| \\nabla _ { x } f ^ { * } ( x ) \\right\\| ^ { 2 } \\mu ( x ) d x = 1\n$$", + "text_format": "latex", + "bbox": [ + 383, + 203, + 580, + 237 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Note that (See for example (Alaoui et al., 2016)) : ", + "bbox": [ + 171, + 239, + 501, + 255 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/b1bef5a7b6eed9ea60d574973bfdbcfd15bb6c6be883f59d037b727efff13616.jpg", + "text": "$$\n\\begin{array} { r } { d i v \\big ( \\mu ( x ) \\nabla _ { x } f ^ { * } ( x ) \\big ) = \\mu ( x ) \\Delta _ { 2 } f ^ { * } ( x ) + \\langle \\nabla _ { x } \\mu ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\rangle , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 257, + 700, + 276 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "since $d i v ( \\nabla _ { x } f ^ { * } ( x ) ) = \\Delta _ { 2 } f ^ { * } ( x )$ . Hence from equation (22) ", + "bbox": [ + 173, + 280, + 571, + 296 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/beeec075d5361465cef2a89020514bc6d424e95834e7833c291d2c7832b2bd2d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mu _ { 1 } ( x ) + \\lambda ^ { * } \\ : d i v \\ : \\left( \\mu ( x ) \\nabla _ { x } f ^ { * } ( x ) \\right) = 0 } \\\\ & { \\Rightarrow \\mu _ { 1 } ( x ) + \\lambda ^ { * } \\ : \\left( \\mu ( x ) \\Delta _ { 2 } f ^ { * } ( x ) + \\left. \\nabla _ { x } \\mu ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\right. \\right) = 0 } \\\\ & { \\Rightarrow \\mu _ { 1 } ( x ) + \\lambda ^ { * } \\ : \\mu ( x ) \\Delta _ { 2 } f ^ { * } ( x ) + \\lambda ^ { * } \\langle \\nabla _ { x } \\mu ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\rangle = 0 } \\\\ & { \\Rightarrow \\Delta _ { 2 } f ^ { * } ( x ) + \\left. \\frac { \\nabla _ { x } \\mu ( x ) } { \\mu ( x ) } , \\nabla _ { x } f ^ { * } ( x ) \\right. + \\frac { \\mu _ { 1 } ( x ) } { \\lambda ^ { * } \\mu ( x ) } = 0 } \\\\ & { \\Rightarrow \\Delta _ { 2 } f ^ { * } ( x ) + \\left. \\nabla _ { x } \\log \\mu ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\right. + \\frac { \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) } { \\lambda ^ { * } \\mu ( x ) } = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 292, + 300, + 697, + 430 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Hence $f ^ { * } , \\lambda ^ { * }$ satisfies : ", + "bbox": [ + 173, + 457, + 328, + 472 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/10dd53780e566aea6ffbd65f50fa596604613800d58ad75d9aadcfe5ab19643d.jpg", + "text": "$$\n\\Delta _ { 2 } f ^ { * } ( x ) + \\langle \\nabla _ { x } \\log \\mu ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\rangle + \\frac { \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) } { \\lambda ^ { * } \\mu ( x ) } = 0\n$$", + "text_format": "latex", + "bbox": [ + 305, + 477, + 692, + 510 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 515, + 202, + 527 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/cca5768b101dcd5a1b17d65ddab7e431bafb05a06387687c1c4c28707fdb121b.jpg", + "text": "$$\n\\int _ { \\mathcal { X } } \\left\\| \\nabla _ { x } f ^ { * } ( x ) \\right\\| ^ { 2 } \\mu ( x ) d x = 1 .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 523, + 599, + 556 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Let us verify that the optimal critic as found in the geometric definition (Theorem 2) of Sobolev IPM that satisfies: ", + "bbox": [ + 174, + 556, + 825, + 585 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/a385e01f2a089778c02d72b0eb70db6f2299d5d8798cac41695c7826794c564f.jpg", + "text": "$$\n\\nabla _ { i } f ^ { * } ( x ) = \\frac { \\partial f ^ { * } ( X ) } { \\partial x _ { i } } = \\frac { D ^ { - i } F _ { \\mathbb { Q } } ( x ) - D ^ { - i } F _ { \\mathbb { P } } ( x ) } { \\lambda ^ { * } d ~ \\mu ( x ) } ~ \\forall ~ i \\in [ d ] ,\n$$", + "text_format": "latex", + "bbox": [ + 295, + 587, + 700, + 621 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "satisfies indeed the PDE. ", + "bbox": [ + 173, + 625, + 338, + 640 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "From equation (27), we want to compute $\\frac { \\partial ^ { 2 } f ( \\boldsymbol { x } ) } { \\partial \\boldsymbol { x } _ { i } ^ { 2 } }$ for all $i$ ", + "bbox": [ + 173, + 643, + 547, + 666 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/0b45fb9b896e9bd7757e9e66019989a5525ac7167f8f974dcd6b262eeadd8230.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { \\partial ^ { 2 } f ( x ) } { \\partial x _ { i } ^ { 2 } } = \\frac { 1 } { \\lambda ^ { * } d } \\Bigg [ \\frac { \\mu ( x ) \\left[ \\frac { \\partial } { \\partial x _ { i } } ( D ^ { - i } F _ { \\mathbb { Q } } ( x ) - D ^ { - i } F _ { \\mathbb { P } } ( x ) ) \\right] - \\left[ D ^ { - i } F _ { \\mathbb { Q } } ( x ) - D ^ { - i } F _ { \\mathbb { P } } ( x ) \\right] \\nabla _ { i } \\mu ( X ) } { \\mu ^ { 2 } ( x ) } \\Bigg ] } \\\\ & { \\quad \\quad \\quad = \\frac { 1 } { \\lambda ^ { * } d } \\Bigg [ \\frac { \\mu ( x ) \\left[ \\mathbb { Q } ( x ) - \\mathbb { P } ( x ) \\right] - \\left[ D ^ { - i } F _ { \\mathbb { Q } } ( x ) - D ^ { - i } F _ { \\mathbb { P } } ( x ) \\right] \\nabla _ { i } \\mu ( X ) } { \\mu ^ { 2 } ( x ) } \\Bigg ] } \\\\ & { \\quad \\quad \\quad = \\frac { \\mathbb { Q } ( x ) - \\mathbb { P } ( x ) } { \\lambda ^ { * } d \\mu ( x ) } - \\frac { \\nabla _ { i } \\mu ( x ) } { \\mu ( x ) } \\nabla _ { i } f ^ { * } ( x ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 186, + 670, + 810, + 792 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Hence, ", + "bbox": [ + 173, + 794, + 222, + 808 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/262116e0058415058c1dd78947f9663e7b7be4922baa2f2fa447f9872c3ef805.jpg", + "text": "$$\n{ \\frac { \\partial ^ { 2 } f ( x ) } { \\partial x _ { i } ^ { 2 } } } + { \\frac { \\nabla _ { i } \\mu ( x ) } { \\mu ( x ) } } \\nabla _ { i } f ( x ) + { \\frac { \\left( \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) \\right) } { \\lambda ^ { * } d \\ \\mu ( x ) } } = 0\n$$", + "text_format": "latex", + "bbox": [ + 330, + 811, + 666, + 848 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Adding equation (28) for all $i \\in [ d ]$ , we get : ", + "bbox": [ + 174, + 859, + 465, + 875 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/3d53af26ce2d5382ee8ad3292746822cbb63110945a572ecdf223f3332304700.jpg", + "text": "$$\n\\sum _ { i = 1 } ^ { d } \\left( \\frac { \\partial ^ { 2 } f ( x ) } { \\partial x _ { i } ^ { 2 } } + \\frac { \\nabla _ { i } \\mu ( x ) } { \\mu ( x ) } \\nabla _ { i } f ( x ) + \\frac { \\left( \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) \\right) } { \\lambda ^ { * } d ~ \\mu ( x ) } \\right) = 0\n$$", + "text_format": "latex", + "bbox": [ + 303, + 878, + 692, + 922 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "As a result, the solution $f ^ { * }$ of the partial differential equation given in equation (25) satisfies the following : ", + "bbox": [ + 173, + 103, + 823, + 131 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/44dbea6a536cb925ecc8309bbfa451d1a5e906fefef60ca00d84d9aed3904417.jpg", + "text": "$$\n\\frac { \\partial f ^ { * } ( x ) } { \\partial x _ { i } } = \\frac { D ^ { - i } F _ { \\mathbb { Q } } ( x ) - D ^ { - i } F _ { \\mathbb { P } } ( x ) } { \\lambda ^ { * } d ~ \\mu ( x ) } ~ \\forall i \\in [ d ]\n$$", + "text_format": "latex", + "bbox": [ + 341, + 128, + 656, + 165 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Using the constraint in (26) we can get the value of $\\lambda ^ { * }$ : ", + "bbox": [ + 173, + 174, + 540, + 189 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/c823ecf201c6656506826bf064a968d261c6cd69c715fcf224521bb05e3a3655.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\displaystyle \\int \\| \\nabla f ^ { * } ( \\boldsymbol { x } ) \\| ^ { 2 } \\mu ( \\boldsymbol { x } ) d \\boldsymbol { x } = 1 } \\\\ & { \\Rightarrow \\displaystyle \\int \\sum _ { i = 1 } ^ { d } \\left( \\frac { \\partial f ^ { * } ( \\boldsymbol { x } ) } { \\partial x _ { i } } \\right) ^ { 2 } \\mu ( \\boldsymbol { x } ) d \\boldsymbol { x } = 1 } \\\\ & { \\Rightarrow \\lambda ^ { * } = \\displaystyle \\frac { 1 } { d } \\sqrt { \\displaystyle \\sum _ { i = 1 } ^ { d } \\int \\frac { \\left( D ^ { - i } F _ { \\mathbb { Q } } ( \\boldsymbol { x } ) - D ^ { - i } F _ { \\mathbb { P } } ( \\boldsymbol { x } ) \\right) ^ { 2 } } { \\mu ( \\boldsymbol { x } ) } d \\boldsymbol { x } } = S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 195, + 714, + 324 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proof of Theorem 4. Define the Stein operator (Oates et al., 2017): ", + "bbox": [ + 173, + 364, + 612, + 381 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/6d3416316aee793bd2d6dc18e1f1704ec9b17d4ddc80d99466a1814424dc821d.jpg", + "text": "$$\n\\begin{array} { l l l } { { T ( \\mu ) [ \\nabla _ { x } f ( x ) ] } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } \\log \\mu ( x ) \\right. + \\displaystyle \\frac { 1 } { 2 } \\left. \\nabla _ { x } , \\nabla _ { x } f ( x ) \\right. } } \\\\ { { } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } \\log \\mu ( x ) \\right. + \\displaystyle \\frac { 1 } { 2 } \\Delta _ { 2 } f ( x ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 274, + 386, + 720, + 449 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "This operator was later used in defining the Stein discrepancy (Gorham & Mackey, 2015; Liu et al., 2016; Chwialkowski et al., 2016; Liu, 2017). ", + "bbox": [ + 171, + 450, + 823, + 479 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Recall that Barbour generator theory provides us a way of constructing such operators that produce mean zero function under $\\mu$ . It is easy to verify that: ", + "bbox": [ + 174, + 487, + 823, + 515 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/f8c347b195c8c08fb3d2aa3ef252e2e9c2891841e85998a89e67c1105cece808.jpg", + "text": "$$\n\\mathbb { E } _ { x \\sim \\mu } T ( \\mu ) \\nabla _ { x } f ( x ) = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 416, + 521, + 581, + 539 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Recall that this operator arises from the overdamped Langevin diffusion, defined by the stochastic differential equation: ", + "bbox": [ + 174, + 551, + 825, + 579 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/9a775c228f7a2803176a2f73ae2d6c0b7dd4ab179b80e48268df058f76dd8724.jpg", + "text": "$$\nd x _ { t } = \\frac { 1 } { 2 } \\nabla _ { x } \\log \\mu ( x _ { t } ) + d W _ { t }\n$$", + "text_format": "latex", + "bbox": [ + 400, + 578, + 596, + 608 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where $( W _ { t } ) _ { t \\geq 0 }$ is a Wiener process. This is related to plug and play networks for generating samples if the distribution is known, using the stochastic differential equation. ", + "bbox": [ + 174, + 611, + 826, + 640 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "From Theorem 3, it is easy to see that the PDE the Sobolev Critic $( f ^ { * } , \\lambda ^ { * } = { \\cal S } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) )$ can be written in term of Stein Operator as follows: ", + "bbox": [ + 174, + 659, + 823, + 688 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/13a63ed56bfbc92d7850451825a15da185431897b980e264e1a4742e8525a7e6.jpg", + "text": "$$\nT ( \\mu ) [ \\nabla _ { x } f ^ { * } ] ( x ) = \\frac { 1 } { 2 \\lambda ^ { * } } \\frac { \\mathbb { Q } ( x ) - \\mathbb { P } ( x ) } { \\mu ( x ) }\n$$", + "text_format": "latex", + "bbox": [ + 374, + 694, + 624, + 728 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Taking absolute values and the expectation with respect to $\\mathbb { Q }$ : ", + "bbox": [ + 173, + 733, + 576, + 748 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/3fd5d739cc15b4036ab687140a42ca69ddf4b38c4231467557e8a98af40f2ed6.jpg", + "text": "$$\n\\left| \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ T ( \\mu ) \\nabla _ { x } f ^ { * } ( x ) \\right] \\right| = { \\frac { 1 } { 2 S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } } \\left| \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ { \\frac { \\mathbb { Q } ( x ) - \\mathbb { P } ( x ) } { \\mu ( x ) } } \\right] \\right|\n$$", + "text_format": "latex", + "bbox": [ + 290, + 753, + 709, + 790 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Recall that the definition of Stein Discrepancy : ", + "bbox": [ + 174, + 801, + 486, + 818 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/831616c30044b22fdf8bee49a8388a5f44daad6efb93275abce91af2a02ed63b.jpg", + "text": "$$\n{ \\mathbb S } ( \\mathbb Q , \\mu ) = \\operatorname* { s u p } _ { \\vec { g } \\in \\mathcal L _ { 2 } ( \\mathcal X , \\mu ) ^ { \\otimes d } } \\left| \\mathbb E _ { \\boldsymbol { x } \\sim \\mathbb Q } \\left[ T ( \\mu ) \\vec { g } ( \\boldsymbol { x } ) \\right] \\right|\n$$", + "text_format": "latex", + "bbox": [ + 356, + 823, + 642, + 852 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "It follows that Sobolev IPM critic satisfies: ", + "bbox": [ + 174, + 864, + 455, + 880 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/a190696d8767374228628efbf39585e8252f102eb0821ad165a16738f674c1e2.jpg", + "text": "$$\n| \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ T ( \\mu ) \\nabla _ { x } f ^ { * } ( x ) \\right] | \\leq \\mathbb { S } ( \\mathbb { Q } , \\mu ) ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 886, + 614, + 904 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Hence we have the following inequality: ", + "bbox": [ + 174, + 909, + 441, + 924 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/9cc512735ea0762daf1b551fd65435bd573c1723f1adfe4a1105d23ea41cc7a5.jpg", + "text": "$$\n\\frac { 1 } { 2 S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } \\left| \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { Q } } \\left[ \\frac { \\mathbb { Q } ( \\boldsymbol { x } ) - \\mathbb { P } ( \\boldsymbol { x } ) } { \\mu ( \\boldsymbol { x } ) } \\right] \\right| \\leq \\mathbb { S } ( \\mathbb { Q } , \\mu )\n$$", + "text_format": "latex", + "bbox": [ + 343, + 118, + 656, + 154 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "This is equivalent to: ", + "bbox": [ + 173, + 160, + 312, + 175 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/97c900eaf9b2de3c35f5d3e37769a081bce9215000b150e7a7c095b7fbee1211.jpg", + "text": "$$\n\\left| \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ { \\frac { \\mathbb { Q } ( x ) - \\mathbb { P } ( x ) } { \\mu ( x ) } } \\right] \\right| \\leq 2 \\qquad { \\underset { \\Delta \\in { \\mathbb { Q } } , \\mu } { \\underbrace { \\mathbb { S } ( \\mathbb { Q } , \\mu ) } } } \\qquad { \\underset { \\Delta \\in { \\mathbb { P } } } { \\underbrace { { \\mathcal { S } } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } } }\n$$", + "text_format": "latex", + "bbox": [ + 251, + 184, + 735, + 220 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Similarly we obtain: ", + "bbox": [ + 173, + 239, + 308, + 255 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/42bbfe8f5bb6cfe693c611d66a286804d28581cfbe8ba2c7a8ae2257f5bfe235.jpg", + "text": "$$\n\\left| \\mathbb { E } _ { x \\sim \\mathbb { P } } \\left[ { \\frac { \\mathbb { Q } ( x ) - \\mathbb { P } ( x ) } { \\mu ( x ) } } \\right] \\right| \\leq 2 \\qquad \\underbrace { { \\mathbb { S } } ( \\mathbb { P } , \\mu ) } _ { \\in { \\mathcal { S } } } \\qquad \\underbrace { { \\mathcal { S } } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } _ { \\in { \\mathcal { S } } }\n$$", + "text_format": "latex", + "bbox": [ + 277, + 265, + 714, + 300 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "For instance consider $\\mu = \\mathbb { P }$ , we have therefore: ", + "bbox": [ + 173, + 328, + 490, + 343 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/51ead85e6389951586621f04237342e0df5fc9c07e81281e20e78e0ac065053c.jpg", + "text": "$$\n{ \\frac { 1 } { 2 } } \\left| \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ { \\frac { \\mathbb { Q } ( x ) } { \\mathbb { P } ( x ) } } \\right] - 1 \\right| \\leq \\mathbb { S } ( \\mathbb { Q } , \\mathbb { P } ) S _ { \\mathbb { P } } ( \\mathbb { P } , \\mathbb { Q } ) .\n$$", + "text_format": "latex", + "bbox": [ + 352, + 352, + 645, + 387 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Note that the left hand side of the inequality is not the total variation distance. ", + "bbox": [ + 173, + 404, + 687, + 419 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Hence for a sequence $\\mathbb { Q } _ { n }$ if the Sobolev distance goes $S _ { \\mathbb { P } } ( \\mathbb { P } , \\mathbb { Q } _ { n } ) 0$ , the ratio $\\begin{array} { r } { r _ { n } ( x ) = \\frac { \\mathbb Q _ { n } ( x ) } { \\mathbb P ( x ) } } \\end{array}$ converges in expectation (w.r.t to $\\mathbb { Q }$ ) to 1. The speed of the convergence is given by the Stein Discrepancy $\\mathbb { S } ( \\mathbb { Q } _ { n } , \\mathbb { P } )$ . ", + "bbox": [ + 174, + 426, + 825, + 474 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "One important observation here is that convergence of PDF ratio is weaker than the conditional CDF as given by the Sobolev distance and of the good fitness of score function as given by Stein discrepancy. ", + "bbox": [ + 174, + 479, + 825, + 522 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "C TEXT EXPERIMENTS: ADDITIONAL PLOTS ", + "text_level": 1, + "bbox": [ + 173, + 566, + 560, + 583 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Comparison of annealed versus non annealed smoothing of $\\mathbb { P } _ { r }$ in Sobolev GAN. ", + "text_level": 1, + "bbox": [ + 173, + 601, + 727, + 616 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/9bc07e8ad8977d3d3e0990973f1d012506b6beae1acd22e818815329398759c9.jpg", + "image_caption": [ + "Figure 6: Comparison of annealed versus non annealed smoothing of $\\mathbb { P } _ { r }$ in Sobolev GAN. We see that annealed smoothing outperforms the non annealed smoothing experiments. " + ], + "image_footnote": [], + "bbox": [ + 367, + 645, + 614, + 767 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Sobolev GAN versus WGAN-GP with RNN. We fix the generator architecture to Resnets. The experiments of using RNN (GRU) as the critic architecture for WGAN-GP and Sobolev is shown in Figure 7 where we used $\\mu = \\mu _ { G P }$ for both cases. We only apply gradient clipping to stabilize the performance without other tricks. We can observe that using RNN degrades the performance. We think that this is due to an optimization issue and a difficulty in training RNN under the GAN objective without any pre-training or conditioning. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/b33296dfeec538aff71afa249d3c78cee91fb6d6002cb89df8dd5f0cad50bf6f.jpg", + "image_caption": [ + "Figure 7: Result of WGAN-GP and Sobolev with RNNs. " + ], + "image_footnote": [], + "bbox": [ + 240, + 114, + 745, + 258 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/22b283fdcd36a259ef1de1da1ae8a6d80587aa0ad141b9ca955c8082e7f17492.jpg", + "image_caption": [ + "Figure 8: Text samples from various GANs considered in this paper. " + ], + "image_footnote": [], + "bbox": [ + 367, + 314, + 614, + 542 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "D SSL: HYPERPARAMETERS AND ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 173, + 602, + 602, + 617 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "For our SSL experiments on CIFAR-10, we use Adam with learning rate $\\eta = 2 \\mathrm { { e } - 4 }$ , $\\beta _ { 1 } ~ = ~ 0 . 5$ and $\\beta _ { 2 } = 0 . 9 9 9$ , both for critic $f$ (without BN) and Generator (with BN). We selected $\\lambda _ { C E } = 1 . 5$ from [0.8, 1.5, 3.0, 5.0]. We train all models for 350 epochs. We used some L2 weight decay: 1e−6 on $\\omega , S$ (i.e. all layers except last) and 1e 3 weight decay on the last layer $v$ . For formulation 1 (Fisher only) we have $\\rho _ { F } = 1 \\mathrm { e } { - 7 }$ , modified critic learning rate $\\eta _ { D } = 1 \\mathrm { e } { - 4 }$ , critic iters $n _ { c } = 2$ . For formulation 2 (Sobolev $^ +$ Fisher) we have $\\rho _ { F } = 5 \\mathrm { e } - 8$ , $\\rho _ { S } = 2 \\mathrm { e } - 8$ , critic iters $n _ { c } = 1$ . For the WGAN-GP (Gulrajani et al., 2017) baseline SSL experiment we followed the original paper with critic iters $n _ { c } = 5$ , $\\eta _ { G } = \\eta _ { D } = 1 \\mathrm { e } { - 4 }$ , Adam $\\beta _ { 2 } { = } 0 . 9$ and GP weight $\\lambda _ { G P } = 1 0 . 0$ . Architectures are as below. We determined $\\lambda _ { C E } = 0 . 3$ to be optimal from $[ 0 . 0 3 , 0 . 1 , 0 . 3 , 1 . 0 , 3 . 0 ]$ . As mentioned in Section 6.2, the $\\mathrm { K } { + } 1$ critic formulation is not able to fit the training set with the GP constraint, so we fall back to the plain critic formulation where the critic $\\langle v , \\Phi _ { \\omega } ( \\bar { x } ) \\rangle$ does not interact with the classifier $\\langle S , \\Phi _ { \\omega } ( x ) \\rangle$ . ", + "bbox": [ + 173, + 633, + 825, + 801 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Architecture: ", + "text_level": 1, + "bbox": [ + 174, + 808, + 261, + 821 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "(6): ConvTranspose2d(128, 64, kernel_size=(4, 4), stride=(2, 2), padding=(1, 1), bias=False) (7): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (8): ReLU (inplace) (9): Conv2d(64, 64, kernel_siz $\\div =$ (3, 3), stride=(1, 1), padding=(1, 1), bias=False) (10): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (11): ReLU (inplace) (12): Conv2d(64, 64, kernel_size $: =$ (3, 3), strid $\\div =$ (1, 1), padding $=$ (1, 1), bias=False) (13): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True) (14): ReLU (inplace) (15): ConvTranspose2d(64, 3, kernel_size $: =$ (4, 4), strid $: =$ (2, 2), padding $^ { \\ast = }$ (1, 1), bias=False) (16): Tanh () ) ) D ( (main): Sequential ( (0): Dropout $\\mathrm { ~ \\bf ~ p ~ } = \\mathrm { ~ \\bf ~ 0 ~ } . 2 \\mathrm { ~ \\bf ~ }$ ) (1): Conv2d(3, 96, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1)) (2): LeakyReLU (0.2, inplace) (3): Conv2d(96, 96, kernel_siz $\\div =$ (3, 3), stride $\\div =$ (1, 1), padding=(1, 1), bias=False) (5): LeakyReLU (0.2, inplace) (6): Conv2d(96, 96, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), bias=False) (8): LeakyReLU (0.2, inplace) (9): Dropout $( \\mathrm { p } ~ = ~ 0 . 5 )$ (10): Conv2d(96, 192, kernel_size $=$ (3, 3), stride $=$ (1, 1), padding $=$ (1, 1), bias=False) (12): LeakyReLU (0.2, inplace) (13): Conv2d(192, 192, kernel_size $=$ (3, 3), stride $: =$ (1, 1), padding $=$ (1, 1), bias $\\scriptstyle = \\mathbf { E }$ alse) (15): LeakyReLU (0.2, inplace) (16): Conv2d(192, 192, kernel_size $: =$ (3, 3), stride $: =$ (2, 2), padding $=$ (1, 1), bias=False) (18): LeakyReLU (0.2, inplace) (19): Dropout ( $\\mathrm { ~ \\cdot ~ p ~ } = \\mathrm { ~ 0 ~ . ~ 5 ~ }$ (20): Conv2d(192, 384, kernel_size $: =$ (3, 3), stride $=$ (1, 1), bias=False) (22): LeakyReLU (0.2, inplace) (23): Dropout (p = 0.5) (24): Conv2d(384, 384, kernel_size $=$ (3, 3), stride $=$ (1, 1), bias=False) (26): LeakyReLU (0.2, inplace) (27): Dropout $( \\mathrm { p } ~ = ~ 0 . 5 )$ (28): Conv2d(384, 384, kernel_size=(1, 1), stride=(1, 1), bias=False) (30): LeakyReLU (0.2, inplace) (31): Dropout $( \\mathrm { p } ~ = ~ 0 . 5 )$ ) ) (V): Linear (6144 -> 1) (S): Linear (6144 -> 10) ) ", + "bbox": [ + 169, + 108, + 728, + 491 + ], + "page_idx": 26 + } +] \ No newline at end of file diff --git a/parse/train/SJA7xfb0b/SJA7xfb0b_middle.json b/parse/train/SJA7xfb0b/SJA7xfb0b_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..f338d2feb2ff77e745eabbbf8b863edc9d1dd84b --- /dev/null +++ b/parse/train/SJA7xfb0b/SJA7xfb0b_middle.json @@ -0,0 +1,74352 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 222, + 96 + ], + "lines": [ + { + "bbox": [ + 104, + 78, + 226, + 98 + ], + "spans": [ + { + "bbox": [ + 104, + 78, + 226, + 98 + ], + "score": 1.0, + "content": "SOBOLEV GAN", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 113, + 114, + 448, + 127 + ], + "lines": [ + { + "bbox": [ + 111, + 112, + 449, + 130 + ], + "spans": [ + { + "bbox": [ + 111, + 112, + 244, + 130 + ], + "score": 1.0, + "content": "Youssef Mroueh†, Chun-Liang", + "type": "text" + }, + { + "bbox": [ + 245, + 115, + 266, + 126 + ], + "score": 0.72, + "content": "\\mathbf { L i } ^ { \\circ , \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 112, + 449, + 130 + ], + "score": 1.0, + "content": ", Tom Sercu†,?, Anant Raj♦,? 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We show how Sobolev ´", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 346, + 469, + 358 + ], + "spans": [ + { + "bbox": [ + 141, + 346, + 469, + 358 + ], + "score": 1.0, + "content": "IPM can be used to train Generative Adversarial Networks (GANs). We then ex-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 358, + 469, + 370 + ], + "spans": [ + { + "bbox": [ + 142, + 358, + 469, + 370 + ], + "score": 1.0, + "content": "ploit the intrinsic conditioning implied by Sobolev IPM in text generation. Finally", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 369, + 469, + 380 + ], + "spans": [ + { + "bbox": [ + 142, + 369, + 469, + 380 + ], + "score": 1.0, + "content": "we show that a variant of Sobolev GAN achieves competitive results in semi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 379, + 469, + 391 + ], + "spans": [ + { + "bbox": [ + 141, + 379, + 469, + 391 + ], + "score": 1.0, + "content": "supervised learning on CIFAR-10, thanks to the smoothness enforced on the critic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 390, + 382, + 403 + ], + "spans": [ + { + "bbox": [ + 141, + 390, + 382, + 403 + ], + "score": 1.0, + "content": "by Sobolev GAN which relates to Laplacian regularization.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 426, + 206, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 208, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 208, + 442 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 463 + ], + "score": 1.0, + "content": "In order to learn Generative Adversarial Networks (Goodfellow et al., 2014), it is now well estab-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "score": 1.0, + "content": "lished that the generator should mimic the distribution of real data, in the sense of a certain dis-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "crepancy measure. Discrepancies between distributions that measure the goodness of the fit of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "neural generator to the real data distribution has been the subject of many recent studies (Arjovsky", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "& Bottou, 2017; Nowozin et al., 2016; Kaae Sønderby et al., 2017; Mao et al., 2017; Arjovsky et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "2017; Gulrajani et al., 2017; Mroueh et al., 2017; Mroueh & Sercu, 2017; Li et al., 2017), most of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 518, + 240, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 240, + 530 + ], + "score": 1.0, + "content": "which focus on training stability.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "In terms of data modalities, most success was booked in plausible natural image generation after the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 546, + 504, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 504, + 557 + ], + "score": 1.0, + "content": "introduction of Deep Convolutional Generative Adversarial Networks (DCGAN) (Radford et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "2015). This success is not only due to advances in training generative adversarial networks in terms", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "of loss functions (Arjovsky et al., 2017) and stable algorithms, but also to the representation power", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "of convolutional neural networks in modeling images and in finding sufficient statistics that capture", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "the continuous density function of natural images. 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We show that the Sobolev IPM compares two distributions", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 291, + 469, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 469, + 304 + ], + "score": 1.0, + "content": "in high dimensions based on weighted conditional Cumulative Distribution Func-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 303, + 469, + 314 + ], + "spans": [ + { + "bbox": [ + 142, + 303, + 469, + 314 + ], + "score": 1.0, + "content": "tions (CDF) of each coordinate on a leave one out basis. The Dominant measure", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 313, + 470, + 325 + ], + "spans": [ + { + "bbox": [ + 142, + 315, + 150, + 325 + ], + "score": 0.73, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 313, + 470, + 325 + ], + "score": 1.0, + "content": "plays a crucial role as it defines the support on which conditional CDFs are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 325, + 469, + 336 + ], + "spans": [ + { + "bbox": [ + 141, + 325, + 469, + 336 + ], + "score": 1.0, + "content": "compared. Sobolev IPM can be seen as an extension of the one dimensional Von-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 336, + 469, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 336, + 469, + 347 + ], + "score": 1.0, + "content": "Mises Cramer statistics to high dimensional distributions. We show how Sobolev ´", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 346, + 469, + 358 + ], + "spans": [ + { + "bbox": [ + 141, + 346, + 469, + 358 + ], + "score": 1.0, + "content": "IPM can be used to train Generative Adversarial Networks (GANs). We then ex-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 358, + 469, + 370 + ], + "spans": [ + { + "bbox": [ + 142, + 358, + 469, + 370 + ], + "score": 1.0, + "content": "ploit the intrinsic conditioning implied by Sobolev IPM in text generation. Finally", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 369, + 469, + 380 + ], + "spans": [ + { + "bbox": [ + 142, + 369, + 469, + 380 + ], + "score": 1.0, + "content": "we show that a variant of Sobolev GAN achieves competitive results in semi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 379, + 469, + 391 + ], + "spans": [ + { + "bbox": [ + 141, + 379, + 469, + 391 + ], + "score": 1.0, + "content": "supervised learning on CIFAR-10, thanks to the smoothness enforced on the critic", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 390, + 382, + 403 + ], + "spans": [ + { + "bbox": [ + 141, + 390, + 382, + 403 + ], + "score": 1.0, + "content": "by Sobolev GAN which relates to Laplacian regularization.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15.5, + "bbox_fs": [ + 141, + 247, + 470, + 403 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 426, + 206, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 208, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 208, + 442 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 463 + ], + "score": 1.0, + "content": "In order to learn Generative Adversarial Networks (Goodfellow et al., 2014), it is now well estab-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "score": 1.0, + "content": "lished that the generator should mimic the distribution of real data, in the sense of a certain dis-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "crepancy measure. Discrepancies between distributions that measure the goodness of the fit of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "neural generator to the real data distribution has been the subject of many recent studies (Arjovsky", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "& Bottou, 2017; Nowozin et al., 2016; Kaae Sønderby et al., 2017; Mao et al., 2017; Arjovsky et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "2017; Gulrajani et al., 2017; Mroueh et al., 2017; Mroueh & Sercu, 2017; Li et al., 2017), most of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 518, + 240, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 240, + 530 + ], + "score": 1.0, + "content": "which focus on training stability.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 452, + 506, + 530 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 699 + ], + "lines": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "In terms of data modalities, most success was booked in plausible natural image generation after the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 546, + 504, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 504, + 557 + ], + "score": 1.0, + "content": "introduction of Deep Convolutional Generative Adversarial Networks (DCGAN) (Radford et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "2015). This success is not only due to advances in training generative adversarial networks in terms", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "of loss functions (Arjovsky et al., 2017) and stable algorithms, but also to the representation power", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "of convolutional neural networks in modeling images and in finding sufficient statistics that capture", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 602 + ], + "score": 1.0, + "content": "the continuous density function of natural images. When moving to neural generators of discrete", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "sequences generative adversarial networks theory and practice are still not very well understood.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "Maximum likelihood pre-training or augmentation, in conjunction with the use of reinforcement", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "learning techniques were proposed in many recent works for training GAN for discrete sequences", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "generation (Yu et al., 2016; Che et al., 2017; Hjelm et al., 2017; Rajeswar et al., 2017). Other", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "methods included using the Gumbel Softmax trick (Kusner & Hernandez-Lobato ´ , 2016) and the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 656, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 667 + ], + "score": 1.0, + "content": "use of auto-encoders to generate adversarially discrete sequences from a continuous space (Zhao", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 666, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 504, + 678 + ], + "score": 1.0, + "content": "et al., 2017). End to end training of GANs for discrete sequence generation is still an open prob-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "lem (Press et al., 2017). Empirical successes of end to end training have been reported within the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "framework of WGAN-GP (Gulrajani et al., 2017), using a proxy for the Wasserstein distance via a", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 104, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "pointwise gradient penalty on the critic. Inspired by this success, we propose in this paper a new In-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "tegral Probability Metric (IPM) between distributions that we coin Sobolev IPM. Intuitively an IPM", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 440, + 117 + ], + "score": 1.0, + "content": "(Muller ¨ , 1997) between two probability distributions looks for a witness function", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 441, + 105, + 448, + 116 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 448, + 104, + 506, + 117 + ], + "score": 1.0, + "content": ", called critic,", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 436, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 436, + 127 + ], + "score": 1.0, + "content": "that maximally discriminates between samples coming from the two distributions:", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 534, + 506, + 701 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 104, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 104, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "pointwise gradient penalty on the critic. Inspired by this success, we propose in this paper a new In-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "tegral Probability Metric (IPM) between distributions that we coin Sobolev IPM. Intuitively an IPM", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 440, + 117 + ], + "score": 1.0, + "content": "(Muller ¨ , 1997) between two probability distributions looks for a witness function", + "type": "text" + }, + { + "bbox": [ + 441, + 105, + 448, + 116 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 104, + 506, + 117 + ], + "score": 1.0, + "content": ", called critic,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 436, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 436, + 127 + ], + "score": 1.0, + "content": "that maximally discriminates between samples coming from the two distributions:", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 132, + 366, + 154 + ], + "lines": [ + { + "bbox": [ + 243, + 132, + 366, + 154 + ], + "spans": [ + { + "bbox": [ + 243, + 132, + 366, + 154 + ], + "score": 0.91, + "content": "\\operatorname* { s u p } _ { f \\in \\mathcal { F } } \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) .", + "type": "interline_equation", + "image_path": "a98ebe639cd3d8ea740d650c0644ca19dff3a339cd72867878bbf88dd81cb39e.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 243, + 132, + 366, + 154 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 505, + 216 + ], + "lines": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 211, + 172 + ], + "score": 1.0, + "content": "Traditionally, the function", + "type": "text" + }, + { + "bbox": [ + 212, + 161, + 218, + 172 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 160, + 341, + 172 + ], + "score": 1.0, + "content": "is defined over a function class", + "type": "text" + }, + { + "bbox": [ + 341, + 161, + 352, + 171 + ], + "score": 0.86, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "that is independent to the distributions", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 172, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 183 + ], + "score": 1.0, + "content": "at hand (Sriperumbudur et al., 2012). The Wasserstein-1 distance corresponds for instance to an IPM", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "where the witness functions are defined over the space of Lipschitz functions; The MMD distance", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 207 + ], + "score": 1.0, + "content": "(Gretton et al., 2012) corresponds to witness functions defined over a ball in a Reproducing Kernel", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 201, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 201, + 217 + ], + "score": 1.0, + "content": "Hilbert Space (RKHS).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 221, + 505, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 221, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 233 + ], + "score": 1.0, + "content": "We will revisit in this paper Fisher IPM defined in (Mroueh & Sercu, 2017), which extends the IPM", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "definition to function classes defined with norms that depend on the distributions. Fisher IPM can", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 494, + 257 + ], + "score": 1.0, + "content": "be seen as restricting the critic to a Lebsegue ball defined with respect to a dominant measure", + "type": "text" + }, + { + "bbox": [ + 494, + 246, + 501, + 255 + ], + "score": 0.75, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 243, + 505, + 257 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 254, + 275, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 275, + 266 + ], + "score": 1.0, + "content": "The Lebsegue norm is defined as follows:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 268, + 272, + 342, + 298 + ], + "lines": [ + { + "bbox": [ + 268, + 272, + 342, + 298 + ], + "spans": [ + { + "bbox": [ + 268, + 272, + 342, + 298 + ], + "score": 0.94, + "content": "\\int _ { \\mathcal { X } } f ^ { 2 } ( x ) \\mu ( x ) d x .", + "type": "interline_equation", + "image_path": "966178bc96ad159d2cef7de229090b6fdbbd04e35a7b168d54122f65f3b2fe91.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 268, + 272, + 342, + 298 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 303, + 281, + 315 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 279, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 133, + 317 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 306, + 141, + 315 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 302, + 244, + 317 + ], + "score": 1.0, + "content": "is a dominant measure of", + "type": "text" + }, + { + "bbox": [ + 245, + 304, + 252, + 313 + ], + "score": 0.82, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 302, + 270, + 317 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 270, + 304, + 279, + 315 + ], + "score": 0.86, + "content": "\\mathbb { Q }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 442, + 332 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 443, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 443, + 333 + ], + "score": 1.0, + "content": "In this paper we extend the IPM framework to critics bounded in the Sobolev norm:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 337, + 354, + 364 + ], + "lines": [ + { + "bbox": [ + 255, + 337, + 354, + 364 + ], + "spans": [ + { + "bbox": [ + 255, + 337, + 354, + 364 + ], + "score": 0.93, + "content": "\\int _ { \\mathcal { X } } \\| \\nabla _ { \\boldsymbol { x } } f ( \\boldsymbol { x } ) \\| _ { 2 } ^ { 2 } \\mu ( \\boldsymbol { x } ) d x ,", + "type": "interline_equation", + "image_path": "9acae480abe15bd92f2858f3db977054cb0898e2e6eca672ab7245aeab1e28a2.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 255, + 337, + 354, + 350.5 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 255, + 350.5, + 354, + 364.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 369, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 104, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "In contrast to Fisher IPM, which compares joint probability density functions of all coordinates", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 381, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 392 + ], + "score": 1.0, + "content": "between two distributions, we will show that Sobolev IPM compares weighted (coordinate-wise)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "conditional Cumulative Distribution Functions for all coordinates on a leave on out basis. Matching", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 402, + 432, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 432, + 416 + ], + "score": 1.0, + "content": "conditional dependencies between coordinates is crucial for sequence modeling.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "Our analysis and empirical verification show that the modeling of the conditional dependencies can", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "score": 1.0, + "content": "be built in to the metric used to learn GANs as in Sobolev IPM. For instance, this gives an advantage", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 104, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "to Sobolev IPM in comparing sequences over Fisher IPM. Nevertheless, in sequence modeling when", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "we parametrize the critic and the generator with a neural network, we find an interesting tradeoff", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "between the metric used and the architectures used to parametrize the critic and the generator as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "well as the conditioning used in the generator. The burden of modeling the conditional long term", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "dependencies can be handled by the IPM loss function as in Sobolev IPM (more accurately the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "choice of the data dependent function class of the critic) or by a simpler metric such as Fisher IPM", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 508, + 504, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 504, + 519 + ], + "score": 1.0, + "content": "together with a powerful architecture for the critic that models conditional long term dependencies", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "such as LSTM or GRUs in conjunction with a curriculum conditioning of the generator as done", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 528, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 505, + 542 + ], + "score": 1.0, + "content": "in (Press et al., 2017). Highlighting those interesting tradeoffs between metrics, data dependent", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 539, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 554 + ], + "score": 1.0, + "content": "functions classes for the critic (Fisher or Sobolev) and architectures is crucial to advance sequence", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 551, + 380, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 380, + 563 + ], + "score": 1.0, + "content": "modeling and more broadly structured data generation using GANs.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 579 + ], + "score": 1.0, + "content": "On the other hand, Sobolev norms have been widely used in manifold regularization in the so called", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "score": 1.0, + "content": "Laplacian framework for semi-supervised learning (SSL) (Belkin et al., 2006). GANs have shown", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 590, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 600 + ], + "score": 1.0, + "content": "success in semi-supervised learning (Salimans et al., 2016; Dumoulin et al., 2017; Dai et al., 2017;", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Kumar et al., 2017). Nevertheless, many normalizations and additional tricks were needed. We show", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "in this paper that a variant of Sobolev GAN achieves strong results in semi-supervised learning on", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 623, + 399, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 399, + 634 + ], + "score": 1.0, + "content": "CIFAR-10, without the need of any activation normalization in the critic.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 378, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 379, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 379, + 652 + ], + "score": 1.0, + "content": "The main contributions of this paper can be summarized as follows:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 130, + 660, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 129, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 129, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "1. We overview in Section 2 different metrics between distribution used in the GAN literature.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 141, + 672, + 456, + 684 + ], + "score": 1.0, + "content": "We then generalize Fisher IPM in Section 3 with a general dominant measure", + "type": "text" + }, + { + "bbox": [ + 456, + 673, + 464, + 684 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "and show", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 683, + 348, + 695 + ], + "spans": [ + { + "bbox": [ + 141, + 683, + 348, + 695 + ], + "score": 1.0, + "content": "how it compares distributions based on their PDFs.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 130, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 130, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "2. 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The Wasserstein-1 distance corresponds for instance to an IPM", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 182, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 505, + 195 + ], + "score": 1.0, + "content": "where the witness functions are defined over the space of Lipschitz functions; The MMD distance", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 505, + 207 + ], + "score": 1.0, + "content": "(Gretton et al., 2012) corresponds to witness functions defined over a ball in a Reproducing Kernel", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 204, + 201, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 201, + 217 + ], + "score": 1.0, + "content": "Hilbert Space (RKHS).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 160, + 505, + 217 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 221, + 505, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 221, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 233 + ], + "score": 1.0, + "content": "We will revisit in this paper Fisher IPM defined in (Mroueh & Sercu, 2017), which extends the IPM", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "definition to function classes defined with norms that depend on the distributions. 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Matching", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 402, + 432, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 432, + 416 + ], + "score": 1.0, + "content": "conditional dependencies between coordinates is crucial for sequence modeling.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 369, + 505, + 416 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "Our analysis and empirical verification show that the modeling of the conditional dependencies can", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "score": 1.0, + "content": "be built in to the metric used to learn GANs as in Sobolev IPM. For instance, this gives an advantage", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 104, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "to Sobolev IPM in comparing sequences over Fisher IPM. Nevertheless, in sequence modeling when", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "we parametrize the critic and the generator with a neural network, we find an interesting tradeoff", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "between the metric used and the architectures used to parametrize the critic and the generator as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "well as the conditioning used in the generator. The burden of modeling the conditional long term", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "dependencies can be handled by the IPM loss function as in Sobolev IPM (more accurately the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "score": 1.0, + "content": "choice of the data dependent function class of the critic) or by a simpler metric such as Fisher IPM", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 508, + 504, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 504, + 519 + ], + "score": 1.0, + "content": "together with a powerful architecture for the critic that models conditional long term dependencies", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "such as LSTM or GRUs in conjunction with a curriculum conditioning of the generator as done", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 528, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 505, + 542 + ], + "score": 1.0, + "content": "in (Press et al., 2017). Highlighting those interesting tradeoffs between metrics, data dependent", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 539, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 554 + ], + "score": 1.0, + "content": "functions classes for the critic (Fisher or Sobolev) and architectures is crucial to advance sequence", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 551, + 380, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 380, + 563 + ], + "score": 1.0, + "content": "modeling and more broadly structured data generation using GANs.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 419, + 506, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 568, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 579 + ], + "score": 1.0, + "content": "On the other hand, Sobolev norms have been widely used in manifold regularization in the so called", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "score": 1.0, + "content": "Laplacian framework for semi-supervised learning (SSL) (Belkin et al., 2006). GANs have shown", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 590, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 600 + ], + "score": 1.0, + "content": "success in semi-supervised learning (Salimans et al., 2016; Dumoulin et al., 2017; Dai et al., 2017;", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "Kumar et al., 2017). Nevertheless, many normalizations and additional tricks were needed. We show", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "in this paper that a variant of Sobolev GAN achieves strong results in semi-supervised learning on", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 623, + 399, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 399, + 634 + ], + "score": 1.0, + "content": "CIFAR-10, without the need of any activation normalization in the critic.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 568, + 505, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 378, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 379, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 379, + 652 + ], + "score": 1.0, + "content": "The main contributions of this paper can be summarized as follows:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 639, + 379, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 660, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 129, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 129, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "1. We overview in Section 2 different metrics between distribution used in the GAN literature.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 141, + 672, + 456, + 684 + ], + "score": 1.0, + "content": "We then generalize Fisher IPM in Section 3 with a general dominant measure", + "type": "text" + }, + { + "bbox": [ + 456, + 673, + 464, + 684 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "and show", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 141, + 683, + 348, + 695 + ], + "spans": [ + { + "bbox": [ + 141, + 683, + 348, + 695 + ], + "score": 1.0, + "content": "how it compares distributions based on their PDFs.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 130, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 130, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "2. We introduce Sobolev IPM in Section 4 by restricting the critic of an IPM to a Sobolev ball", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 142, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 142, + 710, + 321, + 722 + ], + "score": 1.0, + "content": "defined with respect to a dominant measure", + "type": "text" + }, + { + "bbox": [ + 321, + 713, + 328, + 721 + ], + "score": 0.74, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 710, + 506, + 722 + ], + "score": 1.0, + "content": ". 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The intrinsic conditioning and the CDF matching make Sobolev IPM suitable for discrete", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 141, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 141, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "sequence matching and explain the success of the gradient pernalty in WGAN-GP and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 103, + 331, + 118 + ], + "spans": [ + { + "bbox": [ + 141, + 103, + 331, + 118 + ], + "score": 1.0, + "content": "Sobolev GAN in discrete sequence generation.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 128, + 118, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 128, + 118, + 505, + 132 + ], + "score": 1.0, + "content": "4. We give in Section 5 an ALM (Augmented Lagrangian Multiplier) algorithm for training", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 141, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "Sobolev GAN. Similar to Fisher GAN, this algorithm is stable and does not compromise", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 141, + 243, + 153 + ], + "spans": [ + { + "bbox": [ + 141, + 141, + 243, + 153 + ], + "score": 1.0, + "content": "the capacity of the critic.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 128, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 128, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "5. We show in Appendix A that the critic of Sobolev IPM satisfies an elliptic Partial Differ-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 142, + 166, + 505, + 178 + ], + "score": 1.0, + "content": "ential Equation (PDE). 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We", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 142, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "validate that the conditioning implied by Sobolev GAN is crucial for the success and sta-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "bility of GAN in text generation. As a take home message from this study, we see that text", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 140, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 140, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "generation succeeds either by implicit conditioning i.e using Sobolev GAN (or WGAN-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 244, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 244, + 505, + 258 + ], + "score": 1.0, + "content": "GP) together with convolutional critics and generators, or by explicit conditioning i.e using", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 255, + 465, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 465, + 270 + ], + "score": 1.0, + "content": "Fisher IPM together with recurrent critic and generator and curriculum learning.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 129, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 129, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "7. We finally show in Section 6.2 that a variant of Sobolev GAN achieves competitive semi-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 141, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "supervised learning results on CIFAR-10, thanks to the smoothness implied by the Sobolev", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 292, + 190, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 292, + 190, + 304 + ], + "score": 1.0, + "content": "regularizer.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 9 + }, + { + "type": "title", + "bbox": [ + 106, + 318, + 387, + 331 + ], + "lines": [ + { + "bbox": [ + 104, + 317, + 388, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 388, + 333 + ], + "score": 1.0, + "content": "2 OVERVIEW OF METRICS BETWEEN DISTRIBUTIONS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 505, + 409 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "In this Section, we review different representations of probability distributions and metrics for com-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "paring distributions that use those representations. 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We see from Table 1 that", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "Sobolev IPM, compared to Wasserstein Distance, imposes a tractable smoothness constraint on the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 288, + 613 + ], + "score": 1.0, + "content": "critic on points sampled from a distribution", + "type": "text" + }, + { + "bbox": [ + 288, + 602, + 295, + 612 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 600, + 505, + 613 + ], + "score": 1.0, + "content": ", rather then imposing a Lipschitz constraint on all", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 186, + 623 + ], + "score": 1.0, + "content": "points in the space", + "type": "text" + }, + { + "bbox": [ + 186, + 611, + 196, + 621 + ], + "score": 0.78, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 610, + 506, + 623 + ], + "score": 1.0, + "content": ". We also see that Sobolev IPM is the natural generalization of the Cramer´", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "Von-Mises Distance from one dimension to high dimensions. We note that the Energy Distance, a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "form of Maximum Mean Discrepancy for a special kernel, was used in (Bellemare et al., 2017b)", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "as a generalization of the Cramer distance in GAN training but still needed a gradient penalty in ´", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "its algorithmic counterpart leading to a mis-specified distance between distributions. Finally it is", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "worth noting that when comparing Fisher IPM and Sobolev IPM we see that while Fisher IPM com-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "pares joint PDF of the distributions, Sobolev IPM compares weighted (coordinate-wise) conditional", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "CDFs. As we will see later, this conditioning nature of the metric makes Sobolev IPM suitable for", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "comparing sequences. Note that the Stein metric (Liu et al., 2016; Liu, 2017) uses the score func-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "tion to match distributions. 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The intrinsic conditioning and the CDF matching make Sobolev IPM suitable for discrete", + "type": "text" + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 141, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "sequence matching and explain the success of the gradient pernalty in WGAN-GP and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 103, + 331, + 118 + ], + "spans": [ + { + "bbox": [ + 141, + 103, + 331, + 118 + ], + "score": 1.0, + "content": "Sobolev GAN in discrete sequence generation.", + "type": "text" + } + ], + "index": 2, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 118, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 128, + 118, + 505, + 132 + ], + "score": 1.0, + "content": "4. We give in Section 5 an ALM (Augmented Lagrangian Multiplier) algorithm for training", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 141, + 129, + 505, + 142 + ], + "score": 1.0, + "content": "Sobolev GAN. Similar to Fisher GAN, this algorithm is stable and does not compromise", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 141, + 243, + 153 + ], + "spans": [ + { + "bbox": [ + 141, + 141, + 243, + 153 + ], + "score": 1.0, + "content": "the capacity of the critic.", + "type": "text" + } + ], + "index": 5, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 128, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "5. We show in Appendix A that the critic of Sobolev IPM satisfies an elliptic Partial Differ-", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 142, + 166, + 505, + 178 + ], + "score": 1.0, + "content": "ential Equation (PDE). We relate this diffusion to the Fokker-Planck equation and show", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 140, + 176, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 140, + 176, + 505, + 190 + ], + "score": 1.0, + "content": "the behavior of the gradient of the optimal Sobolev critic as a transportation plan between", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 187, + 197, + 199 + ], + "spans": [ + { + "bbox": [ + 141, + 187, + 197, + 199 + ], + "score": 1.0, + "content": "distributions.", + "type": "text" + } + ], + "index": 9, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 201, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 129, + 201, + 505, + 213 + ], + "score": 1.0, + "content": "6. We empirically study Sobolev GAN in character level text generation (Section 6.1). We", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 142, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "validate that the conditioning implied by Sobolev GAN is crucial for the success and sta-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 141, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "bility of GAN in text generation. As a take home message from this study, we see that text", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 140, + 234, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 140, + 234, + 506, + 247 + ], + "score": 1.0, + "content": "generation succeeds either by implicit conditioning i.e using Sobolev GAN (or WGAN-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 244, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 244, + 505, + 258 + ], + "score": 1.0, + "content": "GP) together with convolutional critics and generators, or by explicit conditioning i.e using", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 255, + 465, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 465, + 270 + ], + "score": 1.0, + "content": "Fisher IPM together with recurrent critic and generator and curriculum learning.", + "type": "text" + } + ], + "index": 15, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 129, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "7. 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Those metrics are at the core of training GAN.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "In what follows, we consider probability measures with a positive weakly differentiable probabil-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 375, + 504, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 239, + 389 + ], + "score": 1.0, + "content": "ity density functions (PDF). 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Finally it is", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "worth noting that when comparing Fisher IPM and Sobolev IPM we see that while Fisher IPM com-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "pares joint PDF of the distributions, Sobolev IPM compares weighted (coordinate-wise) conditional", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "CDFs. As we will see later, this conditioning nature of the metric makes Sobolev IPM suitable for", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "comparing sequences. Note that the Stein metric (Liu et al., 2016; Liu, 2017) uses the score func-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "tion to match distributions. 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△(f;P,Q)FFunction classdg(P,Q)Closed Form
-Divergence(Goodfellow et al., 2014)(Nowozin et al., 2016)Ex~Pf(x)-Ex~Q*(f(x))*Fenchel Conjugate{f:x→R,f∈domp*}E~[(]
Wasserstein -1(Arjovsky et al., 2017)(Gulrajani et al., 2017)Ex~Pf(x)-Ex~qf(x){f : x→R,fli ≤1}infπ∈(P,Q) Sx |lx-ylli dπ(x,y)Sinkhorn Divergence(Genevay et al., 2017)
MMD(Li et al., 2017)(Li et al., 2015)(Dziugaite et al., 2015)Ex~Pf(x)-Ex~qf(x){f:x→R,/fl/x ≤1}Ex~Pkx -Ex~Qr llyex
SteinDiscrepancy(Wang & Liu, 2016)Ex~Q[T(P)f(x)]T(P)=(Vxlog(P(x))T+Vx.{f:x→Rdf smooth with zeroboundary conditionNA in generalhas a closed formin RKHS
Cramerfor d =1(Bellemare et al., 2017a)Ex~Pf(x)-Ex~Qf(x){f:x→R,Ea~P(）²≤1,f smooth with zeroboundary conditionEx~P（Fp(2）-F0（））²P(x）xER
μ-FisherIPM(Mroueh & Sercu,2017)Ex~Pf(x)-Ex~Qf(x){f:X→R,f∈L(x,μ),Ex~μf²(x)≤1}Ex~u（P()-Q(x））²2μ(x）
μ-SobolevIPM(This work)Ex~Pf(x)-Ex~qf(x){f:X→R,f eW²(x,μ),Ex~μ|/Vxf(x)²≤1,with zero boundary condition }1VE~u∑i-(（中i(P）-Φi(Q）μ（x）where 𝜙(P) =Px-i(x-i)FP[xx-i-ij(mi)x−=(x1,..xi-1,Ti+1,..d)
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References", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 457, + 318, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 318, + 472 + ], + "score": 1.0, + "content": "are for papers using those metrics for GAN training.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 106, + 492, + 376, + 507 + ], + "lines": [ + { + "bbox": [ + 104, + 492, + 377, + 509 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 377, + 509 + ], + "score": 1.0, + "content": "3 GENERALIZING FISHER IPM: PDF COMPARISON", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "Imposing data-independent constraints on the function class in the IPM framework, such as the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "Lipschitz constraint in the Wasserstein distance is computationally challenging and intractable for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "score": 1.0, + "content": "the general case. 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△(f;P,Q)FFunction classdg(P,Q)Closed Form
-Divergence(Goodfellow et al., 2014)(Nowozin et al., 2016)Ex~Pf(x)-Ex~Q*(f(x))*Fenchel Conjugate{f:x→R,f∈domp*}E~[(]
Wasserstein -1(Arjovsky et al., 2017)(Gulrajani et al., 2017)Ex~Pf(x)-Ex~qf(x){f : x→R,fli ≤1}infπ∈(P,Q) Sx |lx-ylli dπ(x,y)Sinkhorn Divergence(Genevay et al., 2017)
MMD(Li et al., 2017)(Li et al., 2015)(Dziugaite et al., 2015)Ex~Pf(x)-Ex~qf(x){f:x→R,/fl/x ≤1}Ex~Pkx -Ex~Qr llyex
SteinDiscrepancy(Wang & Liu, 2016)Ex~Q[T(P)f(x)]T(P)=(Vxlog(P(x))T+Vx.{f:x→Rdf smooth with zeroboundary conditionNA in generalhas a closed formin RKHS
Cramerfor d =1(Bellemare et al., 2017a)Ex~Pf(x)-Ex~Qf(x){f:x→R,Ea~P(）²≤1,f smooth with zeroboundary conditionEx~P（Fp(2）-F0（））²P(x）xER
μ-FisherIPM(Mroueh & Sercu,2017)Ex~Pf(x)-Ex~Qf(x){f:X→R,f∈L(x,μ),Ex~μf²(x)≤1}Ex~u（P()-Q(x））²2μ(x）
μ-SobolevIPM(This work)Ex~Pf(x)-Ex~qf(x){f:X→R,f eW²(x,μ),Ex~μ|/Vxf(x)²≤1,with zero boundary condition }1VE~u∑i-(（中i(P）-Φi(Q）μ（x）where 𝜙(P) =Px-i(x-i)FP[xx-i-ij(mi)x−=(x1,..xi-1,Ti+1,..d)
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References", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 457, + 318, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 318, + 472 + ], + "score": 1.0, + "content": "are for papers using those metrics for GAN training.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 106, + 492, + 376, + 507 + ], + "lines": [ + { + "bbox": [ + 104, + 492, + 377, + 509 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 377, + 509 + ], + "score": 1.0, + "content": "3 GENERALIZING FISHER IPM: PDF COMPARISON", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "Imposing data-independent constraints on the function class in the IPM framework, such as the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "Lipschitz constraint in the Wasserstein distance is computationally challenging and intractable for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "score": 1.0, + "content": "the general case. 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) g ( x ) \\mu ( x ) d x , \\| f \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } = \\sqrt { \\int _ { \\mathcal { X } } f ^ { 2 } ( x ) \\mu ( x ) d x } .", + "type": "interline_equation", + "image_path": "88f56b94989940a85facb05c3b4699c2785ac41c8649ec20ce8494b9a619d6be.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 154, + 664, + 457, + 675.3333333333334 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 154, + 675.3333333333334, + 457, + 686.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 154, + 686.6666666666667, + 457, + 698.0000000000001 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 702, + 333, + 715 + ], + "lines": [ + { + "bbox": [ + 106, + 702, + 334, + 715 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 146, + 715 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 703, + 187, + 715 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 702, + 334, + 715 + ], + "score": 1.0, + "content": ", can be formally defined as follows:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 702, + 334, + 715 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 719, + 404, + 735 + ], + "lines": [ + { + "bbox": [ + 204, + 719, + 404, + 735 + ], + "spans": [ + { + "bbox": [ + 204, + 719, + 404, + 735 + ], + "score": 0.82, + "content": "\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) = \\{ f : \\mathcal { X } \\to \\mathbb { R } \\mathrm { s . t } \\ \\| f \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } < \\infty \\} .", + "type": "interline_equation", + "image_path": "57c9c82b706a6af55890433de3272cf1166b92ac86003a5135e9b3124436ccfa.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 204, + 719, + 404, + 735 + ], + "spans": [], + 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Fisher GAN (Mroueh & Sercu,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 133, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 303, + 147 + ], + "score": 1.0, + "content": "2017) was originally formulated specifically for", + "type": "text" + }, + { + "bbox": [ + 303, + 134, + 366, + 147 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mu = \\frac 1 2 ( { \\mathbb P } + { \\mathbb Q } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 133, + 485, + 147 + ], + "score": 1.0, + "content": ". We consider here a general", + "type": "text" + }, + { + "bbox": [ + 485, + 136, + 492, + 146 + ], + "score": 0.79, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 133, + 506, + 147 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 414, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 189, + 156 + ], + "score": 1.0, + "content": "long as it dominates", + "type": "text" + }, + { + "bbox": [ + 189, + 146, + 197, + 155 + ], + "score": 0.84, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 145, + 214, + 156 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 146, + 223, + 156 + ], + "score": 0.85, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 145, + 414, + 156 + ], + "score": 1.0, + "content": ". 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The Fisher distance is given by:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 306, + 450, + 341 + ], + "lines": [ + { + "bbox": [ + 198, + 306, + 450, + 341 + ], + "spans": [ + { + "bbox": [ + 198, + 306, + 450, + 341 + ], + "score": 0.93, + "content": "\\mathcal { F } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\left\\| \\frac { \\mathbb { P } - \\mathbb { Q } } { \\mu } \\right\\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } = \\sqrt { \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mu } \\left( \\frac { \\mathbb { P } ( \\boldsymbol { x } ) - \\mathbb { Q } ( \\boldsymbol { x } ) } { \\mu ( \\boldsymbol { x } ) } \\right) ^ { 2 } } .", + "type": "interline_equation", + "image_path": "3db2e30f980cd8a9477d69935b4e5387d1876b47cb6bc9a7613e57d66273b858.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 198, + 306, + 450, + 317.6666666666667 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 198, + 317.6666666666667, + 450, + 329.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 198, + 329.33333333333337, + 450, + 341.00000000000006 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 129, + 348, + 380, + 360 + ], + "lines": [ + { + "bbox": [ + 129, + 347, + 380, + 362 + ], + "spans": [ + { + "bbox": [ + 129, + 347, + 192, + 362 + ], + "score": 1.0, + "content": "2. The optimal", + "type": "text" + }, + { + "bbox": [ + 193, + 349, + 204, + 360 + ], + "score": 0.88, + "content": "f _ { \\chi }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 347, + 325, + 362 + ], + "score": 1.0, + "content": "achieving the Fisher distance", + "type": "text" + }, + { + "bbox": [ + 325, + 348, + 366, + 361 + ], + "score": 0.93, + "content": "\\mathcal { F } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 347, + 380, + 362 + ], + "score": 1.0, + "content": "is:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 302, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 391, + 302, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 302, + 405 + ], + "score": 1.0, + "content": "Proof of Theorem 1. From Equation (2), the op-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 302, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 131, + 416 + ], + "score": 1.0, + "content": "timal", + "type": "text" + }, + { + "bbox": [ + 131, + 404, + 143, + 416 + ], + "score": 0.9, + "content": "f _ { \\chi }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 403, + 302, + 416 + ], + "score": 1.0, + "content": "belong to the intersection of the hy-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 415, + 303, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 215, + 428 + ], + "score": 1.0, + "content": "perplane that has normal", + "type": "text" + }, + { + "bbox": [ + 216, + 415, + 262, + 430 + ], + "score": 0.86, + "content": "n ~ = ~ \\textstyle \\frac { \\mathbb { P } - \\mathbb { Q } } { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 415, + 303, + 428 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 428, + 303, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 229, + 444 + ], + "score": 1.0, + "content": "ball B2(X , µ), hence fχ =", + "type": "text" + }, + { + "bbox": [ + 196, + 429, + 266, + 446 + ], + "score": 0.9, + "content": "\\begin{array} { r } { f _ { \\chi } = \\frac { n ^ { \\cdot } } { \\| n \\| _ { \\mathcal { L } _ { 2 } ( \\chi , \\mu ) } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 428, + 303, + 442 + ], + "score": 1.0, + "content": "Hence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 444, + 302, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 205, + 460 + ], + "score": 0.81, + "content": "\\mathcal { F } ( \\mathbb { P } , \\mathbb { Q } ) = \\| n \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } .", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 446, + 302, + 455 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "image", + "bbox": [ + 358, + 389, + 453, + 455 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 358, + 389, + 453, + 455 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 358, + 389, + 453, + 455 + ], + "spans": [ + { + "bbox": [ + 358, + 389, + 453, + 455 + ], + "score": 0.935, + "type": "image", + "image_path": "39105c967ba0488a4607950203039a45c6b2ef8745052cceb191e400af567c21.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 358, + 389, + 453, + 422.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 358, + 422.0, + 453, + 455.0 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 334, + 486 + ], + "score": 1.0, + "content": "We see from Theorem 1 the role of the dominant measure", + "type": "text" + }, + { + "bbox": [ + 334, + 476, + 341, + 486 + ], + "score": 0.75, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 473, + 505, + 486 + ], + "score": 1.0, + "content": ": the optimal critic is defined with respect", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 486, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 497 + ], + "score": 1.0, + "content": "to this measure and the overall Fisher distance can be seen as an average weighted distance between", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 496, + 504, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 436, + 508 + ], + "score": 1.0, + "content": "probability density functions, where the average is taken on points sampled from", + "type": "text" + }, + { + "bbox": [ + 436, + 498, + 443, + 507 + ], + "score": 0.77, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 496, + 504, + 508 + ], + "score": 1.0, + "content": ". We give here", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 507, + 185, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 173, + 520 + ], + "score": 1.0, + "content": "some choices of", + "type": "text" + }, + { + "bbox": [ + 173, + 509, + 181, + 519 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 507, + 185, + 520 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 129, + 525, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 130, + 525, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 130, + 525, + 158, + 538 + ], + "score": 1.0, + "content": "1. For", + "type": "text" + }, + { + "bbox": [ + 159, + 525, + 217, + 539 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mu = \\frac 1 2 ( { \\mathbb P } + { \\mathbb Q } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 525, + 505, + 538 + ], + "score": 1.0, + "content": ", we obtain the symmetric chi-squared distance as defined in (Mroueh &", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 536, + 197, + 548 + ], + "spans": [ + { + "bbox": [ + 141, + 536, + 197, + 548 + ], + "score": 1.0, + "content": "Sercu, 2017).", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 128, + 550, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 128, + 550, + 142, + 564 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 142, + 552, + 162, + 563 + ], + "score": 0.81, + "content": "\\mu _ { G P }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 550, + 437, + 564 + ], + "score": 1.0, + "content": ", the implicit distribution defined by the interpolation lines between", + "type": "text" + }, + { + "bbox": [ + 437, + 552, + 449, + 562 + ], + "score": 0.89, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 550, + 468, + 564 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 468, + 551, + 482, + 563 + ], + "score": 0.89, + "content": "\\mathbb { Q } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 550, + 505, + 564 + ], + "score": 1.0, + "content": "as in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 142, + 562, + 238, + 574 + ], + "spans": [ + { + "bbox": [ + 142, + 562, + 238, + 574 + ], + "score": 1.0, + "content": "(Gulrajani et al., 2017).", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 128, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 128, + 575, + 168, + 589 + ], + "score": 1.0, + "content": "3. When", + "type": "text" + }, + { + "bbox": [ + 169, + 578, + 176, + 588 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 575, + 252, + 589 + ], + "score": 1.0, + "content": "does not dominate", + "type": "text" + }, + { + "bbox": [ + 253, + 577, + 260, + 586 + ], + "score": 0.77, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 575, + 281, + 589 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 281, + 577, + 290, + 587 + ], + "score": 0.83, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 575, + 505, + 589 + ], + "score": 1.0, + "content": ", we obtain a non symmetric divergence. For example", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 139, + 586, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 139, + 588, + 156, + 606 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 157, + 590, + 184, + 603 + ], + "score": 0.87, + "content": "\\mu = \\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 588, + 187, + 606 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 187, + 588, + 318, + 605 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathcal { F } _ { \\mathbb { P } } ^ { 2 } ( \\mathbb { P } , \\mathbb { Q } ) = \\int _ { \\mathcal { X } } \\frac { ( \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) ) ^ { 2 } } { \\mathbb { P } ( x ) } d x } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 586, + 507, + 606 + ], + "score": 1.0, + "content": "(P(x)−Q(x))2P(x) dx. We see here that for this particular choice we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 602, + 267, + 617 + ], + "spans": [ + { + "bbox": [ + 142, + 602, + 267, + 617 + ], + "score": 1.0, + "content": "obtain the Pearson divergence.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 107, + 630, + 203, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 204, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 204, + 645 + ], + "score": 1.0, + "content": "4 SOBOLEV IPM", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "In this Section, we introduce the Sobolev IPM. In a nutshell, the Sobolev IPM constrains the critic", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "function to belong to a ball in the restricted Sobolev Space. In other words we constrain the norm", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 222, + 690 + ], + "score": 1.0, + "content": "of the gradient of the critic", + "type": "text" + }, + { + "bbox": [ + 222, + 677, + 256, + 689 + ], + "score": 0.92, + "content": "\\nabla _ { x } f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 676, + 506, + 690 + ], + "score": 1.0, + "content": ". We will show that by moving from a Lebesgue constraint", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "as in Fisher IPM to a Sobolev constraint as in Sobolev IPM, the metric changes from a joint PDF", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "matching to weighted (ccordinate-wise) conditional CDFs matching. The intrinsic conditioning built", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "in to the Sobolev IPM and the comparison of cumulative distributions makes Sobolev IPM suitable", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 245, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 245, + 734 + ], + "score": 1.0, + "content": "for comparing discrete sequences.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "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": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 285, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 286, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 286, + 96 + ], + "score": 1.0, + "content": "We define the unit Lebesgue ball as follows:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 286, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 207, + 95, + 402, + 111 + ], + "lines": [ + { + "bbox": [ + 207, + 95, + 402, + 111 + ], + "spans": [ + { + "bbox": [ + 207, + 95, + 402, + 111 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\mathbb { B } _ { 2 } ( \\mathcal { X } , \\mu ) = \\{ f \\in \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) , \\| f \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } \\leq 1 \\} . } \\end{array}", + "type": "interline_equation", + "image_path": "ed5b076ecf167418ab81fd98dc01d6d5f7e66b35768fe8f111ec1304aecbf9ce.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 207, + 95, + 402, + 111 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 112, + 505, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 111, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 124 + ], + "score": 1.0, + "content": "Fisher IPM defined in (Mroueh & Sercu, 2017), searches for the critic function in the Lebesgue Ball", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 123, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 123, + 146, + 135 + ], + "score": 0.92, + "content": "\\mathbb { B } _ { 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 123, + 338, + 135 + ], + "score": 1.0, + "content": "that maximizes the mean discrepancy between", + "type": "text" + }, + { + "bbox": [ + 338, + 124, + 346, + 133 + ], + "score": 0.82, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 123, + 364, + 135 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 364, + 124, + 373, + 134 + ], + "score": 0.84, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 123, + 505, + 135 + ], + "score": 1.0, + "content": ". Fisher GAN (Mroueh & Sercu,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 133, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 303, + 147 + ], + "score": 1.0, + "content": "2017) was originally formulated specifically for", + "type": "text" + }, + { + "bbox": [ + 303, + 134, + 366, + 147 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\mu = \\frac 1 2 ( { \\mathbb P } + { \\mathbb Q } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 133, + 485, + 147 + ], + "score": 1.0, + "content": ". We consider here a general", + "type": "text" + }, + { + "bbox": [ + 485, + 136, + 492, + 146 + ], + "score": 0.79, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 133, + 506, + 147 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 414, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 189, + 156 + ], + "score": 1.0, + "content": "long as it dominates", + "type": "text" + }, + { + "bbox": [ + 189, + 146, + 197, + 155 + ], + "score": 0.84, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 145, + 214, + 156 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 146, + 223, + 156 + ], + "score": 0.85, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 145, + 414, + 156 + ], + "score": 1.0, + "content": ". We define Generalized Fisher IPM as follows:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 111, + 506, + 156 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 158, + 402, + 180 + ], + "lines": [ + { + "bbox": [ + 209, + 158, + 402, + 180 + ], + "spans": [ + { + "bbox": [ + 209, + 158, + 402, + 180 + ], + "score": 0.93, + "content": "\\mathcal { F } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\operatorname* { s u p } _ { f \\in \\mathbb { B } _ { 2 } ( \\mathcal { X } , \\mu ) } \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x )", + "type": "interline_equation", + "image_path": "0459b6283064953f4a579db933a7da8d2f8da64146d6cb7f61d718f7edc339ed.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 209, + 158, + 402, + 180 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 182, + 147, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 149, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 149, + 194 + ], + "score": 1.0, + "content": "Note that:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 181, + 149, + 194 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 189, + 406, + 219 + ], + "lines": [ + { + "bbox": [ + 204, + 189, + 406, + 219 + ], + "spans": [ + { + "bbox": [ + 204, + 189, + 406, + 219 + ], + "score": 0.9, + "content": "\\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) = \\left. f , { \\frac { \\mathbb { P } - \\mathbb { Q } } { \\mu } } \\right. _ { { \\mathcal { L } } _ { 2 } ( \\mathcal { X } , \\mu ) } .", + "type": "interline_equation", + "image_path": "7adc5ae6d5564e7bdcb40573887203e12aae54584adacb2f75345dce0941e0c9.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 189, + 406, + 204.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 204, + 204.0, + 406, + 219.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 285, + 230 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 286, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 286, + 231 + ], + "score": 1.0, + "content": "Hence Fisher IPM can be written as follows:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 217, + 286, + 231 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 229, + 396, + 259 + ], + "lines": [ + { + "bbox": [ + 215, + 229, + 396, + 259 + ], + "spans": [ + { + "bbox": [ + 215, + 229, + 396, + 259 + ], + "score": 0.92, + "content": "\\mathcal { F } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\operatorname* { s u p } _ { f \\in \\mathbb { B } _ { 2 } ( \\mathcal { X } , \\mu ) } \\left. f , \\frac { \\mathbb { P } - \\mathbb { Q } } { \\mu } \\right. _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) }", + "type": "interline_equation", + "image_path": "cf2ca2c1ee9e66d8cc139d9ab4818c3ad24e905d70ad8d497fd2b16dffeacf48.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 215, + 229, + 396, + 244.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 215, + 244.0, + 396, + 259.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 260, + 225, + 271 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 226, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 226, + 273 + ], + "score": 1.0, + "content": "We have the following result:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 258, + 226, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 273, + 493, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 495, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 495, + 286 + ], + "score": 1.0, + "content": "Theorem 1 (Generalized Fisher IPM). The Fisher distance and the optimal critic are as follows:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 271, + 495, + 286 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 292, + 271, + 304 + ], + "lines": [ + { + "bbox": [ + 129, + 291, + 272, + 307 + ], + "spans": [ + { + "bbox": [ + 129, + 291, + 272, + 307 + ], + "score": 1.0, + "content": "1. The Fisher distance is given by:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 129, + 291, + 272, + 307 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 306, + 450, + 341 + ], + "lines": [ + { + "bbox": [ + 198, + 306, + 450, + 341 + ], + "spans": [ + { + "bbox": [ + 198, + 306, + 450, + 341 + ], + "score": 0.93, + "content": "\\mathcal { F } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\left\\| \\frac { \\mathbb { P } - \\mathbb { Q } } { \\mu } \\right\\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } = \\sqrt { \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mu } \\left( \\frac { \\mathbb { P } ( \\boldsymbol { x } ) - \\mathbb { Q } ( \\boldsymbol { x } ) } { \\mu ( \\boldsymbol { x } ) } \\right) ^ { 2 } } .", + "type": "interline_equation", + "image_path": "3db2e30f980cd8a9477d69935b4e5387d1876b47cb6bc9a7613e57d66273b858.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 198, + 306, + 450, + 317.6666666666667 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 198, + 317.6666666666667, + 450, + 329.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 198, + 329.33333333333337, + 450, + 341.00000000000006 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 129, + 348, + 380, + 360 + ], + "lines": [ + { + "bbox": [ + 129, + 347, + 380, + 362 + ], + "spans": [ + { + "bbox": [ + 129, + 347, + 192, + 362 + ], + "score": 1.0, + "content": "2. The optimal", + "type": "text" + }, + { + "bbox": [ + 193, + 349, + 204, + 360 + ], + "score": 0.88, + "content": "f _ { \\chi }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 347, + 325, + 362 + ], + "score": 1.0, + "content": "achieving the Fisher distance", + "type": "text" + }, + { + "bbox": [ + 325, + 348, + 366, + 361 + ], + "score": 0.93, + "content": "\\mathcal { F } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 347, + 380, + 362 + ], + "score": 1.0, + "content": "is:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 129, + 347, + 380, + 362 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 302, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 391, + 302, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 302, + 405 + ], + "score": 1.0, + "content": "Proof of Theorem 1. From Equation (2), the op-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 302, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 131, + 416 + ], + "score": 1.0, + "content": "timal", + "type": "text" + }, + { + "bbox": [ + 131, + 404, + 143, + 416 + ], + "score": 0.9, + "content": "f _ { \\chi }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 403, + 302, + 416 + ], + "score": 1.0, + "content": "belong to the intersection of the hy-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 415, + 303, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 215, + 428 + ], + "score": 1.0, + "content": "perplane that has normal", + "type": "text" + }, + { + "bbox": [ + 216, + 415, + 262, + 430 + ], + "score": 0.86, + "content": "n ~ = ~ \\textstyle \\frac { \\mathbb { P } - \\mathbb { Q } } { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 415, + 303, + 428 + ], + "score": 1.0, + "content": "and the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 428, + 303, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 229, + 444 + ], + "score": 1.0, + "content": "ball B2(X , µ), hence fχ =", + "type": "text" + }, + { + "bbox": [ + 196, + 429, + 266, + 446 + ], + "score": 0.9, + "content": "\\begin{array} { r } { f _ { \\chi } = \\frac { n ^ { \\cdot } } { \\| n \\| _ { \\mathcal { L } _ { 2 } ( \\chi , \\mu ) } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 428, + 303, + 442 + ], + "score": 1.0, + "content": "Hence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 444, + 302, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 205, + 460 + ], + "score": 0.81, + "content": "\\mathcal { F } ( \\mathbb { P } , \\mathbb { Q } ) = \\| n \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) } .", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 446, + 302, + 455 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 391, + 303, + 460 + ] + }, + { + "type": "image", + "bbox": [ + 358, + 389, + 453, + 455 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 358, + 389, + 453, + 455 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 358, + 389, + 453, + 455 + ], + "spans": [ + { + "bbox": [ + 358, + 389, + 453, + 455 + ], + "score": 0.935, + "type": "image", + "image_path": "39105c967ba0488a4607950203039a45c6b2ef8745052cceb191e400af567c21.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 358, + 389, + 453, + 422.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 358, + 422.0, + 453, + 455.0 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 473, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 334, + 486 + ], + "score": 1.0, + "content": "We see from Theorem 1 the role of the dominant measure", + "type": "text" + }, + { + "bbox": [ + 334, + 476, + 341, + 486 + ], + "score": 0.75, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 473, + 505, + 486 + ], + "score": 1.0, + "content": ": the optimal critic is defined with respect", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 486, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 497 + ], + "score": 1.0, + "content": "to this measure and the overall Fisher distance can be seen as an average weighted distance between", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 496, + 504, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 436, + 508 + ], + "score": 1.0, + "content": "probability density functions, where the average is taken on points sampled from", + "type": "text" + }, + { + "bbox": [ + 436, + 498, + 443, + 507 + ], + "score": 0.77, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 496, + 504, + 508 + ], + "score": 1.0, + "content": ". We give here", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 507, + 185, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 173, + 520 + ], + "score": 1.0, + "content": "some choices of", + "type": "text" + }, + { + "bbox": [ + 173, + 509, + 181, + 519 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 507, + 185, + 520 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 106, + 473, + 505, + 520 + ] + }, + { + "type": "list", + "bbox": [ + 129, + 525, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 130, + 525, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 130, + 525, + 158, + 538 + ], + "score": 1.0, + "content": "1. For", + "type": "text" + }, + { + "bbox": [ + 159, + 525, + 217, + 539 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mu = \\frac 1 2 ( { \\mathbb P } + { \\mathbb Q } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 525, + 505, + 538 + ], + "score": 1.0, + "content": ", we obtain the symmetric chi-squared distance as defined in (Mroueh &", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 536, + 197, + 548 + ], + "spans": [ + { + "bbox": [ + 141, + 536, + 197, + 548 + ], + "score": 1.0, + "content": "Sercu, 2017).", + "type": "text" + } + ], + "index": 32, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 550, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 128, + 550, + 142, + 564 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 142, + 552, + 162, + 563 + ], + "score": 0.81, + "content": "\\mu _ { G P }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 550, + 437, + 564 + ], + "score": 1.0, + "content": ", the implicit distribution defined by the interpolation lines between", + "type": "text" + }, + { + "bbox": [ + 437, + 552, + 449, + 562 + ], + "score": 0.89, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 550, + 468, + 564 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 468, + 551, + 482, + 563 + ], + "score": 0.89, + "content": "\\mathbb { Q } _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 550, + 505, + 564 + ], + "score": 1.0, + "content": "as in", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 562, + 238, + 574 + ], + "spans": [ + { + "bbox": [ + 142, + 562, + 238, + 574 + ], + "score": 1.0, + "content": "(Gulrajani et al., 2017).", + "type": "text" + } + ], + "index": 34, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 128, + 575, + 168, + 589 + ], + "score": 1.0, + "content": "3. When", + "type": "text" + }, + { + "bbox": [ + 169, + 578, + 176, + 588 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 575, + 252, + 589 + ], + "score": 1.0, + "content": "does not dominate", + "type": "text" + }, + { + "bbox": [ + 253, + 577, + 260, + 586 + ], + "score": 0.77, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 575, + 281, + 589 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 281, + 577, + 290, + 587 + ], + "score": 0.83, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 575, + 505, + 589 + ], + "score": 1.0, + "content": ", we obtain a non symmetric divergence. For example", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 139, + 586, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 139, + 588, + 156, + 606 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 157, + 590, + 184, + 603 + ], + "score": 0.87, + "content": "\\mu = \\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 588, + 187, + 606 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 187, + 588, + 318, + 605 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathcal { F } _ { \\mathbb { P } } ^ { 2 } ( \\mathbb { P } , \\mathbb { Q } ) = \\int _ { \\mathcal { X } } \\frac { ( \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) ) ^ { 2 } } { \\mathbb { P } ( x ) } d x } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 586, + 507, + 606 + ], + "score": 1.0, + "content": "(P(x)−Q(x))2P(x) dx. We see here that for this particular choice we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 602, + 267, + 617 + ], + "spans": [ + { + "bbox": [ + 142, + 602, + 267, + 617 + ], + "score": 1.0, + "content": "obtain the Pearson divergence.", + "type": "text" + } + ], + "index": 37, + "is_list_end_line": true + } + ], + "index": 34, + "bbox_fs": [ + 128, + 525, + 507, + 617 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 630, + 203, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 204, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 204, + 645 + ], + "score": 1.0, + "content": "4 SOBOLEV IPM", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "In this Section, we introduce the Sobolev IPM. In a nutshell, the Sobolev IPM constrains the critic", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "function to belong to a ball in the restricted Sobolev Space. In other words we constrain the norm", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 222, + 690 + ], + "score": 1.0, + "content": "of the gradient of the critic", + "type": "text" + }, + { + "bbox": [ + 222, + 677, + 256, + 689 + ], + "score": 0.92, + "content": "\\nabla _ { x } f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 676, + 506, + 690 + ], + "score": 1.0, + "content": ". We will show that by moving from a Lebesgue constraint", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "as in Fisher IPM to a Sobolev constraint as in Sobolev IPM, the metric changes from a joint PDF", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "matching to weighted (ccordinate-wise) conditional CDFs matching. The intrinsic conditioning built", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "in to the Sobolev IPM and the comparison of cumulative distributions makes Sobolev IPM suitable", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 245, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 245, + 734 + ], + "score": 1.0, + "content": "for comparing discrete sequences.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 655, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 457, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 458, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 458, + 94 + ], + "score": 1.0, + "content": "4.1 DEFINITION AND EXPRESSION OF SOBOLEV IPM IN TERMS OF COORDINATE", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 130, + 93, + 222, + 104 + ], + "spans": [ + { + "bbox": [ + 130, + 93, + 222, + 104 + ], + "score": 1.0, + "content": "CONDITIONAL CDFS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 104, + 113, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 113, + 504, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 493, + 126 + ], + "score": 1.0, + "content": "We will start by recalling some definitions on Sobolev Spaces. We assume in the following that", + "type": "text" + }, + { + "bbox": [ + 494, + 114, + 504, + 124 + ], + "score": 0.8, + "content": "\\mathcal { X }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 123, + 384, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 329, + 139 + ], + "score": 1.0, + "content": "is compact and consider functions in the Sobolev space", + "type": "text" + }, + { + "bbox": [ + 329, + 124, + 379, + 137 + ], + "score": 0.93, + "content": "\\mathring { W } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 123, + 384, + 139 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 140, + 430, + 168 + ], + "lines": [ + { + "bbox": [ + 180, + 140, + 430, + 168 + ], + "spans": [ + { + "bbox": [ + 180, + 140, + 430, + 168 + ], + "score": 0.94, + "content": "W ^ { 1 , 2 } ( \\mathcal { X } , \\mu ) = \\left\\{ f : \\mathcal { X } \\to \\mathbb { R } , \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) \\right. ^ { 2 } \\mu ( x ) d x < \\infty \\right\\} ,", + "type": "interline_equation", + "image_path": "4dd3c4c9510738e81204ba0d088fe7d352e34da54909b8462ff90e72252d37bb.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 180, + 140, + 430, + 168 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 504, + 196 + ], + "lines": [ + { + "bbox": [ + 106, + 169, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 259, + 185 + ], + "score": 1.0, + "content": "We restrict ourselves to functions in", + "type": "text" + }, + { + "bbox": [ + 259, + 171, + 309, + 183 + ], + "score": 0.93, + "content": "W ^ { 1 , 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 169, + 505, + 185 + ], + "score": 1.0, + "content": "vanishing at the boundary, and note this space", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 181, + 251, + 198 + ], + "spans": [ + { + "bbox": [ + 107, + 182, + 156, + 197 + ], + "score": 0.92, + "content": "W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 181, + 251, + 198 + ], + "score": 1.0, + "content": ". Note that in this case:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 199, + 393, + 234 + ], + "lines": [ + { + "bbox": [ + 219, + 199, + 393, + 234 + ], + "spans": [ + { + "bbox": [ + 219, + 199, + 393, + 234 + ], + "score": 0.94, + "content": "\\| f \\| _ { W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu ) } = \\sqrt { \\int _ { \\mathcal { X } } \\| \\nabla _ { x } f ( x ) \\| ^ { 2 } \\mu ( x ) d x }", + "type": "interline_equation", + "image_path": "2214e21f4eadc9152ee5680c70949b5c27ead062efe29705cb6cd7ad541fef3b.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 199, + 393, + 216.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 219, + 216.5, + 393, + 234.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 503, + 251 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 355, + 252 + ], + "score": 1.0, + "content": "defines a semi-norm. We can similarly define a dot product in", + "type": "text" + }, + { + "bbox": [ + 355, + 237, + 405, + 251 + ], + "score": 0.93, + "content": "W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 236, + 423, + 252 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 423, + 236, + 500, + 251 + ], + "score": 0.93, + "content": "f , g \\in W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 236, + 505, + 252 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 253, + 412, + 280 + ], + "lines": [ + { + "bbox": [ + 199, + 253, + 412, + 280 + ], + "spans": [ + { + "bbox": [ + 199, + 253, + 412, + 280 + ], + "score": 0.92, + "content": "\\langle f , g \\rangle _ { W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu ) } = \\int _ { \\mathcal { X } } \\langle \\nabla _ { x } f ( x ) , \\nabla _ { x } g ( x ) \\rangle _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x .", + "type": "interline_equation", + "image_path": "519d221b59a2972c2c722e9abe4f0070a191217e27f4769d20b3d3ecb4b62046.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 199, + 253, + 412, + 280 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 288, + 504, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 302 + ], + "score": 1.0, + "content": "Hence we define the following Sobolev IPM, by restricting the critic of the mean discrepancy to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 299, + 183, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 183, + 311 + ], + "score": 1.0, + "content": "Sobolev unit ball :", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 312, + 434, + 344 + ], + "lines": [ + { + "bbox": [ + 177, + 312, + 434, + 344 + ], + "spans": [ + { + "bbox": [ + 177, + 312, + 434, + 344 + ], + "score": 0.94, + "content": "\\mathcal { S } _ { \\boldsymbol { \\mu } } ( \\mathbb { P } , \\mathbb { Q } ) = \\operatorname* { s u p } _ { \\substack { f \\in W _ { 0 } ^ { 1 , 2 } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\boldsymbol { \\mu } ) } \\leq 1 } } \\bigg \\{ \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { P } } f ( \\boldsymbol { x } ) - \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { Q } } f ( \\boldsymbol { x } ) \\bigg \\} .", + "type": "interline_equation", + "image_path": "2164ff51845e7b061208c655341d466218cfe9d7307e1638f2149974e75927b1.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 177, + 312, + 434, + 322.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 177, + 322.6666666666667, + 434, + 333.33333333333337 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 177, + 333.33333333333337, + 434, + 344.00000000000006 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 506, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "score": 1.0, + "content": "When compared to the Wasserstein distance, the Sobolev IPM given in Equation (3) uses a data", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 276, + 370 + ], + "score": 1.0, + "content": "dependent gradient constraint (depends on", + "type": "text" + }, + { + "bbox": [ + 276, + 360, + 284, + 370 + ], + "score": 0.77, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 358, + 506, + 370 + ], + "score": 1.0, + "content": ") rather than a data independent Lipchitz constraint. Let", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 369, + 451, + 381 + ], + "spans": [ + { + "bbox": [ + 107, + 370, + 119, + 380 + ], + "score": 0.87, + "content": "F _ { \\mathbb { P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 369, + 137, + 381 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 137, + 369, + 151, + 381 + ], + "score": 0.89, + "content": "F _ { \\mathbb { Q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 369, + 323, + 381 + ], + "score": 1.0, + "content": "be the cumulative distribution functions of", + "type": "text" + }, + { + "bbox": [ + 323, + 370, + 331, + 379 + ], + "score": 0.83, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 369, + 348, + 381 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 349, + 369, + 358, + 380 + ], + "score": 0.86, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 369, + 451, + 381 + ], + "score": 1.0, + "content": "respectively. We have:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 384, + 361, + 411 + ], + "lines": [ + { + "bbox": [ + 249, + 384, + 361, + 411 + ], + "spans": [ + { + "bbox": [ + 249, + 384, + 361, + 411 + ], + "score": 0.94, + "content": "\\mathbb { P } ( x ) = \\frac { \\partial ^ { d } } { \\partial x _ { 1 } \\hdots \\partial x _ { d } } F _ { \\mathbb { P } } ( x ) ,", + "type": "interline_equation", + "image_path": "12e013d292fb43ace5cb893d767b9deaa3a5d0371cc443ca3e7ec2fd7bab3c03.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 249, + 384, + 361, + 411 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 164, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 164, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 164, + 426 + ], + "score": 1.0, + "content": "and we define", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 421, + 412, + 450 + ], + "lines": [ + { + "bbox": [ + 198, + 421, + 412, + 450 + ], + "spans": [ + { + "bbox": [ + 198, + 421, + 412, + 450 + ], + "score": 0.94, + "content": "D ^ { - i } = \\frac { \\partial ^ { d - 1 } } { \\partial x _ { 1 } \\dots \\partial x _ { i - 1 } \\partial x _ { i + 1 } \\dots \\partial x _ { d } } , \\mathrm { f o r } i = 1 \\dots d .", + "type": "interline_equation", + "image_path": "743b961821257fa0fa92772b952b707d1896db052f5fae4f4bb446ccb49fcefd.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 421, + 412, + 435.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 198, + 435.5, + 412, + 450.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 427, + 470 + ], + "lines": [ + { + "bbox": [ + 107, + 455, + 424, + 473 + ], + "spans": [ + { + "bbox": [ + 107, + 457, + 126, + 468 + ], + "score": 0.87, + "content": "D ^ { - i }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 455, + 183, + 473 + ], + "score": 1.0, + "content": "computes the", + "type": "text" + }, + { + "bbox": [ + 183, + 457, + 213, + 470 + ], + "score": 0.91, + "content": "( d - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 455, + 419, + 473 + ], + "score": 1.0, + "content": "high-order partial derivative excluding the variable", + "type": "text" + }, + { + "bbox": [ + 420, + 459, + 424, + 468 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 108, + 484, + 504, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "Our main result is presented in Theorem 2. Additional theoretical results are given in Appendix A.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 496, + 253, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 253, + 509 + ], + "score": 1.0, + "content": "All proofs are given in Appendix B.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 503, + 534 + ], + "lines": [ + { + "bbox": [ + 107, + 509, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 107, + 509, + 271, + 523 + ], + "score": 1.0, + "content": "Theorem 2 (Sobolev IPM). Assume that", + "type": "text" + }, + { + "bbox": [ + 272, + 511, + 284, + 522 + ], + "score": 0.88, + "content": "F _ { \\mathbb { P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 509, + 305, + 523 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 306, + 511, + 319, + 523 + ], + "score": 0.89, + "content": "F _ { \\mathbb { Q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 509, + 504, + 523 + ], + "score": 1.0, + "content": "and its d derivatives exist and are continuous:", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 521, + 347, + 536 + ], + "spans": [ + { + "bbox": [ + 107, + 523, + 119, + 534 + ], + "score": 0.86, + "content": "F _ { \\mathbb { P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 521, + 138, + 536 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 138, + 522, + 192, + 535 + ], + "score": 0.93, + "content": "F _ { \\mathbb { Q } } \\in C ^ { d } ( \\mathcal { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 521, + 324, + 536 + ], + "score": 1.0, + "content": ". Define the differential operator", + "type": "text" + }, + { + "bbox": [ + 324, + 522, + 340, + 532 + ], + "score": 0.89, + "content": "D ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 521, + 347, + 536 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 256, + 538, + 353, + 552 + ], + "lines": [ + { + "bbox": [ + 256, + 538, + 353, + 552 + ], + "spans": [ + { + "bbox": [ + 256, + 538, + 353, + 552 + ], + "score": 0.91, + "content": "D ^ { - } = ( D ^ { - 1 } , \\ldots D ^ { - d } ) .", + "type": "interline_equation", + "image_path": "3c7fdaadd09bf295ec830d3bc5cb7007418d6ca293309fe91d976278733e575d.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 256, + 538, + 353, + 552 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 555, + 418, + 568 + ], + "lines": [ + { + "bbox": [ + 123, + 555, + 418, + 568 + ], + "spans": [ + { + "bbox": [ + 123, + 555, + 418, + 568 + ], + "score": 0.35, + "content": "x = ( x _ { 1 } , \\ldots x _ { i - 1 } , x _ { i } , x _ { i + 1 } , \\ldots x _ { d } ) , l e t x ^ { - i } = ( x _ { 1 } , \\ldots x _ { i - 1 } , x _ { i + 1 } , \\ldots x _ { d } ) .", + "type": "interline_equation", + "image_path": "63d9ab9f3fb9fe1c2a6a7b82d836d425d046d82439e3a55bea0eb26f1ab45225.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 123, + 555, + 418, + 568 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 572, + 409, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 409, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 409, + 587 + ], + "score": 1.0, + "content": "The Sobolev IPM given in Equation (3) has the following equivalent forms:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 130, + 592, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 129, + 592, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 129, + 592, + 506, + 606 + ], + "score": 1.0, + "content": "1. Sobolev IPM as comparison of high order partial derivatives of CDFs. The Sobolev IPM", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 603, + 238, + 617 + ], + "spans": [ + { + "bbox": [ + 141, + 603, + 238, + 617 + ], + "score": 1.0, + "content": "has the following form:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 619, + 432, + 653 + ], + "lines": [ + { + "bbox": [ + 195, + 619, + 432, + 653 + ], + "spans": [ + { + "bbox": [ + 195, + 619, + 432, + 653 + ], + "score": 0.94, + "content": "S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\frac { 1 } { d } \\sqrt { \\int _ { \\mathcal { X } } \\frac { \\sum _ { i = 1 } ^ { d } ( D ^ { - i } F _ { \\mathbb { P } } ( x ) - D ^ { - i } F _ { \\mathbb { Q } } ( x ) ) ^ { 2 } } { \\mu ( x ) } d x } .", + "type": "interline_equation", + "image_path": "1b1c47f69d225b36578c4457fe34066d17c0087bbe99bfe90899a175ce3b816c.jpg" + } + ] + } + ], + "index": 33.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 619, + 432, + 636.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 195, + 636.0, + 432, + 653.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 129, + 661, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 129, + 661, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 129, + 661, + 505, + 674 + ], + "score": 1.0, + "content": "2. Sobolev IPM as comparison of weighted (coordinate-wise) conditional CDFs. The Sobolev", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 671, + 356, + 685 + ], + "spans": [ + { + "bbox": [ + 141, + 671, + 356, + 685 + ], + "score": 1.0, + "content": "IPM can be written in the following equivalent form:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 689, + 533, + 723 + ], + "lines": [ + { + "bbox": [ + 142, + 689, + 533, + 723 + ], + "spans": [ + { + "bbox": [ + 142, + 689, + 533, + 723 + ], + "score": 0.94, + "content": "S _ { \\mu } ^ { 2 } ( { \\mathbb { P } } , { \\mathbb { Q } } ) = \\frac { 1 } { d ^ { 2 } } { \\mathbb { E } } _ { x \\sim \\mu } \\sum _ { i = 1 } ^ { d } \\left( \\frac { { \\mathbb { P } } _ { X ^ { - i } } ( x ^ { - i } ) F _ { { \\mathbb { P } } _ { [ X _ { i } ] X ^ { - i } = x ^ { - i } ] } } ( x _ { i } ) - { \\mathbb { Q } } _ { X ^ { - i } } ( x ^ { - i } ) F _ { { \\mathbb { Q } } _ { [ X _ { i } ] X ^ { - i } = x ^ { - i } ] } } ( x _ { i } ) } { \\mu ( x ) } \\right) ^ { 2 } .", + "type": "interline_equation", + "image_path": "2a5b2bc7e573d2b9c2a0fd824ac10bb207b79aeeac1c362884b6ce9aaa3aa8a2.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 142, + 689, + 533, + 700.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 142, + 700.3333333333334, + 533, + 711.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 142, + 711.6666666666667, + 533, + 723.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 457, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 458, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 458, + 94 + ], + "score": 1.0, + "content": "4.1 DEFINITION AND EXPRESSION OF SOBOLEV IPM IN TERMS OF COORDINATE", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 130, + 93, + 222, + 104 + ], + "spans": [ + { + "bbox": [ + 130, + 93, + 222, + 104 + ], + "score": 1.0, + "content": "CONDITIONAL CDFS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 104, + 113, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 113, + 504, + 126 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 493, + 126 + ], + "score": 1.0, + "content": "We will start by recalling some definitions on Sobolev Spaces. We assume in the following that", + "type": "text" + }, + { + "bbox": [ + 494, + 114, + 504, + 124 + ], + "score": 0.8, + "content": "\\mathcal { X }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 123, + 384, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 329, + 139 + ], + "score": 1.0, + "content": "is compact and consider functions in the Sobolev space", + "type": "text" + }, + { + "bbox": [ + 329, + 124, + 379, + 137 + ], + "score": 0.93, + "content": "\\mathring { W } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 123, + 384, + 139 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 113, + 504, + 139 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 140, + 430, + 168 + ], + "lines": [ + { + "bbox": [ + 180, + 140, + 430, + 168 + ], + "spans": [ + { + "bbox": [ + 180, + 140, + 430, + 168 + ], + "score": 0.94, + "content": "W ^ { 1 , 2 } ( \\mathcal { X } , \\mu ) = \\left\\{ f : \\mathcal { X } \\to \\mathbb { R } , \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) \\right. ^ { 2 } \\mu ( x ) d x < \\infty \\right\\} ,", + "type": "interline_equation", + "image_path": "4dd3c4c9510738e81204ba0d088fe7d352e34da54909b8462ff90e72252d37bb.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 180, + 140, + 430, + 168 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 171, + 504, + 196 + ], + "lines": [ + { + "bbox": [ + 106, + 169, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 259, + 185 + ], + "score": 1.0, + "content": "We restrict ourselves to functions in", + "type": "text" + }, + { + "bbox": [ + 259, + 171, + 309, + 183 + ], + "score": 0.93, + "content": "W ^ { 1 , 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 169, + 505, + 185 + ], + "score": 1.0, + "content": "vanishing at the boundary, and note this space", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 181, + 251, + 198 + ], + "spans": [ + { + "bbox": [ + 107, + 182, + 156, + 197 + ], + "score": 0.92, + "content": "W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 181, + 251, + 198 + ], + "score": 1.0, + "content": ". Note that in this case:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 106, + 169, + 505, + 198 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 199, + 393, + 234 + ], + "lines": [ + { + "bbox": [ + 219, + 199, + 393, + 234 + ], + "spans": [ + { + "bbox": [ + 219, + 199, + 393, + 234 + ], + "score": 0.94, + "content": "\\| f \\| _ { W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu ) } = \\sqrt { \\int _ { \\mathcal { X } } \\| \\nabla _ { x } f ( x ) \\| ^ { 2 } \\mu ( x ) d x }", + "type": "interline_equation", + "image_path": "2214e21f4eadc9152ee5680c70949b5c27ead062efe29705cb6cd7ad541fef3b.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 199, + 393, + 216.5 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 219, + 216.5, + 393, + 234.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 236, + 503, + 251 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 355, + 252 + ], + "score": 1.0, + "content": "defines a semi-norm. We can similarly define a dot product in", + "type": "text" + }, + { + "bbox": [ + 355, + 237, + 405, + 251 + ], + "score": 0.93, + "content": "W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 236, + 423, + 252 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 423, + 236, + 500, + 251 + ], + "score": 0.93, + "content": "f , g \\in W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu )", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 236, + 505, + 252 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 236, + 505, + 252 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 253, + 412, + 280 + ], + "lines": [ + { + "bbox": [ + 199, + 253, + 412, + 280 + ], + "spans": [ + { + "bbox": [ + 199, + 253, + 412, + 280 + ], + "score": 0.92, + "content": "\\langle f , g \\rangle _ { W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu ) } = \\int _ { \\mathcal { X } } \\langle \\nabla _ { x } f ( x ) , \\nabla _ { x } g ( x ) \\rangle _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x .", + "type": "interline_equation", + "image_path": "519d221b59a2972c2c722e9abe4f0070a191217e27f4769d20b3d3ecb4b62046.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 199, + 253, + 412, + 280 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 288, + 504, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 505, + 302 + ], + "score": 1.0, + "content": "Hence we define the following Sobolev IPM, by restricting the critic of the mean discrepancy to the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 299, + 183, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 183, + 311 + ], + "score": 1.0, + "content": "Sobolev unit ball :", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 287, + 505, + 311 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 312, + 434, + 344 + ], + "lines": [ + { + "bbox": [ + 177, + 312, + 434, + 344 + ], + "spans": [ + { + "bbox": [ + 177, + 312, + 434, + 344 + ], + "score": 0.94, + "content": "\\mathcal { S } _ { \\boldsymbol { \\mu } } ( \\mathbb { P } , \\mathbb { Q } ) = \\operatorname* { s u p } _ { \\substack { f \\in W _ { 0 } ^ { 1 , 2 } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\boldsymbol { \\mu } ) } \\leq 1 } } \\bigg \\{ \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { P } } f ( \\boldsymbol { x } ) - \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { Q } } f ( \\boldsymbol { x } ) \\bigg \\} .", + "type": "interline_equation", + "image_path": "2164ff51845e7b061208c655341d466218cfe9d7307e1638f2149974e75927b1.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 177, + 312, + 434, + 322.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 177, + 322.6666666666667, + 434, + 333.33333333333337 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 177, + 333.33333333333337, + 434, + 344.00000000000006 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 506, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "score": 1.0, + "content": "When compared to the Wasserstein distance, the Sobolev IPM given in Equation (3) uses a data", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 276, + 370 + ], + "score": 1.0, + "content": "dependent gradient constraint (depends on", + "type": "text" + }, + { + "bbox": [ + 276, + 360, + 284, + 370 + ], + "score": 0.77, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 358, + 506, + 370 + ], + "score": 1.0, + "content": ") rather than a data independent Lipchitz constraint. Let", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 369, + 451, + 381 + ], + "spans": [ + { + "bbox": [ + 107, + 370, + 119, + 380 + ], + "score": 0.87, + "content": "F _ { \\mathbb { P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 369, + 137, + 381 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 137, + 369, + 151, + 381 + ], + "score": 0.89, + "content": "F _ { \\mathbb { Q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 369, + 323, + 381 + ], + "score": 1.0, + "content": "be the cumulative distribution functions of", + "type": "text" + }, + { + "bbox": [ + 323, + 370, + 331, + 379 + ], + "score": 0.83, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 369, + 348, + 381 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 349, + 369, + 358, + 380 + ], + "score": 0.86, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 369, + 451, + 381 + ], + "score": 1.0, + "content": "respectively. We have:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 346, + 506, + 381 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 384, + 361, + 411 + ], + "lines": [ + { + "bbox": [ + 249, + 384, + 361, + 411 + ], + "spans": [ + { + "bbox": [ + 249, + 384, + 361, + 411 + ], + "score": 0.94, + "content": "\\mathbb { P } ( x ) = \\frac { \\partial ^ { d } } { \\partial x _ { 1 } \\hdots \\partial x _ { d } } F _ { \\mathbb { P } } ( x ) ,", + "type": "interline_equation", + "image_path": "12e013d292fb43ace5cb893d767b9deaa3a5d0371cc443ca3e7ec2fd7bab3c03.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 249, + 384, + 361, + 411 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 164, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 164, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 164, + 426 + ], + "score": 1.0, + "content": "and we define", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 413, + 164, + 426 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 421, + 412, + 450 + ], + "lines": [ + { + "bbox": [ + 198, + 421, + 412, + 450 + ], + "spans": [ + { + "bbox": [ + 198, + 421, + 412, + 450 + ], + "score": 0.94, + "content": "D ^ { - i } = \\frac { \\partial ^ { d - 1 } } { \\partial x _ { 1 } \\dots \\partial x _ { i - 1 } \\partial x _ { i + 1 } \\dots \\partial x _ { d } } , \\mathrm { f o r } i = 1 \\dots d .", + "type": "interline_equation", + "image_path": "743b961821257fa0fa92772b952b707d1896db052f5fae4f4bb446ccb49fcefd.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 198, + 421, + 412, + 435.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 198, + 435.5, + 412, + 450.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 427, + 470 + ], + "lines": [ + { + "bbox": [ + 107, + 455, + 424, + 473 + ], + "spans": [ + { + "bbox": [ + 107, + 457, + 126, + 468 + ], + "score": 0.87, + "content": "D ^ { - i }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 455, + 183, + 473 + ], + "score": 1.0, + "content": "computes the", + "type": "text" + }, + { + "bbox": [ + 183, + 457, + 213, + 470 + ], + "score": 0.91, + "content": "( d - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 455, + 419, + 473 + ], + "score": 1.0, + "content": "high-order partial derivative excluding the variable", + "type": "text" + }, + { + "bbox": [ + 420, + 459, + 424, + 468 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 107, + 455, + 424, + 473 + ] + }, + { + "type": "list", + "bbox": [ + 108, + 484, + 504, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "Our main result is presented in Theorem 2. Additional theoretical results are given in Appendix A.", + "type": "text" + } + ], + "index": 24, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 496, + 253, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 253, + 509 + ], + "score": 1.0, + "content": "All proofs are given in Appendix B.", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 485, + 505, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 503, + 534 + ], + "lines": [ + { + "bbox": [ + 107, + 509, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 107, + 509, + 271, + 523 + ], + "score": 1.0, + "content": "Theorem 2 (Sobolev IPM). Assume that", + "type": "text" + }, + { + "bbox": [ + 272, + 511, + 284, + 522 + ], + "score": 0.88, + "content": "F _ { \\mathbb { P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 509, + 305, + 523 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 306, + 511, + 319, + 523 + ], + "score": 0.89, + "content": "F _ { \\mathbb { Q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 509, + 504, + 523 + ], + "score": 1.0, + "content": "and its d derivatives exist and are continuous:", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 521, + 347, + 536 + ], + "spans": [ + { + "bbox": [ + 107, + 523, + 119, + 534 + ], + "score": 0.86, + "content": "F _ { \\mathbb { P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 521, + 138, + 536 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 138, + 522, + 192, + 535 + ], + "score": 0.93, + "content": "F _ { \\mathbb { Q } } \\in C ^ { d } ( \\mathcal { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 521, + 324, + 536 + ], + "score": 1.0, + "content": ". Define the differential operator", + "type": "text" + }, + { + "bbox": [ + 324, + 522, + 340, + 532 + ], + "score": 0.89, + "content": "D ^ { - }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 521, + 347, + 536 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 107, + 509, + 504, + 536 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 256, + 538, + 353, + 552 + ], + "lines": [ + { + "bbox": [ + 256, + 538, + 353, + 552 + ], + "spans": [ + { + "bbox": [ + 256, + 538, + 353, + 552 + ], + "score": 0.91, + "content": "D ^ { - } = ( D ^ { - 1 } , \\ldots D ^ { - d } ) .", + "type": "interline_equation", + "image_path": "3c7fdaadd09bf295ec830d3bc5cb7007418d6ca293309fe91d976278733e575d.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 256, + 538, + 353, + 552 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 555, + 418, + 568 + ], + "lines": [ + { + "bbox": [ + 123, + 555, + 418, + 568 + ], + "spans": [ + { + "bbox": [ + 123, + 555, + 418, + 568 + ], + "score": 0.35, + "content": "x = ( x _ { 1 } , \\ldots x _ { i - 1 } , x _ { i } , x _ { i + 1 } , \\ldots x _ { d } ) , l e t x ^ { - i } = ( x _ { 1 } , \\ldots x _ { i - 1 } , x _ { i + 1 } , \\ldots x _ { d } ) .", + "type": "interline_equation", + "image_path": "63d9ab9f3fb9fe1c2a6a7b82d836d425d046d82439e3a55bea0eb26f1ab45225.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 123, + 555, + 418, + 568 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 572, + 409, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 409, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 409, + 587 + ], + "score": 1.0, + "content": "The Sobolev IPM given in Equation (3) has the following equivalent forms:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 572, + 409, + 587 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 592, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 129, + 592, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 129, + 592, + 506, + 606 + ], + "score": 1.0, + "content": "1. Sobolev IPM as comparison of high order partial derivatives of CDFs. The Sobolev IPM", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 603, + 238, + 617 + ], + "spans": [ + { + "bbox": [ + 141, + 603, + 238, + 617 + ], + "score": 1.0, + "content": "has the following form:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 129, + 592, + 506, + 617 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 619, + 432, + 653 + ], + "lines": [ + { + "bbox": [ + 195, + 619, + 432, + 653 + ], + "spans": [ + { + "bbox": [ + 195, + 619, + 432, + 653 + ], + "score": 0.94, + "content": "S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\frac { 1 } { d } \\sqrt { \\int _ { \\mathcal { X } } \\frac { \\sum _ { i = 1 } ^ { d } ( D ^ { - i } F _ { \\mathbb { P } } ( x ) - D ^ { - i } F _ { \\mathbb { Q } } ( x ) ) ^ { 2 } } { \\mu ( x ) } d x } .", + "type": "interline_equation", + "image_path": "1b1c47f69d225b36578c4457fe34066d17c0087bbe99bfe90899a175ce3b816c.jpg" + } + ] + } + ], + "index": 33.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 619, + 432, + 636.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 195, + 636.0, + 432, + 653.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 129, + 661, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 129, + 661, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 129, + 661, + 505, + 674 + ], + "score": 1.0, + "content": "2. Sobolev IPM as comparison of weighted (coordinate-wise) conditional CDFs. The Sobolev", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 671, + 356, + 685 + ], + "spans": [ + { + "bbox": [ + 141, + 671, + 356, + 685 + ], + "score": 1.0, + "content": "IPM can be written in the following equivalent form:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 129, + 661, + 505, + 685 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 689, + 533, + 723 + ], + "lines": [ + { + "bbox": [ + 142, + 689, + 533, + 723 + ], + "spans": [ + { + "bbox": [ + 142, + 689, + 533, + 723 + ], + "score": 0.94, + "content": "S _ { \\mu } ^ { 2 } ( { \\mathbb { P } } , { \\mathbb { Q } } ) = \\frac { 1 } { d ^ { 2 } } { \\mathbb { E } } _ { x \\sim \\mu } \\sum _ { i = 1 } ^ { d } \\left( \\frac { { \\mathbb { P } } _ { X ^ { - i } } ( x ^ { - i } ) F _ { { \\mathbb { P } } _ { [ X _ { i } ] X ^ { - i } = x ^ { - i } ] } } ( x _ { i } ) - { \\mathbb { Q } } _ { X ^ { - i } } ( x ^ { - i } ) F _ { { \\mathbb { Q } } _ { [ X _ { i } ] X ^ { - i } = x ^ { - i } ] } } ( x _ { i } ) } { \\mu ( x ) } \\right) ^ { 2 } .", + "type": "interline_equation", + "image_path": "2a5b2bc7e573d2b9c2a0fd824ac10bb207b79aeeac1c362884b6ce9aaa3aa8a2.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 142, + 689, + 533, + 700.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 142, + 700.3333333333334, + 533, + 711.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 142, + 711.6666666666667, + 533, + 723.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 129, + 82, + 353, + 95 + ], + "lines": [ + { + "bbox": [ + 128, + 81, + 354, + 96 + ], + "spans": [ + { + "bbox": [ + 128, + 81, + 216, + 96 + ], + "score": 1.0, + "content": "3. The optimal critic", + "type": "text" + }, + { + "bbox": [ + 216, + 83, + 228, + 94 + ], + "score": 0.88, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 81, + 354, + 96 + ], + "score": 1.0, + "content": "satisfies the following identity:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 103, + 460, + 131 + ], + "lines": [ + { + "bbox": [ + 186, + 103, + 460, + 131 + ], + "spans": [ + { + "bbox": [ + 186, + 103, + 460, + 131 + ], + "score": 0.93, + "content": "\\nabla _ { x } f ^ { * } ( x ) = \\frac { 1 } { d S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } \\frac { D ^ { - } F _ { \\mathbb { Q } } ( x ) - D ^ { - } F _ { \\mathbb { P } } ( x ) } { \\mu ( x ) } , \\mu - a l m o s t s u r e l y .", + "type": "interline_equation", + "image_path": "297f36e6e4f5011107b075eb04b65fc03f0bdbf7bb22ec038cd57b8be573697b.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 186, + 103, + 460, + 112.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 186, + 112.33333333333333, + 460, + 121.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 186, + 121.66666666666666, + 460, + 131.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 152, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 101, + 149, + 507, + 173 + ], + "spans": [ + { + "bbox": [ + 101, + 149, + 383, + 173 + ], + "score": 1.0, + "content": "Sobolev IPM Approximation. Learning in the whole Sobolev space", + "type": "text" + }, + { + "bbox": [ + 383, + 152, + 406, + 166 + ], + "score": 0.92, + "content": "W _ { 0 } ^ { 1 , 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 149, + 507, + 173 + ], + "score": 1.0, + "content": "is challenging hence we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 333, + 178 + ], + "score": 1.0, + "content": "need to restrict our function class to a hypothesis class", + "type": "text" + }, + { + "bbox": [ + 347, + 164, + 506, + 178 + ], + "score": 1.0, + "content": ", such as neural networks. We assume", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 245, + 189 + ], + "score": 1.0, + "content": "in the following that functions in", + "type": "text" + }, + { + "bbox": [ + 245, + 176, + 259, + 186 + ], + "score": 0.86, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 176, + 372, + 189 + ], + "score": 1.0, + "content": "vanish on the boundary of", + "type": "text" + }, + { + "bbox": [ + 372, + 176, + 383, + 186 + ], + "score": 0.84, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 176, + 505, + 189 + ], + "score": 1.0, + "content": ", and restrict the optimization", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 194, + 200 + ], + "score": 1.0, + "content": "to the function space", + "type": "text" + }, + { + "bbox": [ + 194, + 187, + 208, + 197 + ], + "score": 0.84, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 186, + 213, + 200 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 214, + 187, + 228, + 197 + ], + "score": 0.81, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "can be a Reproducing Kernel Hilbert Space as in the MMD case or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 197, + 434, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 415, + 210 + ], + "score": 1.0, + "content": "parametrized by a neural network. Define the Sobolev IPM approximation in", + "type": "text" + }, + { + "bbox": [ + 416, + 198, + 429, + 208 + ], + "score": 0.86, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 197, + 434, + 210 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 185, + 218, + 425, + 248 + ], + "lines": [ + { + "bbox": [ + 185, + 218, + 425, + 248 + ], + "spans": [ + { + "bbox": [ + 185, + 218, + 425, + 248 + ], + "score": 0.94, + "content": "\\mathcal { S } _ { \\mathcal { H } , \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\operatorname* { s u p } _ { \\substack { f \\in \\mathcal { H } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } } \\leq 1 } } \\left\\{ \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) \\right\\}", + "type": "interline_equation", + "image_path": "971ea3456676bcb111ab4fbee39726afebd8782ade48f246922be856df9f0f05.jpg" + } + ] + } + ], + "index": 9.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 218, + 425, + 233.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 185, + 233.0, + 425, + 248.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 257, + 505, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 384, + 271 + ], + "score": 1.0, + "content": "The following Lemma shows that the Sobolev IPM approximation in", + "type": "text" + }, + { + "bbox": [ + 384, + 259, + 398, + 268 + ], + "score": 0.86, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "is proportional to Sobolev", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "score": 1.0, + "content": "IPM. The tightness of the approximation of the Sobolev IPM is governed by the tightness of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 294, + 293 + ], + "score": 1.0, + "content": "approximation of the optimal Sobolev Critic", + "type": "text" + }, + { + "bbox": [ + 294, + 280, + 306, + 292 + ], + "score": 0.88, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 280, + 320, + 293 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 320, + 280, + 334, + 290 + ], + "score": 0.88, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 280, + 505, + 293 + ], + "score": 1.0, + "content": ". This approximation is measured in the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 290, + 293, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 293, + 304 + ], + "score": 1.0, + "content": "Sobolev sense, using the Sobolev dot product.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 105, + 318, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 388, + 331 + ], + "score": 1.0, + "content": "Lemma 1 (Sobolev IPM Approximation in a Hypothesis Class). Let", + "type": "text" + }, + { + "bbox": [ + 388, + 319, + 402, + 329 + ], + "score": 0.74, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "be a function space with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 102, + 327, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 102, + 327, + 492, + 347 + ], + "score": 1.0, + "content": "functions vanishing at the boundary. For any f ∈ H and for f ∗ the optimal critic in W 1,20 ,", + "type": "text" + }, + { + "bbox": [ + 479, + 332, + 505, + 343 + ], + "score": 1.0, + "content": "we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 342, + 132, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 132, + 356 + ], + "score": 1.0, + "content": "have:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 360, + 469, + 393 + ], + "lines": [ + { + "bbox": [ + 142, + 360, + 469, + 393 + ], + "spans": [ + { + "bbox": [ + 142, + 360, + 469, + 393 + ], + "score": 0.93, + "content": "\\mathcal { S } _ { \\mathcal { H } , \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\mathcal { S } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\operatorname* { s u p } _ { \\substack { f \\in \\mathcal { H } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } ( \\chi , \\mu ) } \\leq 1 } } \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\right. _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x .", + "type": "interline_equation", + "image_path": "854756f3154ceda8fb70a3a4edd5a69c5e21b3651ce7d7acdb0f36d3e2488644.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 142, + 360, + 469, + 371.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 142, + 371.0, + 469, + 382.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 142, + 382.0, + 469, + 393.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 460, + 416 + ], + "score": 1.0, + "content": "Note that this Lemma means that the Sobolev IPM is well approximated if the space", + "type": "text" + }, + { + "bbox": [ + 460, + 404, + 474, + 414 + ], + "score": 0.83, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "has an", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 267, + 427 + ], + "score": 1.0, + "content": "enough representation power to express", + "type": "text" + }, + { + "bbox": [ + 267, + 415, + 305, + 427 + ], + "score": 0.92, + "content": "\\nabla _ { x } f ^ { * } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 415, + 505, + 427 + ], + "score": 1.0, + "content": ". This is parallel to the Fisher IPM approximation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "(Mroueh & Sercu, 2017) where it is shown that the Fisher IPM approximation error is proportional", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "to the critic approximation in the Lebesgue sense. Having in mind that the gradient of the critic is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "the information that is passed on to the generator, we see that this convergence in the Sobolev sense", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 360, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 360, + 472 + ], + "score": 1.0, + "content": "to the optimal critic is an important property for GAN training.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 105, + 475, + 504, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "Relation to Fokker-Planck Diffusion. We show in Appendix A that the optimal Sobolev critic is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 486, + 407, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 407, + 499 + ], + "score": 1.0, + "content": "the solution of the following elliptic PDE (with zero boundary conditions):", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 506, + 375, + 534 + ], + "lines": [ + { + "bbox": [ + 236, + 506, + 375, + 534 + ], + "spans": [ + { + "bbox": [ + 236, + 506, + 375, + 534 + ], + "score": 0.95, + "content": "\\frac { \\mathbb { P } - \\mathbb { Q } } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } = - \\mathrm { d i v } ( \\mu ( x ) \\nabla _ { x } f ( x ) ) .", + "type": "interline_equation", + "image_path": "130523609727ae305fca8ff6534076a07e4f5637a5d4dc5e460b99ebf3723f21.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 236, + 506, + 375, + 520.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 236, + 520.0, + 375, + 534.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 504, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 504, + 555 + ], + "score": 1.0, + "content": "We further link the elliptic PDE given in Equation (8) and the Fokker-Planck diffusion. As we illus-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "score": 1.0, + "content": "trate in Figure 2(b) the gradient of the critic defines a transportation plan for moving the distribution", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 565, + 182, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 151, + 577 + ], + "score": 1.0, + "content": "mass from", + "type": "text" + }, + { + "bbox": [ + 151, + 565, + 160, + 576 + ], + "score": 0.86, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 565, + 171, + 577 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 172, + 565, + 178, + 575 + ], + "score": 0.78, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 565, + 182, + 577 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 592, + 413, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 414, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 414, + 606 + ], + "score": 1.0, + "content": "Discussion of Theorem 2. We make the following remarks on Theorem 2:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 108, + 615, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 108, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 108, + 615, + 358, + 628 + ], + "score": 1.0, + "content": "1. From Theorem 2, we see that the Sobolev IPM compares", + "type": "text" + }, + { + "bbox": [ + 358, + 617, + 365, + 626 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "higher order partial derivatives of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 119, + 625, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 119, + 625, + 238, + 641 + ], + "score": 1.0, + "content": "the cumulative distributions", + "type": "text" + }, + { + "bbox": [ + 238, + 627, + 251, + 638 + ], + "score": 0.89, + "content": "F _ { \\mathbb { P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 625, + 271, + 641 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 272, + 627, + 285, + 639 + ], + "score": 0.89, + "content": "F _ { \\mathbb { Q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 625, + 505, + 641 + ], + "score": 1.0, + "content": ", while Fisher IPM compares the probability density", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 119, + 636, + 164, + 650 + ], + "spans": [ + { + "bbox": [ + 119, + 636, + 164, + 650 + ], + "score": 1.0, + "content": "functions.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 108, + 656, + 342, + 669 + ], + "lines": [ + { + "bbox": [ + 107, + 656, + 342, + 669 + ], + "spans": [ + { + "bbox": [ + 107, + 656, + 214, + 669 + ], + "score": 1.0, + "content": "2. The dominant measure", + "type": "text" + }, + { + "bbox": [ + 214, + 659, + 222, + 668 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 656, + 342, + 669 + ], + "score": 1.0, + "content": "plays a similar role to Fisher:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 677, + 428, + 712 + ], + "lines": [ + { + "bbox": [ + 196, + 677, + 428, + 712 + ], + "spans": [ + { + "bbox": [ + 196, + 677, + 428, + 712 + ], + "score": 0.94, + "content": "S _ { \\mu } ^ { 2 } ( { \\mathbb { P } } , \\mathbb { Q } ) = \\frac 1 { d ^ { 2 } } \\sum _ { i = 1 } ^ { d } { \\mathbb { E } } _ { x \\sim \\mu } \\left( \\frac { D ^ { - i } F _ { \\mathbb { P } } ( x ) - D ^ { - i } F _ { \\mathbb { Q } } ( x ) } { \\mu ( x ) } \\right) ^ { 2 } ,", + "type": "interline_equation", + "image_path": "74e0ea07b0892100bb72453fb981347f5d3b84bb4cb932af144d8ebd56ed9107.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 677, + 428, + 694.5 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 196, + 694.5, + 428, + 712.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 120, + 720, + 401, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 718, + 402, + 735 + ], + "spans": [ + { + "bbox": [ + 119, + 718, + 390, + 735 + ], + "score": 1.0, + "content": "the average distance is defined with respect to points sampled from", + "type": "text" + }, + { + "bbox": [ + 391, + 724, + 397, + 732 + ], + "score": 0.77, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 718, + 402, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 129, + 82, + 353, + 95 + ], + "lines": [ + { + "bbox": [ + 128, + 81, + 354, + 96 + ], + "spans": [ + { + "bbox": [ + 128, + 81, + 216, + 96 + ], + "score": 1.0, + "content": "3. The optimal critic", + "type": "text" + }, + { + "bbox": [ + 216, + 83, + 228, + 94 + ], + "score": 0.88, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 81, + 354, + 96 + ], + "score": 1.0, + "content": "satisfies the following identity:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 128, + 81, + 354, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 103, + 460, + 131 + ], + "lines": [ + { + "bbox": [ + 186, + 103, + 460, + 131 + ], + "spans": [ + { + "bbox": [ + 186, + 103, + 460, + 131 + ], + "score": 0.93, + "content": "\\nabla _ { x } f ^ { * } ( x ) = \\frac { 1 } { d S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } \\frac { D ^ { - } F _ { \\mathbb { Q } } ( x ) - D ^ { - } F _ { \\mathbb { P } } ( x ) } { \\mu ( x ) } , \\mu - a l m o s t s u r e l y .", + "type": "interline_equation", + "image_path": "297f36e6e4f5011107b075eb04b65fc03f0bdbf7bb22ec038cd57b8be573697b.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 186, + 103, + 460, + 112.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 186, + 112.33333333333333, + 460, + 121.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 186, + 121.66666666666666, + 460, + 131.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 152, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 101, + 149, + 507, + 173 + ], + "spans": [ + { + "bbox": [ + 101, + 149, + 383, + 173 + ], + "score": 1.0, + "content": "Sobolev IPM Approximation. Learning in the whole Sobolev space", + "type": "text" + }, + { + "bbox": [ + 383, + 152, + 406, + 166 + ], + "score": 0.92, + "content": "W _ { 0 } ^ { 1 , 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 149, + 507, + 173 + ], + "score": 1.0, + "content": "is challenging hence we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 333, + 178 + ], + "score": 1.0, + "content": "need to restrict our function class to a hypothesis class", + "type": "text" + }, + { + "bbox": [ + 347, + 164, + 506, + 178 + ], + "score": 1.0, + "content": ", such as neural networks. We assume", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 245, + 189 + ], + "score": 1.0, + "content": "in the following that functions in", + "type": "text" + }, + { + "bbox": [ + 245, + 176, + 259, + 186 + ], + "score": 0.86, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 176, + 372, + 189 + ], + "score": 1.0, + "content": "vanish on the boundary of", + "type": "text" + }, + { + "bbox": [ + 372, + 176, + 383, + 186 + ], + "score": 0.84, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 176, + 505, + 189 + ], + "score": 1.0, + "content": ", and restrict the optimization", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 194, + 200 + ], + "score": 1.0, + "content": "to the function space", + "type": "text" + }, + { + "bbox": [ + 194, + 187, + 208, + 197 + ], + "score": 0.84, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 186, + 213, + 200 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 214, + 187, + 228, + 197 + ], + "score": 0.81, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "can be a Reproducing Kernel Hilbert Space as in the MMD case or", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 197, + 434, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 415, + 210 + ], + "score": 1.0, + "content": "parametrized by a neural network. 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The tightness of the approximation of the Sobolev IPM is governed by the tightness of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 294, + 293 + ], + "score": 1.0, + "content": "approximation of the optimal Sobolev Critic", + "type": "text" + }, + { + "bbox": [ + 294, + 280, + 306, + 292 + ], + "score": 0.88, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 280, + 320, + 293 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 320, + 280, + 334, + 290 + ], + "score": 0.88, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 280, + 505, + 293 + ], + "score": 1.0, + "content": ". This approximation is measured in the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 290, + 293, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 293, + 304 + ], + "score": 1.0, + "content": "Sobolev sense, using the Sobolev dot product.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 258, + 505, + 304 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 318, + 505, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 388, + 331 + ], + "score": 1.0, + "content": "Lemma 1 (Sobolev IPM Approximation in a Hypothesis Class). Let", + "type": "text" + }, + { + "bbox": [ + 388, + 319, + 402, + 329 + ], + "score": 0.74, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "be a function space with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 102, + 327, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 102, + 327, + 492, + 347 + ], + "score": 1.0, + "content": "functions vanishing at the boundary. For any f ∈ H and for f ∗ the optimal critic in W 1,20 ,", + "type": "text" + }, + { + "bbox": [ + 479, + 332, + 505, + 343 + ], + "score": 1.0, + "content": "we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 342, + 132, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 132, + 356 + ], + "score": 1.0, + "content": "have:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 102, + 318, + 505, + 356 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 360, + 469, + 393 + ], + "lines": [ + { + "bbox": [ + 142, + 360, + 469, + 393 + ], + "spans": [ + { + "bbox": [ + 142, + 360, + 469, + 393 + ], + "score": 0.93, + "content": "\\mathcal { S } _ { \\mathcal { H } , \\mu } ( \\mathbb { P } , \\mathbb { Q } ) = \\mathcal { S } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\operatorname* { s u p } _ { \\substack { f \\in \\mathcal { H } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } ( \\chi , \\mu ) } \\leq 1 } } \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\right. _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x .", + "type": "interline_equation", + "image_path": "854756f3154ceda8fb70a3a4edd5a69c5e21b3651ce7d7acdb0f36d3e2488644.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 142, + 360, + 469, + 371.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 142, + 371.0, + 469, + 382.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 142, + 382.0, + 469, + 393.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 460, + 416 + ], + "score": 1.0, + "content": "Note that this Lemma means that the Sobolev IPM is well approximated if the space", + "type": "text" + }, + { + "bbox": [ + 460, + 404, + 474, + 414 + ], + "score": 0.83, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "has an", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 267, + 427 + ], + "score": 1.0, + "content": "enough representation power to express", + "type": "text" + }, + { + "bbox": [ + 267, + 415, + 305, + 427 + ], + "score": 0.92, + "content": "\\nabla _ { x } f ^ { * } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 415, + 505, + 427 + ], + "score": 1.0, + "content": ". This is parallel to the Fisher IPM approximation", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "(Mroueh & Sercu, 2017) where it is shown that the Fisher IPM approximation error is proportional", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "to the critic approximation in the Lebesgue sense. Having in mind that the gradient of the critic is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "the information that is passed on to the generator, we see that this convergence in the Sobolev sense", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 459, + 360, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 360, + 472 + ], + "score": 1.0, + "content": "to the optimal critic is an important property for GAN training.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 402, + 506, + 472 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 475, + 504, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "Relation to Fokker-Planck Diffusion. We show in Appendix A that the optimal Sobolev critic is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 486, + 407, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 407, + 499 + ], + "score": 1.0, + "content": "the solution of the following elliptic PDE (with zero boundary conditions):", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 475, + 505, + 499 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 506, + 375, + 534 + ], + "lines": [ + { + "bbox": [ + 236, + 506, + 375, + 534 + ], + "spans": [ + { + "bbox": [ + 236, + 506, + 375, + 534 + ], + "score": 0.95, + "content": "\\frac { \\mathbb { P } - \\mathbb { Q } } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } = - \\mathrm { d i v } ( \\mu ( x ) \\nabla _ { x } f ( x ) ) .", + "type": "interline_equation", + "image_path": "130523609727ae305fca8ff6534076a07e4f5637a5d4dc5e460b99ebf3723f21.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 236, + 506, + 375, + 520.0 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 236, + 520.0, + 375, + 534.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 504, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 504, + 555 + ], + "score": 1.0, + "content": "We further link the elliptic PDE given in Equation (8) and the Fokker-Planck diffusion. As we illus-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 566 + ], + "score": 1.0, + "content": "trate in Figure 2(b) the gradient of the critic defines a transportation plan for moving the distribution", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 565, + 182, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 151, + 577 + ], + "score": 1.0, + "content": "mass from", + "type": "text" + }, + { + "bbox": [ + 151, + 565, + 160, + 576 + ], + "score": 0.86, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 565, + 171, + 577 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 172, + 565, + 178, + 575 + ], + "score": 0.78, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 565, + 182, + 577 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 542, + 505, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 592, + 413, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 414, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 414, + 606 + ], + "score": 1.0, + "content": "Discussion of Theorem 2. We make the following remarks on Theorem 2:", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 591, + 414, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 615, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 108, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 108, + 615, + 358, + 628 + ], + "score": 1.0, + "content": "1. From Theorem 2, we see that the Sobolev IPM compares", + "type": "text" + }, + { + "bbox": [ + 358, + 617, + 365, + 626 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "higher order partial derivatives of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 119, + 625, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 119, + 625, + 238, + 641 + ], + "score": 1.0, + "content": "the cumulative distributions", + "type": "text" + }, + { + "bbox": [ + 238, + 627, + 251, + 638 + ], + "score": 0.89, + "content": "F _ { \\mathbb { P } }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 625, + 271, + 641 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 272, + 627, + 285, + 639 + ], + "score": 0.89, + "content": "F _ { \\mathbb { Q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 625, + 505, + 641 + ], + "score": 1.0, + "content": ", while Fisher IPM compares the probability density", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 119, + 636, + 164, + 650 + ], + "spans": [ + { + "bbox": [ + 119, + 636, + 164, + 650 + ], + "score": 1.0, + "content": "functions.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 108, + 615, + 505, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 656, + 342, + 669 + ], + "lines": [ + { + "bbox": [ + 107, + 656, + 342, + 669 + ], + "spans": [ + { + "bbox": [ + 107, + 656, + 214, + 669 + ], + "score": 1.0, + "content": "2. The dominant measure", + "type": "text" + }, + { + "bbox": [ + 214, + 659, + 222, + 668 + ], + "score": 0.82, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 656, + 342, + 669 + ], + "score": 1.0, + "content": "plays a similar role to Fisher:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 107, + 656, + 342, + 669 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 677, + 428, + 712 + ], + "lines": [ + { + "bbox": [ + 196, + 677, + 428, + 712 + ], + "spans": [ + { + "bbox": [ + 196, + 677, + 428, + 712 + ], + "score": 0.94, + "content": "S _ { \\mu } ^ { 2 } ( { \\mathbb { P } } , \\mathbb { Q } ) = \\frac 1 { d ^ { 2 } } \\sum _ { i = 1 } ^ { d } { \\mathbb { E } } _ { x \\sim \\mu } \\left( \\frac { D ^ { - i } F _ { \\mathbb { P } } ( x ) - D ^ { - i } F _ { \\mathbb { Q } } ( x ) } { \\mu ( x ) } \\right) ^ { 2 } ,", + "type": "interline_equation", + "image_path": "74e0ea07b0892100bb72453fb981347f5d3b84bb4cb932af144d8ebd56ed9107.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 196, + 677, + 428, + 694.5 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 196, + 694.5, + 428, + 712.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 120, + 720, + 401, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 718, + 402, + 735 + ], + "spans": [ + { + "bbox": [ + 119, + 718, + 390, + 735 + ], + "score": 1.0, + "content": "the average distance is defined with respect to points sampled from", + "type": "text" + }, + { + "bbox": [ + 391, + 724, + 397, + 732 + ], + "score": 0.77, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 718, + 402, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 119, + 718, + 402, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 107, + 81, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 81, + 471, + 95 + ], + "score": 1.0, + "content": "3. Comparison of coordinate-wise Conditional CDFs. We note in the following", + "type": "text" + }, + { + "bbox": [ + 472, + 82, + 504, + 95 + ], + "score": 0.84, + "content": "\\begin{array} { r l } { x ^ { - i } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 121, + 93, + 310, + 106 + ], + "spans": [ + { + "bbox": [ + 121, + 94, + 228, + 106 + ], + "score": 0.87, + "content": "( x _ { 1 } , \\ldots x _ { i - 1 } , x _ { i + 1 } , \\ldots x _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 93, + 310, + 106 + ], + "score": 1.0, + "content": ". 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This leads us to the form", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 119, + 255, + 204, + 267 + ], + "spans": [ + { + "bbox": [ + 119, + 255, + 204, + 267 + ], + "score": 1.0, + "content": "given in Equation 5.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 120, + 268, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 120, + 268, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 120, + 268, + 362, + 280 + ], + "score": 1.0, + "content": "We see that the Sobolev IPM compares for each dimension", + "type": "text" + }, + { + "bbox": [ + 363, + 269, + 367, + 278 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 268, + 505, + 280 + ], + "score": 1.0, + "content": "the conditional cumulative distri-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 119, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 119, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "bution of each variable given the other variables, weighted by their density function. We refer to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 119, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 119, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "this as comparison of coordinate-wise CDFs on a leave one out basis. From this we see that we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 119, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 119, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "are comparing CDFs, which are better behaved on discrete distributions. Moreover, the condi-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 119, + 310, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 119, + 310, + 506, + 326 + ], + "score": 1.0, + "content": "tioning built in to this metric will play a crucial role in comparing sequences as the conditioning", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 120, + 323, + 301, + 334 + ], + "spans": [ + { + "bbox": [ + 120, + 323, + 301, + 334 + ], + "score": 1.0, + "content": "is important in this context (See section 6.1).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 107, + 347, + 244, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 245, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 245, + 360 + ], + "score": 1.0, + "content": "4.2 ILLUSTRATIVE EXAMPLES", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "Sobolev IPM / Cramer Distance and Wasserstein-1 in one Dimension. ´ In one dimension,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 277, + 391 + ], + "score": 1.0, + "content": "Sobolev IPM is the Cramer Distance (for ´", + "type": "text" + }, + { + "bbox": [ + 277, + 381, + 285, + 390 + ], + "score": 0.8, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 378, + 335, + 391 + ], + "score": 1.0, + "content": "uniform on", + "type": "text" + }, + { + "bbox": [ + 335, + 379, + 345, + 389 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 378, + 402, + 391 + ], + "score": 1.0, + "content": ", we note this", + "type": "text" + }, + { + "bbox": [ + 402, + 380, + 433, + 390 + ], + "score": 0.89, + "content": "\\mu : = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "). While Sobolev", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 389, + 504, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 497, + 403 + ], + "score": 1.0, + "content": "IPM in one dimension measures the discrepancy between CDFs, the one dimensional Wasserstein-", + "type": "text" + }, + { + "bbox": [ + 498, + 392, + 504, + 402 + ], + "score": 0.76, + "content": "p", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 401, + 340, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 340, + 413 + ], + "score": 1.0, + "content": "distance measures the discrepancy between inverse CDFs:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 417, + 487, + 445 + ], + "lines": [ + { + "bbox": [ + 122, + 417, + 487, + 445 + ], + "spans": [ + { + "bbox": [ + 122, + 417, + 487, + 445 + ], + "score": 0.92, + "content": "S _ { \\mu : = 1 } ^ { 2 } ( { \\mathbb { P } } , \\mathbb { Q } ) = \\int _ { \\chi } ( F _ { \\mathbb { P } } ( x ) - F _ { \\mathbb { Q } } ( x ) ) ^ { 2 } d x { \\mathrm { ~ v e r s u s ~ } } W _ { p } ^ { p } ( { \\mathbb { P } } , \\mathbb { Q } ) = \\int _ { 0 } ^ { 1 } | F _ { \\mathbb { P } } ^ { - 1 } ( u ) - F _ { \\mathbb { Q } } ^ { - 1 } ( u ) | ^ { p } d u ,", + "type": "interline_equation", + "image_path": "183d4cd42789e60279bafd5256498ce1453a649fdcef170b5027e44a88bb104d.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 122, + 417, + 487, + 426.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 122, + 426.3333333333333, + 487, + 435.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 122, + 435.66666666666663, + 487, + 444.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 449, + 341, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 342, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 281, + 464 + ], + "score": 1.0, + "content": "Recall also that the Fisher IPM for uniform", + "type": "text" + }, + { + "bbox": [ + 282, + 452, + 289, + 461 + ], + "score": 0.81, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 446, + 342, + 464 + ], + "score": 1.0, + "content": "is given by :", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 465, + 385, + 491 + ], + "lines": [ + { + "bbox": [ + 225, + 465, + 385, + 491 + ], + "spans": [ + { + "bbox": [ + 225, + 465, + 385, + 491 + ], + "score": 0.94, + "content": "\\mathcal { F } _ { \\mu : = 1 } ^ { 2 } ( \\mathbb { P } , \\mathbb { Q } ) = \\int _ { \\mathcal { X } } ( \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) ) ^ { 2 } d x .", + "type": "interline_equation", + "image_path": "bc4a4fd656eb1ed525ce0227233b10fb01960a523616219858e1b67184ca0376.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 225, + 465, + 385, + 491 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 263, + 507 + ], + "score": 1.0, + "content": "Consider for instance two point masses", + "type": "text" + }, + { + "bbox": [ + 264, + 496, + 297, + 507 + ], + "score": 0.92, + "content": "\\mathbb { P } = \\delta _ { a _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 495, + 315, + 507 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 316, + 496, + 350, + 507 + ], + "score": 0.91, + "content": "\\mathbb { Q } = \\delta _ { a _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 495, + 371, + 507 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 372, + 496, + 416, + 507 + ], + "score": 0.92, + "content": "a _ { 1 } , a _ { 2 } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 495, + 506, + 507 + ], + "score": 1.0, + "content": ". The rationale behind", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "using Wasserstein distance for GAN training is that since it is a weak metric, for far distributions", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 517, + 509, + 553 + ], + "spans": [ + { + "bbox": [ + 107, + 528, + 242, + 542 + ], + "score": 0.91, + "content": "W _ { 1 } ^ { 1 } ( \\mathbb { P } , \\mathbb { Q } ) = S _ { \\mu : = 1 } ^ { 2 } \\stackrel { - } { = } | a _ { 1 } - a _ { 2 } |", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 517, + 272, + 553 + ], + "score": 1.0, + "content": "me sign, while e than", + "type": "text" + }, + { + "bbox": [ + 272, + 529, + 346, + 542 + ], + "score": 0.9, + "content": "\\mathcal { F } _ { \\mu : = 1 } ^ { 2 } ( \\mathbb { P } , \\mathbb { Q } ) = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 517, + 509, + 553 + ], + "score": 1.0, + "content": "2017). In this case, it is easy to see that. As we see from this simple example,g distributions on discrete spaces. See", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 540, + 128, + 550 + ], + "spans": [ + { + "bbox": [ + 107, + 540, + 128, + 550 + ], + "score": 0.74, + "content": "C D F", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 376, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 376, + 563 + ], + "score": 1.0, + "content": "Figure 1, for a further discussion of this effect in the GAN context.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 338, + 586 + ], + "score": 1.0, + "content": "Sobolev IPM between two 2D Gaussians. 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We know from", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 607, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 622 + ], + "score": 1.0, + "content": "Theorem 2 that the gradient of the Sobolev optimal critic is proportional to the following vector", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 619, + 130, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 130, + 631 + ], + "score": 1.0, + "content": "field:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 626, + 474, + 685 + ], + "lines": [ + { + "bbox": [ + 203, + 626, + 474, + 685 + ], + "spans": [ + { + "bbox": [ + 203, + 626, + 474, + 685 + ], + "score": 0.69, + "content": "\\begin{array} { l } { \\displaystyle \\nabla f ^ { * } ( x , y ) \\alpha \\frac { 1 } { \\mu ( x , y ) } \\left[ \\frac { \\partial } { \\partial x } ( F _ { \\mathbb { Q } } ( x , y ) - F _ { \\mathbb { P } } ( x , y ) ) \\right] } \\\\ { \\displaystyle \\mu _ { 1 } = [ 1 , 0 ] , \\Sigma _ { 1 } = \\left[ \\frac { 1 . 9 } { 0 . 8 } \\quad 0 . 8 \\right] \\mu _ { 2 } = [ 1 , - 2 ] , \\Sigma _ { 2 } = \\left[ \\frac { 1 . 9 } { - 0 . 8 } \\quad - 0 . 8 \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "e50ee0347c9ba8d47bdefce58ca0db56fec3e968585997acd5e08b9f5538b96a.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 203, + 626, + 474, + 645.6666666666666 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 203, + 645.6666666666666, + 474, + 665.3333333333333 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 203, + 665.3333333333333, + 474, + 684.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 657, + 471, + 683 + ], + "lines": [ + { + "bbox": [ + 102, + 652, + 203, + 690 + ], + "spans": [ + { + "bbox": [ + 102, + 652, + 203, + 690 + ], + "score": 1.0, + "content": "In Figure 2 we consider", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 735 + ], + "lines": [ + { + "bbox": [ + 104, + 686, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 505, + 700 + ], + "score": 1.0, + "content": "In Figure 2(a) we plot the numerical solution of the PDE satisfied by the optimal Sobolev", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "critic given in Equation (8), using MATLAB solver for elliptic PDEs (more accurately we solve", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 260, + 721 + ], + "score": 0.9, + "content": "- d i v ( \\bar { \\mu } ( x ) \\nabla _ { x } f ( \\bar { x } ) ) = \\mathbb { P } ( x ) - \\bar { \\mathbb { Q } } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", hence we obtain the solution of Equation (8) up to a nor-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 103, + 718, + 507, + 738 + ], + "spans": [ + { + "bbox": [ + 103, + 718, + 189, + 738 + ], + "score": 1.0, + "content": "malization constant", + "type": "text" + }, + { + "bbox": [ + 190, + 720, + 227, + 735 + ], + "score": 0.86, + "content": "( \\frac { 1 } { S _ { \\mu } ( { \\mathbb P } , { \\mathbb Q } ) } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 718, + 507, + 738 + ], + "score": 1.0, + "content": ". We numerically solve the PDE on a rectangle with zero boundary", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 107, + 81, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 81, + 471, + 95 + ], + "score": 1.0, + "content": "3. Comparison of coordinate-wise Conditional CDFs. We note in the following", + "type": "text" + }, + { + "bbox": [ + 472, + 82, + 504, + 95 + ], + "score": 0.84, + "content": "\\begin{array} { r l } { x ^ { - i } } & { { } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 121, + 93, + 310, + 106 + ], + "spans": [ + { + "bbox": [ + 121, + 94, + 228, + 106 + ], + "score": 0.87, + "content": "( x _ { 1 } , \\ldots x _ { i - 1 } , x _ { i + 1 } , \\ldots x _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 93, + 310, + 106 + ], + "score": 1.0, + "content": ". Note that we have:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 107, + 81, + 504, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 109, + 524, + 230 + ], + "lines": [ + { + "bbox": [ + 124, + 109, + 524, + 230 + ], + "spans": [ + { + "bbox": [ + 124, + 109, + 524, + 230 + ], + "score": 0.87, + "content": "\\begin{array} { r l } & { \\mathsf { P } ^ { - i } F _ { \\mathbb { P } } ( x ) = \\frac { \\partial ^ { d - 1 } } { \\partial x _ { 1 } \\hdots \\partial x _ { i - 1 } \\partial x _ { i + 1 } \\hdots \\partial x _ { d } } \\displaystyle \\int _ { - \\infty } ^ { x _ { 1 } } \\hdots \\int _ { - \\infty } ^ { x _ { d } } \\mathbb { P } ( u _ { 1 } \\hdots u _ { d } ) d u _ { 1 } \\hdots d u _ { d } } \\\\ & { \\hphantom { \\mathsf { P } ^ { - i } F _ { \\mathbb { P } } ( x ) = } = \\displaystyle \\int _ { - \\infty } ^ { x _ { i } } \\mathbb { P } ( x _ { 1 } , \\hdots , x _ { i - 1 } , u , x _ { i + 1 } , \\hdots , x _ { d } ) d u } \\\\ & { \\hphantom { \\mathsf { P } ^ { - i } F _ { \\mathbb { P } } ( x ) = } = \\mathbb { P } _ { X ^ { - i } } ( x _ { 1 } , \\hdots , x _ { i - 1 } , x _ { i + 1 } , \\hdots x _ { d } ) \\displaystyle \\int _ { - \\infty } ^ { x _ { i } } \\mathbb { P } _ { [ X _ { i } | X ^ { - i } = x ^ { - i } ] } ( u | x _ { 1 } , \\hdots , x _ { i - 1 } , x _ { i + 1 } , \\hdots x _ { d } ) d u } \\\\ & { \\displaystyle ( \\mathrm { U s i n g ~ B a y e s ~ r u l e } ) } \\\\ & { \\hphantom { \\mathsf { P } ^ { - i } F _ { \\mathbb { P } } ( x ) = } = \\mathbb { P } _ { X ^ { - i } } ( x ^ { - i } ) F _ { \\mathbb { P } _ { [ X _ { i } | X ^ { - i } = x ^ { - i } ] } } ( x _ { i } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "18bd3f92f5c34e7173ad84a6c2be2d7bf1c6ec7e29506402937c180d48994612.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 124, + 109, + 524, + 149.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 124, + 149.33333333333334, + 524, + 189.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 124, + 189.66666666666669, + 524, + 230.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 120, + 232, + 506, + 266 + ], + "lines": [ + { + "bbox": [ + 119, + 231, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 119, + 231, + 195, + 245 + ], + "score": 1.0, + "content": "Note that for each", + "type": "text" + }, + { + "bbox": [ + 195, + 234, + 200, + 243 + ], + "score": 0.38, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 231, + 204, + 245 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 204, + 232, + 248, + 245 + ], + "score": 0.92, + "content": "D ^ { - i } F _ { \\mathbb { P } } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 231, + 428, + 245 + ], + "score": 1.0, + "content": "is the cumulative distribution of the variable", + "type": "text" + }, + { + "bbox": [ + 429, + 234, + 441, + 244 + ], + "score": 0.88, + "content": "X _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 231, + 506, + 245 + ], + "score": 1.0, + "content": "given the other", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 120, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 120, + 243, + 159, + 256 + ], + "score": 1.0, + "content": "variables", + "type": "text" + }, + { + "bbox": [ + 159, + 244, + 207, + 254 + ], + "score": 0.91, + "content": "X ^ { - i } = x ^ { - i }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 243, + 353, + 256 + ], + "score": 1.0, + "content": ", weighted by the density function of", + "type": "text" + }, + { + "bbox": [ + 354, + 244, + 373, + 254 + ], + "score": 0.91, + "content": "X ^ { - i }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 243, + 384, + 256 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 384, + 244, + 400, + 254 + ], + "score": 0.89, + "content": "x ^ { - i }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 243, + 505, + 256 + ], + "score": 1.0, + "content": ". This leads us to the form", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 119, + 255, + 204, + 267 + ], + "spans": [ + { + "bbox": [ + 119, + 255, + 204, + 267 + ], + "score": 1.0, + "content": "given in Equation 5.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 119, + 231, + 506, + 267 + ] + }, + { + "type": "text", + "bbox": [ + 120, + 268, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 120, + 268, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 120, + 268, + 362, + 280 + ], + "score": 1.0, + "content": "We see that the Sobolev IPM compares for each dimension", + "type": "text" + }, + { + "bbox": [ + 363, + 269, + 367, + 278 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 268, + 505, + 280 + ], + "score": 1.0, + "content": "the conditional cumulative distri-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 119, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 119, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "bution of each variable given the other variables, weighted by their density function. 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We consider in this example", + "type": "text" + }, + { + "bbox": [ + 398, + 596, + 437, + 610 + ], + "score": 0.93, + "content": "\\textstyle \\mu = { \\frac { \\mathbb { P } + \\mathbb { Q } } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 593, + 506, + 611 + ], + "score": 1.0, + "content": ". We know from", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 607, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 622 + ], + "score": 1.0, + "content": "Theorem 2 that the gradient of the Sobolev optimal critic is proportional to the following vector", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 619, + 130, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 130, + 631 + ], + "score": 1.0, + "content": "field:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 574, + 506, + 631 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 626, + 474, + 685 + ], + "lines": [ + { + "bbox": [ + 203, + 626, + 474, + 685 + ], + "spans": [ + { + "bbox": [ + 203, + 626, + 474, + 685 + ], + "score": 0.69, + "content": "\\begin{array} { l } { \\displaystyle \\nabla f ^ { * } ( x , y ) \\alpha \\frac { 1 } { \\mu ( x , y ) } \\left[ \\frac { \\partial } { \\partial x } ( F _ { \\mathbb { Q } } ( x , y ) - F _ { \\mathbb { P } } ( x , y ) ) \\right] } \\\\ { \\displaystyle \\mu _ { 1 } = [ 1 , 0 ] , \\Sigma _ { 1 } = \\left[ \\frac { 1 . 9 } { 0 . 8 } \\quad 0 . 8 \\right] \\mu _ { 2 } = [ 1 , - 2 ] , \\Sigma _ { 2 } = \\left[ \\frac { 1 . 9 } { - 0 . 8 } \\quad - 0 . 8 \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "e50ee0347c9ba8d47bdefce58ca0db56fec3e968585997acd5e08b9f5538b96a.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 203, + 626, + 474, + 645.6666666666666 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 203, + 645.6666666666666, + 474, + 665.3333333333333 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 203, + 665.3333333333333, + 474, + 684.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 657, + 471, + 683 + ], + "lines": [ + { + "bbox": [ + 102, + 652, + 203, + 690 + ], + "spans": [ + { + "bbox": [ + 102, + 652, + 203, + 690 + ], + "score": 1.0, + "content": "In Figure 2 we consider", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 102, + 652, + 203, + 690 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 735 + ], + "lines": [ + { + "bbox": [ + 104, + 686, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 505, + 700 + ], + "score": 1.0, + "content": "In Figure 2(a) we plot the numerical solution of the PDE satisfied by the optimal Sobolev", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "critic given in Equation (8), using MATLAB solver for elliptic PDEs (more accurately we solve", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 260, + 721 + ], + "score": 0.9, + "content": "- d i v ( \\bar { \\mu } ( x ) \\nabla _ { x } f ( \\bar { x } ) ) = \\mathbb { P } ( x ) - \\bar { \\mathbb { Q } } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", hence we obtain the solution of Equation (8) up to a nor-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 103, + 718, + 507, + 738 + ], + "spans": [ + { + "bbox": [ + 103, + 718, + 189, + 738 + ], + "score": 1.0, + "content": "malization constant", + "type": "text" + }, + { + "bbox": [ + 190, + 720, + 227, + 735 + ], + "score": 0.86, + "content": "( \\frac { 1 } { S _ { \\mu } ( { \\mathbb P } , { \\mathbb Q } ) } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 718, + 507, + 738 + ], + "score": 1.0, + "content": ". We numerically solve the PDE on a rectangle with zero boundary", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "conditions. We see that the optimal Sobolev critic separates the two distributions well. In Figure", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "2(b) we then numerically compute the gradient of the optimal Sobolev critic on a 2D grid as given", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "in Equation 9 (using numerical evaluation of the CDF and finite difference for the evaluation of the", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 438, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 369, + 450 + ], + "score": 1.0, + "content": "partial derivatives). 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As discussed in Section A.1, we see that the gradient", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 458, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 331, + 475 + ], + "score": 1.0, + "content": "of the critic (wrt to the input), defines on the support of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 332, + 460, + 371, + 474 + ], + "score": 0.94, + "content": "\\textstyle \\mu = { \\frac { \\mathbb { P } + \\mathbb { Q } } { 2 } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 371, + 458, + 506, + 475 + ], + "score": 1.0, + "content": "a transportation plan for moving", + "type": "text", + "cross_page": true + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 473, + 244, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 213, + 484 + ], + "score": 1.0, + "content": "the distribution mass from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 214, + 473, + 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match a (smoothed) discrete", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 350, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 104, + 350, + 505, + 363 + ], + "score": 1.0, + "content": "real distribution and a continuous generator. In this case, the CDF matching enabled by Sobolev", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "IPM gives non zero discrepancy between a (smoothed) discrete and a continuous density even if the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 372, + 438, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 438, + 385 + ], + "score": 1.0, + "content": "densities have disjoint supports. This ensures non vanishing gradients of the critic.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 418 + ], + "score": 1.0, + "content": "conditions. We see that the optimal Sobolev critic separates the two distributions well. In Figure", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 505, + 429 + ], + "score": 1.0, + "content": "2(b) we then numerically compute the gradient of the optimal Sobolev critic on a 2D grid as given", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "score": 1.0, + "content": "in Equation 9 (using numerical evaluation of the CDF and finite difference for the evaluation of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 438, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 369, + 450 + ], + "score": 1.0, + "content": "partial derivatives). We plot in Figure 2(b) the density functions of", + "type": "text" + }, + { + "bbox": [ + 370, + 439, + 377, + 448 + ], + "score": 0.81, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 438, + 394, + 450 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 394, + 438, + 403, + 450 + ], + "score": 0.86, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 438, + 506, + 450 + ], + "score": 1.0, + "content": "as well as the vector field", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 461 + ], + "score": 1.0, + "content": "of the gradient of the optimal Sobolev critic. As discussed in Section A.1, we see that the gradient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 458, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 331, + 475 + ], + "score": 1.0, + "content": "of the critic (wrt to the input), defines on the support of", + "type": "text" + }, + { + "bbox": [ + 332, + 460, + 371, + 474 + ], + "score": 0.94, + "content": "\\textstyle \\mu = { \\frac { \\mathbb { P } + \\mathbb { Q } } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 458, + 506, + 475 + ], + "score": 1.0, + "content": "a transportation plan for moving", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 473, + 244, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 213, + 484 + ], + "score": 1.0, + "content": "the distribution mass from", + "type": "text" + }, + { + "bbox": [ + 214, + 473, + 223, + 484 + ], + "score": 0.87, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 473, + 234, + 484 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 234, + 473, + 241, + 482 + ], + "score": 0.79, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 473, + 244, + 484 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 500, + 207, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 208, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 208, + 516 + ], + "score": 1.0, + "content": "5 SOBOLEV GAN", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 506, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "Now we turn to the problem of learning GANs with Sobolev IPM. 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See Algorithm 1.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 107, + 499, + 219, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 498, + 220, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 220, + 513 + ], + "score": 1.0, + "content": "Algorithm 1 Sobolev GAN", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "table", + "bbox": [ + 109, + 513, + 505, + 683 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 513, + 505, + 683 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 513, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 109, + 513, + 505, + 683 + ], + "score": 0.87, + "html": "
Input: ρ penalty weight, n Learning rate, nc number of iterations for training the critic, N batch size
Initialize p, 0,λ=0
repeat
for j = 1 to nc do
Samplea minibatch xi,i=1...N,xi~ Pr
Sample a minibatch zi,i = 1...N,zi ~ Pz
(gp,gx)←(VpLs,VxLs)(p,0,λ)
p ←p+η ADAM(p,9p)
λ ← λ- pgx {SGD rule on 入with learning rate ρ} end for
Sample zi,i= 1...N, zi ~ Pz
d←Vθ8(fp,g0）=-VθN∑=1fp(g0(zi）） N
0←0-nADAM(0,dθ)
until θ converges
", + "type": "table", + "image_path": "b7766853bc2e593a135cf95879ea92915912856229217e062dbd9939f34d0ad6.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 109, + 513, + 505, + 569.6666666666666 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 109, + 569.6666666666666, + 505, + 626.3333333333333 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 109, + 626.3333333333333, + 505, + 682.9999999999999 + ], + "spans": [], + "index": 33 + } + ] + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 249, + 711 + ], + "score": 1.0, + "content": "Remark 1. Note that in Algorithm", + "type": "text" + }, + { + "bbox": [ + 249, + 700, + 254, + 709 + ], + "score": 0.37, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 698, + 505, + 711 + ], + "score": 1.0, + "content": ", we obtain a biased estimate since we are using same samples", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "for the cost function and the constraint, but the incurred bias can be shown to be small and vanishing", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 721, + 476, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 476, + 732 + ], + "score": 1.0, + "content": "as the number of samples increases as shown and justified in (Shivaswamy & Jebara, 2010).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "10", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 122, + 98, + 286, + 217 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 122, + 98, + 286, + 217 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 98, + 286, + 217 + ], + "spans": [ + { + "bbox": [ + 122, + 98, + 286, + 217 + ], + "score": 0.946, + "type": "image", + "image_path": "7e31e2430319983c215121a0e70ffc66b6fbcb2eef7bf131143c1e35323dcb1d.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 122, + 98, + 286, + 111.22222222222223 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 122, + 111.22222222222223, + 286, + 124.44444444444446 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 122, + 124.44444444444446, + 286, + 137.66666666666669 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 122, + 137.66666666666669, + 286, + 150.8888888888889 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 122, + 150.8888888888889, + 286, + 164.11111111111114 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 122, + 164.11111111111114, + 286, + 177.33333333333337 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 122, + 177.33333333333337, + 286, + 190.5555555555556 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 122, + 190.5555555555556, + 286, + 203.77777777777783 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 122, + 203.77777777777783, + 286, + 217.00000000000006 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 307, + 228, + 491, + 269 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 307, + 227, + 492, + 239 + ], + "spans": [ + { + "bbox": [ + 307, + 227, + 492, + 239 + ], + "score": 1.0, + "content": "(b) Optimal Sobolev Transport Vector Field", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 308, + 236, + 490, + 250 + ], + "spans": [ + { + "bbox": [ + 308, + 238, + 344, + 249 + ], + "score": 0.92, + "content": "\\nabla _ { x } f ^ { * } { \\bar { ( x ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 236, + 454, + 250 + ], + "score": 1.0, + "content": "(arrows are the vector field", + "type": "text" + }, + { + "bbox": [ + 454, + 238, + 490, + 249 + ], + "score": 0.92, + "content": "\\nabla _ { x } f ^ { * } ( x )", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 307, + 248, + 492, + 259 + ], + "spans": [ + { + "bbox": [ + 307, + 248, + 492, + 259 + ], + "score": 1.0, + "content": "evaluated on the 2D grid. 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Then given the critic parameters", + "type": "text" + }, + { + "bbox": [ + 483, + 460, + 489, + 469 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 469, + 481, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 232, + 484 + ], + "score": 1.0, + "content": "optimize the generator weights", + "type": "text" + }, + { + "bbox": [ + 232, + 471, + 239, + 481 + ], + "score": 0.81, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 470, + 366, + 484 + ], + "score": 1.0, + "content": "to minimize the objective minθ", + "type": "text" + }, + { + "bbox": [ + 366, + 469, + 405, + 483 + ], + "score": 0.89, + "content": "\\hat { \\mathcal { E } } ( f _ { p } , g _ { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 470, + 481, + 484 + ], + "score": 1.0, + "content": ". See Algorithm 1.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 435, + 505, + 484 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 499, + 219, + 511 + ], + "lines": [ + { + "bbox": [ + 106, + 498, + 220, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 220, + 513 + ], + "score": 1.0, + "content": "Algorithm 1 Sobolev GAN", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "table", + "bbox": [ + 109, + 513, + 505, + 683 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 513, + 505, + 683 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 513, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 109, + 513, + 505, + 683 + ], + "score": 0.87, + "html": "
Input: ρ penalty weight, n Learning rate, nc number of iterations for training the critic, N batch size
Initialize p, 0,λ=0
repeat
for j = 1 to nc do
Samplea minibatch xi,i=1...N,xi~ Pr
Sample a minibatch zi,i = 1...N,zi ~ Pz
(gp,gx)←(VpLs,VxLs)(p,0,λ)
p ←p+η ADAM(p,9p)
λ ← λ- pgx {SGD rule on 入with learning rate ρ} end for
Sample zi,i= 1...N, zi ~ Pz
d←Vθ8(fp,g0）=-VθN∑=1fp(g0(zi）） N
0←0-nADAM(0,dθ)
until θ converges
", + "type": "table", + "image_path": "b7766853bc2e593a135cf95879ea92915912856229217e062dbd9939f34d0ad6.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 109, + 513, + 505, + 569.6666666666666 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 109, + 569.6666666666666, + 505, + 626.3333333333333 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 109, + 626.3333333333333, + 505, + 682.9999999999999 + ], + "spans": [], + "index": 33 + } + ] + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 249, + 711 + ], + "score": 1.0, + "content": "Remark 1. Note that in Algorithm", + "type": "text" + }, + { + "bbox": [ + 249, + 700, + 254, + 709 + ], + "score": 0.37, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 698, + 505, + 711 + ], + "score": 1.0, + "content": ", we obtain a biased estimate since we are using same samples", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "for the cost function and the constraint, but the incurred bias can be shown to be small and vanishing", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 721, + 476, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 476, + 732 + ], + "score": 1.0, + "content": "as the number of samples increases as shown and justified in (Shivaswamy & Jebara, 2010).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 698, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 370, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 371, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 371, + 96 + ], + "score": 1.0, + "content": "Relation to WGAN-GP. WGAN-GP can be written as follows:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 96, + 457, + 130 + ], + "lines": [ + { + "bbox": [ + 153, + 96, + 457, + 130 + ], + "spans": [ + { + "bbox": [ + 153, + 96, + 457, + 130 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\substack { g _ { \\theta } } } \\operatorname* { s u p } _ { f , \\| \\nabla _ { x } f _ { p } ( \\bar { x } _ { i } ) \\| = 1 , \\bar { x } _ { i } \\sim \\mu _ { G P } } \\hat { \\bar { \\mathcal { E } } } ( f _ { p } , g _ { \\theta } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } f _ { p } ( x _ { i } ) - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } f _ { p } ( g _ { \\theta } ( z _ { i } ) )", + "type": "interline_equation", + "image_path": "791192fb27625a05cd429c211071280256274c29486f742719fb77e1e0b8c129.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 153, + 96, + 457, + 107.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 153, + 107.33333333333333, + 457, + 118.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 153, + 118.66666666666666, + 457, + 130.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 505, + 146 + ], + "score": 1.0, + "content": "The main difference between WGAN-GP and our setting, is that WGAN-GP enforces pointwise con-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 102, + 143, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 102, + 143, + 225, + 162 + ], + "score": 1.0, + "content": "straints on points drawn from", + "type": "text" + }, + { + "bbox": [ + 225, + 147, + 264, + 158 + ], + "score": 0.88, + "content": "\\mu = \\mu _ { G P }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 143, + 402, + 162 + ], + "score": 1.0, + "content": "via a point-wise quadratic penalty", + "type": "text" + }, + { + "bbox": [ + 402, + 144, + 506, + 159 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { { \\big ( \\hat { \\mathcal { E } } ( f _ { p } , g _ { \\theta } ) - \\lambda \\sum _ { i = 1 } ^ { N } \\bigl ( 1 - \\bigl ) } \\quad } & { { } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 158, + 504, + 170 + ], + "spans": [ + { + "bbox": [ + 107, + 158, + 165, + 170 + ], + "score": 0.91, + "content": "\\| \\nabla _ { x } f ( \\widetilde { x } _ { i } ) \\| ) ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 158, + 504, + 170 + ], + "score": 1.0, + "content": "while we enforce that constraint on average as a Sobolev norm, allowing us the co-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 354, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 354, + 181 + ], + "score": 1.0, + "content": "ordinate weighted conditional CDF interpretation of the IPM.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 196, + 301, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 195, + 303, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 303, + 211 + ], + "score": 1.0, + "content": "6 APPLICATIONS OF SOBOLEV GAN", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 220, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 220, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 234 + ], + "score": 1.0, + "content": "Sobolev IPM has two important properties; The first stems from the conditioning built in to the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "metric through the weighted conditional CDF interpretation. The second stems from the diffusion", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 242, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 104, + 242, + 506, + 256 + ], + "score": 1.0, + "content": "properties that the critic of Sobolev IPM satisfies (Appendix A) that has theoretical and practical ties", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 253, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 267 + ], + "score": 1.0, + "content": "to the Laplacian regularizer and diffusion on manifolds used in semi-supervised learning (Belkin", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 263, + 159, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 159, + 277 + ], + "score": 1.0, + "content": "et al., 2006).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 294 + ], + "score": 1.0, + "content": "In this Section, we exploit those two important properties in two applications of Sobolev GAN: Text", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 293, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 305 + ], + "score": 1.0, + "content": "generation and semi-supervised learning. First in text generation, which can be seen as a discrete", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "sequence generation, Sobolev GAN (and WGAN-GP) enable training GANs without need to do", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "explicit brute-force conditioning. We attribute this to the built-in conditioning in Sobolev IPM (for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "the sequence aspect) and to the CDF matching (for the discrete aspect). Secondly using GANs in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 336, + 507, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 507, + 350 + ], + "score": 1.0, + "content": "semi-supervised learning is a promising avenue for learning using unlabeled data. We show that a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "variant of Sobolev GAN can achieve strong SSL results on the CIFAR-10 dataset, without the need", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 360, + 435, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 435, + 370 + ], + "score": 1.0, + "content": "of any form of activation normalization in the networks or any extra ad hoc tricks.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 106, + 383, + 309, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 383, + 311, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 311, + 396 + ], + "score": 1.0, + "content": "6.1 TEXT GENERATION WITH SOBOLEV GAN", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 504, + 415 + ], + "score": 1.0, + "content": "In this Section, we present an empirical study of Sobolev GAN in character level text generation.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "Our empirical study on end to end training of character-level GAN for text generation is articulated", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 281, + 438 + ], + "score": 1.0, + "content": "on four dimensions (loss, critic, generator,", + "type": "text" + }, + { + "bbox": [ + 282, + 427, + 290, + 437 + ], + "score": 0.73, + "content": "\\pmb { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "). (1) the loss used (GP: WGAN-GP (Gulrajani et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "2017), S: Sobolev or F: Fisher) (2) the architecture of the critic (Resnets or RNN) (3) the architecture", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 495, + 461 + ], + "score": 1.0, + "content": "of the generator (Resnets or RNN or RNN with curriculum learning) (4) the sampling distribution", + "type": "text" + }, + { + "bbox": [ + 496, + 449, + 504, + 459 + ], + "score": 0.77, + "content": "\\pmb { \\mu }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 459, + 176, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 176, + 470 + ], + "score": 1.0, + "content": "in the constraint.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 487 + ], + "score": 1.0, + "content": "Text Generation Experiments. We train a character-level GAN on Google Billion Word dataset and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 485, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 104, + 485, + 506, + 500 + ], + "score": 1.0, + "content": "follow the same experimental setup used in (Gulrajani et al., 2017). The generated sequence length", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "score": 1.0, + "content": "is 32 and the evaluation is based on Jensen-Shannon divergence on empirical 4-gram probabilities", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "(JS-4) of validation data and generated data. JS-4 may not be an ideal evaluation criteria, but it", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 518, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 518, + 505, + 533 + ], + "score": 1.0, + "content": "is a reasonable metric for current character-level GAN results, which is still far from generating", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 531, + 198, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 198, + 542 + ], + "score": 1.0, + "content": "meaningful sentences.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 248, + 559 + ], + "score": 1.0, + "content": "Annealed Smoothing of discrete", + "type": "text" + }, + { + "bbox": [ + 248, + 548, + 261, + 558 + ], + "score": 0.88, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 547, + 338, + 559 + ], + "score": 1.0, + "content": "in the constraint", + "type": "text" + }, + { + "bbox": [ + 339, + 549, + 347, + 559 + ], + "score": 0.77, + "content": "\\pmb { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 547, + 505, + 559 + ], + "score": 1.0, + "content": ". Since the generator distribution will", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 464, + 570 + ], + "score": 1.0, + "content": "always be defined on a continuous space, we can replace the discrete “real” distribution", + "type": "text" + }, + { + "bbox": [ + 464, + 558, + 475, + 569 + ], + "score": 0.89, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "with a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 569, + 504, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 301, + 581 + ], + "score": 1.0, + "content": "smoothed version (Gaussian kernel smoothing)", + "type": "text" + }, + { + "bbox": [ + 302, + 569, + 373, + 581 + ], + "score": 0.91, + "content": "\\mathbb { P } _ { r } \\star \\mathcal { N } ( 0 , \\sigma ^ { 2 } I _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 569, + 504, + 581 + ], + "score": 1.0, + "content": ". This corresponds to doing the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 200, + 593 + ], + "score": 1.0, + "content": "following sampling for", + "type": "text" + }, + { + "bbox": [ + 201, + 580, + 279, + 591 + ], + "score": 0.92, + "content": "\\mathbb { P } _ { r } : x + \\xi , x \\sim \\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 579, + 299, + 593 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 300, + 580, + 368, + 592 + ], + "score": 0.91, + "content": "\\xi \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } I _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 579, + 506, + 593 + ], + "score": 1.0, + "content": ". Note that we only inject noise to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "the “real” distribution with the goal of smoothing the support of the discrete distribution, as opposed", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 603, + 504, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 504, + 613 + ], + "score": 1.0, + "content": "to instance noise on both “real” and “fake” to stabilize the training, as introduced in (Kaae Sønderby", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "score": 1.0, + "content": "et al., 2017; Arjovsky & Bottou, 2017). As it is common in optimization by continuation (Mobahi", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 623, + 478, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 280, + 637 + ], + "score": 1.0, + "content": "& III, 2015), we also anneal the noise level", + "type": "text" + }, + { + "bbox": [ + 281, + 626, + 288, + 634 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 623, + 478, + 637 + ], + "score": 1.0, + "content": "as the training progresses on a linear schedule.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 106, + 640, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 653 + ], + "score": 1.0, + "content": "Sobolev GAN versus WGAN-GP with Resnets. 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First in text generation, which can be seen as a discrete", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "sequence generation, Sobolev GAN (and WGAN-GP) enable training GANs without need to do", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "explicit brute-force conditioning. We attribute this to the built-in conditioning in Sobolev IPM (for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "the sequence aspect) and to the CDF matching (for the discrete aspect). Secondly using GANs in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 336, + 507, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 507, + 350 + ], + "score": 1.0, + "content": "semi-supervised learning is a promising avenue for learning using unlabeled data. We show that a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "variant of Sobolev GAN can achieve strong SSL results on the CIFAR-10 dataset, without the need", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 360, + 435, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 435, + 370 + ], + "score": 1.0, + "content": "of any form of activation normalization in the networks or any extra ad hoc tricks.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 282, + 507, + 370 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 383, + 309, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 383, + 311, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 311, + 396 + ], + "score": 1.0, + "content": "6.1 TEXT GENERATION WITH SOBOLEV GAN", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 504, + 415 + ], + "score": 1.0, + "content": "In this Section, we present an empirical study of Sobolev GAN in character level text generation.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 426 + ], + "score": 1.0, + "content": "Our empirical study on end to end training of character-level GAN for text generation is articulated", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 281, + 438 + ], + "score": 1.0, + "content": "on four dimensions (loss, critic, generator,", + "type": "text" + }, + { + "bbox": [ + 282, + 427, + 290, + 437 + ], + "score": 0.73, + "content": "\\pmb { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "). (1) the loss used (GP: WGAN-GP (Gulrajani et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "2017), S: Sobolev or F: Fisher) (2) the architecture of the critic (Resnets or RNN) (3) the architecture", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 495, + 461 + ], + "score": 1.0, + "content": "of the generator (Resnets or RNN or RNN with curriculum learning) (4) the sampling distribution", + "type": "text" + }, + { + "bbox": [ + 496, + 449, + 504, + 459 + ], + "score": 0.77, + "content": "\\pmb { \\mu }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 459, + 176, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 176, + 470 + ], + "score": 1.0, + "content": "in the constraint.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 404, + 505, + 470 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 487 + ], + "score": 1.0, + "content": "Text Generation Experiments. We train a character-level GAN on Google Billion Word dataset and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 485, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 104, + 485, + 506, + 500 + ], + "score": 1.0, + "content": "follow the same experimental setup used in (Gulrajani et al., 2017). The generated sequence length", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "score": 1.0, + "content": "is 32 and the evaluation is based on Jensen-Shannon divergence on empirical 4-gram probabilities", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "(JS-4) of validation data and generated data. 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Since the generator distribution will", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 464, + 570 + ], + "score": 1.0, + "content": "always be defined on a continuous space, we can replace the discrete “real” distribution", + "type": "text" + }, + { + "bbox": [ + 464, + 558, + 475, + 569 + ], + "score": 0.89, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "with a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 569, + 504, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 301, + 581 + ], + "score": 1.0, + "content": "smoothed version (Gaussian kernel smoothing)", + "type": "text" + }, + { + "bbox": [ + 302, + 569, + 373, + 581 + ], + "score": 0.91, + "content": "\\mathbb { P } _ { r } \\star \\mathcal { N } ( 0 , \\sigma ^ { 2 } I _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 569, + 504, + 581 + ], + "score": 1.0, + "content": ". 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For WGAN-GP we used", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 305, + 372 + ], + "score": 1.0, + "content": "the open source implementation with the penalty", + "type": "text" + }, + { + "bbox": [ + 306, + 362, + 337, + 371 + ], + "score": 0.85, + "content": "\\lambda = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "as in (Gulrajani et al., 2017). Results are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 369, + 507, + 385 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 460, + 385 + ], + "score": 1.0, + "content": "given in Figure 3(a) for the JS-4 evaluation of both WGAN-GP and Sobolev GAN for", + "type": "text" + }, + { + "bbox": [ + 460, + 373, + 501, + 383 + ], + "score": 0.88, + "content": "\\mu = \\mu _ { G P }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 369, + 507, + 385 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "score": 1.0, + "content": "In Figure 3(b) we show the JS-4 evaluation of Sobolev GAN with the annealed noise smoothing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 136, + 405 + ], + "score": 0.92, + "content": "\\mu _ { s } ^ { a } ( \\bar { \\sigma _ { 0 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 393, + 316, + 406 + ], + "score": 1.0, + "content": ", for various values of the initial noise level", + "type": "text" + }, + { + "bbox": [ + 317, + 394, + 328, + 405 + ], + "score": 0.86, + "content": "\\sigma _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 393, + 506, + 406 + ], + "score": 1.0, + "content": ". We see that the training succeeds in both", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "cases. Sobolev GAN achieves slightly better results than WGAN-GP for the annealing that starts", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 103, + 415, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 103, + 415, + 191, + 430 + ], + "score": 1.0, + "content": "with high noise level", + "type": "text" + }, + { + "bbox": [ + 191, + 417, + 228, + 428 + ], + "score": 0.89, + "content": "\\sigma _ { 0 } = 1 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 415, + 454, + 430 + ], + "score": 1.0, + "content": ". We note that without smoothing and annealing i.e using", + "type": "text" + }, + { + "bbox": [ + 455, + 415, + 501, + 429 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\bar { \\boldsymbol { \\mu } } = \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 415, + 505, + 430 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 310, + 439 + ], + "score": 1.0, + "content": "Sobolev GAN is behind. Annealed smoothing of", + "type": "text" + }, + { + "bbox": [ + 311, + 428, + 322, + 438 + ], + "score": 0.88, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 427, + 505, + 439 + ], + "score": 1.0, + "content": ", helps the training as the real distribution is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "slowly going from a continuous distribution to a discrete distribution. See Appendix C (Figure 6)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 449, + 369, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 369, + 462 + ], + "score": 1.0, + "content": "for a comparison between annealed and non annealed smoothing.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 108, + 466, + 504, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 465, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 480 + ], + "score": 1.0, + "content": "We give in Appendix C a comparison of WGAN-GP and Sobolev GAN for a Resnet generator", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 476, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 490 + ], + "score": 1.0, + "content": "architecture and an RNN critic. The RNN has degraded performance due to optimization difficulties.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 493, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "Fisher GAN Curriculum Conditioning versus Sobolev GAN: Explicit versus Implicit condi-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "tioning. We analyze how Fisher GAN behaves under different architectures of generators and critics.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 516, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 527 + ], + "score": 1.0, + "content": "We first fix the generator to be ResNet. We study 3 different architectures of critics: ResNet, GRU", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 527, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 396, + 538 + ], + "score": 1.0, + "content": "(we follow the experimental setup from (Press et al., 2017)), and hybrid", + "type": "text" + }, + { + "bbox": [ + 396, + 527, + 453, + 538 + ], + "score": 0.39, + "content": "\\mathrm { R e s N e t + G R U }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 527, + 504, + 538 + ], + "score": 1.0, + "content": "(Reed et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 457, + 550 + ], + "score": 1.0, + "content": "2016). We notice that RNN is unstable, we need to clip the gradient values of critics in", + "type": "text" + }, + { + "bbox": [ + 457, + 538, + 501, + 550 + ], + "score": 0.59, + "content": "[ - 0 . 5 , 0 . 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 537, + 505, + 550 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 548, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 288, + 562 + ], + "score": 1.0, + "content": "and the gradient of the Lagrange multiplier", + "type": "text" + }, + { + "bbox": [ + 288, + 549, + 302, + 560 + ], + "score": 0.88, + "content": "\\lambda _ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 548, + 316, + 562 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 316, + 549, + 363, + 560 + ], + "score": 0.91, + "content": "[ - 1 0 ^ { 4 } , \\bar { 1 } 0 ^ { 4 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 548, + 402, + 562 + ], + "score": 1.0, + "content": ". We fix", + "type": "text" + }, + { + "bbox": [ + 402, + 549, + 453, + 560 + ], + "score": 0.92, + "content": "\\rho _ { F } = 1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 548, + 506, + 562 + ], + "score": 1.0, + "content": "and we use", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 558, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 146, + 571 + ], + "score": 0.88, + "content": "\\mu = \\mu _ { G P }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 558, + 326, + 573 + ], + "score": 1.0, + "content": ". We search the value for the learning rate in", + "type": "text" + }, + { + "bbox": [ + 327, + 560, + 379, + 572 + ], + "score": 0.88, + "content": "[ 1 0 ^ { - 5 } , 1 \\dot { 0 } ^ { - 4 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 558, + 446, + 573 + ], + "score": 1.0, + "content": ". We see that for", + "type": "text" + }, + { + "bbox": [ + 447, + 561, + 487, + 572 + ], + "score": 0.9, + "content": "\\mu = \\mu _ { G P }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 558, + 506, + 573 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 133, + 581 + ], + "score": 0.62, + "content": "G = \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "Resnet for various critic architectures, Fisher GAN fails at the task of text generation (Figure", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 128, + 592 + ], + "score": 0.25, + "content": "{ \\mathfrak { 4 } } \\ { \\mathfrak { a - c } } { \\mathrm { , } }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 581, + 290, + 595 + ], + "score": 1.0, + "content": "). Nevertheless, when using RNN critics", + "type": "text" + }, + { + "bbox": [ + 290, + 582, + 334, + 593 + ], + "score": 0.39, + "content": "( \\mathrm { F i g } \\quad \\mathsf { b } , \\mathsf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "a marginal improvement happens over the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "score": 1.0, + "content": "fully collapsed state when using a resnet critic (Fig 4 a). We hypothesize that RNN critics enable", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 443, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 443, + 615 + ], + "score": 1.0, + "content": "some conditioning and factoring of the distribution, which is lacking in Fisher IPM.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 620, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "Finally Figure 4 (d) shows the result of training with recurrent generator and critic. We follow (Press", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "et al., 2017) in terms of GRU architecture, but differ by using Fisher GAN rather than WGAN-GP.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 639, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 138, + 658 + ], + "score": 1.0, + "content": "We use", + "type": "text" + }, + { + "bbox": [ + 139, + 642, + 186, + 657 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mu = { \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 639, + 474, + 658 + ], + "score": 1.0, + "content": "i.e. without annealed noise smoothing. We train (F, D=RNN,G=RNN,", + "type": "text" + }, + { + "bbox": [ + 474, + 642, + 501, + 657 + ], + "score": 0.85, + "content": "{ \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } .", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 639, + 507, + 658 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 356, + 667 + ], + "score": 1.0, + "content": "using curriculum conditioning of the generator for all lengths", + "type": "text" + }, + { + "bbox": [ + 357, + 655, + 363, + 665 + ], + "score": 0.73, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "as done in (Press et al., 2017): the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 217, + 678 + ], + "score": 1.0, + "content": "generator is conditioned on", + "type": "text" + }, + { + "bbox": [ + 217, + 666, + 244, + 676 + ], + "score": 0.87, + "content": "3 2 - \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 665, + 352, + 678 + ], + "score": 1.0, + "content": "characters and predicts the", + "type": "text" + }, + { + "bbox": [ + 352, + 666, + 358, + 676 + ], + "score": 0.73, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "remaining characters. We increment", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 131, + 687 + ], + "score": 0.9, + "content": "\\ell = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 676, + 271, + 689 + ], + "score": 1.0, + "content": "to 32 on a regular schedule (every", + "type": "text" + }, + { + "bbox": [ + 271, + 677, + 288, + 687 + ], + "score": 0.69, + "content": "1 5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 676, + 444, + 689 + ], + "score": 1.0, + "content": "updates). JS-4 is only computed when", + "type": "text" + }, + { + "bbox": [ + 444, + 677, + 469, + 687 + ], + "score": 0.89, + "content": "\\ell > 4", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 676, + 506, + 689 + ], + "score": 1.0, + "content": ". 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The JS-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 309, + 214, + 419, + 226 + ], + "spans": [ + { + "bbox": [ + 309, + 214, + 327, + 226 + ], + "score": 1.0, + "content": "4 of", + "type": "text" + }, + { + "bbox": [ + 327, + 215, + 378, + 226 + ], + "score": 0.92, + "content": "\\mu _ { s } ^ { a } ( \\sigma _ { 0 } = 1 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 214, + 419, + 226 + ], + "score": 1.0, + "content": "is 0.3268.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "image_caption", + "bbox": [ + 106, + 239, + 506, + 261 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 374, + 252 + ], + "score": 1.0, + "content": "Figure 3: Result of Sobolev GAN for various dominating measure", + "type": "text" + }, + { + "bbox": [ + 374, + 241, + 381, + 250 + ], + "score": 0.78, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 239, + 505, + 252 + ], + "score": 1.0, + "content": ", for resnets as architectures of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 249, + 218, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 218, + 262 + ], + "score": 1.0, + "content": "the critic and the generator.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 210, + 339 + ], + "score": 1.0, + "content": "use softmax to normalize", + "type": "text" + }, + { + "bbox": [ + 210, + 328, + 248, + 339 + ], + "score": 0.9, + "content": "\\log p + \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 327, + 505, + 339 + ], + "score": 1.0, + "content": ". 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The noise level", + "type": "text" + }, + { + "bbox": [ + 264, + 340, + 271, + 348 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 337, + 505, + 351 + ], + "score": 1.0, + "content": "was annealed following a linear schedule starting from an", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 103, + 347, + 507, + 366 + ], + "spans": [ + { + "bbox": [ + 103, + 347, + 177, + 366 + ], + "score": 1.0, + "content": "initial noise level", + "type": "text" + }, + { + "bbox": [ + 177, + 351, + 189, + 361 + ], + "score": 0.85, + "content": "\\sigma _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 347, + 238, + 366 + ], + "score": 1.0, + "content": "(at iteration", + "type": "text" + }, + { + "bbox": [ + 238, + 351, + 243, + 360 + ], + "score": 0.55, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 347, + 246, + 366 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 246, + 349, + 338, + 362 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\sigma _ { i } = \\sigma _ { 0 } ( 1 - \\frac { i } { M a x i t e r } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 347, + 373, + 366 + ], + "score": 1.0, + "content": ", Maxite", + "type": "text" + }, + { + "bbox": [ + 374, + 350, + 398, + 360 + ], + "score": 0.64, + "content": "{ \\displaystyle = 3 0 \\mathsf K }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 347, + 507, + 366 + ], + "score": 1.0, + "content": "). For WGAN-GP we used", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 305, + 372 + ], + "score": 1.0, + "content": "the open source implementation with the penalty", + "type": "text" + }, + { + "bbox": [ + 306, + 362, + 337, + 371 + ], + "score": 0.85, + "content": "\\lambda = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "as in (Gulrajani et al., 2017). Results are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 369, + 507, + 385 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 460, + 385 + ], + "score": 1.0, + "content": "given in Figure 3(a) for the JS-4 evaluation of both WGAN-GP and Sobolev GAN for", + "type": "text" + }, + { + "bbox": [ + 460, + 373, + 501, + 383 + ], + "score": 0.88, + "content": "\\mu = \\mu _ { G P }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 369, + 507, + 385 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "score": 1.0, + "content": "In Figure 3(b) we show the JS-4 evaluation of Sobolev GAN with the annealed noise smoothing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 136, + 405 + ], + "score": 0.92, + "content": "\\mu _ { s } ^ { a } ( \\bar { \\sigma _ { 0 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 393, + 316, + 406 + ], + "score": 1.0, + "content": ", for various values of the initial noise level", + "type": "text" + }, + { + "bbox": [ + 317, + 394, + 328, + 405 + ], + "score": 0.86, + "content": "\\sigma _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 393, + 506, + 406 + ], + "score": 1.0, + "content": ". We see that the training succeeds in both", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "cases. Sobolev GAN achieves slightly better results than WGAN-GP for the annealing that starts", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 103, + 415, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 103, + 415, + 191, + 430 + ], + "score": 1.0, + "content": "with high noise level", + "type": "text" + }, + { + "bbox": [ + 191, + 417, + 228, + 428 + ], + "score": 0.89, + "content": "\\sigma _ { 0 } = 1 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 415, + 454, + 430 + ], + "score": 1.0, + "content": ". We note that without smoothing and annealing i.e using", + "type": "text" + }, + { + "bbox": [ + 455, + 415, + 501, + 429 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\bar { \\boldsymbol { \\mu } } = \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 415, + 505, + 430 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 310, + 439 + ], + "score": 1.0, + "content": "Sobolev GAN is behind. Annealed smoothing of", + "type": "text" + }, + { + "bbox": [ + 311, + 428, + 322, + 438 + ], + "score": 0.88, + "content": "\\mathbb { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 427, + 505, + 439 + ], + "score": 1.0, + "content": ", helps the training as the real distribution is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "slowly going from a continuous distribution to a discrete distribution. See Appendix C (Figure 6)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 449, + 369, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 369, + 462 + ], + "score": 1.0, + "content": "for a comparison between annealed and non annealed smoothing.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 17.5, + "bbox_fs": [ + 103, + 327, + 507, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 466, + 504, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 465, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 506, + 480 + ], + "score": 1.0, + "content": "We give in Appendix C a comparison of WGAN-GP and Sobolev GAN for a Resnet generator", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 476, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 490 + ], + "score": 1.0, + "content": "architecture and an RNN critic. The RNN has degraded performance due to optimization difficulties.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 465, + 506, + 490 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 493, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "Fisher GAN Curriculum Conditioning versus Sobolev GAN: Explicit versus Implicit condi-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "tioning. We analyze how Fisher GAN behaves under different architectures of generators and critics.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 516, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 527 + ], + "score": 1.0, + "content": "We first fix the generator to be ResNet. We study 3 different architectures of critics: ResNet, GRU", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 527, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 396, + 538 + ], + "score": 1.0, + "content": "(we follow the experimental setup from (Press et al., 2017)), and hybrid", + "type": "text" + }, + { + "bbox": [ + 396, + 527, + 453, + 538 + ], + "score": 0.39, + "content": "\\mathrm { R e s N e t + G R U }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 527, + 504, + 538 + ], + "score": 1.0, + "content": "(Reed et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 457, + 550 + ], + "score": 1.0, + "content": "2016). We notice that RNN is unstable, we need to clip the gradient values of critics in", + "type": "text" + }, + { + "bbox": [ + 457, + 538, + 501, + 550 + ], + "score": 0.59, + "content": "[ - 0 . 5 , 0 . 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 537, + 505, + 550 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 548, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 288, + 562 + ], + "score": 1.0, + "content": "and the gradient of the Lagrange multiplier", + "type": "text" + }, + { + "bbox": [ + 288, + 549, + 302, + 560 + ], + "score": 0.88, + "content": "\\lambda _ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 548, + 316, + 562 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 316, + 549, + 363, + 560 + ], + "score": 0.91, + "content": "[ - 1 0 ^ { 4 } , \\bar { 1 } 0 ^ { 4 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 548, + 402, + 562 + ], + "score": 1.0, + "content": ". We fix", + "type": "text" + }, + { + "bbox": [ + 402, + 549, + 453, + 560 + ], + "score": 0.92, + "content": "\\rho _ { F } = 1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 548, + 506, + 562 + ], + "score": 1.0, + "content": "and we use", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 558, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 146, + 571 + ], + "score": 0.88, + "content": "\\mu = \\mu _ { G P }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 558, + 326, + 573 + ], + "score": 1.0, + "content": ". We search the value for the learning rate in", + "type": "text" + }, + { + "bbox": [ + 327, + 560, + 379, + 572 + ], + "score": 0.88, + "content": "[ 1 0 ^ { - 5 } , 1 \\dot { 0 } ^ { - 4 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 558, + 446, + 573 + ], + "score": 1.0, + "content": ". We see that for", + "type": "text" + }, + { + "bbox": [ + 447, + 561, + 487, + 572 + ], + "score": 0.9, + "content": "\\mu = \\mu _ { G P }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 558, + 506, + 573 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 133, + 581 + ], + "score": 0.62, + "content": "G = \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "Resnet for various critic architectures, Fisher GAN fails at the task of text generation (Figure", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 128, + 592 + ], + "score": 0.25, + "content": "{ \\mathfrak { 4 } } \\ { \\mathfrak { a - c } } { \\mathrm { , } }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 581, + 290, + 595 + ], + "score": 1.0, + "content": "). Nevertheless, when using RNN critics", + "type": "text" + }, + { + "bbox": [ + 290, + 582, + 334, + 593 + ], + "score": 0.39, + "content": "( \\mathrm { F i g } \\quad \\mathsf { b } , \\mathsf { c } )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "a marginal improvement happens over the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "score": 1.0, + "content": "fully collapsed state when using a resnet critic (Fig 4 a). We hypothesize that RNN critics enable", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 443, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 443, + 615 + ], + "score": 1.0, + "content": "some conditioning and factoring of the distribution, which is lacking in Fisher IPM.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 493, + 506, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 620, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "Finally Figure 4 (d) shows the result of training with recurrent generator and critic. We follow (Press", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "et al., 2017) in terms of GRU architecture, but differ by using Fisher GAN rather than WGAN-GP.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 639, + 507, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 138, + 658 + ], + "score": 1.0, + "content": "We use", + "type": "text" + }, + { + "bbox": [ + 139, + 642, + 186, + 657 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mu = { \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 639, + 474, + 658 + ], + "score": 1.0, + "content": "i.e. without annealed noise smoothing. We train (F, D=RNN,G=RNN,", + "type": "text" + }, + { + "bbox": [ + 474, + 642, + 501, + 657 + ], + "score": 0.85, + "content": "{ \\frac { \\mathbb { P } _ { r } + \\mathbb { Q } _ { \\theta } } { 2 } } .", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 639, + 507, + 658 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 356, + 667 + ], + "score": 1.0, + "content": "using curriculum conditioning of the generator for all lengths", + "type": "text" + }, + { + "bbox": [ + 357, + 655, + 363, + 665 + ], + "score": 0.73, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "as done in (Press et al., 2017): the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 217, + 678 + ], + "score": 1.0, + "content": "generator is conditioned on", + "type": "text" + }, + { + "bbox": [ + 217, + 666, + 244, + 676 + ], + "score": 0.87, + "content": "3 2 - \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 665, + 352, + 678 + ], + "score": 1.0, + "content": "characters and predicts the", + "type": "text" + }, + { + "bbox": [ + 352, + 666, + 358, + 676 + ], + "score": 0.73, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "remaining characters. 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Note that the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 707, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 707, + 506, + 723 + ], + "score": 1.0, + "content": "need of this explicit brute force conditioning for Fisher GAN, highlights the implicit conditioning", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "induced by Sobolev GAN via the gradient regularizer, without the need for curriculum conditioning.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 620, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 209, + 80, + 393, + 203 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 209, + 80, + 393, + 203 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 209, + 80, + 393, + 203 + ], + "spans": [ + { + "bbox": [ + 209, + 80, + 393, + 203 + ], + "score": 0.967, + "type": "image", + "image_path": "2cad69ff4d412b18994156d6951f045e5d5629da2e40f46cfb2e8bb9373125ba.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 209, + 80, + 393, + 93.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 209, + 93.66666666666667, + 393, + 107.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 209, + 107.33333333333334, + 393, + 121.00000000000001 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 209, + 121.00000000000001, + 393, + 134.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 209, + 134.66666666666669, + 393, + 148.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 209, + 148.33333333333334, + 393, + 162.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 209, + 162.0, + 393, + 175.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 209, + 175.66666666666666, + 393, + 189.33333333333331 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 209, + 189.33333333333331, + 393, + 202.99999999999997 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 214, + 506, + 314 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 444, + 228 + ], + "score": 1.0, + "content": "Figure 4: Fisher GAN with different architectures for critics: (a-c) We see that for", + "type": "text" + }, + { + "bbox": [ + 445, + 217, + 486, + 226 + ], + "score": 0.89, + "content": "\\mu = \\mu _ { G P }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 131, + 236 + ], + "score": 0.7, + "content": "G =", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "Resnet for various critic architectures, Fisher GAN fails at the task of text generation. We", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "notice small improvements for RNN critics (b-c) due to the conditioning and factoring of the dis-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 246, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 104, + 246, + 506, + 262 + ], + "score": 1.0, + "content": "tribution. (d) Fisher GAN with recurrent generator and critic, trained on a curriculum conditioning", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 197, + 272 + ], + "score": 1.0, + "content": "for increasing lengths", + "type": "text" + }, + { + "bbox": [ + 197, + 259, + 203, + 269 + ], + "score": 0.59, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 258, + 505, + 272 + ], + "score": 1.0, + "content": ", increments indicated by gridlines. 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We have", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 494, + 412, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 290, + 506 + ], + "score": 1.0, + "content": "the following training equations for the (critic", + "type": "text" + }, + { + "bbox": [ + 291, + 495, + 299, + 504 + ], + "score": 0.77, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 494, + 412, + 506 + ], + "score": 1.0, + "content": "classifer) and the generator:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 510, + 464, + 539 + ], + "lines": [ + { + "bbox": [ + 147, + 510, + 464, + 539 + ], + "spans": [ + { + "bbox": [ + 147, + 510, + 464, + 539 + ], + "score": 0.87, + "content": "{ \\mathrm { C r i t i c } } + { \\mathrm { C l a s s i f i e r } } ; \\quad \\operatorname* { m a x } _ { S , \\Phi _ { \\omega } , f } \\mathcal { L } _ { D } = \\mathcal { L } _ { \\mathrm { a l m } } ^ { \\mathrm { G A N } } ( f , g _ { \\theta } ) - \\lambda _ { C E } \\sum _ { ( x , y ) \\in \\mathrm { l a b } } C E ( p ( y | x ) , y )", + "type": "interline_equation", + "image_path": "73a48d9b2d0b771c267d552ceb29b802d35dabab0224cb66b7f31be6684d17f0.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 147, + 510, + 464, + 519.6666666666666 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 147, + 519.6666666666666, + 464, + 529.3333333333333 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 147, + 529.3333333333333, + 464, + 538.9999999999999 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 543, + 372, + 563 + ], + "lines": [ + { + "bbox": [ + 239, + 543, + 372, + 563 + ], + "spans": [ + { + "bbox": [ + 239, + 543, + 372, + 563 + ], + "score": 0.82, + "content": "\\mathrm { G e n e r a t o r : \\ m a x } { \\mathcal { L } } _ { G } = { \\hat { \\mathcal { E } } } ( f , g _ { \\theta } )", + "type": "interline_equation", + "image_path": "ded3d8d7b9bfb1e91382d83e938938173d7ca784cd5c705701bdfb7b402c6e8f.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 239, + 543, + 372, + 563 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 503, + 585 + ], + "lines": [ + { + "bbox": [ + 102, + 563, + 502, + 588 + ], + "spans": [ + { + "bbox": [ + 102, + 563, + 249, + 588 + ], + "score": 1.0, + "content": "where the main IPM objective with", + "type": "text" + }, + { + "bbox": [ + 250, + 570, + 260, + 580 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 563, + 298, + 588 + ], + "score": 1.0, + "content": "samples:", + "type": "text" + }, + { + "bbox": [ + 299, + 566, + 502, + 586 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\hat { \\mathcal { E } } ( f , g _ { \\theta } ) = \\frac { 1 } { N } \\left( \\sum _ { x \\in \\mathrm { u n l } } f ( x ) - \\sum _ { z \\sim p _ { z } } f ( g _ { \\theta } ( z ) ) \\right) . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 504, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 338, + 602 + ], + "score": 1.0, + "content": "Following (Mroueh & Sercu, 2017) we use the following “", + "type": "text" + }, + { + "bbox": [ + 339, + 590, + 364, + 601 + ], + "score": 0.87, + "content": "K + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "parametrization” for the critic (See", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 599, + 152, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 152, + 615 + ], + "score": 1.0, + "content": "Figure 5) :", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 610, + 399, + 663 + ], + "lines": [ + { + "bbox": [ + 212, + 610, + 399, + 663 + ], + "spans": [ + { + "bbox": [ + 212, + 610, + 399, + 663 + ], + "score": 0.93, + "content": "f ( x ) = \\underbrace { \\sum _ { y = 1 } ^ { K } p ( y | x ) \\left. 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We", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 249 + ], + "score": 1.0, + "content": "notice small improvements for RNN critics (b-c) due to the conditioning and factoring of the dis-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 246, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 104, + 246, + 506, + 262 + ], + "score": 1.0, + "content": "tribution. (d) Fisher GAN with recurrent generator and critic, trained on a curriculum conditioning", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 197, + 272 + ], + "score": 1.0, + "content": "for increasing lengths", + "type": "text" + }, + { + "bbox": [ + 197, + 259, + 203, + 269 + ], + "score": 0.59, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 258, + 505, + 272 + ], + "score": 1.0, + "content": ", increments indicated by gridlines. In this curriculum conditioning setup,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "with recurrent critics and generators, the training of Fisher GAN succeeds and reaches similar levels", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "of Sobolev GAN (and WGAN-GP). It is important to note that by doing this explicit curriculum", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "conditioning for Fisher GAN, we highlight the implicit conditioning induced by Sobolev GAN, via", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 302, + 203, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 203, + 316 + ], + "score": 1.0, + "content": "the gradient regularizer.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 107, + 334, + 354, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 355, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 355, + 346 + ], + "score": 1.0, + "content": "6.2 SEMI-SUPERVISED LEARNING WITH SOBOLEV GAN", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 366 + ], + "score": 1.0, + "content": "A proper and promising framework for evaluating GANs consists in using it as a regularizer in the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "semi-supervised learning setting (Salimans et al., 2016; Dumoulin et al., 2017; Kumar et al., 2017).", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 505, + 388 + ], + "score": 1.0, + "content": "As mentioned before, the Sobolev norm as a regularizer for the Sobolev IPM draws connections with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "the Laplacian regularization in manifold learning (Belkin et al., 2006). In the Laplacian framework", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "of semi-supervised learning, the classifier satisfies a smoothness constraint imposed by controlling", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 409, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 179, + 424 + ], + "score": 1.0, + "content": "its Sobolev norm:", + "type": "text" + }, + { + "bbox": [ + 180, + 409, + 277, + 424 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\int _ { \\mathcal { X } } \\left\\| \\nabla _ { x } \\bar { f } ( x ) \\right\\| ^ { 2 } \\mu ^ { 2 } ( x ) d x } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "(Alaoui et al., 2016). In this Section, we present a variant", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Xof Sobolev GAN that achieves competitive performance in semi-supervised learning on the CIFAR-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "10 dataset Krizhevsky & Hinton (2009) without using any internal activation normalization in the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "critic, such as batch normalization (BN) (Ioffe & Szegedy, 2015), layer normalization (LN) (Ba", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 455, + 374, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 374, + 467 + ], + "score": 1.0, + "content": "et al., 2016), or weight normalization (Salimans & Kingma, 2016).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 354, + 506, + 467 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 104, + 470, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 294, + 486 + ], + "score": 1.0, + "content": "In this setting, a convolutional neural network", + "type": "text" + }, + { + "bbox": [ + 294, + 472, + 358, + 483 + ], + "score": 0.9, + "content": "\\Phi _ { \\omega } : \\mathcal { X } \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 470, + 505, + 486 + ], + "score": 1.0, + "content": "is shared between the cross entropy", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 480, + 507, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 181, + 496 + ], + "score": 1.0, + "content": "(CE) training of a", + "type": "text" + }, + { + "bbox": [ + 181, + 483, + 191, + 493 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 480, + 256, + 496 + ], + "score": 1.0, + "content": "-class classifier", + "type": "text" + }, + { + "bbox": [ + 256, + 482, + 308, + 493 + ], + "score": 0.89, + "content": "( S \\in \\mathbb { R } ^ { K \\times m } )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 480, + 507, + 496 + ], + "score": 1.0, + "content": ") and the critic of GAN (See Figure 5). We have", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 494, + 412, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 290, + 506 + ], + "score": 1.0, + "content": "the following training equations for the (critic", + "type": "text" + }, + { + "bbox": [ + 291, + 495, + 299, + 504 + ], + "score": 0.77, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 494, + 412, + 506 + ], + "score": 1.0, + "content": "classifer) and the generator:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 470, + 507, + 506 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 510, + 464, + 539 + ], + "lines": [ + { + "bbox": [ + 147, + 510, + 464, + 539 + ], + "spans": [ + { + "bbox": [ + 147, + 510, + 464, + 539 + ], + "score": 0.87, + "content": "{ \\mathrm { C r i t i c } } + { \\mathrm { C l a s s i f i e r } } ; \\quad \\operatorname* { m a x } _ { S , \\Phi _ { \\omega } , f } \\mathcal { L } _ { D } = \\mathcal { L } _ { \\mathrm { a l m } } ^ { \\mathrm { G A N } } ( f , g _ { \\theta } ) - \\lambda _ { C E } \\sum _ { ( x , y ) \\in \\mathrm { l a b } } C E ( p ( y | x ) , y )", + "type": "interline_equation", + "image_path": "73a48d9b2d0b771c267d552ceb29b802d35dabab0224cb66b7f31be6684d17f0.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 147, + 510, + 464, + 519.6666666666666 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 147, + 519.6666666666666, + 464, + 529.3333333333333 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 147, + 529.3333333333333, + 464, + 538.9999999999999 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 543, + 372, + 563 + ], + "lines": [ + { + "bbox": [ + 239, + 543, + 372, + 563 + ], + "spans": [ + { + "bbox": [ + 239, + 543, + 372, + 563 + ], + "score": 0.82, + "content": "\\mathrm { G e n e r a t o r : \\ m a x } { \\mathcal { L } } _ { G } = { \\hat { \\mathcal { E } } } ( f , g _ { \\theta } )", + "type": "interline_equation", + "image_path": "ded3d8d7b9bfb1e91382d83e938938173d7ca784cd5c705701bdfb7b402c6e8f.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 239, + 543, + 372, + 563 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 503, + 585 + ], + "lines": [ + { + "bbox": [ + 102, + 563, + 502, + 588 + ], + "spans": [ + { + "bbox": [ + 102, + 563, + 249, + 588 + ], + "score": 1.0, + "content": "where the main IPM objective with", + "type": "text" + }, + { + "bbox": [ + 250, + 570, + 260, + 580 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 563, + 298, + 588 + ], + "score": 1.0, + "content": "samples:", + "type": "text" + }, + { + "bbox": [ + 299, + 566, + 502, + 586 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\hat { \\mathcal { E } } ( f , g _ { \\theta } ) = \\frac { 1 } { N } \\left( \\sum _ { x \\in \\mathrm { u n l } } f ( x ) - \\sum _ { z \\sim p _ { z } } f ( g _ { \\theta } ( z ) ) \\right) . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 102, + 563, + 502, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 504, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 338, + 602 + ], + "score": 1.0, + "content": "Following (Mroueh & Sercu, 2017) we use the following “", + "type": "text" + }, + { + "bbox": [ + 339, + 590, + 364, + 601 + ], + "score": 0.87, + "content": "K + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "parametrization” for the critic (See", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 599, + 152, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 152, + 615 + ], + "score": 1.0, + "content": "Figure 5) :", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 589, + 505, + 615 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 610, + 399, + 663 + ], + "lines": [ + { + "bbox": [ + 212, + 610, + 399, + 663 + ], + "spans": [ + { + "bbox": [ + 212, + 610, + 399, + 663 + ], + "score": 0.93, + "content": "f ( x ) = \\underbrace { \\sum _ { y = 1 } ^ { K } p ( y | x ) \\left. S _ { y } , \\Phi _ { \\omega } ( x ) \\right. } _ { f _ { + } : \\mathrm { \\normalfont ~ { \\mathrm { \\fontfamily { ~ \\ddots ~ } \\selectfont ~ { \\ c r i t i c } } } } } - \\underbrace { \\left. v , \\Phi _ { \\omega } ( x ) \\right. } _ { f _ { - } : \\mathrm { \\normalfont ~ { \\cdot \\mathrm { \\sc ~ { \\ c r i t i c } } } } }", + "type": "interline_equation", + "image_path": "d42fbe727de6f14478c28b1bd7f1f5b76b3d1f0e3ed080794746d2e0b574d849.jpg" + } + ] + } + ], + "index": 40.5, + "virtual_lines": [ + { + "bbox": [ + 212, + 610, + 399, + 623.25 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 212, + 623.25, + 399, + 636.5 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 212, + 636.5, + 399, + 649.75 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 212, + 649.75, + 399, + 663.0 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 146, + 678 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 665, + 280, + 678 + ], + "score": 0.91, + "content": "p ( y | x ) = \\mathrm { S o f t m a x } ( \\langle \\mathrm { S } , \\Phi _ { \\omega } ( \\mathrm { x } ) \\rangle ) _ { \\mathrm { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "appears both in the critic formulation and in the Cross-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 361, + 689 + ], + "score": 1.0, + "content": "Entropy term in Equation (11). 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This parametrization adapts the idea of (Salimans et al., 2016), which was formulated", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "specifically for the classic KL / JSD based GANs, to IPM-based GANs. 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This", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 436, + 128 + ], + "score": 1.0, + "content": "leads us to the formulation below, where we apply the Sobolev constraint only on", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 436, + 115, + 450, + 127 + ], + "score": 0.88, + "content": "f _ { - }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 451, + 115, + 505, + 128 + ], + "score": 1.0, + "content": ". 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Number of labeled examples Model100020004000 Misclassification rate8000
CatGAN (Springenberg, 2015) FM (Salimans et al., 2016) ALI (Dumoulin et al., 2017)21.83 ± 2.0119.61 ± 2.0919.58 18.63 ± 2.3217.72 ± 1.82
Tangents Reg (Kumar et al., 2017)19.98 ± 0.3 20.06 ± 0.519.09 ± 0.1517.99 ± 0.54 16.78 ± 0.617.05 ± 0.50
II-model (Laine & Aila, 2016) *16.55 ± 0.29
VAT (Miyato et al., 2017)14.87
Bad Gan (Dai et al., 2017) *14.41 ± 0.30
VAT+EntMin+Large (Miyato et al., 2017) *13.15
Sajadi (Sajjadi et al., 2016) *11.29
WGAN-GP (Gulrajani et al., 2017) t44.85 ± 0.2837.62 ± 0.5632.66 ± 0.4830.38± 0.22
Fisher,layer norm (Mroueh & Sercu,2017)19.74± 0.2117.87 ± 0.3816.13 ± 0.5314.81 ± 0.16
Fisher, no norm (Mroueh & Sercu, 2017)21.49 ± 0.1819.20 ± 0.4617.30 ± 0.3015.57 ± 0.33
Sobolev + Fisher, no norm (This Work)20.14 ± 0.2117.38 ± 0.1015.77 ± 0.1914.20± 0.08
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Number of labeled examples Model100020004000 Misclassification rate8000
CatGAN (Springenberg, 2015) FM (Salimans et al., 2016) ALI (Dumoulin et al., 2017)21.83 ± 2.0119.61 ± 2.0919.58 18.63 ± 2.3217.72 ± 1.82
Tangents Reg (Kumar et al., 2017)19.98 ± 0.3 20.06 ± 0.519.09 ± 0.1517.99 ± 0.54 16.78 ± 0.617.05 ± 0.50
II-model (Laine & Aila, 2016) *16.55 ± 0.29
VAT (Miyato et al., 2017)14.87
Bad Gan (Dai et al., 2017) *14.41 ± 0.30
VAT+EntMin+Large (Miyato et al., 2017) *13.15
Sajadi (Sajjadi et al., 2016) *11.29
WGAN-GP (Gulrajani et al., 2017) t44.85 ± 0.2837.62 ± 0.5632.66 ± 0.4830.38± 0.22
Fisher,layer norm (Mroueh & Sercu,2017)19.74± 0.2117.87 ± 0.3816.13 ± 0.5314.81 ± 0.16
Fisher, no norm (Mroueh & Sercu, 2017)21.49 ± 0.1819.20 ± 0.4617.30 ± 0.3015.57 ± 0.33
Sobolev + Fisher, no norm (This Work)20.14 ± 0.2117.38 ± 0.1015.77 ± 0.1914.20± 0.08
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The optimal critic of Sobolev IPM", + "type": "text" + }, + { + "bbox": [ + 458, + 208, + 469, + 219 + ], + "score": 0.87, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 207, + 505, + 221 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 218, + 187, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 161, + 231 + ], + "score": 1.0, + "content": "the following", + "type": "text" + }, + { + "bbox": [ + 161, + 219, + 182, + 229 + ], + "score": 0.29, + "content": "P D E", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 218, + 187, + 231 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 231, + 425, + 259 + ], + "lines": [ + { + "bbox": [ + 186, + 231, + 425, + 259 + ], + "spans": [ + { + "bbox": [ + 186, + 231, + 425, + 259 + ], + "score": 0.93, + "content": "\\Delta f ^ { * } ( x ) + \\left. \\nabla _ { x } \\log \\mu ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\right. + \\frac { \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\mu ( x ) } = 0 .", + "type": "interline_equation", + "image_path": "11a1a882d27c217eb8c6f794348747448fe25025ec5b8fdb311f21b816fa8589.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 231, + 425, + 245.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 186, + 245.0, + 425, + 259.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 504, + 300 + ], + "lines": [ + { + "bbox": [ + 102, + 266, + 508, + 293 + ], + "spans": [ + { + "bbox": [ + 102, + 266, + 213, + 293 + ], + "score": 1.0, + "content": "Define the Stein Operator:", + "type": "text" + }, + { + "bbox": [ + 213, + 270, + 423, + 290 + ], + "score": 0.89, + "content": "\\begin{array} { r } { T ( \\mu ) \\vec { g } ( x ) = \\frac { 1 } { 2 } \\Big ( \\langle \\nabla _ { x } \\log ( \\mu ( x ) ) , \\vec { g } ( x ) \\rangle + d i v ( \\vec { g } ( x ) ) \\Big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 266, + 508, + 293 + ], + "score": 1.0, + "content": ". Hence we have the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 286, + 271, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 237, + 303 + ], + "score": 1.0, + "content": "following Transport Equation of", + "type": "text" + }, + { + "bbox": [ + 238, + 289, + 245, + 299 + ], + "score": 0.83, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 286, + 256, + 303 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 257, + 289, + 266, + 300 + ], + "score": 0.83, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 286, + 271, + 303 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 302, + 399, + 316 + ], + "lines": [ + { + "bbox": [ + 209, + 302, + 399, + 316 + ], + "spans": [ + { + "bbox": [ + 209, + 302, + 399, + 316 + ], + "score": 0.82, + "content": "\\mathbb { Q } ( x ) = \\mathbb { P } ( x ) + 2 S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\mu ( x ) T ( \\mu ) \\nabla _ { x } f ^ { * } ( x ) .", + "type": "interline_equation", + "image_path": "1e8e1ddb948c1f5c091710119ff5cfe0e18fe54c83f125800249ff258b965310.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 209, + 302, + 399, + 316 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 280, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 281, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 281, + 331 + ], + "score": 1.0, + "content": "Recall the definition of Stein Discrepancy :", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 330, + 404, + 353 + ], + "lines": [ + { + "bbox": [ + 205, + 330, + 404, + 353 + ], + "spans": [ + { + "bbox": [ + 205, + 330, + 404, + 353 + ], + "score": 0.88, + "content": "\\mathbb { S } ( \\mathbb { Q } , \\mu ) = \\operatorname* { s u p } _ { \\vec { g } } | \\mathbb { E } _ { x \\sim \\mathbb { Q } } [ T ( \\mu ) \\vec { g } ( x ) ] | , \\vec { g } : \\mathcal { X } \\mathbb { R } ^ { d } .", + "type": "interline_equation", + "image_path": "2877884eb1971d1d3f73c91b22dfd55168fc0436ac4b2c417bab1a06eeda727c.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 205, + 330, + 404, + 353 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 444, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 446, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 446, + 369 + ], + "score": 1.0, + "content": "Theorem 4 (Sobolev and Stein Discrepanices). The following inequality holds true:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 367, + 450, + 395 + ], + "lines": [ + { + "bbox": [ + 156, + 367, + 450, + 395 + ], + "spans": [ + { + "bbox": [ + 156, + 367, + 450, + 395 + ], + "score": 0.87, + "content": "\\bigg | \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { Q } } \\left[ \\frac { \\mathbb { Q } ( \\boldsymbol { x } ) - \\mathbb { P } ( \\boldsymbol { x } ) } { \\mu ( \\boldsymbol { x } ) } \\right] \\bigg | \\leq 2 \\qquad \\underbrace { \\mathbb { S } ( \\mathbb { Q } , \\mu ) } _ { \\mathrm { S } ( \\mathbb { Q } , \\mu ) } \\qquad \\underbrace { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } _ { \\mathrm { S } _ { \\boldsymbol { \\mu } } ( \\mathbb { Q } ) }", + "type": "interline_equation", + "image_path": "eeb5ec94fe21ee822359870e030f1434ca8ac52a226b712de209aa416c3f67fb.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 156, + 367, + 450, + 376.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 156, + 376.3333333333333, + 450, + 385.66666666666663 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 156, + 385.66666666666663, + 450, + 394.99999999999994 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 506, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 196, + 426 + ], + "score": 1.0, + "content": "Consider for example", + "type": "text" + }, + { + "bbox": [ + 196, + 415, + 224, + 425 + ], + "score": 0.91, + "content": "\\mu = \\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 413, + 284, + 426 + ], + "score": 1.0, + "content": ", and sequence", + "type": "text" + }, + { + "bbox": [ + 285, + 415, + 299, + 425 + ], + "score": 0.9, + "content": "\\mathbb { Q } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 413, + 420, + 426 + ], + "score": 1.0, + "content": ". If the Sobolev distance goes", + "type": "text" + }, + { + "bbox": [ + 421, + 414, + 486, + 426 + ], + "score": 0.93, + "content": "S _ { \\mathbb { P } } ( \\mathbb { P } , \\mathbb { Q } _ { n } ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 413, + 506, + 426 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 423, + 507, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 128, + 440 + ], + "score": 1.0, + "content": "ratio", + "type": "text" + }, + { + "bbox": [ + 128, + 425, + 192, + 443 + ], + "score": 0.94, + "content": "\\begin{array} { r } { r _ { n } ( x ) = \\frac { \\mathbb { Q } _ { n } ( x ) } { \\mathbb { P } ( x ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 423, + 507, + 444 + ], + "score": 1.0, + "content": "Qn(x)P(x) converges in expectation (w.r.t to Q) to 1. The speed of the convergence is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 275, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 232, + 454 + ], + "score": 1.0, + "content": "given by the Stein Discrepancy", + "type": "text" + }, + { + "bbox": [ + 233, + 441, + 271, + 453 + ], + "score": 0.93, + "content": "\\mathbb { S } ( \\mathbb { Q } _ { n } , \\mathbb { P } )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 441, + 275, + 454 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 464, + 502, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 504, + 476 + ], + "score": 1.0, + "content": "Relation to Fokker-Planck Diffusion Equation and Particles dynamics. Note that PDE", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 475, + 432, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 432, + 488 + ], + "score": 1.0, + "content": "satisifed by the Sobolev critic given in Equation (13) can be equivalently written:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 488, + 377, + 515 + ], + "lines": [ + { + "bbox": [ + 234, + 488, + 377, + 515 + ], + "spans": [ + { + "bbox": [ + 234, + 488, + 377, + 515 + ], + "score": 0.94, + "content": "\\frac { \\mathbb { P } - \\mathbb { Q } } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } = - \\mathrm { d i v } ( \\mu ( x ) \\nabla _ { x } f ^ { * } ( x ) ) ,", + "type": "interline_equation", + "image_path": "6c0c12a2a6be6aa27f7a455544535d1930265afa23d226fdc15e72c20e552f62.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 488, + 377, + 501.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 234, + 501.5, + 377, + 515.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 516, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "written in this form, we draw a connection with the Fokker-Planck Equation for the evolution of a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 526, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 174, + 540 + ], + "score": 1.0, + "content": "density function", + "type": "text" + }, + { + "bbox": [ + 175, + 529, + 183, + 538 + ], + "score": 0.82, + "content": "q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 526, + 308, + 540 + ], + "score": 1.0, + "content": "that is the density of particles", + "type": "text" + }, + { + "bbox": [ + 309, + 527, + 348, + 538 + ], + "score": 0.93, + "content": "X _ { t } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 526, + 505, + 540 + ], + "score": 1.0, + "content": "evolving with a drift (a velocity field)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 537, + 221, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 216, + 550 + ], + "score": 0.89, + "content": "V ( x , i ) : \\mathcal { X } \\times [ 0 , \\infty [ \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 537, + 221, + 550 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 160, + 551, + 449, + 564 + ], + "lines": [ + { + "bbox": [ + 163, + 550, + 449, + 567 + ], + "spans": [ + { + "bbox": [ + 163, + 552, + 239, + 564 + ], + "score": 0.9, + "content": "d X _ { t } = V ( X _ { t } , t ) d t", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 550, + 325, + 567 + ], + "score": 1.0, + "content": ", where the density of", + "type": "text" + }, + { + "bbox": [ + 326, + 553, + 339, + 564 + ], + "score": 0.89, + "content": "X _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 550, + 387, + 567 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 387, + 552, + 444, + 565 + ], + "score": 0.93, + "content": "q _ { 0 } ( x ) = \\mathbb { Q } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 550, + 449, + 567 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 130, + 566, + 457, + 578 + ], + "lines": [ + { + "bbox": [ + 128, + 565, + 459, + 579 + ], + "spans": [ + { + "bbox": [ + 128, + 565, + 413, + 579 + ], + "score": 1.0, + "content": "Fokker-Planck Equation states that the evolution of the particles density", + "type": "text" + }, + { + "bbox": [ + 413, + 568, + 422, + 578 + ], + "score": 0.85, + "content": "q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 565, + 459, + 579 + ], + "score": 1.0, + "content": "satisfies:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 579, + 367, + 603 + ], + "lines": [ + { + "bbox": [ + 244, + 579, + 367, + 603 + ], + "spans": [ + { + "bbox": [ + 244, + 579, + 367, + 603 + ], + "score": 0.95, + "content": "\\frac { d \\boldsymbol { q } _ { t } } { d t } ( \\boldsymbol { x } ) = - \\mathrm { d i v } ( \\boldsymbol { q } _ { t } ( \\boldsymbol { x } ) V ( \\boldsymbol { x } , t ) )", + "type": "interline_equation", + "image_path": "775fdf072fd235807b3bfb8f331579429df6aef60d2b70b5948031e0ed865845.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 244, + 579, + 367, + 603 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "Comparing Equation (15) and Equation (16), we identify then the gradient of Sobolev critic as a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "score": 1.0, + "content": "drift. This suggests that one can define “Sobolev descent” as the evolution of particles along the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 626, + 165, + 639 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 165, + 639 + ], + "score": 1.0, + "content": "gradient flow:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 158, + 640, + 452, + 653 + ], + "lines": [ + { + "bbox": [ + 159, + 639, + 452, + 655 + ], + "spans": [ + { + "bbox": [ + 159, + 640, + 243, + 653 + ], + "score": 0.92, + "content": "d X _ { t } = \\nabla _ { x } f _ { t } ^ { * } ( X _ { t } ) d t", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 639, + 329, + 655 + ], + "score": 1.0, + "content": ", where the density of", + "type": "text" + }, + { + "bbox": [ + 330, + 641, + 343, + 652 + ], + "score": 0.89, + "content": "X _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 639, + 390, + 655 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 390, + 640, + 448, + 653 + ], + "score": 0.93, + "content": "q _ { 0 } ( x ) = \\mathbb { Q } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 639, + 452, + 655 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 134, + 667 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 655, + 146, + 667 + ], + "score": 0.89, + "content": "f _ { t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 655, + 269, + 667 + ], + "score": 1.0, + "content": "is the Sobolev critic between", + "type": "text" + }, + { + "bbox": [ + 270, + 657, + 279, + 666 + ], + "score": 0.83, + "content": "q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 655, + 298, + 667 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 298, + 655, + 306, + 665 + ], + "score": 0.79, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 655, + 505, + 667 + ], + "score": 1.0, + "content": ". One can show that the limit distribution of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 153, + 679 + ], + "score": 1.0, + "content": "particles is", + "type": "text" + }, + { + "bbox": [ + 154, + 666, + 161, + 676 + ], + "score": 0.75, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 664, + 505, + 679 + ], + "score": 1.0, + "content": ". The analysis of “Sobolev descent” and its relation to Stein Descent (Liu & Wang,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "2016; Liu, 2017) is beyond the scope of this paper and will be studied in a separate work. Hence", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "we see that the gradient of the Sobolev critic defines a transportation plan to move particles whose", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 164, + 711 + ], + "score": 1.0, + "content": "distribution is", + "type": "text" + }, + { + "bbox": [ + 164, + 699, + 174, + 710 + ], + "score": 0.86, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 698, + 308, + 711 + ], + "score": 1.0, + "content": "to particles whose distribution is", + "type": "text" + }, + { + "bbox": [ + 309, + 699, + 317, + 709 + ], + "score": 0.76, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "(See Figure 2). This highlights the role of the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "gradient of the critic in the context of GAN training in term of transporting the distribution of the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 240, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 240, + 732 + ], + "score": 1.0, + "content": "generator to the real distribution.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41 + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 447, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 449, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 449, + 95 + ], + "score": 1.0, + "content": "A THEORY: APPROXIMATION AND TRANSPORT INTERPRETATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 504, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "In this Section we present the theoretical properties of Sobolev IPM and how it relates to distributions", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 116, + 471, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 471, + 129 + ], + "score": 1.0, + "content": "transport theory and other known metrics between distributions, notably the Stein distance.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 106, + 105, + 505, + 129 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 140, + 393, + 153 + ], + "lines": [ + { + "bbox": [ + 106, + 140, + 394, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 394, + 153 + ], + "score": 1.0, + "content": "A.1 DISTRIBUTION TRANSPORT PERSPECTIVE ON SOBOLEV IPM", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "In this Section, we characterize the optimal critic of the Sobolev IPM as a solution of a non linear", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 172, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 184 + ], + "score": 1.0, + "content": "PDE. The solution of the variational problem of the Sobolev IPM satisfies a non linear PDE that can", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 505, + 196 + ], + "score": 1.0, + "content": "be derived using standard tools from calculus of variations (Ekeland & Turnbull, 1983; Alaoui et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 192, + 136, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 192, + 136, + 207 + ], + "score": 1.0, + "content": "2016).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 160, + 506, + 207 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 207, + 505, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 457, + 221 + ], + "score": 1.0, + "content": "Theorem 3 (PDE satisfied by the Sobolev Critic). The optimal critic of Sobolev IPM", + "type": "text" + }, + { + "bbox": [ + 458, + 208, + 469, + 219 + ], + "score": 0.87, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 207, + 505, + 221 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 218, + 187, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 161, + 231 + ], + "score": 1.0, + "content": "the following", + "type": "text" + }, + { + "bbox": [ + 161, + 219, + 182, + 229 + ], + "score": 0.29, + "content": "P D E", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 218, + 187, + 231 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 207, + 505, + 231 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 231, + 425, + 259 + ], + "lines": [ + { + "bbox": [ + 186, + 231, + 425, + 259 + ], + "spans": [ + { + "bbox": [ + 186, + 231, + 425, + 259 + ], + "score": 0.93, + "content": "\\Delta f ^ { * } ( x ) + \\left. \\nabla _ { x } \\log \\mu ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\right. + \\frac { \\mathbb { P } ( x ) - \\mathbb { Q } ( x ) } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\mu ( x ) } = 0 .", + "type": "interline_equation", + "image_path": "11a1a882d27c217eb8c6f794348747448fe25025ec5b8fdb311f21b816fa8589.jpg" + } + ] + } + ], + "index": 10.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 231, + 425, + 245.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 186, + 245.0, + 425, + 259.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 504, + 300 + ], + "lines": [ + { + "bbox": [ + 102, + 266, + 508, + 293 + ], + "spans": [ + { + "bbox": [ + 102, + 266, + 213, + 293 + ], + "score": 1.0, + "content": "Define the Stein Operator:", + "type": "text" + }, + { + "bbox": [ + 213, + 270, + 423, + 290 + ], + "score": 0.89, + "content": "\\begin{array} { r } { T ( \\mu ) \\vec { g } ( x ) = \\frac { 1 } { 2 } \\Big ( \\langle \\nabla _ { x } \\log ( \\mu ( x ) ) , \\vec { g } ( x ) \\rangle + d i v ( \\vec { g } ( x ) ) \\Big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 266, + 508, + 293 + ], + "score": 1.0, + "content": ". Hence we have the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 286, + 271, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 237, + 303 + ], + "score": 1.0, + "content": "following Transport Equation of", + "type": "text" + }, + { + "bbox": [ + 238, + 289, + 245, + 299 + ], + "score": 0.83, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 286, + 256, + 303 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 257, + 289, + 266, + 300 + ], + "score": 0.83, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 286, + 271, + 303 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 102, + 266, + 508, + 303 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 209, + 302, + 399, + 316 + ], + "lines": [ + { + "bbox": [ + 209, + 302, + 399, + 316 + ], + "spans": [ + { + "bbox": [ + 209, + 302, + 399, + 316 + ], + "score": 0.82, + "content": "\\mathbb { Q } ( x ) = \\mathbb { P } ( x ) + 2 S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\mu ( x ) T ( \\mu ) \\nabla _ { x } f ^ { * } ( x ) .", + "type": "interline_equation", + "image_path": "1e8e1ddb948c1f5c091710119ff5cfe0e18fe54c83f125800249ff258b965310.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 209, + 302, + 399, + 316 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 317, + 280, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 281, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 281, + 331 + ], + "score": 1.0, + "content": "Recall the definition of Stein Discrepancy :", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 316, + 281, + 331 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 330, + 404, + 353 + ], + "lines": [ + { + "bbox": [ + 205, + 330, + 404, + 353 + ], + "spans": [ + { + "bbox": [ + 205, + 330, + 404, + 353 + ], + "score": 0.88, + "content": "\\mathbb { S } ( \\mathbb { Q } , \\mu ) = \\operatorname* { s u p } _ { \\vec { g } } | \\mathbb { E } _ { x \\sim \\mathbb { Q } } [ T ( \\mu ) \\vec { g } ( x ) ] | , \\vec { g } : \\mathcal { X } \\mathbb { R } ^ { d } .", + "type": "interline_equation", + "image_path": "2877884eb1971d1d3f73c91b22dfd55168fc0436ac4b2c417bab1a06eeda727c.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 205, + 330, + 404, + 353 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 444, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 446, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 446, + 369 + ], + "score": 1.0, + "content": "Theorem 4 (Sobolev and Stein Discrepanices). The following inequality holds true:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 352, + 446, + 369 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 367, + 450, + 395 + ], + "lines": [ + { + "bbox": [ + 156, + 367, + 450, + 395 + ], + "spans": [ + { + "bbox": [ + 156, + 367, + 450, + 395 + ], + "score": 0.87, + "content": "\\bigg | \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { Q } } \\left[ \\frac { \\mathbb { Q } ( \\boldsymbol { x } ) - \\mathbb { P } ( \\boldsymbol { x } ) } { \\mu ( \\boldsymbol { x } ) } \\right] \\bigg | \\leq 2 \\qquad \\underbrace { \\mathbb { S } ( \\mathbb { Q } , \\mu ) } _ { \\mathrm { S } ( \\mathbb { Q } , \\mu ) } \\qquad \\underbrace { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } _ { \\mathrm { S } _ { \\boldsymbol { \\mu } } ( \\mathbb { Q } ) }", + "type": "interline_equation", + "image_path": "eeb5ec94fe21ee822359870e030f1434ca8ac52a226b712de209aa416c3f67fb.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 156, + 367, + 450, + 376.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 156, + 376.3333333333333, + 450, + 385.66666666666663 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 156, + 385.66666666666663, + 450, + 394.99999999999994 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 506, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 196, + 426 + ], + "score": 1.0, + "content": "Consider for example", + "type": "text" + }, + { + "bbox": [ + 196, + 415, + 224, + 425 + ], + "score": 0.91, + "content": "\\mu = \\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 413, + 284, + 426 + ], + "score": 1.0, + "content": ", and sequence", + "type": "text" + }, + { + "bbox": [ + 285, + 415, + 299, + 425 + ], + "score": 0.9, + "content": "\\mathbb { Q } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 413, + 420, + 426 + ], + "score": 1.0, + "content": ". If the Sobolev distance goes", + "type": "text" + }, + { + "bbox": [ + 421, + 414, + 486, + 426 + ], + "score": 0.93, + "content": "S _ { \\mathbb { P } } ( \\mathbb { P } , \\mathbb { Q } _ { n } ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 413, + 506, + 426 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 423, + 507, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 128, + 440 + ], + "score": 1.0, + "content": "ratio", + "type": "text" + }, + { + "bbox": [ + 128, + 425, + 192, + 443 + ], + "score": 0.94, + "content": "\\begin{array} { r } { r _ { n } ( x ) = \\frac { \\mathbb { Q } _ { n } ( x ) } { \\mathbb { P } ( x ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 423, + 507, + 444 + ], + "score": 1.0, + "content": "Qn(x)P(x) converges in expectation (w.r.t to Q) to 1. The speed of the convergence is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 275, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 232, + 454 + ], + "score": 1.0, + "content": "given by the Stein Discrepancy", + "type": "text" + }, + { + "bbox": [ + 233, + 441, + 271, + 453 + ], + "score": 0.93, + "content": "\\mathbb { S } ( \\mathbb { Q } _ { n } , \\mathbb { P } )", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 441, + 275, + 454 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 413, + 507, + 454 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 464, + 502, + 487 + ], + "lines": [ + { + "bbox": [ + 106, + 463, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 504, + 476 + ], + "score": 1.0, + "content": "Relation to Fokker-Planck Diffusion Equation and Particles dynamics. Note that PDE", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 475, + 432, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 432, + 488 + ], + "score": 1.0, + "content": "satisifed by the Sobolev critic given in Equation (13) can be equivalently written:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 463, + 504, + 488 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 488, + 377, + 515 + ], + "lines": [ + { + "bbox": [ + 234, + 488, + 377, + 515 + ], + "spans": [ + { + "bbox": [ + 234, + 488, + 377, + 515 + ], + "score": 0.94, + "content": "\\frac { \\mathbb { P } - \\mathbb { Q } } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } = - \\mathrm { d i v } ( \\mu ( x ) \\nabla _ { x } f ^ { * } ( x ) ) ,", + "type": "interline_equation", + "image_path": "6c0c12a2a6be6aa27f7a455544535d1930265afa23d226fdc15e72c20e552f62.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 234, + 488, + 377, + 501.5 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 234, + 501.5, + 377, + 515.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 516, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "written in this form, we draw a connection with the Fokker-Planck Equation for the evolution of a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 526, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 174, + 540 + ], + "score": 1.0, + "content": "density function", + "type": "text" + }, + { + "bbox": [ + 175, + 529, + 183, + 538 + ], + "score": 0.82, + "content": "q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 526, + 308, + 540 + ], + "score": 1.0, + "content": "that is the density of particles", + "type": "text" + }, + { + "bbox": [ + 309, + 527, + 348, + 538 + ], + "score": 0.93, + "content": "X _ { t } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 526, + 505, + 540 + ], + "score": 1.0, + "content": "evolving with a drift (a velocity field)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 537, + 221, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 216, + 550 + ], + "score": 0.89, + "content": "V ( x , i ) : \\mathcal { X } \\times [ 0 , \\infty [ \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 537, + 221, + 550 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 515, + 505, + 550 + ] + }, + { + "type": "text", + "bbox": [ + 160, + 551, + 449, + 564 + ], + "lines": [ + { + "bbox": [ + 163, + 550, + 449, + 567 + ], + "spans": [ + { + "bbox": [ + 163, + 552, + 239, + 564 + ], + "score": 0.9, + "content": "d X _ { t } = V ( X _ { t } , t ) d t", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 550, + 325, + 567 + ], + "score": 1.0, + "content": ", where the density of", + "type": "text" + }, + { + "bbox": [ + 326, + 553, + 339, + 564 + ], + "score": 0.89, + "content": "X _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 550, + 387, + 567 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 387, + 552, + 444, + 565 + ], + "score": 0.93, + "content": "q _ { 0 } ( x ) = \\mathbb { Q } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 550, + 449, + 567 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 163, + 550, + 449, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 566, + 457, + 578 + ], + "lines": [ + { + "bbox": [ + 128, + 565, + 459, + 579 + ], + "spans": [ + { + "bbox": [ + 128, + 565, + 413, + 579 + ], + "score": 1.0, + "content": "Fokker-Planck Equation states that the evolution of the particles density", + "type": "text" + }, + { + "bbox": [ + 413, + 568, + 422, + 578 + ], + "score": 0.85, + "content": "q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 565, + 459, + 579 + ], + "score": 1.0, + "content": "satisfies:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 128, + 565, + 459, + 579 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 579, + 367, + 603 + ], + "lines": [ + { + "bbox": [ + 244, + 579, + 367, + 603 + ], + "spans": [ + { + "bbox": [ + 244, + 579, + 367, + 603 + ], + "score": 0.95, + "content": "\\frac { d \\boldsymbol { q } _ { t } } { d t } ( \\boldsymbol { x } ) = - \\mathrm { d i v } ( \\boldsymbol { q } _ { t } ( \\boldsymbol { x } ) V ( \\boldsymbol { x } , t ) )", + "type": "interline_equation", + "image_path": "775fdf072fd235807b3bfb8f331579429df6aef60d2b70b5948031e0ed865845.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 244, + 579, + 367, + 603 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 637 + ], + "lines": [ + { + "bbox": [ + 106, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 506, + 617 + ], + "score": 1.0, + "content": "Comparing Equation (15) and Equation (16), we identify then the gradient of Sobolev critic as a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 628 + ], + "score": 1.0, + "content": "drift. This suggests that one can define “Sobolev descent” as the evolution of particles along the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 626, + 165, + 639 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 165, + 639 + ], + "score": 1.0, + "content": "gradient flow:", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 604, + 506, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 158, + 640, + 452, + 653 + ], + "lines": [ + { + "bbox": [ + 159, + 639, + 452, + 655 + ], + "spans": [ + { + "bbox": [ + 159, + 640, + 243, + 653 + ], + "score": 0.92, + "content": "d X _ { t } = \\nabla _ { x } f _ { t } ^ { * } ( X _ { t } ) d t", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 639, + 329, + 655 + ], + "score": 1.0, + "content": ", where the density of", + "type": "text" + }, + { + "bbox": [ + 330, + 641, + 343, + 652 + ], + "score": 0.89, + "content": "X _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 639, + 390, + 655 + ], + "score": 1.0, + "content": "is given by", + "type": "text" + }, + { + "bbox": [ + 390, + 640, + 448, + 653 + ], + "score": 0.93, + "content": "q _ { 0 } ( x ) = \\mathbb { Q } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 639, + 452, + 655 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 134, + 667 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 655, + 146, + 667 + ], + "score": 0.89, + "content": "f _ { t } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 655, + 269, + 667 + ], + "score": 1.0, + "content": "is the Sobolev critic between", + "type": "text" + }, + { + "bbox": [ + 270, + 657, + 279, + 666 + ], + "score": 0.83, + "content": "q _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 655, + 298, + 667 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 298, + 655, + 306, + 665 + ], + "score": 0.79, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 655, + 505, + 667 + ], + "score": 1.0, + "content": ". One can show that the limit distribution of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 153, + 679 + ], + "score": 1.0, + "content": "particles is", + "type": "text" + }, + { + "bbox": [ + 154, + 666, + 161, + 676 + ], + "score": 0.75, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 664, + 505, + 679 + ], + "score": 1.0, + "content": ". The analysis of “Sobolev descent” and its relation to Stein Descent (Liu & Wang,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "2016; Liu, 2017) is beyond the scope of this paper and will be studied in a separate work. Hence", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "we see that the gradient of the Sobolev critic defines a transportation plan to move particles whose", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 164, + 711 + ], + "score": 1.0, + "content": "distribution is", + "type": "text" + }, + { + "bbox": [ + 164, + 699, + 174, + 710 + ], + "score": 0.86, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 698, + 308, + 711 + ], + "score": 1.0, + "content": "to particles whose distribution is", + "type": "text" + }, + { + "bbox": [ + 309, + 699, + 317, + 709 + ], + "score": 0.76, + "content": "\\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "(See Figure 2). This highlights the role of the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "gradient of the critic in the context of GAN training in term of transporting the distribution of the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 240, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 240, + 732 + ], + "score": 1.0, + "content": "generator to the real distribution.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 37, + "bbox_fs": [ + 159, + 639, + 452, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [], + "index": 41, + "bbox_fs": [ + 105, + 655, + 505, + 732 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 169, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 171, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 171, + 96 + ], + "score": 1.0, + "content": "B PROOFS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 504, + 134 + ], + "lines": [ + { + "bbox": [ + 105, + 109, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 109, + 208, + 124 + ], + "score": 1.0, + "content": "Proof of Theorem 2. 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For", + "type": "text" + }, + { + "bbox": [ + 474, + 493, + 505, + 505 + ], + "score": 0.85, + "content": "g , h \\in", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 502, + 310, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 159, + 516 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 502, + 310, + 516 + ], + "score": 1.0, + "content": "the dot product is defined as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 527, + 398, + 554 + ], + "lines": [ + { + "bbox": [ + 212, + 527, + 398, + 554 + ], + "spans": [ + { + "bbox": [ + 212, + 527, + 398, + 554 + ], + "score": 0.94, + "content": "\\langle g , h \\rangle _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } } = \\int _ { \\mathcal { X } } \\langle g ( x ) , h ( x ) \\rangle _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x", + "type": "interline_equation", + "image_path": "ff87f31353f9898c7f2bbf29ab57a1bb4118c4fb13ecb3191d07e8140275131d.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 212, + 527, + 398, + 554 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 199, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 200, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 200, + 578 + ], + "score": 1.0, + "content": "and the norm is given :", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 586, + 378, + 613 + ], + "lines": [ + { + "bbox": [ + 232, + 586, + 378, + 613 + ], + "spans": [ + { + "bbox": [ + 232, + 586, + 378, + 613 + ], + "score": 0.94, + "content": "\\| g \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } } = \\int _ { X } \\| g \\| _ { \\mathbb { R } ^ { d } } ^ { 2 } \\mu ( x ) d x .", + "type": "interline_equation", + "image_path": "5f64fccbbf2a9e795760a7de33f747b99bf60130b2d97b7d8b093dfcc1567b13.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 232, + 586, + 378, + 613 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 626, + 457, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 458, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 397, + 641 + ], + "score": 1.0, + "content": "We can write the objective in Equation (18) in term of the dot product in", + "type": "text" + }, + { + "bbox": [ + 397, + 626, + 451, + 639 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 624, + 458, + 641 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 651, + 439, + 681 + ], + "lines": [ + { + "bbox": [ + 172, + 651, + 439, + 681 + ], + "spans": [ + { + "bbox": [ + 172, + 651, + 439, + 681 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - 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For", + "type": "text" + }, + { + "bbox": [ + 474, + 493, + 505, + 505 + ], + "score": 0.85, + "content": "g , h \\in", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 502, + 310, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 159, + 516 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 502, + 310, + 516 + ], + "score": 1.0, + "content": "the dot product is defined as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 491, + 505, + 516 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 527, + 398, + 554 + ], + "lines": [ + { + "bbox": [ + 212, + 527, + 398, + 554 + ], + "spans": [ + { + "bbox": [ + 212, + 527, + 398, + 554 + ], + "score": 0.94, + "content": "\\langle g , h \\rangle _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } } = \\int _ { \\mathcal { X } } \\langle g ( x ) , h ( x ) \\rangle _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x", + "type": "interline_equation", + "image_path": "ff87f31353f9898c7f2bbf29ab57a1bb4118c4fb13ecb3191d07e8140275131d.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 212, + 527, + 398, + 554 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 199, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 565, + 200, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 200, + 578 + ], + "score": 1.0, + "content": "and the norm is given :", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 565, + 200, + 578 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 586, + 378, + 613 + ], + "lines": [ + { + "bbox": [ + 232, + 586, + 378, + 613 + ], + "spans": [ + { + "bbox": [ + 232, + 586, + 378, + 613 + ], + "score": 0.94, + "content": "\\| g \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } } = \\int _ { X } \\| g \\| _ { \\mathbb { R } ^ { d } } ^ { 2 } \\mu ( x ) d x .", + "type": "interline_equation", + "image_path": "5f64fccbbf2a9e795760a7de33f747b99bf60130b2d97b7d8b093dfcc1567b13.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 232, + 586, + 378, + 613 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 626, + 457, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 458, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 397, + 641 + ], + "score": 1.0, + "content": "We can write the objective in Equation (18) in term of the dot product in", + "type": "text" + }, + { + "bbox": [ + 397, + 626, + 451, + 639 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 624, + 458, + 641 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 624, + 458, + 641 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 651, + 439, + 681 + ], + "lines": [ + { + "bbox": [ + 172, + 651, + 439, + 681 + ], + "spans": [ + { + "bbox": [ + 172, + 651, + 439, + 681 + ], + "score": 0.94, + "content": "\\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) = \\frac { 1 } { d } \\left. \\nabla _ { x } f , \\frac { D ^ { - } ( F _ { \\mathbb { Q } } - F _ { \\mathbb { P } } ) } { \\mu } \\right. _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } } .", + "type": "interline_equation", + "image_path": "7719e18f2f858292d82f2903671b0aa608a4b3cd4a510f662d0d485f621360a4.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 172, + 651, + 439, + 681 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 694, + 505, + 707 + ], + "lines": [ + { + "bbox": [ + 105, + 692, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 448, + 708 + ], + "score": 1.0, + "content": "On the other hand the constraint in Equation (3) can be written in terms of the norm in", + "type": "text" + }, + { + "bbox": [ + 448, + 694, + 501, + 707 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 692, + 505, + 708 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 692, + 505, + 708 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 719, + 376, + 735 + ], + "lines": [ + { + "bbox": [ + 236, + 719, + 376, + 735 + ], + "spans": [ + { + "bbox": [ + 236, + 719, + 376, + 735 + ], + "score": 0.92, + "content": "\\| f \\| _ { W _ { 0 } ^ { 1 , 2 } ( \\mathcal { X } , \\mu ) } = \\| \\nabla _ { x } f \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } }", + "type": "interline_equation", + "image_path": "a2717f1a2dd74db0025a8c09ce269305bd1bf18898a3c09957b249f7f8b5c19b.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 236, + 719, + 376, + 735 + ], + "spans": [], + "index": 23 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 502, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 502, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 502, + 96 + ], + "score": 1.0, + "content": "Replacing the objective and constraint given in Equations (19) and (20) in Equation (3), we obtain:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 96, + 507, + 274 + ], + "lines": [ + { + "bbox": [ + 111, + 96, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 111, + 96, + 507, + 274 + ], + "score": 0.87, + "content": "\\begin{array} { r l } & { \\mathfrak { H } ( \\mathbb { P } , \\mathfrak { G } ) = \\frac { 1 } { d } _ { f , \\vert \\mathcal { C } _ { 2 } , \\vert \\mathcal { C } _ { 2 } , \\alpha , \\beta , 0 ; 0 } \\le \\mathfrak { C } _ { \\ge } f , \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } \\vert \\mathfrak { F } ) } { \\mu } \\Bigr \\rangle _ { \\mathcal { B } ( \\mathfrak { x } , \\mu ) \\le \\mathfrak { c } } } \\\\ & { \\ = \\frac { 1 } { d } _ { g \\le \\mathcal { G } _ { 2 } ( x , \\beta ) \\le \\mathfrak { c } , \\vert \\mathfrak { H } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { x } , \\mu } ) \\le \\mathfrak { c } } \\Big \\langle g , \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } ) } { \\mu } \\Big \\rangle _ { \\mathcal { B } _ { \\ge } ( x , \\mu ) \\ge \\mathfrak { c } } } \\\\ & { \\ = \\frac { 1 } { d } \\bigg \\Vert \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } ) } { \\mu } \\bigg \\Vert _ { \\mathcal { B } _ { \\ge } ( x , \\mu ) \\le \\mathfrak { c } } } \\\\ & { \\ \\left( \\mathtt { B y ~ d e n i n i o n ~ o f ~ } \\Vert \\cdot \\Vert _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) \\le \\mathfrak { c } } , g ^ { - } = \\frac { D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) - D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) } { \\mu ( \\mathfrak { x } ) } \\frac { 1 } { \\left. \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } ) } { \\mu } \\right. _ { \\mathcal { B } _ { \\ge } ( \\mathcal { X } , \\mu ) \\le \\mathfrak { c } } } \\right) } \\\\ & \\ = \\frac { 1 } { d } \\sqrt { \\int _ { x } \\frac { \\Vert D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) - D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) \\Vert _ { \\mathcal { A } ( \\mathcal { X } , \\mu ) } ^ { 2 } } { \\mu ( \\mathfrak { x } ) } d \\mathfrak { x } . } \\end{array}", + "type": "interline_equation", + "image_path": "97b0a5f9eab2359015dacdd9aac8c371e326ca18071ef98d8578c2a27cdeec4e.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 111, + 96, + 507, + 155.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 155.33333333333334, + 507, + 214.66666666666669 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 111, + 214.66666666666669, + 507, + 274.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 323, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 323, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 273, + 288 + ], + "score": 1.0, + "content": "Hence we find also that the optimal critic", + "type": "text" + }, + { + "bbox": [ + 273, + 275, + 285, + 286 + ], + "score": 0.89, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 273, + 323, + 288 + ], + "score": 1.0, + "content": "satisfies:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 288, + 421, + 326 + ], + "lines": [ + { + "bbox": [ + 170, + 288, + 421, + 326 + ], + "spans": [ + { + "bbox": [ + 170, + 288, + 421, + 326 + ], + "score": 0.94, + "content": "\\nabla _ { x } f ^ { * } ( x ) = \\frac { D ^ { - } F _ { \\mathbb { Q } } ( x ) - D ^ { - } F _ { \\mathbb { P } } ( x ) } { \\mu ( x ) } \\frac { 1 } { \\Big \\| \\frac { D ^ { - } ( F _ { \\mathbb { Q } } - F _ { \\mathbb { P } } ) } { \\mu } \\Big \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } } } .", + "type": "interline_equation", + "image_path": "e2a8405da9d66cf04b46a09981465e33ed699ddef6794bf122d29007c27d98f4.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 170, + 288, + 421, + 300.6666666666667 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 170, + 300.6666666666667, + 421, + 313.33333333333337 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 170, + 313.33333333333337, + 421, + 326.00000000000006 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 182, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 183, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 173, + 365 + ], + "score": 1.0, + "content": "Proof of Lemma", + "type": "text" + }, + { + "bbox": [ + 174, + 353, + 179, + 362 + ], + "score": 0.53, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 351, + 183, + 365 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 366, + 487, + 466 + ], + "lines": [ + { + "bbox": [ + 125, + 366, + 487, + 466 + ], + "spans": [ + { + "bbox": [ + 125, + 366, + 487, + 466 + ], + "score": 0.95, + "content": "\\begin{array} { r c l } { \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) } & { = } & { \\displaystyle \\frac { 1 } { d } \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , D ^ { - } ( F _ { \\mathbb { Q } } ( x ) - F _ { \\mathbb { P } } ( x ) ) \\right. _ { \\mathbb { R } ^ { d } } d x } \\\\ & { = } & { \\displaystyle S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , \\frac { D ^ { - } ( F _ { \\mathbb { Q } } ( x ) - F _ { \\mathbb { P } } ( x ) ) } { \\mu ( x ) d S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } \\right. _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x } \\\\ & { = } & { \\displaystyle S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\right. \\mu ( x ) d x } \\\\ & { = } & { \\displaystyle S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\left. f , f ^ { * } \\right. _ { W _ { 0 } ^ { 1 , 2 } } } \\end{array}", + "type": "interline_equation", + "image_path": "0611e104c0c0ee07559606e0f2c96ba20e8c410008e3997a61e6623b0731c165.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 125, + 366, + 487, + 399.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 125, + 399.3333333333333, + 487, + 432.66666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 125, + 432.66666666666663, + 487, + 465.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 171, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 173, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 173, + 479 + ], + "score": 1.0, + "content": "Hence we have:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 139, + 480, + 470, + 507 + ], + "lines": [ + { + "bbox": [ + 139, + 480, + 470, + 507 + ], + "spans": [ + { + "bbox": [ + 139, + 480, + 470, + 507 + ], + "score": 0.89, + "content": "\\operatorname* { s u p } _ { f \\in \\mathcal { M } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } } \\leq 1 } \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) = S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\operatorname* { s u p } _ { f \\in \\mathcal { M } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } } \\leq 1 } \\langle f , f ^ { * } \\rangle _ { W _ { 0 } ^ { 1 , 2 } } ,", + "type": "interline_equation", + "image_path": "22fb9c98448596c9e95e9dea3a007cdce4c4dcc55ba8d44c667739f3e4943ab5.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 139, + 480, + 470, + 507 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 205, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 206, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 206, + 522 + ], + "score": 1.0, + "content": "It follows therefore that:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 523, + 409, + 550 + ], + "lines": [ + { + "bbox": [ + 200, + 523, + 409, + 550 + ], + "spans": [ + { + "bbox": [ + 200, + 523, + 409, + 550 + ], + "score": 0.91, + "content": "\\mathcal { S } _ { \\mathcal { H } } ( \\mathbb { P } , \\mathbb { Q } ) = S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\operatorname* { s u p } _ { f \\in \\mathcal { H } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } } \\leq 1 } \\langle f , f ^ { * } \\rangle _ { W _ { 0 } ^ { 1 , 2 } }", + "type": "interline_equation", + "image_path": "4a1fe5d02a45604832f0c2d765a821ad936cac8baa7aad2cafc27a21eacf0ab5.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 200, + 523, + 409, + 550 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "We conclude that the Sobolev IPM can be approximated in arbitrary space as long as it has enough", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 588, + 504, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 504, + 599 + ], + "score": 1.0, + "content": "capacity to approximate the optimal critic. Interestingly the approximation error is measured now", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 597, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 612 + ], + "score": 1.0, + "content": "with the Sobolev semi-norm, while in Fisher it was measured with the Lebesgue norm. Approx-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 608, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 623 + ], + "score": 1.0, + "content": "imations with Sobolev Semi-norms are stronger then Lebesgue norms as given by the Poincare", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 149, + 635 + ], + "score": 1.0, + "content": "inequality", + "type": "text" + }, + { + "bbox": [ + 149, + 620, + 242, + 635 + ], + "score": 0.9, + "content": "( \\vert \\vert f \\vert \\vert _ { \\mathcal { L } _ { 2 } } \\le C \\left. f \\right. _ { W _ { 0 } ^ { 1 , 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 620, + 506, + 635 + ], + "score": 1.0, + "content": ", meaning if the error goes to zero in Sobolev sense it also goes to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 633, + 332, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 332, + 646 + ], + "score": 1.0, + "content": "zero in the Lebesgue sense , but the converse is not true.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 105, + 656, + 504, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "Proof of Theorem 3. The proof follows similar arguments in the proofs of the analysis of Laplacian", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 668, + 405, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 405, + 680 + ], + "score": 1.0, + "content": "regularization in semi-supervised learning studied by (Alaoui et al., 2016).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 697, + 417, + 735 + ], + "lines": [ + { + "bbox": [ + 183, + 697, + 417, + 735 + ], + "spans": [ + { + "bbox": [ + 183, + 697, + 417, + 735 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } & { { } = \\operatorname* { s u p } _ { f \\in W _ { 0 } ^ { 1 , 2 } } \\Big \\{ \\mathbb { E } _ { x \\sim \\mathbb { P } } \\left[ f ( x ) \\right] - \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ f ( x ) \\right] \\Big \\} } \\\\ { s . t . } & { { } \\quad \\mathbb { E } _ { x \\sim \\mu } \\| \\nabla f ( x ) \\| _ { 2 } ^ { 2 } \\leq 1 , } \\end{array}", + "type": "interline_equation", + "image_path": "16204a8fe673ff37deede84c74672bcba84a9389a4840469eb842215b7d60971.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 183, + 697, + 417, + 716.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 183, + 716.0, + 417, + 735.0 + ], + "spans": [], + "index": 25 + } + ] + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 328, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 496, + 330, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 496, + 330, + 505, + 340 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 553, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 495, + 555, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 495, + 555, + 505, + 564 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 502, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 502, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 502, + 96 + ], + "score": 1.0, + "content": "Replacing the objective and constraint given in Equations (19) and (20) in Equation (3), we obtain:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 502, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 96, + 507, + 274 + ], + "lines": [ + { + "bbox": [ + 111, + 96, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 111, + 96, + 507, + 274 + ], + "score": 0.87, + "content": "\\begin{array} { r l } & { \\mathfrak { H } ( \\mathbb { P } , \\mathfrak { G } ) = \\frac { 1 } { d } _ { f , \\vert \\mathcal { C } _ { 2 } , \\vert \\mathcal { C } _ { 2 } , \\alpha , \\beta , 0 ; 0 } \\le \\mathfrak { C } _ { \\ge } f , \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } \\vert \\mathfrak { F } ) } { \\mu } \\Bigr \\rangle _ { \\mathcal { B } ( \\mathfrak { x } , \\mu ) \\le \\mathfrak { c } } } \\\\ & { \\ = \\frac { 1 } { d } _ { g \\le \\mathcal { G } _ { 2 } ( x , \\beta ) \\le \\mathfrak { c } , \\vert \\mathfrak { H } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { g } } \\vert _ { \\mathfrak { x } , \\mu } ) \\le \\mathfrak { c } } \\Big \\langle g , \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } ) } { \\mu } \\Big \\rangle _ { \\mathcal { B } _ { \\ge } ( x , \\mu ) \\ge \\mathfrak { c } } } \\\\ & { \\ = \\frac { 1 } { d } \\bigg \\Vert \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } ) } { \\mu } \\bigg \\Vert _ { \\mathcal { B } _ { \\ge } ( x , \\mu ) \\le \\mathfrak { c } } } \\\\ & { \\ \\left( \\mathtt { B y ~ d e n i n i o n ~ o f ~ } \\Vert \\cdot \\Vert _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) \\le \\mathfrak { c } } , g ^ { - } = \\frac { D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) - D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) } { \\mu ( \\mathfrak { x } ) } \\frac { 1 } { \\left. \\frac { D ^ { - } ( F _ { \\mathfrak { g } } - F _ { \\mathfrak { g } } ) } { \\mu } \\right. _ { \\mathcal { B } _ { \\ge } ( \\mathcal { X } , \\mu ) \\le \\mathfrak { c } } } \\right) } \\\\ & \\ = \\frac { 1 } { d } \\sqrt { \\int _ { x } \\frac { \\Vert D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) - D ^ { - } F _ { \\mathfrak { g } } ( \\mathfrak { x } ) \\Vert _ { \\mathcal { A } ( \\mathcal { X } , \\mu ) } ^ { 2 } } { \\mu ( \\mathfrak { x } ) } d \\mathfrak { x } . } \\end{array}", + "type": "interline_equation", + "image_path": "97b0a5f9eab2359015dacdd9aac8c371e326ca18071ef98d8578c2a27cdeec4e.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 111, + 96, + 507, + 155.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 155.33333333333334, + 507, + 214.66666666666669 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 111, + 214.66666666666669, + 507, + 274.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 323, + 286 + ], + "lines": [ + { + "bbox": [ + 105, + 273, + 323, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 273, + 288 + ], + "score": 1.0, + "content": "Hence we find also that the optimal critic", + "type": "text" + }, + { + "bbox": [ + 273, + 275, + 285, + 286 + ], + "score": 0.89, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 273, + 323, + 288 + ], + "score": 1.0, + "content": "satisfies:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 273, + 323, + 288 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 288, + 421, + 326 + ], + "lines": [ + { + "bbox": [ + 170, + 288, + 421, + 326 + ], + "spans": [ + { + "bbox": [ + 170, + 288, + 421, + 326 + ], + "score": 0.94, + "content": "\\nabla _ { x } f ^ { * } ( x ) = \\frac { D ^ { - } F _ { \\mathbb { Q } } ( x ) - D ^ { - } F _ { \\mathbb { P } } ( x ) } { \\mu ( x ) } \\frac { 1 } { \\Big \\| \\frac { D ^ { - } ( F _ { \\mathbb { Q } } - F _ { \\mathbb { P } } ) } { \\mu } \\Big \\| _ { \\mathcal { L } _ { 2 } ( \\mathcal { X } , \\mu ) ^ { \\otimes d } } } .", + "type": "interline_equation", + "image_path": "e2a8405da9d66cf04b46a09981465e33ed699ddef6794bf122d29007c27d98f4.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 170, + 288, + 421, + 300.6666666666667 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 170, + 300.6666666666667, + 421, + 313.33333333333337 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 170, + 313.33333333333337, + 421, + 326.00000000000006 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 182, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 351, + 183, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 173, + 365 + ], + "score": 1.0, + "content": "Proof of Lemma", + "type": "text" + }, + { + "bbox": [ + 174, + 353, + 179, + 362 + ], + "score": 0.53, + "content": "\\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 351, + 183, + 365 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 351, + 183, + 365 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 366, + 487, + 466 + ], + "lines": [ + { + "bbox": [ + 125, + 366, + 487, + 466 + ], + "spans": [ + { + "bbox": [ + 125, + 366, + 487, + 466 + ], + "score": 0.95, + "content": "\\begin{array} { r c l } { \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) } & { = } & { \\displaystyle \\frac { 1 } { d } \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , D ^ { - } ( F _ { \\mathbb { Q } } ( x ) - F _ { \\mathbb { P } } ( x ) ) \\right. _ { \\mathbb { R } ^ { d } } d x } \\\\ & { = } & { \\displaystyle S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , \\frac { D ^ { - } ( F _ { \\mathbb { Q } } ( x ) - F _ { \\mathbb { P } } ( x ) ) } { \\mu ( x ) d S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } \\right. _ { \\mathbb { R } ^ { d } } \\mu ( x ) d x } \\\\ & { = } & { \\displaystyle S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\int _ { \\mathcal { X } } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } f ^ { * } ( x ) \\right. \\mu ( x ) d x } \\\\ & { = } & { \\displaystyle S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\left. f , f ^ { * } \\right. _ { W _ { 0 } ^ { 1 , 2 } } } \\end{array}", + "type": "interline_equation", + "image_path": "0611e104c0c0ee07559606e0f2c96ba20e8c410008e3997a61e6623b0731c165.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 125, + 366, + 487, + 399.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 125, + 399.3333333333333, + 487, + 432.66666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 125, + 432.66666666666663, + 487, + 465.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 171, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 173, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 173, + 479 + ], + "score": 1.0, + "content": "Hence we have:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 465, + 173, + 479 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 139, + 480, + 470, + 507 + ], + "lines": [ + { + "bbox": [ + 139, + 480, + 470, + 507 + ], + "spans": [ + { + "bbox": [ + 139, + 480, + 470, + 507 + ], + "score": 0.89, + "content": "\\operatorname* { s u p } _ { f \\in \\mathcal { M } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } } \\leq 1 } \\mathbb { E } _ { x \\sim \\mathbb { P } } f ( x ) - \\mathbb { E } _ { x \\sim \\mathbb { Q } } f ( x ) = S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\operatorname* { s u p } _ { f \\in \\mathcal { M } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } } \\leq 1 } \\langle f , f ^ { * } \\rangle _ { W _ { 0 } ^ { 1 , 2 } } ,", + "type": "interline_equation", + "image_path": "22fb9c98448596c9e95e9dea3a007cdce4c4dcc55ba8d44c667739f3e4943ab5.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 139, + 480, + 470, + 507 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 205, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 206, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 206, + 522 + ], + "score": 1.0, + "content": "It follows therefore that:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 509, + 206, + 522 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 523, + 409, + 550 + ], + "lines": [ + { + "bbox": [ + 200, + 523, + 409, + 550 + ], + "spans": [ + { + "bbox": [ + 200, + 523, + 409, + 550 + ], + "score": 0.91, + "content": "\\mathcal { S } _ { \\mathcal { H } } ( \\mathbb { P } , \\mathbb { Q } ) = S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) \\operatorname* { s u p } _ { f \\in \\mathcal { H } , \\| f \\| _ { W _ { 0 } ^ { 1 , 2 } } \\leq 1 } \\langle f , f ^ { * } \\rangle _ { W _ { 0 } ^ { 1 , 2 } }", + "type": "interline_equation", + "image_path": "4a1fe5d02a45604832f0c2d765a821ad936cac8baa7aad2cafc27a21eacf0ab5.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 200, + 523, + 409, + 550 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "We conclude that the Sobolev IPM can be approximated in arbitrary space as long as it has enough", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 588, + 504, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 504, + 599 + ], + "score": 1.0, + "content": "capacity to approximate the optimal critic. Interestingly the approximation error is measured now", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 597, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 612 + ], + "score": 1.0, + "content": "with the Sobolev semi-norm, while in Fisher it was measured with the Lebesgue norm. Approx-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 608, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 623 + ], + "score": 1.0, + "content": "imations with Sobolev Semi-norms are stronger then Lebesgue norms as given by the Poincare", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 149, + 635 + ], + "score": 1.0, + "content": "inequality", + "type": "text" + }, + { + "bbox": [ + 149, + 620, + 242, + 635 + ], + "score": 0.9, + "content": "( \\vert \\vert f \\vert \\vert _ { \\mathcal { L } _ { 2 } } \\le C \\left. f \\right. _ { W _ { 0 } ^ { 1 , 2 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 620, + 506, + 635 + ], + "score": 1.0, + "content": ", meaning if the error goes to zero in Sobolev sense it also goes to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 633, + 332, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 332, + 646 + ], + "score": 1.0, + "content": "zero in the Lebesgue sense , but the converse is not true.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 576, + 506, + 646 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 656, + 504, + 680 + ], + "lines": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "Proof of Theorem 3. The proof follows similar arguments in the proofs of the analysis of Laplacian", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 668, + 405, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 405, + 680 + ], + "score": 1.0, + "content": "regularization in semi-supervised learning studied by (Alaoui et al., 2016).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 657, + 505, + 680 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 183, + 697, + 417, + 735 + ], + "lines": [ + { + "bbox": [ + 183, + 697, + 417, + 735 + ], + "spans": [ + { + "bbox": [ + 183, + 697, + 417, + 735 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } & { { } = \\operatorname* { s u p } _ { f \\in W _ { 0 } ^ { 1 , 2 } } \\Big \\{ \\mathbb { E } _ { x \\sim \\mathbb { P } } \\left[ f ( x ) \\right] - \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ f ( x ) \\right] \\Big \\} } \\\\ { s . t . } & { { } \\quad \\mathbb { E } _ { x \\sim \\mu } \\| \\nabla f ( x ) \\| _ { 2 } ^ { 2 } \\leq 1 , } \\end{array}", + "type": "interline_equation", + "image_path": "16204a8fe673ff37deede84c74672bcba84a9389a4840469eb842215b7d60971.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 183, + 697, + 417, + 716.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 183, + 716.0, + 417, + 735.0 + ], + "spans": [], + "index": 25 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 258, + 95 + ], + "score": 1.0, + "content": "Note that this problem is convex in", + "type": "text" + }, + { + "bbox": [ + 258, + 83, + 266, + 94 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "(Ekeland & Turnbull, 1983). 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\\int _ { \\mathcal { X } } \\| \\nabla _ { x } f ( x ) \\| _ { 2 } ^ { 2 } \\ \\mu ( x ) d x { \\Big ) }", + "type": "interline_equation", + "image_path": "4b3acba07029e2f916f9fafefde867c118b2f61a87a2ee021db7672ffdae327e.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 171, + 223, + 439, + 250 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 255, + 235, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 235, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 235, + 267 + ], + "score": 1.0, + "content": "From the calculus of variations:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 255, + 235, + 267 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 271, + 514, + 433 + ], + "lines": [ + { + "bbox": [ + 111, + 271, + 514, + 433 + ], + "spans": [ + { + "bbox": [ + 111, + 271, + 514, + 433 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\displaystyle \\sum _ { k = 1 } ^ { \\infty } ( f + \\epsilon h , \\lambda ) = \\int _ { x } ( f + \\epsilon h ) ( x ) \\mu _ { 1 } ( x ) d x + \\frac { \\lambda } { 2 } \\Bigl ( 1 - 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i } F _ { \\mathbb { Q } } ( x ) - D ^ { - i } F _ { \\mathbb { P } } ( x ) } { \\lambda ^ { * } d ~ \\mu ( x ) } ~ \\forall ~ i \\in [ d ] ,", + "type": "interline_equation", + "image_path": "a385e01f2a089778c02d72b0eb70db6f2299d5d8798cac41695c7826794c564f.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 181, + 465, + 429, + 492 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 207, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 208, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 208, + 507 + ], + "score": 1.0, + "content": "satisfies indeed the PDE.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 495, + 208, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 335, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 332, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 272, + 531 + ], + "score": 1.0, + "content": "From equation (27), we want to compute", + "type": "text" + }, + { + "bbox": [ + 273, + 510, + 299, + 529 + ], + "score": 0.93, + "content": "\\frac { \\partial ^ { 2 } f ( \\boldsymbol { x } ) } { \\partial \\boldsymbol { x } _ { i } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 507, + 327, + 531 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 327, + 515, + 332, + 523 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 507, + 332, + 531 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 114, + 531, + 496, + 628 + ], + "lines": [ + { + "bbox": [ + 114, + 531, + 496, + 628 + ], + "spans": [ + { + "bbox": [ + 114, + 531, + 496, + 628 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\frac { \\partial ^ { 2 } f ( x ) } { \\partial x _ { i } ^ { 2 } } = \\frac { 1 } { \\lambda ^ { * } d } \\Bigg [ \\frac { \\mu ( x ) \\left[ \\frac { \\partial } { \\partial x _ { i } } ( D ^ { - 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i } F _ { \\mathbb { Q } } ( x ) - D ^ { - i } F _ { \\mathbb { P } } ( x ) } { \\lambda ^ { * } d ~ \\mu ( x ) } ~ \\forall i \\in [ d ]", + "type": "interline_equation", + "image_path": "44dbea6a536cb925ecc8309bbfa451d1a5e906fefef60ca00d84d9aed3904417.jpg" + } + ] + } + ], + "index": 2.5, + "virtual_lines": [ + { + "bbox": [ + 209, + 102, + 402, + 116.5 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 209, + 116.5, + 402, + 131.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 331, + 150 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 331, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 313, + 151 + ], + "score": 1.0, + "content": "Using the constraint in (26) we can get the value of", + "type": "text" + }, + { + "bbox": [ + 313, + 139, + 325, + 149 + ], + "score": 0.87, + "content": "\\lambda ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 138, + 331, + 151 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 155, + 437, + 257 + ], + "lines": [ + { + "bbox": [ + 171, + 155, + 437, + 257 + ], + "spans": [ + { + "bbox": [ + 171, + 155, + 437, + 257 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad \\displaystyle \\int \\| \\nabla f ^ { * } ( \\boldsymbol { x } ) \\| ^ { 2 } \\mu ( \\boldsymbol { x } ) d \\boldsymbol { x } = 1 } \\\\ & { \\Rightarrow \\displaystyle \\int \\sum _ { i = 1 } ^ { d } \\left( \\frac { \\partial f ^ { * } ( \\boldsymbol { x } ) } { \\partial x _ { i } } \\right) ^ { 2 } \\mu ( \\boldsymbol { x } ) d \\boldsymbol { x } = 1 } \\\\ & { \\Rightarrow \\lambda ^ { * } = \\displaystyle \\frac { 1 } { d } \\sqrt { \\displaystyle \\sum _ { i = 1 } ^ { d } \\int \\frac { \\left( D ^ { - i } F _ { \\mathbb { Q } } ( \\boldsymbol { x } ) - D ^ { - i } F _ { \\mathbb { P } } ( \\boldsymbol { x } ) \\right) ^ { 2 } } { \\mu ( \\boldsymbol { x } ) } d \\boldsymbol { x } } = S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "c823ecf201c6656506826bf064a968d261c6cd69c715fcf224521bb05e3a3655.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 171, + 155, + 437, + 189.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 171, + 189.0, + 437, + 223.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 171, + 223.0, + 437, + 257.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 375, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 289, + 374, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 374, + 302 + ], + "score": 1.0, + "content": "Proof of Theorem 4. Define the Stein operator (Oates et al., 2017):", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 306, + 441, + 356 + ], + "lines": [ + { + "bbox": [ + 168, + 306, + 441, + 356 + ], + "spans": [ + { + "bbox": [ + 168, + 306, + 441, + 356 + ], + "score": 0.94, + "content": "\\begin{array} { l l l } { { T ( \\mu ) [ \\nabla _ { x } f ( x ) ] } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } \\log \\mu ( x ) \\right. + \\displaystyle \\frac { 1 } { 2 } \\left. \\nabla _ { x } , \\nabla _ { x } f ( x ) \\right. } } \\\\ { { } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } \\log \\mu ( x ) \\right. + \\displaystyle \\frac { 1 } { 2 } \\Delta _ { 2 } f ( x ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "6d3416316aee793bd2d6dc18e1f1704ec9b17d4ddc80d99466a1814424dc821d.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 168, + 306, + 441, + 322.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 168, + 322.6666666666667, + 441, + 339.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 168, + 339.33333333333337, + 441, + 356.00000000000006 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 357, + 504, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "This operator was later used in defining the Stein discrepancy (Gorham & Mackey, 2015; Liu et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 368, + 288, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 288, + 381 + ], + "score": 1.0, + "content": "2016; Chwialkowski et al., 2016; Liu, 2017).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 504, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "score": 1.0, + "content": "Recall that Barbour generator theory provides us a way of constructing such operators that produce", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 397, + 317, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 211, + 408 + ], + "score": 1.0, + "content": "mean zero function under", + "type": "text" + }, + { + "bbox": [ + 211, + 398, + 218, + 408 + ], + "score": 0.79, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 397, + 317, + 408 + ], + "score": 1.0, + "content": ". It is easy to verify that:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 413, + 356, + 427 + ], + "lines": [ + { + "bbox": [ + 255, + 413, + 356, + 427 + ], + "spans": [ + { + "bbox": [ + 255, + 413, + 356, + 427 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { x \\sim \\mu } T ( \\mu ) \\nabla _ { x } f ( x ) = 0 .", + "type": "interline_equation", + "image_path": "f8c347b195c8c08fb3d2aa3ef252e2e9c2891841e85998a89e67c1105cece808.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 255, + 413, + 356, + 427 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "score": 1.0, + "content": "Recall that this operator arises from the overdamped Langevin diffusion, defined by the stochastic", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 447, + 192, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 192, + 461 + ], + "score": 1.0, + "content": "differential equation:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 458, + 365, + 482 + ], + "lines": [ + { + "bbox": [ + 245, + 458, + 365, + 482 + ], + "spans": [ + { + "bbox": [ + 245, + 458, + 365, + 482 + ], + "score": 0.94, + "content": "d x _ { t } = \\frac { 1 } { 2 } \\nabla _ { x } \\log \\mu ( x _ { t } ) + d W _ { t }", + "type": "interline_equation", + "image_path": "9a775c228f7a2803176a2f73ae2d6c0b7dd4ab179b80e48268df058f76dd8724.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 245, + 458, + 365, + 482 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 506, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 135, + 498 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 136, + 484, + 171, + 496 + ], + "score": 0.92, + "content": "( W _ { t } ) _ { t \\geq 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 482, + 505, + 498 + ], + "score": 1.0, + "content": "is a Wiener process. This is related to plug and play networks for generating", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 495, + 420, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 420, + 507 + ], + "score": 1.0, + "content": "samples if the distribution is known, using the stochastic differential equation.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 504, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 385, + 536 + ], + "score": 1.0, + "content": "From Theorem 3, it is easy to see that the PDE the Sobolev Critic", + "type": "text" + }, + { + "bbox": [ + 385, + 523, + 473, + 536 + ], + "score": 0.92, + "content": "( f ^ { * } , \\lambda ^ { * } = { \\cal S } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 521, + 505, + 536 + ], + "score": 1.0, + "content": "can be", 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149 + ], + "score": 0.87, + "content": "\\lambda ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 138, + 331, + 151 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 106, + 138, + 331, + 151 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 155, + 437, + 257 + ], + "lines": [ + { + "bbox": [ + 171, + 155, + 437, + 257 + ], + "spans": [ + { + "bbox": [ + 171, + 155, + 437, + 257 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\quad \\displaystyle \\int \\| \\nabla f ^ { * } ( \\boldsymbol { x } ) \\| ^ { 2 } \\mu ( \\boldsymbol { x } ) d \\boldsymbol { x } = 1 } \\\\ & { \\Rightarrow \\displaystyle \\int \\sum _ { i = 1 } ^ { d } \\left( \\frac { \\partial f ^ { * } ( \\boldsymbol { x } ) } { \\partial x _ { i } } \\right) ^ { 2 } \\mu ( \\boldsymbol { x } ) d \\boldsymbol { x } = 1 } \\\\ & { \\Rightarrow \\lambda ^ { * } = \\displaystyle \\frac { 1 } { d } \\sqrt { \\displaystyle \\sum _ { i = 1 } ^ { d } \\int \\frac { \\left( D ^ { - i } F _ { \\mathbb { Q } } ( \\boldsymbol { x } ) - D ^ { - i } F _ { \\mathbb { P } } ( \\boldsymbol { x } ) \\right) ^ { 2 } } { \\mu ( \\boldsymbol { x } ) } d \\boldsymbol { x } } = S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "c823ecf201c6656506826bf064a968d261c6cd69c715fcf224521bb05e3a3655.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 171, + 155, + 437, + 189.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 171, + 189.0, + 437, + 223.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 171, + 223.0, + 437, + 257.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 289, + 375, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 289, + 374, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 374, + 302 + ], + "score": 1.0, + "content": "Proof of Theorem 4. Define the Stein operator (Oates et al., 2017):", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 289, + 374, + 302 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 168, + 306, + 441, + 356 + ], + "lines": [ + { + "bbox": [ + 168, + 306, + 441, + 356 + ], + "spans": [ + { + "bbox": [ + 168, + 306, + 441, + 356 + ], + "score": 0.94, + "content": "\\begin{array} { l l l } { { T ( \\mu ) [ \\nabla _ { x } f ( x ) ] } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } \\log \\mu ( x ) \\right. + \\displaystyle \\frac { 1 } { 2 } \\left. \\nabla _ { x } , \\nabla _ { x } f ( x ) \\right. } } \\\\ { { } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\left. \\nabla _ { x } f ( x ) , \\nabla _ { x } \\log \\mu ( x ) \\right. + \\displaystyle \\frac { 1 } { 2 } \\Delta _ { 2 } f ( x ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "6d3416316aee793bd2d6dc18e1f1704ec9b17d4ddc80d99466a1814424dc821d.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 168, + 306, + 441, + 322.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 168, + 322.6666666666667, + 441, + 339.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 168, + 339.33333333333337, + 441, + 356.00000000000006 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 357, + 504, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "This operator was later used in defining the Stein discrepancy (Gorham & Mackey, 2015; Liu et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 368, + 288, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 288, + 381 + ], + "score": 1.0, + "content": "2016; Chwialkowski et al., 2016; Liu, 2017).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 106, + 357, + 505, + 381 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 504, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "score": 1.0, + "content": "Recall that Barbour generator theory provides us a way of constructing such operators that produce", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 397, + 317, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 211, + 408 + ], + "score": 1.0, + "content": "mean zero function under", + "type": "text" + }, + { + "bbox": [ + 211, + 398, + 218, + 408 + ], + "score": 0.79, + "content": "\\mu", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 397, + 317, + 408 + ], + "score": 1.0, + "content": ". It is easy to verify that:", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 384, + 506, + 408 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 413, + 356, + 427 + ], + "lines": [ + { + "bbox": [ + 255, + 413, + 356, + 427 + ], + "spans": [ + { + "bbox": [ + 255, + 413, + 356, + 427 + ], + "score": 0.91, + "content": "\\mathbb { E } _ { x \\sim \\mu } T ( \\mu ) \\nabla _ { x } f ( x ) = 0 .", + "type": "interline_equation", + "image_path": "f8c347b195c8c08fb3d2aa3ef252e2e9c2891841e85998a89e67c1105cece808.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 255, + 413, + 356, + 427 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "score": 1.0, + "content": "Recall that this operator arises from the overdamped Langevin diffusion, defined by the stochastic", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 447, + 192, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 192, + 461 + ], + "score": 1.0, + "content": "differential equation:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 437, + 505, + 461 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 458, + 365, + 482 + ], + "lines": [ + { + "bbox": [ + 245, + 458, + 365, + 482 + ], + "spans": [ + { + "bbox": [ + 245, + 458, + 365, + 482 + ], + "score": 0.94, + "content": "d x _ { t } = \\frac { 1 } { 2 } \\nabla _ { x } \\log \\mu ( x _ { t } ) + d W _ { t }", + "type": "interline_equation", + "image_path": "9a775c228f7a2803176a2f73ae2d6c0b7dd4ab179b80e48268df058f76dd8724.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 245, + 458, + 365, + 482 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 506, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 135, + 498 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 136, + 484, + 171, + 496 + ], + "score": 0.92, + "content": "( W _ { t } ) _ { t \\geq 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 482, + 505, + 498 + ], + "score": 1.0, + "content": "is a Wiener process. This is related to plug and play networks for generating", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 495, + 420, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 420, + 507 + ], + "score": 1.0, + "content": "samples if the distribution is known, using the stochastic differential equation.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 482, + 505, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 504, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 385, + 536 + ], + "score": 1.0, + "content": "From Theorem 3, it is easy to see that the PDE the Sobolev Critic", + "type": "text" + }, + { + "bbox": [ + 385, + 523, + 473, + 536 + ], + "score": 0.92, + "content": "( f ^ { * } , \\lambda ^ { * } = { \\cal S } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 521, + 505, + 536 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 534, + 285, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 285, + 547 + ], + "score": 1.0, + "content": "written in term of Stein Operator as follows:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 521, + 505, + 547 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 550, + 382, + 577 + ], + "lines": [ + { + "bbox": [ + 229, + 550, + 382, + 577 + ], + "spans": [ + { + "bbox": [ + 229, + 550, + 382, + 577 + ], + "score": 0.93, + "content": "T ( \\mu ) [ \\nabla _ { x } f ^ { * } ] ( x ) = \\frac { 1 } { 2 \\lambda ^ { * } } \\frac { \\mathbb { Q } ( x ) - \\mathbb { P } ( x ) } { \\mu ( x ) }", + "type": "interline_equation", + "image_path": "13a63ed56bfbc92d7850451825a15da185431897b980e264e1a4742e8525a7e6.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 229, + 550, + 382, + 577 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 581, + 353, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 354, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 340, + 596 + ], + "score": 1.0, + "content": "Taking absolute values and the expectation with respect to", + "type": "text" + }, + { + "bbox": [ + 341, + 582, + 350, + 593 + ], + "score": 0.85, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 579, + 354, + 596 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 579, + 354, + 596 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 597, + 434, + 626 + ], + "lines": [ + { + "bbox": [ + 178, + 597, + 434, + 626 + ], + "spans": [ + { + "bbox": [ + 178, + 597, + 434, + 626 + ], + "score": 0.94, + "content": "\\left| \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ T ( \\mu ) \\nabla _ { x } f ^ { * } ( x ) \\right] \\right| = { \\frac { 1 } { 2 S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } } \\left| \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ { \\frac { \\mathbb { Q } ( x ) - \\mathbb { P } ( x ) } { \\mu ( x ) } } \\right] \\right|", + "type": "interline_equation", + "image_path": "3fd5d739cc15b4036ab687140a42ca69ddf4b38c4231467557e8a98af40f2ed6.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 178, + 597, + 434, + 606.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 178, + 606.6666666666666, + 434, + 616.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 178, + 616.3333333333333, + 434, + 625.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 298, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 299, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 299, + 650 + ], + "score": 1.0, + "content": "Recall that the definition of Stein Discrepancy :", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 633, + 299, + 650 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 652, + 393, + 675 + ], + "lines": [ + { + "bbox": [ + 218, + 652, + 393, + 675 + ], + "spans": [ + { + "bbox": [ + 218, + 652, + 393, + 675 + ], + "score": 0.93, + "content": "{ \\mathbb S } ( \\mathbb Q , \\mu ) = \\operatorname* { s u p } _ { \\vec { g } \\in \\mathcal L _ { 2 } ( \\mathcal X , \\mu ) ^ { \\otimes d } } \\left| \\mathbb E _ { \\boldsymbol { x } \\sim \\mathbb Q } \\left[ T ( \\mu ) \\vec { g } ( \\boldsymbol { x } ) \\right] \\right|", + "type": "interline_equation", + "image_path": "831616c30044b22fdf8bee49a8388a5f44daad6efb93275abce91af2a02ed63b.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 218, + 652, + 393, + 675 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 685, + 279, + 697 + ], + "lines": [ + { + "bbox": [ + 105, + 684, + 280, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 280, + 698 + ], + "score": 1.0, + "content": "It follows that Sobolev IPM critic satisfies:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 684, + 280, + 698 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 702, + 376, + 716 + ], + "lines": [ + { + "bbox": [ + 234, + 702, + 376, + 716 + ], + "spans": [ + { + "bbox": [ + 234, + 702, + 376, + 716 + ], + "score": 0.89, + "content": "| \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ T ( \\mu ) \\nabla _ { x } f ^ { * } ( x ) \\right] | \\leq \\mathbb { S } ( \\mathbb { Q } , \\mu ) ,", + "type": "interline_equation", + "image_path": "a190696d8767374228628efbf39585e8252f102eb0821ad165a16738f674c1e2.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 234, + 702, + 376, + 716 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 720, + 270, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 718, + 270, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 270, + 734 + ], + "score": 1.0, + "content": "Hence we have the following inequality:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 718, + 270, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 210, + 94, + 402, + 122 + ], + "lines": [ + { + "bbox": [ + 210, + 94, + 402, + 122 + ], + "spans": [ + { + "bbox": [ + 210, + 94, + 402, + 122 + ], + "score": 0.91, + "content": "\\frac { 1 } { 2 S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } \\left| \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { Q } } \\left[ \\frac { \\mathbb { Q } ( \\boldsymbol { x } ) - \\mathbb { P } ( \\boldsymbol { x } ) } { \\mu ( \\boldsymbol { x } ) } \\right] \\right| \\leq \\mathbb { S } ( \\mathbb { Q } , \\mu )", + "type": "interline_equation", + "image_path": "9cc512735ea0762daf1b551fd65435bd573c1723f1adfe4a1105d23ea41cc7a5.jpg" + } + ] + } + ], + "index": 0, + "virtual_lines": [ + { + "bbox": [ + 210, + 94, + 402, + 122 + ], + "spans": [], + "index": 0 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 127, + 191, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 126, + 193, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 193, + 140 + ], + "score": 1.0, + "content": "This is equivalent to:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 146, + 450, + 175 + ], + "lines": [ + { + "bbox": [ + 154, + 146, + 450, + 175 + ], + "spans": [ + { + "bbox": [ + 154, + 146, + 450, + 175 + ], + "score": 0.82, + "content": "\\left| \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ { \\frac { \\mathbb { Q } ( x ) - \\mathbb { P } ( x ) } { \\mu ( x ) } } \\right] \\right| \\leq 2 \\qquad { \\underset { \\Delta \\in { \\mathbb { Q } } , \\mu } { \\underbrace { \\mathbb { S } ( \\mathbb { Q } , \\mu ) } } } \\qquad { \\underset { \\Delta \\in { \\mathbb { P } } } { \\underbrace { { \\mathcal { S } } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } } }", + "type": "interline_equation", + "image_path": "97c900eaf9b2de3c35f5d3e37769a081bce9215000b150e7a7c095b7fbee1211.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 154, + 146, + 450, + 155.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 154, + 155.66666666666666, + 450, + 165.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 154, + 165.33333333333331, + 450, + 174.99999999999997 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 190, + 189, + 202 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 191, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 191, + 203 + ], + "score": 1.0, + "content": "Similarly we obtain:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 210, + 437, + 238 + ], + "lines": [ + { + "bbox": [ + 170, + 210, + 437, + 238 + ], + "spans": [ + { + "bbox": [ + 170, + 210, + 437, + 238 + ], + "score": 0.86, + "content": "\\left| \\mathbb { E } _ { x \\sim \\mathbb { P } } \\left[ { \\frac { \\mathbb { Q } ( x ) - \\mathbb { P } ( x ) } { \\mu ( x ) } } \\right] \\right| \\leq 2 \\qquad \\underbrace { { \\mathbb { S } } ( \\mathbb { P } , \\mu ) } _ { \\in { \\mathcal { S } } } \\qquad \\underbrace { { \\mathcal { S } } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } _ { \\in { \\mathcal { S } } }", + "type": "interline_equation", + "image_path": "42bbfe8f5bb6cfe693c611d66a286804d28581cfbe8ba2c7a8ae2257f5bfe235.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 170, + 210, + 437, + 238 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 260, + 300, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 300, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 194, + 273 + ], + "score": 1.0, + "content": "For instance consider", + "type": "text" + }, + { + "bbox": [ + 194, + 261, + 221, + 272 + ], + "score": 0.92, + "content": "\\mu = \\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 259, + 300, + 273 + ], + "score": 1.0, + "content": ", we have therefore:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 279, + 395, + 307 + ], + "lines": [ + { + "bbox": [ + 216, + 279, + 395, + 307 + ], + "spans": [ + { + "bbox": [ + 216, + 279, + 395, + 307 + ], + "score": 0.93, + "content": "{ \\frac { 1 } { 2 } } \\left| \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ { \\frac { \\mathbb { Q } ( x ) } { \\mathbb { P } ( x ) } } \\right] - 1 \\right| \\leq \\mathbb { S } ( \\mathbb { Q } , \\mathbb { P } ) S _ { \\mathbb { P } } ( \\mathbb { P } , \\mathbb { Q } ) .", + "type": "interline_equation", + "image_path": "51ead85e6389951586621f04237342e0df5fc9c07e81281e20e78e0ac065053c.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 216, + 279, + 395, + 307 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 421, + 332 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 420, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 420, + 333 + ], + "score": 1.0, + "content": "Note that the left hand side of the inequality is not the total variation distance.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 504, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 197, + 355 + ], + "score": 1.0, + "content": "Hence for a sequence", + "type": "text" + }, + { + "bbox": [ + 197, + 340, + 212, + 351 + ], + "score": 0.9, + "content": "\\mathbb { Q } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 334, + 331, + 355 + ], + "score": 1.0, + "content": "if the Sobolev distance goes", + "type": "text" + }, + { + "bbox": [ + 331, + 339, + 398, + 352 + ], + "score": 0.93, + "content": "S _ { \\mathbb { P } } ( \\mathbb { P } , \\mathbb { Q } _ { n } ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 334, + 440, + 355 + ], + "score": 1.0, + "content": ", the ratio", + "type": "text" + }, + { + "bbox": [ + 440, + 337, + 504, + 354 + ], + "score": 0.92, + "content": "\\begin{array} { r } { r _ { n } ( x ) = \\frac { \\mathbb Q _ { n } ( x ) } { \\mathbb P ( x ) } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 247, + 365 + ], + "score": 1.0, + "content": "converges in expectation (w.r.t to", + "type": "text" + }, + { + "bbox": [ + 247, + 353, + 257, + 364 + ], + "score": 0.78, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 352, + 506, + 365 + ], + "score": 1.0, + "content": ") to 1. 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We see", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 632, + 425, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 425, + 644 + ], + "score": 1.0, + "content": "that annealed smoothing outperforms the non annealed smoothing experiments.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "Sobolev GAN versus WGAN-GP with RNN. We fix the generator architecture to Resnets. The", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "experiments of using RNN (GRU) as the critic architecture for WGAN-GP and Sobolev is shown", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 686, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 217, + 701 + ], + "score": 1.0, + "content": "in Figure 7 where we used", + "type": "text" + }, + { + "bbox": [ + 218, + 689, + 259, + 699 + ], + "score": 0.89, + "content": "\\mu = \\mu _ { G P }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 686, + 505, + 701 + ], + "score": 1.0, + "content": "for both cases. We only apply gradient clipping to stabilize", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "the performance without other tricks. We can observe that using RNN degrades the performance.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "We think that this is due to an optimization issue and a difficulty in training RNN under the GAN", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 720, + 310, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 310, + 734 + ], + "score": 1.0, + "content": "objective without any pre-training or conditioning.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + } + ], + "page_idx": 24, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 419, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 495, + 421, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 495, + 421, + 505, + 431 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "interline_equation", + "bbox": [ + 210, + 94, + 402, + 122 + ], + "lines": [ + { + "bbox": [ + 210, + 94, + 402, + 122 + ], + "spans": [ + { + "bbox": [ + 210, + 94, + 402, + 122 + ], + "score": 0.91, + "content": "\\frac { 1 } { 2 S _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } \\left| \\mathbb { E } _ { \\boldsymbol { x } \\sim \\mathbb { Q } } \\left[ \\frac { \\mathbb { Q } ( \\boldsymbol { x } ) - 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\\mathbb { P } ( x ) } { \\mu ( x ) } } \\right] \\right| \\leq 2 \\qquad \\underbrace { { \\mathbb { S } } ( \\mathbb { P } , \\mu ) } _ { \\in { \\mathcal { S } } } \\qquad \\underbrace { { \\mathcal { S } } _ { \\mu } ( \\mathbb { P } , \\mathbb { Q } ) } _ { \\in { \\mathcal { S } } }", + "type": "interline_equation", + "image_path": "42bbfe8f5bb6cfe693c611d66a286804d28581cfbe8ba2c7a8ae2257f5bfe235.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 170, + 210, + 437, + 238 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 260, + 300, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 300, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 194, + 273 + ], + "score": 1.0, + "content": "For instance consider", + "type": "text" + }, + { + "bbox": [ + 194, + 261, + 221, + 272 + ], + "score": 0.92, + "content": "\\mu = \\mathbb { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 259, + 300, + 273 + ], + "score": 1.0, + "content": ", we have therefore:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 259, + 300, + 273 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 216, + 279, + 395, + 307 + ], + "lines": [ + { + "bbox": [ + 216, + 279, + 395, + 307 + ], + "spans": [ + { + "bbox": [ + 216, + 279, + 395, + 307 + ], + "score": 0.93, + "content": "{ \\frac { 1 } { 2 } } \\left| \\mathbb { E } _ { x \\sim \\mathbb { Q } } \\left[ { \\frac { \\mathbb { Q } ( x ) } { \\mathbb { P } ( x ) } } \\right] - 1 \\right| \\leq \\mathbb { S } ( \\mathbb { Q } , \\mathbb { P } ) S _ { \\mathbb { P } } ( \\mathbb { P } , \\mathbb { Q } ) .", + "type": "interline_equation", + "image_path": "51ead85e6389951586621f04237342e0df5fc9c07e81281e20e78e0ac065053c.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 216, + 279, + 395, + 307 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 320, + 421, + 332 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 420, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 420, + 333 + ], + "score": 1.0, + "content": "Note that the left hand side of the inequality is not the total variation distance.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 320, + 420, + 333 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 504, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 197, + 355 + ], + "score": 1.0, + "content": "Hence for a sequence", + "type": "text" + }, + { + "bbox": [ + 197, + 340, + 212, + 351 + ], + "score": 0.9, + "content": "\\mathbb { Q } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 334, + 331, + 355 + ], + "score": 1.0, + "content": "if the Sobolev distance goes", + "type": "text" + }, + { + "bbox": [ + 331, + 339, + 398, + 352 + ], + "score": 0.93, + "content": "S _ { \\mathbb { P } } ( \\mathbb { P } , \\mathbb { Q } _ { n } ) 0", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 334, + 440, + 355 + ], + "score": 1.0, + "content": ", the ratio", + "type": "text" + }, + { + "bbox": [ + 440, + 337, + 504, + 354 + ], + "score": 0.92, + "content": "\\begin{array} { r } { r _ { n } ( x ) = \\frac { \\mathbb Q _ { n } ( x ) } { \\mathbb P ( x ) } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 247, + 365 + ], + "score": 1.0, + "content": "converges in expectation (w.r.t to", + "type": "text" + }, + { + "bbox": [ + 247, + 353, + 257, + 364 + ], + "score": 0.78, + "content": "\\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 352, + 506, + 365 + ], + "score": 1.0, + "content": ") to 1. 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We only apply gradient clipping to stabilize", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "the performance without other tricks. 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We selected", + "type": "text" + }, + { + "bbox": [ + 457, + 514, + 504, + 525 + ], + "score": 0.88, + "content": "\\lambda _ { C E } = 1 . 5", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "from [0.8, 1.5, 3.0, 5.0]. We train all models for 350 epochs. 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For formulation 1", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 198, + 560 + ], + "score": 1.0, + "content": "(Fisher only) we have", + "type": "text" + }, + { + "bbox": [ + 199, + 547, + 249, + 558 + ], + "score": 0.9, + "content": "\\rho _ { F } = 1 \\mathrm { e } { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 546, + 370, + 560 + ], + "score": 1.0, + "content": ", modified critic learning rate", + "type": "text" + }, + { + "bbox": [ + 370, + 547, + 421, + 558 + ], + "score": 0.9, + "content": "\\eta _ { D } = 1 \\mathrm { e } { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 546, + 469, + 560 + ], + "score": 1.0, + "content": ", critic iters", + "type": "text" + }, + { + "bbox": [ + 469, + 547, + 501, + 558 + ], + "score": 0.9, + "content": "n _ { c } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 546, + 506, + 560 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 104, + 556, + 217, + 570 + ], + "score": 1.0, + "content": "For formulation 2 (Sobolev", + "type": "text" + }, + { + "bbox": [ + 218, + 559, + 226, + 567 + ], + "score": 0.77, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 556, + 292, + 570 + ], + "score": 1.0, + "content": "Fisher) we have", + "type": "text" + }, + { + "bbox": [ + 293, + 558, + 341, + 569 + ], + "score": 0.85, + "content": "\\rho _ { F } = 5 \\mathrm { e } - 8", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 556, + 345, + 570 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 345, + 558, + 393, + 569 + ], + "score": 0.85, + "content": "\\rho _ { S } = 2 \\mathrm { e } - 8", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 556, + 439, + 570 + ], + "score": 1.0, + "content": ", critic iters", + "type": "text" + }, + { + "bbox": [ + 439, + 558, + 469, + 568 + ], + "score": 0.91, + "content": "n _ { c } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 556, + 506, + 570 + ], + "score": 1.0, + "content": ". 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△(f;P,Q)FFunction classdg(P,Q)Closed Form
-Divergence(Goodfellow et al., 2014)(Nowozin et al., 2016)Ex~Pf(x)-Ex~Q*(f(x))*Fenchel Conjugate{f:x→R,f∈domp*}E~[(]
Wasserstein -1(Arjovsky et al., 2017)(Gulrajani et al., 2017)Ex~Pf(x)-Ex~qf(x){f : x→R,fli ≤1}infπ∈(P,Q) Sx |lx-ylli dπ(x,y)Sinkhorn Divergence(Genevay et al., 2017)
MMD(Li et al., 2017)(Li et al., 2015)(Dziugaite et al., 2015)Ex~Pf(x)-Ex~qf(x){f:x→R,/fl/x ≤1}Ex~Pkx -Ex~Qr llyex
SteinDiscrepancy(Wang & Liu, 2016)Ex~Q[T(P)f(x)]T(P)=(Vxlog(P(x))T+Vx.{f:x→Rdf smooth with zeroboundary conditionNA in generalhas a closed formin RKHS
Cramerfor d =1(Bellemare et al., 2017a)Ex~Pf(x)-Ex~Qf(x){f:x→R,Ea~P(）²≤1,f smooth with zeroboundary conditionEx~P（Fp(2）-F0（））²P(x）xER
μ-FisherIPM(Mroueh & Sercu,2017)Ex~Pf(x)-Ex~Qf(x){f:X→R,f∈L(x,μ),Ex~μf²(x)≤1}Ex~u（P()-Q(x））²2μ(x）
μ-SobolevIPM(This work)Ex~Pf(x)-Ex~qf(x){f:X→R,f eW²(x,μ),Ex~μ|/Vxf(x)²≤1,with zero boundary condition }1VE~u∑i-(（中i(P）-Φi(Q）μ（x）where 𝜙(P) =Px-i(x-i)FP[xx-i-ij(mi)x−=(x1,..xi-1,Ti+1,..d)
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Input: ρ penalty weight, n Learning rate, nc number of iterations for training the critic, N batch size
Initialize p, 0,λ=0
repeat
for j = 1 to nc do
Samplea minibatch xi,i=1...N,xi~ Pr
Sample a minibatch zi,i = 1...N,zi ~ Pz
(gp,gx)←(VpLs,VxLs)(p,0,λ)
p ←p+η ADAM(p,9p)
λ ← λ- pgx {SGD rule on 入with learning rate ρ} end for
Sample zi,i= 1...N, zi ~ Pz
d←Vθ8(fp,g0）=-VθN∑=1fp(g0(zi）） N
0←0-nADAM(0,dθ)
until θ converges
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Number of labeled examples Model100020004000 Misclassification rate8000
CatGAN (Springenberg, 2015) FM (Salimans et al., 2016) ALI (Dumoulin et al., 2017)21.83 ± 2.0119.61 ± 2.0919.58 18.63 ± 2.3217.72 ± 1.82
Tangents Reg (Kumar et al., 2017)19.98 ± 0.3 20.06 ± 0.519.09 ± 0.1517.99 ± 0.54 16.78 ± 0.617.05 ± 0.50
II-model (Laine & Aila, 2016) *16.55 ± 0.29
VAT (Miyato et al., 2017)14.87
Bad Gan (Dai et al., 2017) *14.41 ± 0.30
VAT+EntMin+Large (Miyato et al., 2017) *13.15
Sajadi (Sajjadi et al., 2016) *11.29
WGAN-GP (Gulrajani et al., 2017) t44.85 ± 0.2837.62 ± 0.5632.66 ± 0.4830.38± 0.22
Fisher,layer norm (Mroueh & Sercu,2017)19.74± 0.2117.87 ± 0.3816.13 ± 0.5314.81 ± 0.16
Fisher, no norm (Mroueh & Sercu, 2017)21.49 ± 0.1819.20 ± 0.4617.30 ± 0.3015.57 ± 0.33
Sobolev + Fisher, no norm (This Work)20.14 ± 0.2117.38 ± 0.1015.77 ± 0.1914.20± 0.08
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sha256:b862a438e5486844a9c47b3f92e810b4a7fc05184898f2fcc276b0513d2e2530 +size 49732 diff --git a/parse/train/SJl98sR5tX/SJl98sR5tX.md b/parse/train/SJl98sR5tX/SJl98sR5tX.md new file mode 100644 index 0000000000000000000000000000000000000000..e669ccff55847ab81aa98863e8fb54e9eab73e21 --- /dev/null +++ b/parse/train/SJl98sR5tX/SJl98sR5tX.md @@ -0,0 +1,423 @@ +# INTERACTIVE AGENT MODELING BY LEARNING TOPROBE + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +The ability of modeling the other agents, such as understanding their intentions and skills, is essential to an agent’s interactions with other agents. Conventional agent modeling relies on passive observation from demonstrations. In this work, we propose an interactive agent modeling scheme enabled by encouraging an agent to learn to probe. In particular, the probing agent (i.e., a learner) learns to interact with the environment and with a target agent (i.e., a demonstrator) to maximize the change in the observed behaviors of that agent. Through probing, rich behaviors can be observed and are used for enhancing the agent modeling to learn a more accurate mind model of the target agent. Our framework consists of two learning processes: i) imitation learning for an approximated agent model and ii) pure curiosity-driven reinforcement learning for an efficient probing policy to discover new behaviors that otherwise can not be observed. We have validated our approach in four different tasks. The experimental results suggest that the agent model learned by our approach i) generalizes better in novel scenarios than the ones learned by passive observation, random probing, and other curiositydriven approaches do, and ii) can be used for enhancing performance in multiple applications including distilling optimal planning to a policy net, collaboration, and competition. A video demo is available at https://www.dropbox.com/ s/8mz6rd3349tso67/Probing_Demo.mov?dl $= 0$ . + +# 1 INTRODUCTION + +An accurate understanding of other agents is essential to many multi-agent problems, such as collaboration, competition, and learning from an expert agent. Humans achieve this not only by passively observing others’ behaviors, but also by actively probing others including interacting with them or changing the environment and conditions so that they can understand others’ intentions, skills, and capabilities better. For instance, when working with a colleague for the first time, one may intentionally create diverse situations where the true intention and skill set of that colleague can be clearly revealed, which in turn helps improve the collaboration. + +Inspired by this observation, in this work, we try to enable a probing agent (i.e., a learner) to automatically learn a good policy for probing in a way that helps it discover new behaviors of a target agent (i.e., a demonstrator) and thus learn a better model of the target agent that is generalizable to unseen environments or settings. Different from common task-oriented policy training, the learning of our probing policy is purely driven by the motivation of maximizing the knowledge about the target agent’s model. We show a simple case in Figure 1 to illustrate this idea, where the learner and the demonstrator are initially located in the upper part and the lower part of the room respectively. The true policy of the demonstrator is trying to go to the upper part by finding the shortest path. However, since the room layout is fixed, the learner may overfit the only path observed from the demonstrator. By actively creating new gaps, the learner is able to discover various paths, which will greatly improve the accuracy of the approximated model of the demonstrator. + +We consider the following setting for probing-based interactive agent modeling. In an environment, there are two general types of agents: i) a demonstrator who possesses certain skills for a single task or multiple tasks, and ii) a learner who has no prior knowledge of the environment and the demonstrator’s skills. The purpose of the learner is to efficiently and thoroughly learn all of the demonstrator’s skills and goals by not only passively watching the demonstrations but also actively interacting with the environment and/or the demonstrator. This learning process entails both imitation learning (IL) for modeling the demonstrator’s skills and goals, and reinforcement learning (RL) + +![](images/3c1afb04daa88192307507702776c69b03f1854302bfa148df1e924a35747df7.jpg) +Figure 1: Illustration of our probing-based interactive agent modeling. Here, the demonstrator tries to go from the bottom-right corner to the upper part of the room. The passive learner (left) only observes one path in the fixed environment while the probing learner (right) removes a wall block to create a new gap so that the demonstrator will change its path accordingly. + +for optimizing probing policy to diversify the task settings and the demonstrations to facilitate the imitation learning. Note that we assume that the demonstrator will always truthfully reveal its skills and intentions in any scenarios. + +A key idea in our approach is to use task independent RL training purely driven by a curiosity reward. For this, we represent the demonstrator’s mind by i) a latent vector to encode and track an agent’s intention and belief, and ii) a policy conditioned on the latent vector and the agent’s observed state for action prediction (i.e., the agent’s skills). By introducing this latent vector, we are able to characterize an agent’s policy by a low dimensional representation, and to reflect the change of policy by the change of this latent vector. Since the goal of probing policy is to cause the demonstrator to change its policy so that the learner may observe diverse demonstrations, it is natural to apply the change in the latent mind representation as the curiosity-driven incentive for the learner. + +We evaluate our approach on four tasks in different domains (grid worlds and algorithmic problems). The experimental results indicate that our probing-based interactive agent modeling framework can: i) efficiently model the demonstrator’s mind that is generalizable in unseen scenarios, and ii) can be applied to several applications including distilling optimal plans to a policy net by automatically diversifying task settings, and improving multi-agent collaboration as well as competition using the learned agent model. + +# 2 RELATED WORK + +In multi-agent reinforcement learning (MARL), agent modeling or opponent modeling plays an essential role as the ability of understanding other agent’s goals and predicting their actions can greatly facilitate both collaborative and competitive purposes (Busoniu et al., 2008; Albrecht & Stone, 2018). Previous work has attempted to achieve this by task-oriented learning for maximizing a specified collaborative or competitive reward in the given tasks. For instance, inspired by game theory, there have been approaches aiming at finding Nash equilibira in multi-agent games, where agents’ models are represented by their utilities (Littman, 1994; Hu & Wellman, 2003) and strategies (Claus & Boutilier, 1998; Tesauro, 2004; Powers & Shoham, 2005; Heinrich et al., 2015; Heinrich & Silver, 2016; Lanctot et al., 2017). Recently, some deep RL methods have incorporated simple agent modeling into Q-learning (Lowe et al., 2017) or policy updates (Foerster et al., 2018). Auxiliary tasks like explicitly predicting other agents’ goals have also been applied to MARL (Mordatch & Abbeel, 2018). In previous work, the agents’ incentive of modeling other agents comes from reaching a common goal or conflicting goals. In contrast, we never define a task-specific reward for the learner since the goal of our probing-based interactive agent modeling is not to reach a predefined goal but rather to learn a good mind model of the demonstrator which can be generalized to unseen settings and transferred to multi-agent tasks afterwards when a task-dependent reward is given. + +Our work is greatly inspired by Theory of Mind (ToM) (Premack & Woodruff, 1978), which is a general and powerful framework to model an agent’s mind and use the mind model to better explain or predict the agent’s behaviors. Baker et al. (2009) has proposed a Bayesian formulation to incorporate an agent’s desires, intentions, and belief about the world into the agent’s policy in order to predict the agent’s goals and actions via inverse planning. Rabinowitz et al. (2018) adopts a simpler mind representation (i.e., a latent vector) learned by a neural net (ToMnet). In our work, the learner is also trying to learn the policy of the demonstrator with a simple mind modeling. However, instead of only serving as a passive observer, we encourage the learner to probe so that it will learn to interact with the environment and with the demonstrator to quickly and continuously discover new behaviors of the demonstrator, which in turn helps learning a better mind model. + +![](images/84aa49337f3c7c08fcf23511a43c0078a3fcfdc51c45b78d41146d298740de5e.jpg) +Figure 2: An overview of our model. Architecture details are in Appendix C. Note that the modules do not share weights, and the dashed line indicates that it is a feed forward only path (no back propagation through this path to update the mind model). + +Our task-independent reward is related to the curiosity-driven reward applied to an RL agent for encouraging exploration (Strehl & Littman, 2008; Bellemare et al., 2016; Tang et al., 2017; Pathak et al., 2017). Our probing framework differs from this in two ways: i) instead of exploring the world states, we encourage the learner to discover new behaviors of the demonstrator to learn a better model of its mind; ii) the curiosity-driven rewards in previous work only serve as auxiliary rewards for achieving specific goals, whereas in our case, the sole motivation of our learner agent is from curiosity, and we demonstrate that this type of pure curiosity-driven learning can actually yield rich behaviors and general agent modeling. + +There is certain similarity between active learning and our learning to prob mechanism. As Yang & Shafto (2017) shows, active learning is more effective than optimal teaching when the learner and teacher are not conceptually aligned, which is exactly the scenario in our problem setting (the learner does do not share any conceptual common ground with the demonstrator at the beginning). However, active learning typically addresses problems such as classification (Tong & Koller, 2001; Kapoor et al., 2007) by generating queries to an oracle to get additional ground-truth supervision. There are have been work on active imitation learning (Shon et al., 2007; nd Geoffrey J. Gordon & Bagnell, 2011; Judah et al., 2012) and active inverse reinforcement learning (Lopes et al., 2009) utilizing the similar concept, where the learner asks quires at certain states to a human oracle for guidance on what actions to take at those states. In contrast, our work goes beyond the scope of the existing work on active learning – we aim at training a learner agent to directly interact with the environment and with the target agent in order to automatically diversify the task settings and learn a better agent model without any task-dependent training objectives so that the learned agent models can be applied to improve the learner’s performance in various applications. + +Lastly, our task-independent learning objective can also be connected with meta-learning (Wang et al., 2016; Finn et al., 2017), which is to learn a meta strategy that can conduct efficient multi-task learning (Maclaurin et al., 2015; Duan et al., 2017; Hariharan & Girshick, 2017; Wichrowska et al., 2017; Yu et al., 2018; Baker et al., 2017) or adapt an agent’s policy to its opponent’s policy (Al-Shedivat et al., 2018) in a competitive setting. In this work, the purpose of our task-independent learning is to learn to probe a demonstrator for a better modeling of its mind, which is different from existing meta-learning approaches. + +# 3 APPROACH + +# 3.1 MODEL + +We assume a Markov Decision Process (MDP) framework for both the demonstrator and the learner, where their behaviors at time $t$ are denoted by a pair of state and action $( s _ { d } ^ { t } , a _ { d } ^ { t } )$ and $( s _ { l } ^ { t } , a _ { l } ^ { t } )$ respectively. The history of their behaviors upon time $t$ is represented by trajectories $\Gamma _ { d } ^ { \acute { t } } = \big \{ ( s _ { d } ^ { \tau } , a _ { d } ^ { \tau } ) : \tau = 1 , \cdot \cdot \cdot , t \big \}$ and $\Gamma _ { l } ^ { t } = \{ ( s _ { l } ^ { \tau } , a _ { l } ^ { \tau } ) : \tau = 1 , \cdot \cdot \cdot , t \}$ respectively. + +Our interactive agent modeling framework is illustrated in Figure 2, which consists of two parts: i) learner’s estimation of the demonstrator’s model and ii) the learner’s probing policy for a better understanding of the demonstrator’s model. + +To estimate the demonstrator’s model, the learner maintains a behavior tracker, $\mathcal { M } ( \cdot )$ , to encode the observed trajectory of the demonstrator, which generates a latent vector, $m ^ { t } = \mathcal { M } ( \Gamma _ { d } ^ { t } )$ . This latent vector can be viewed as a simplified representation of the demonstrator’s mind upon time $t$ , hence the learner may use it to characterize the demonstrator’s policy, $\pi _ { d } ( a _ { d } ^ { t } \vert s _ { d } ^ { t } , m ^ { t - 1 } )$ , from which the learner may predict the demonstrator’s future action $\hat { a } _ { d } ^ { t }$ . Note that for each demonstration, $m ^ { t }$ always starts from the same constant, ${ \bf m } ^ { 0 } = { \bf 0 }$ . This particular definition of the demonstrator’s policy may also be connected with the option framework in hierarchical RL (Sutton et al., 1999), where the behavior tracker serves as a global policy to update the temporal abstraction $m ^ { t }$ and consequently changes the local policy $\pi _ { d }$ . + +In this work, we require a learner to interact with the environment and/or with the demonstrator instead of passively watching the demonstrations. We enable this by learning a probing policy for the learner, $\pi _ { l } ( a _ { l } ^ { t } | \dot { s _ { l } ^ { t } } , m ^ { t - 1 } )$ , where $m ^ { t - 1 }$ is from the current demonstration. The main purpose of the probing policy is to incite new behaviors of the demonstrator, thus we adopt a curiosity-driven reward to train this policy. Particularly, we define the reward function as + +$$ +\begin{array} { r } { r ^ { t } = R ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } , m ^ { t } ) = | | m ^ { t } - m ^ { t - 1 } | | ^ { 2 } , } \end{array} +$$ + +where $m ^ { t }$ is the successive output of the behavior tracker after observing $( s _ { d } ^ { t } , a _ { d } ^ { t } )$ . + +Finally, based on the probing policy, the learner can perform the probing as the rollout procedure outlined in Algorithm 1 (see Appendix A). + +In summary, there are four key components in our probing-based interactive agent modeling: + +• A behavior tracker $\mathcal { M } ( \Gamma _ { d } ^ { t } ; \theta _ { M } )$ ; +• The approximated demonstrator’s policy $\pi _ { d } ( a _ { d } ^ { t } \vert s _ { d } ^ { t } , m ^ { t - 1 } ; \theta _ { d } )$ ; +• A probing policy for the learner $\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , m ^ { t - 1 } ; \theta _ { l } )$ ; +• A value function for the probing policy $V ( s _ { l } ^ { t } , m ^ { t - 1 } ; \theta _ { V } )$ . + +Please refer to Appendix C for the details of the network architecture. + +# 3.2 LEARNING + +As discussed above, we have two main learning objectives corresponding to the two parts in our model respectively: i) minimizing imitation error (i.e., cross-entropy loss for action prediction) and ii) maximizing accumulated probing reward (i.e., probing policy optimization). Consequently, our approach includes an imitation learning process for recovering demonstrator’s policy and a reinforcement learning process for optimizing the probing policy. These two processes are intertwined and influenced by each other: the IL process provides the behavior tracker guiding the probing policy while the RL process helps $\mathrm { I L }$ to observe more diverse behaviors from the demonstrator, thus enabling an interactive learning scheme. Algorithm 2 in Appendix A summarizes the overall learning approach, where $N$ is the total number of training iterations. The optimization details for the two learning processes are introduced as follows. + +# 3.2.1 IMITATION LEARNING + +For $\mathrm { I L }$ , we want to learn a good behavior tracker as well as the demonstrator’s policy. For this, we minimize a cross-entropy loss for predicting demonstrator’s actions: + +$$ +\mathcal { L } ( \theta _ { M } , \theta _ { d } ) = \mathbb { E } \left[ - \log \pi _ { d } ( a _ { d } ^ { t } | s _ { d } ^ { t } , m ^ { t - 1 } ) \right] . +$$ + +# 3.2.2 REINFORCEMENT LEARNING + +The goal of RL is to train a good probing policy that will maximize the change of behavior and/or discover new behaviors of the demonstrator to facilitate the imitation learning. Based on the reward $\begin{array} { r } { \mathbb { E } \left[ \sum _ { \tau = 0 } ^ { \infty } \gamma ^ { \tau } r ^ { t + \tau } \right] } \end{array}$ n Eq. (, where $\gamma$ this goal is equivalent to maximizing the accumulated reward, is the discounted factor. $J ( \theta _ { l } ) =$ + +For the policy optimization, we use Advantage Actor-Critic (A2C) (Mnih et al., 2016) to conduct on-policy training. The policy gradient is + +$$ +\nabla _ { \theta _ { l } } J ( \theta _ { l } ) = \nabla _ { \theta _ { l } } \left[ \log \pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , m ^ { t - 1 } ; \theta _ { l } ) A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } ) + \lambda \mathcal { H } ( \pi _ { l } ( \cdot | s _ { l } ^ { t } , m ^ { t - 1 } ; \theta _ { l } ) ) \right] , +$$ + +where $A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } )$ is the advantage estimation defined as $\begin{array} { r } { A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } ) = \sum _ { \tau = 0 } ^ { \infty } \gamma ^ { \tau } r ^ { t + \tau } \ - } \end{array}$ $V ( s _ { l } ^ { t } , m ^ { t - 1 } )$ and $\mathcal { H } ( \cdot )$ is the entropy regularization weighted by the constant $\lambda = 0 . 0 1$ for encouraging exploration. The value function is updated by the following gradient: + +$$ +\nabla _ { \theta _ { V } } \frac { 1 } { 2 } \left( \sum _ { \tau = 0 } ^ { \infty } \gamma ^ { \tau } r ^ { t + \tau } - V ( s _ { l } ^ { t } , m ^ { t - 1 } ; \theta _ { V } ) \right) ^ { 2 } . +$$ + +Note that when we update the probing policy and the value function, the behavior tracker is fixed (i.e., no back propagation through the dashed path in Figure 2). Thus $\theta _ { M }$ will only be updated by the IL loss in Eq. (2). This is to ensure that the change of $\mathrm { \Sigma } _ { m } \bar { t }$ is only caused by the change in policy or in behaviors, and not by the change of the parameters of the mind model, $\theta _ { M }$ . + +# 4 EXPERIMENTS + +To evaluate our approach, we introduce four tasks as shown in Figure 3, including three grid world tasks (passing through obstacles, maze navigation, construction) and an algorithmic problem (sorting). + +In order to test the generalization ability of the learned agent model, we adopt a strict training procedure, where only one particular environment and task design is given during training. Specially, for grid world tasks, we fix the environment layout and/or item placement in each demonstration, whereas for the sorting task, we use the exact same input array throughout the training. At testing time, we randomize the task settings to an extent to create novel environments/inputs that have never been seen during training. We provide the specific settings in Section 4.1. + +We implement rule-based policies for the demonstrator i) by searching the best plan from the initial state to the goal state for the grid world tasks or ii) by the bubble sort algorithm for sorting. When there is no possible path to reach the goal (e.g., blocked by the learner), the demonstrator will stop until a viable path appears. + +# 4.1 TASKS + +![](images/3b9d1a1dd367e5002b7172940644e658adf68fc36a8b0e845a272a5d536c86d3.jpg) +Figure 3: Illustration of the evaluated tasks. + +Passing. In this task modified from Baker et al. (2009), the demonstrator is initially located at the bottom-right corner and is trying to pass through the closest gap to get into the upper part of the room. The demonstrator can take 5 actions including moving in four directions and stopping, whereas the learner can move in four directions, stop, and also pickup or put down a wall block. The training environment is shown in Figure 3, where the gap is always located at the left end of the wall in the middle. In testing cases, we randomly place the location of the gap and the initial position of the demonstrator. + +Maze Navigation. Inspired by similar tasks in recent literature (Andreas et al., 2017), we place a few door blocks and tools (a key and a hammer) in a four-room maze, where the key can be used to open the yellow door but has no effect on the blue door, which must be broken by the hammer. The demonstrator is trying to go from the top-right room to the top-left room. The two agents share the same action space including moving in four directions, picking up an item, and putting down an item. Also, they can only carry one item at a time. In the training setting, the initial positions of both agents and the door blocks are fixed as shown in Figure 3, whereas the tools may be randomly placed at only a few locations. The rules in this environment are in fact fairly complex compared to other grid world tasks in previous work, where multiple sub-goals such as getting the tools, getting the door blocks, placing the door blocks, and walking through the doors are involved. + +Construction. We adapt the stacking tasks in Shu et al. (2018) into a grid world, where the demonstrator has a latent goal invisible to the learner, which is to construct a new block by putting two blocks with a specific color combination together. Three items are present in a room and they are assigned with different colors randomly. In each episode, the demonstrator is randomly assigned with a goal (i.e., a pair of colors). It then seeks the needed blocks and puts one of them beside the other one. In order to predict the demonstrator’s actions precisely, the learner must infer the correct goal first, which requires a sophisticated and dynamic agent modeling. Both agents share the same action space as in Maze Navigation. In training, there are no obstacles in the room. To increase the difficulty of goal inference, in testing scenarios, we randomly place a few wall blocks as obstacles around the colored blocks. + +Sorting. Compared to a grid world, algorithmic problems are less visually informative and entail more abstract reasoning. For this, we design a sorting task where an array with certain length is given at the beginning. In our experiments, we set the length to be 10 and restrain the size of numbers in the array to be 4 bits (i.e., from 0 to 15). The demonstrator is able to perform a bubble sort algorithm to rearrange the input array in an ascending order. Its action at each step is to select a pair of numbers to swap. For every 5 steps done by the demonstrator, the learner can select a number and flipping one of the bit of that number. Both agents can choose to do nothing for a step. During training, we only provide one constant array so that the sorting always starts from the same initial array. This is a very challenging setting as only 10 out of 16 possible numbers are present in the training example and the fixed ordering may also easily cause overfitting. For testing, we generate random arrays as inputs. + +For more details about the task settings, please refer to Appendix D. + +# 4.2 GENERALIZATION IN UNSEEN TASK SETTINGS + +![](images/1871cb7eec22fea8ea3cee03fbce6bab4a89e5ce0e304bc11a26ace233854a63.jpg) +Figure 4: Action prediction accuracies in novel testing settings over numbers of training iterations. + +![](images/013eb5aa25e3450135a48626efaa1b683544a12479c08644269d9f5f88ec7693.jpg) +Figure 5: Action prediction accuracies in novel testing settings over numbers of training iterations with $10 \%$ random actions. + +One of the main goals of learning to probe is to learn a good agent model that can be generalized to unseen scenarios. To evaluate how accurate our agent model is for approximating the true mind of the demonstrator, we may test the accuracy of predicting the demonstrator’s actions using the learned $\pi _ { d }$ and behavior tracker $\mathcal { M } ( \cdot )$ in testing task settings unseen by the learner during training. A high + +prediction accuracy in unseen settings will indicate good generalization of the learned agent model. +To eliminate the effects from probing, we remove the learner from the environment during testing. + +We compare our model with six baselines: i) ToMnet in Rabinowitz et al. (2018), which learns the demonstrator’s model by only observing the given demonstrations without interactions, ii) our model without training the probing policy using RL (i.e., the leaner always takes random actions), iii), ours without the attention-based fusion (concatenating state feature and $m ^ { t - 1 }$ instead), iv) using two LSTMs for the estimated demonstrator’s policy and the probing policy respectively (Figure 16), v) using count-based bonus as reward (Strehl & Littman, 2008; Bellemare et al., 2016; Tang et al., 2017), and vi) using cross-entropy loss for action prediction as reward (i.e., exploration by self-supervised prediction in Pathak et al. (2017)). To ensure fair comparison, the training settings and the testing settings are shared by all methods. We provide more details of the baselines in Appendix E. + +Figure 4 shows the predication accuracy in testing settings of the three approaches based on the models from different training iterations. It is clear that with more iterations, our probing policy can greatly help increase the accuracy by discovering new behaviors, and consequently yields much higher testing accuracy than the baselines do. The results of “ours w/o fusion” and “2-LSTM” baselines further demonstrates the importance of our attention-based fusion layer and the use of a separate behavior tracker. By randomizing $10 \%$ of demonstrator’s actions (Figure 5), we show that the probing policy can also handle stochastic and sub-optimal policies. It can be clearly seen from the results that the performance of the two baselines based on different curiosity rewards is clearly inferior to ours, which demonstrates the advantage of defining the behavioral change as the intrinsic reward for the purpose of agent modeling. We have also evaluated the robustness of our approach by showing the standard deviation from multiple runs as shown in Figure 9, which demonstrates a reasonably low variance across multiple runs. + +We demonstrate the effect of dimensionality of the latent vector $m ^ { t }$ (i.e., the complexity of the agent model) in Figure 4. In simple environments, our approach still outperforms the baselines even when the dimensionality is decreased from 8 to 2 or 4. In more complex tasks like Sorting, a higher dimension is necessary for the agent modeling. + +# 4.3 EVOLUTION OF LEARNED PROBING STRATEGY + +As training progresses, we observe that our probing policy is able to progressively discover new behaviors through interactions that are adapted to the demonstrator’s policy. For instance, in Maze Navigation, we find that the learner first learns to place one door, then gradually learns to place two doors at the appropriate moments to force the demonstrator to go back and forth to get the needed tools for opening the doors. Finally, the probing policy will even blocks the demonstrator for a while before it goes through the last door. Due to the space limit, we show this in the demo video. + +We also provide more analysis and visualization of the probing behavior and the resulting latent vectors in Appendix B.2 and Appendix B.3. + +# 4.4 EMERGENCE OF OBSTRUCTIVE BEHAVIORS FROM PROBING + +Although we never explicitly set an adversarial goal for the learner, we do observe a natural emergence of obstructive behaviors caused by the probing, which can be quantitatively measured by the success rate of the demonstrator within a time limit as shown in Figure 6. This phenomenon is aligned with common sense that the optimal probing policy to discover new behaviors of the demonstrator should constantly force the demonstrator to change its plan, which will eventually delay the time when the demonstrator finishes the task. Because of the reward defined in Eq. (1), the probing policy learned from RL is also maximizing the accumulated behavioral change of the demonstrator just like the common sense. This further justifies our simple yet effective reward design. + +![](images/f79f849d4628693a408ffe9f369a47a5064b5907c7894cb647ef97d04aad2d23.jpg) +Figure 6: The average success rate of the demonstrator within the given time limit. + +![](images/22c56a09143e8dc063fbd0fed96feb71578073b1b84e64124f4c3a82addce74e.jpg) +Figure 7: The learning curves of the collaborative task (reward is rescaled). + +![](images/b2076ef0ebd4994b1be4f0abf9861f2baf22ffd6fcc9e875892dea55ad60f109.jpg) +Figure 8: The learning curves of the competitive task (reward is rescaled). + +4.5 APPLICATION 1: DISTILLING OPTIMAL PLANS TO A POLICY NET + +Table 1: Success rates using the learned demonstrator’s policy in unseen tasking settings. Unless specified, the evaluated policies were learned from demonstrations without random actions. + +
MethodPassingMaze NavigationConstructionSorting
Ours0.710.600.480.82
Ours (from sub-optimal demo)0.680.510.47N/A
Ours w/o RL0.130.310.230.80
Ours w/o fusion0.110.190.290.10
Ours (2-LSTM)0.170.020.170
Passive (ToMnet)0.1100.120
Count-based0.2200.310.39
Self-supervised0.2300.360.56
+ +Optimal planning sometimes requires a long computational time. For acceleration, it is common to distill the optimal plans to a policy net (Lazaric et al., 2010; Guo et al., 2014). However, the distilled policy net may not generalize well in new scenarios if the training settings are not diverse enough. Thus, the nature of our approach makes it suitable for improving the generalization without manually designing a large number of diverse settings. + +For this, we evaluate the success rates when the learner directly uses $\pi _ { d }$ (with an 8-dim latent vector) to perform tasks in testing settings without finetuning. The results summarized in Table 1 are consistent with the findings based on action predictions. We have also tested the success rate of the policy learned from sub-optimal demonstrations with $10 \%$ random actions. Its performance is comparable to the one learned from perfect demonstrations by our approach. It also outperforms the baselines trained from optimal demonstrations. We didn’t test the randomized demonstrations for Sorting as it is unnecessary to randomize bubble sort algorithm. Note that since the learner is unaware of the goal in Construction, we let the learner take over the task after the first block has been picked up. + +# 4.6 APPLICATION 2: COLLABORATION + +To test whether the improved agent modeling by learning a probing policy can facilitate multi-agent collaboration, we modify the Construction task to be a collaborative task, where the learner is trying to help the demonstrator (fixed policy) to finish the task. For every step the demonstrator takes, the learner will get a $- 0 . 0 5$ penalty. When the goal is reached, it will be given a reward of 1. This setting is difficult for training a collaborative policy since the demonstrator is capable of finishing the task by itself. In order to help finish the task faster, the learner must infer the true goal of the demonstrator quickly and shares part of the labor accordingly. + +We fix the behavior tracker module, $\mathcal { M } ( \cdot )$ , trained from our interactive agent modeling and retrain the learner’s policy $\pi _ { l }$ using the task reward defined above. For comparison, we implement two baselines: i) retraining $\pi _ { l }$ based on $\mathcal { M }$ learned from passive agent modeling (i.e., ToMnet) and ii) training a policy without agent modeling, i.e., $\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , s _ { d } ^ { t } )$ . + +Figure 7 demonstrates the learning curves, where the reward is rescaled so that the theoretical maximum reward from a perfect policy is 1. From the curves, we may see that the policy trained with our interactively learned agent model significantly outperforms both baselines. + +# 4.7 APPLICATION 3: COMPETITION + +Similar to Section 4.6, we design a competitive task based on the Construction task, where the learner gets a 0.05 reward for every step and a -1.0 penalty if the opponent achieves its goal. We adopt the same training procedure as in Section 4.6, and also rescale the reward according to the maximum reward. As Figure 8 shows, the mind model learned by our approach improves the learning efficiency and the converged reward by a large margin. + +# 5 CONCLUSIONS + +In this work, we have proposed a novel agent modeling approach, i.e., probing-based interactive agent modeling. The core idea is to learn a probing policy using only a curiosity-driven reward, which is able to discover new behaviors of the target agent. We achieve this by incorporating two learning processes $\mathrm { I L }$ and RL) together. 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In Robotics: Science and Systems (RSS), 2018. + +# A PSEUDO CODE OF OUR ALGORITHMS + +# Algorithm 1 Rollout $ { T _ { \mathrm { m a x } } } )$ + +Input: Maximum steps $T _ { \mathrm { m a x } }$ +Output: Episode length $T$ , trajectories $\Gamma _ { d } ^ { T }$ and $\Gamma _ { l } ^ { T }$ , and the latent vector sequence $M$ +1: Initialize the environment +2: $\Gamma _ { d } ^ { 0 } \gets \emptyset , \Gamma _ { l } ^ { 0 } \gets \emptyset , M \gets \emptyset , m ^ { 0 } \gets \mathbf { 0 } , t \gets 0$ +3: repeat +4: $t \gets t + 1$ +5: Observe $s _ { d } ^ { t }$ and $a _ { d } ^ { t }$ from the demonstrator +6: Observe $s _ { l } ^ { t }$ from the environment +7: Sample and execute the learner’s action $a _ { l } ^ { t } \sim \pi _ { l } ( s _ { l } ^ { t } , m ^ { t - 1 } )$ +8: $\Gamma _ { d } ^ { t } \Gamma _ { d } ^ { t - 1 } \cup \{ ( s _ { d } ^ { t } , a _ { d } ^ { t } ) \} , \Gamma _ { l } ^ { t } \Gamma _ { l } ^ { t - 1 } \cup \{ ( s _ { l } ^ { t } , a _ { l } ^ { t } ) \}$ +9: $m ^ { t } \gets \mathcal { M } ( \Gamma _ { d } ^ { t } ) , M \gets M \cup \{ m ^ { t } \}$ + +# Algorithm 2 Learning Algorithm + +1: Initialize parameters $\Theta = \left. \theta _ { M } , \theta _ { b } , \theta _ { l } , \theta _ { V } \right.$ +2: Set $T _ { \mathrm { m a x } }$ (the maximum steps in an episode) and $N$ (the number of training iterations) +3: $i \gets 1$ +4: repeat +5: $T , \Gamma _ { d } ^ { T } , \Gamma _ { l } ^ { T } , M \gets \mathrm { R o l l o u t } ( T _ { \mathrm { m a x } } )$ +6: IL: Update $\theta _ { M }$ and $\theta _ { d }$ based on Eq. (2) using $\Gamma _ { d } ^ { T }$ +7: RL: Update $\theta _ { l }$ and $\theta _ { V }$ based on Eq. (3 and Eq. (4) respectively using $\Gamma _ { l } ^ { T }$ , and $M$ +8: $i \gets i + 1$ +9: until $i = N$ + +# B MORE RESULTS + +# B.1 ROBUSTNESS EVALUATION + +We show the mean and standard deviation of the prediction accuracy from 5 runs by our full model in Figure 9 for Maze Navigation to validate the robustness of our approach. + +![](images/d3766e4783ca2f1226bf81c173ed6690f9a07ff438e980d834c86b081f958f61.jpg) +10: until $t = T _ { \operatorname* { m a x } }$ or the task is finished 11: $T \gets t$ +Figure 9: Mean and standard deviation of multiple runs in Maze Navigation. + +# B.2 VISUALIZATION OF LATENT VECTORS + +Figure 10 visualizes the latent vectors obtained from demonstrations with probing and without probing, where the latent vectors were computed by the same behavior tracker in both cases. This provides empirical evidences that by finding new latent vectors, we are able to discover new demonstrations with probing. + +# B.3 VISUALIZATION OF THE CHANGE IN $m ^ { t }$ AND THE CHANGE IN POLICY + +To show that the change in $m ^ { t }$ indeed indicates the change in policy, we compute the correlation of $| | m ^ { t } - m ^ { t - 1 } | | ^ { 2 }$ and $\tilde { K L } ( \pi _ { d } ( \cdot | s ^ { t + 1 } , m ^ { t } ) | | \pi _ { d } ( \cdot | s ^ { t + 1 } , m ^ { t - 1 } ) )$ (i.e., how different the policy conditioned on the new latent vector $m ^ { t }$ is compared to the one with the old latent vector $m ^ { t - 1 }$ ). Figure 11 demonstrates the correlation between the change in $m ^ { t }$ and the corresponding change in policy in testing settings. The high correlation validates our hypothesis that the distance between consecutive latent vectors $m ^ { t }$ and $\bar { m } ^ { t - 1 }$ reflects the policy change of the demonstrator. + +![](images/f82b7a583fb0ccf3c5a6ef501adc2d8413d68eb07d6414f87e572db80aaacff7.jpg) +Figure 10: t-SNE embedding of $m ^ { t }$ . + +![](images/069435882d499904bb6f9b71b6a7dc219cc94a8b05cbac2d28d7728bc6f34cb1.jpg) +Figure 11: Correlation between the change in $m ^ { t }$ and the change in policy in testing settings $\dot { \boldsymbol { r } }$ is Pearson correlation coefficient). + +# C NETWORK ARCHITECTURE OF OUR MODEL + +State Encoder. The input of the state encoder is a multi-channel tensor. For the grid world case, the input dimension is $1 1 \bar { \times } 1 1 \times ( N _ { \mathrm { b l o c k s } } + 1 )$ , where $1 1 \times 1 1$ is the size of the grid world, $N _ { \mathrm { i t e m s } }$ is the number of types of blocks, and the additional channel is to show the position of the corresponding agent, i.e., the position of the demonstrator for $s _ { d } ^ { t }$ or the position of the learner for $s _ { l } ^ { t }$ . The other agent is treated as an obstacle and its position is encoded into the channel corresponded to the wall block. In the case of Sorting task, the input dimension is $1 0 \times 1 \times 4$ , representing 10 numbers in an array where the size of each number is 4 bits. The state encoder has one convolutional layer which consists of 32 filters with kernel size of $1 \times 1$ and stride of 1. + +Behavior Tracker. Assuming the action space is $A$ , we combine the state input and the action input by augmenting the state input with $A$ channels, each of which corresponds to an action. We set the channel of the observed action to be all ones and set the remaining $A - 1$ channels to be zeros. This combined state and action input is then fed into a convolutional layer with 32 filters (the kernel size is $1 \times 1$ and the stride is 1). The output is flatten into a vector and passed through two fully connected (FC) layers (all have 128 dimensions). The resulting 128-dim vector serves as the input of an LSTM with 128 hidden units. Finally, an FC layer takes in the hidden state from the LSTM and outputs the latent vector $m ^ { t }$ as the mind representation. + +![](images/9601707c6b27d9000ede144af1654db03552ded43028f6a05face04cd1b1a543.jpg) +Figure 12: The attention-based fusion module. + +Fusion. As shown in Figure 12, we design our fusion module using an attention based mechanism similar to the one introduced by Chaplot et al. (2017), where the latent vector $m ^ { t - 1 }$ is fed into an FC layer outputting an $N$ -dim attention vector (each element is from 0 to 1) corresponding to the $N$ feature maps from the state encoder (here $N = 3 2$ ). Formally, we have an attention vector $h = \sigma ( m ^ { t - 1 } ) \in \dot { \mathbb { R } } ^ { 3 2 }$ , where $\sigma ( \cdot )$ is an FC layer with sigmoid activation. $h$ is spatially expanded to a $H \times W \times 3 2$ tensor, $\pmb { H } ( h ) \in \mathbb { R } ^ { H \times W \times 3 2 }$ , where the elements in $k$ -th channel correspond to the $k$ -th element in $h$ . We then reweight each feature maps using the attention vector, which becomes the fusion output. I.e., $f ( \phi ( s ^ { t } ) , m ^ { t - \tilde { 1 } } ) = \phi ( s ^ { t } ) \odot H ( \sigma \tilde { ( } m ^ { t - 1 } ) \mathbf { \tilde { ) } }$ , where $\phi ( s ^ { t } )$ are the feature maps from the state encoder, $f ( \cdot )$ is the fusion layer, and $\odot$ is element-wise product. + +Policy. The input of this module is the flattened output from the fusion module, and is fed to an LSTM with 128 hidden units followed by an FC layer with softmax activation. The resulting output is an action distribution representing the policy (either $\pi _ { d }$ or $\pi _ { l }$ ). For Sorting task, we slightly modify the output to fit the problem. We decompose the demonstrator’s policy as ${ \pi } _ { d } ( a _ { d } ^ { t } \vert s _ { d } ^ { t } , m ^ { t - 1 } ) ~ = ~ { \pi } _ { d } ^ { ( 1 ) } ( a _ { d } ^ { \bar { t } , 1 } \vert s _ { d } ^ { t } , m ^ { t - 1 } ) { \pi } _ { d } ^ { ( 2 ) } ( a _ { d } ^ { t , 2 } \vert s _ { d } ^ { t } , m ^ { t - 1 } )$ , where $a _ { d } ^ { t , 1 }$ and $a _ { d } ^ { t , 2 }$ are the indices of the numbers the demonstrator chooses to swap. For the learner’s policy, it is decomposed as $\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , m ^ { t - 1 } ) = \pi _ { l } ^ { \mathrm { i d } } ( a _ { l } ^ { t , 1 } | s _ { l } ^ { t } , m ^ { t - 1 } ) \pi _ { l } ^ { \mathrm { b i t } } ( a _ { l } ^ { t , 2 } | s _ { l } ^ { t } , m ^ { t - 1 } )$ instead, where $a _ { l } ^ { t , 1 }$ indicates the number that the learner selects to change and $a _ { l } ^ { t , 2 }$ is the bit of that number that needs to be flipped. When $a _ { d } ^ { t , 1 }$ or $a _ { l } ^ { t , 1 }$ is larger than the length of the array, it means that the demonstrator or the learner is choosing to do nothing respectively. + +Value. We also have a value net designed for training the learner’s policy using A2C (i.e., $V ( s _ { l } ^ { t } , m ^ { t - 1 } ) )$ , which takes in the hidden state from the LSTM in the learner’s policy module and outputs a scalar value after an FC layer. + +The network is trained with RMSProp (Tieleman & Hinto, 2012) using a learning rate of 0.001. +During training, $\epsilon$ -greedy is applied to the rollout, where the $\epsilon$ gradually decreases from 0.1 to 0.01. + +# D TASK SETTINGS + +We assume full observations of the world state for both agents in all tasks but the internal state of an agent (e.g., goals) is unobservable to another agent. The discounted factor is set to be $\Gamma = 0 . 9 5$ . + +For the demonstrator in the grid world tasks, we implemented search based path planning and used simple heuristics to perform branch and bound for acceleration. In particular, the state for the search algorithm in Passing is the map status, whereas the state in Maze Navigation and Construction is the combination of map status and the demonstrator’s inventory. + +# D.1 PASSING + +![](images/104fb1162f260d597db6e2b8b643e42e84d9eb60187fab5116d2e2aba9e01f85.jpg) +Figure 13: The training setting and examples of testing settings for Passing. + +Figure 13 shows the training setting where the locations of the gap and the staring point of the demonstrator are fixed, and the examples of testing settings where the placement of the gap and the initial position of the demonstrator is randomized. We terminate a training episode if the demonstrator has not passed the obstacle after 15 steps. + +# D.2 MAZE NAVIGATION + +The training setting in Maze Navigation is designed as shown in Figure 14a, where the placement of the tools is restrained in the purple region, and the positions of the demonstrator’s starting point and the doors are fixed. For testing, we randomly put one or two doors to fill the gaps; the demonstrator + +![](images/7e7f142e00277a702495ff544dfc962f5c601cf88c03a1f6483f2f64ce64d982.jpg) +Figure 14: The training setting and examples of testing settings for Maze Navigation. + +and the tools can be randomly placed in the purple region. The demonstrator is always guaranteed to be able to find a path from its starting point to the destination (the top-left room). A training episode has a time limit of 60 steps. + +# D.3 CONSTRUCTION + +![](images/16b7b3aa92dee0ccd1525db6154d8b34c50807af6f08c7694f6469a3309c6d55.jpg) +Figure 15: The training setting and examples of testing settings for Construction. + +The room layout in training setting is fixed and shown in Figure 15a. In testing settings, we randomly put six wall blocks around the three colored blocks to create obstacles. Figure 15b displays a few examples of testing scenarios. Note that in both training and testing, we allow randomized coloring as long as the goal can be achieved. + +For each episode, we assign a random goal (a pair of colors) for the demonstrator. The maximum episode length is 30 steps during training. + +# D.4 SORTING + +# Algorithm 3 Modified Bubble Sort + +Input: Initial array $X = [ x _ { 1 } , x _ { 2 } , \cdot \cdot \cdot , x _ { n } ]$ , where $_ n$ is the length. +Output: Sorted array +1: Last position $i \gets 0$ +2: Steps $t \gets 0$ +3: while $X$ is not in an ascending order do +4: $c \gets 0$ +5: while $c < n - 1$ do +6: if $x _ { i } > x _ { i + 1 }$ then +7: Swap $x _ { i }$ and $x _ { i + 1 }$ , i.e., the demonstrator’s $t$ -th action is $( i , i + 1 )$ +8: $t \gets \bar { t } + 1$ +9: break +10: end if +11: $\begin{array} { l } { c c + 1 } \\ { i ( i + 1 ) \% ( n - 1 ) } \end{array}$ +12: +13: end while +14: end while + +In training, there is only one sequence as the initial state, i.e., [2, 0, 5, 12, 14, 10, 3, 11, 9, 7]. The testing settings include 100 randomly generated initial sequences. Training episodes have a 30-step time limit. + +Since the learner may change certain numbers during the process of sorting, the original bubble sort may fail to finish the sorting successfully since it will not look back at the sorted part of the array. To address this, we modify the original bubble sort algorithm so that it will continue to sort the sequence until it is in an ascending order. Algorithm 3 outlines how the demonstrator swaps the numbers. + +# E DETAILS OF BASELINES + +# E.1 REWARD FUNCTIONS IN BASELINES + +We define the reward functions used for baselines, count-based reward and self-supervised prediction here. + +# Count-based reward: + +$$ +r ^ { t } = R ( s _ { d } ^ { t } ) = \frac { \beta } { \sqrt { N ( s _ { d } ^ { t } ) } } , +$$ + +where $\beta$ is a constant (we set $\beta = 1$ , which gives the best results in our experiments), and $N ( s _ { d } ^ { t } )$ is the counts of state visitation. In our experiments, the counting can be efficiently implemented by hashing. This reward encourages the learner to push the demonstrator to new states in order to incite new demonstrations. + +# Self-supervised prediction: + +$$ +\begin{array} { r } { r ^ { t } = R ( s _ { d } ^ { t } , m ^ { t - 1 } , a _ { d } ^ { t } ) = - \log \pi _ { d } ( a _ { d } ^ { t } | s _ { d } ^ { t } , m ^ { t - 1 } ) , } \end{array} +$$ + +where $a _ { d } ^ { t }$ is the ground-truth action from the demonstrator. This reward essentially measures action prediction loss of the estimated demonstrator’s policy, which is designed to encourage the learner to find new scenarios where the previously learned demonstrator’s policy becomes less accurate. + +E.2 NETWORK ARCHITECTURE OF THE 2-LSTM BASELINE + +![](images/4d94346a5d39928d4bc7f89cf3ad6b330ebd1c289bd68248bddb18577350c585.jpg) +Figure 16: The network architecture of the 2-LSTM baseline. + +Figure 16 illustrates the network architecture of the 2-LSTM baseline, where two LSTMs all have 128 hidden units. \ No newline at end of file diff --git a/parse/train/SJl98sR5tX/SJl98sR5tX_content_list.json b/parse/train/SJl98sR5tX/SJl98sR5tX_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..13ac3cc34c89485e6eb4cd9c5217fad5f88aa417 --- /dev/null +++ b/parse/train/SJl98sR5tX/SJl98sR5tX_content_list.json @@ -0,0 +1,2045 @@ +[ + { + "type": "text", + "text": "INTERACTIVE AGENT MODELING BY LEARNING TOPROBE", + "text_level": 1, + "bbox": [ + 176, + 99, + 823, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 196 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 233, + 544, + 248 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The ability of modeling the other agents, such as understanding their intentions and skills, is essential to an agent’s interactions with other agents. Conventional agent modeling relies on passive observation from demonstrations. In this work, we propose an interactive agent modeling scheme enabled by encouraging an agent to learn to probe. In particular, the probing agent (i.e., a learner) learns to interact with the environment and with a target agent (i.e., a demonstrator) to maximize the change in the observed behaviors of that agent. Through probing, rich behaviors can be observed and are used for enhancing the agent modeling to learn a more accurate mind model of the target agent. Our framework consists of two learning processes: i) imitation learning for an approximated agent model and ii) pure curiosity-driven reinforcement learning for an efficient probing policy to discover new behaviors that otherwise can not be observed. We have validated our approach in four different tasks. The experimental results suggest that the agent model learned by our approach i) generalizes better in novel scenarios than the ones learned by passive observation, random probing, and other curiositydriven approaches do, and ii) can be used for enhancing performance in multiple applications including distilling optimal planning to a policy net, collaboration, and competition. A video demo is available at https://www.dropbox.com/ s/8mz6rd3349tso67/Probing_Demo.mov?dl $= 0$ . ", + "bbox": [ + 233, + 263, + 766, + 517 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 544, + 336, + 560 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "An accurate understanding of other agents is essential to many multi-agent problems, such as collaboration, competition, and learning from an expert agent. Humans achieve this not only by passively observing others’ behaviors, but also by actively probing others including interacting with them or changing the environment and conditions so that they can understand others’ intentions, skills, and capabilities better. For instance, when working with a colleague for the first time, one may intentionally create diverse situations where the true intention and skill set of that colleague can be clearly revealed, which in turn helps improve the collaboration. ", + "bbox": [ + 174, + 575, + 825, + 669 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Inspired by this observation, in this work, we try to enable a probing agent (i.e., a learner) to automatically learn a good policy for probing in a way that helps it discover new behaviors of a target agent (i.e., a demonstrator) and thus learn a better model of the target agent that is generalizable to unseen environments or settings. Different from common task-oriented policy training, the learning of our probing policy is purely driven by the motivation of maximizing the knowledge about the target agent’s model. We show a simple case in Figure 1 to illustrate this idea, where the learner and the demonstrator are initially located in the upper part and the lower part of the room respectively. The true policy of the demonstrator is trying to go to the upper part by finding the shortest path. However, since the room layout is fixed, the learner may overfit the only path observed from the demonstrator. By actively creating new gaps, the learner is able to discover various paths, which will greatly improve the accuracy of the approximated model of the demonstrator. ", + "bbox": [ + 174, + 676, + 825, + 823 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We consider the following setting for probing-based interactive agent modeling. In an environment, there are two general types of agents: i) a demonstrator who possesses certain skills for a single task or multiple tasks, and ii) a learner who has no prior knowledge of the environment and the demonstrator’s skills. The purpose of the learner is to efficiently and thoroughly learn all of the demonstrator’s skills and goals by not only passively watching the demonstrations but also actively interacting with the environment and/or the demonstrator. This learning process entails both imitation learning (IL) for modeling the demonstrator’s skills and goals, and reinforcement learning (RL) ", + "bbox": [ + 174, + 829, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/3c1afb04daa88192307507702776c69b03f1854302bfa148df1e924a35747df7.jpg", + "image_caption": [ + "Figure 1: Illustration of our probing-based interactive agent modeling. Here, the demonstrator tries to go from the bottom-right corner to the upper part of the room. The passive learner (left) only observes one path in the fixed environment while the probing learner (right) removes a wall block to create a new gap so that the demonstrator will change its path accordingly. " + ], + "image_footnote": [], + "bbox": [ + 269, + 99, + 727, + 185 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "for optimizing probing policy to diversify the task settings and the demonstrations to facilitate the imitation learning. Note that we assume that the demonstrator will always truthfully reveal its skills and intentions in any scenarios. ", + "bbox": [ + 174, + 276, + 821, + 318 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A key idea in our approach is to use task independent RL training purely driven by a curiosity reward. For this, we represent the demonstrator’s mind by i) a latent vector to encode and track an agent’s intention and belief, and ii) a policy conditioned on the latent vector and the agent’s observed state for action prediction (i.e., the agent’s skills). By introducing this latent vector, we are able to characterize an agent’s policy by a low dimensional representation, and to reflect the change of policy by the change of this latent vector. Since the goal of probing policy is to cause the demonstrator to change its policy so that the learner may observe diverse demonstrations, it is natural to apply the change in the latent mind representation as the curiosity-driven incentive for the learner. ", + "bbox": [ + 174, + 324, + 825, + 433 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We evaluate our approach on four tasks in different domains (grid worlds and algorithmic problems). The experimental results indicate that our probing-based interactive agent modeling framework can: i) efficiently model the demonstrator’s mind that is generalizable in unseen scenarios, and ii) can be applied to several applications including distilling optimal plans to a policy net by automatically diversifying task settings, and improving multi-agent collaboration as well as competition using the learned agent model. ", + "bbox": [ + 174, + 439, + 825, + 518 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 539, + 344, + 554 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In multi-agent reinforcement learning (MARL), agent modeling or opponent modeling plays an essential role as the ability of understanding other agent’s goals and predicting their actions can greatly facilitate both collaborative and competitive purposes (Busoniu et al., 2008; Albrecht & Stone, 2018). Previous work has attempted to achieve this by task-oriented learning for maximizing a specified collaborative or competitive reward in the given tasks. For instance, inspired by game theory, there have been approaches aiming at finding Nash equilibira in multi-agent games, where agents’ models are represented by their utilities (Littman, 1994; Hu & Wellman, 2003) and strategies (Claus & Boutilier, 1998; Tesauro, 2004; Powers & Shoham, 2005; Heinrich et al., 2015; Heinrich & Silver, 2016; Lanctot et al., 2017). Recently, some deep RL methods have incorporated simple agent modeling into Q-learning (Lowe et al., 2017) or policy updates (Foerster et al., 2018). Auxiliary tasks like explicitly predicting other agents’ goals have also been applied to MARL (Mordatch & Abbeel, 2018). In previous work, the agents’ incentive of modeling other agents comes from reaching a common goal or conflicting goals. In contrast, we never define a task-specific reward for the learner since the goal of our probing-based interactive agent modeling is not to reach a predefined goal but rather to learn a good mind model of the demonstrator which can be generalized to unseen settings and transferred to multi-agent tasks afterwards when a task-dependent reward is given. ", + "bbox": [ + 174, + 569, + 825, + 784 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our work is greatly inspired by Theory of Mind (ToM) (Premack & Woodruff, 1978), which is a general and powerful framework to model an agent’s mind and use the mind model to better explain or predict the agent’s behaviors. Baker et al. (2009) has proposed a Bayesian formulation to incorporate an agent’s desires, intentions, and belief about the world into the agent’s policy in order to predict the agent’s goals and actions via inverse planning. Rabinowitz et al. (2018) adopts a simpler mind representation (i.e., a latent vector) learned by a neural net (ToMnet). In our work, the learner is also trying to learn the policy of the demonstrator with a simple mind modeling. However, instead of only serving as a passive observer, we encourage the learner to probe so that it will learn to interact with the environment and with the demonstrator to quickly and continuously discover new behaviors of the demonstrator, which in turn helps learning a better mind model. ", + "bbox": [ + 174, + 789, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/84aa49337f3c7c08fcf23511a43c0078a3fcfdc51c45b78d41146d298740de5e.jpg", + "image_caption": [ + "Figure 2: An overview of our model. Architecture details are in Appendix C. Note that the modules do not share weights, and the dashed line indicates that it is a feed forward only path (no back propagation through this path to update the mind model). " + ], + "image_footnote": [], + "bbox": [ + 336, + 101, + 660, + 306 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our task-independent reward is related to the curiosity-driven reward applied to an RL agent for encouraging exploration (Strehl & Littman, 2008; Bellemare et al., 2016; Tang et al., 2017; Pathak et al., 2017). Our probing framework differs from this in two ways: i) instead of exploring the world states, we encourage the learner to discover new behaviors of the demonstrator to learn a better model of its mind; ii) the curiosity-driven rewards in previous work only serve as auxiliary rewards for achieving specific goals, whereas in our case, the sole motivation of our learner agent is from curiosity, and we demonstrate that this type of pure curiosity-driven learning can actually yield rich behaviors and general agent modeling. ", + "bbox": [ + 174, + 385, + 825, + 492 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "There is certain similarity between active learning and our learning to prob mechanism. As Yang & Shafto (2017) shows, active learning is more effective than optimal teaching when the learner and teacher are not conceptually aligned, which is exactly the scenario in our problem setting (the learner does do not share any conceptual common ground with the demonstrator at the beginning). However, active learning typically addresses problems such as classification (Tong & Koller, 2001; Kapoor et al., 2007) by generating queries to an oracle to get additional ground-truth supervision. There are have been work on active imitation learning (Shon et al., 2007; nd Geoffrey J. Gordon & Bagnell, 2011; Judah et al., 2012) and active inverse reinforcement learning (Lopes et al., 2009) utilizing the similar concept, where the learner asks quires at certain states to a human oracle for guidance on what actions to take at those states. In contrast, our work goes beyond the scope of the existing work on active learning – we aim at training a learner agent to directly interact with the environment and with the target agent in order to automatically diversify the task settings and learn a better agent model without any task-dependent training objectives so that the learned agent models can be applied to improve the learner’s performance in various applications. ", + "bbox": [ + 174, + 500, + 825, + 686 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Lastly, our task-independent learning objective can also be connected with meta-learning (Wang et al., 2016; Finn et al., 2017), which is to learn a meta strategy that can conduct efficient multi-task learning (Maclaurin et al., 2015; Duan et al., 2017; Hariharan & Girshick, 2017; Wichrowska et al., 2017; Yu et al., 2018; Baker et al., 2017) or adapt an agent’s policy to its opponent’s policy (Al-Shedivat et al., 2018) in a competitive setting. In this work, the purpose of our task-independent learning is to learn to probe a demonstrator for a better modeling of its mind, which is different from existing meta-learning approaches. ", + "bbox": [ + 174, + 693, + 825, + 786 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 APPROACH", + "text_level": 1, + "bbox": [ + 176, + 808, + 297, + 824 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 MODEL ", + "text_level": 1, + "bbox": [ + 174, + 840, + 267, + 854 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We assume a Markov Decision Process (MDP) framework for both the demonstrator and the learner, where their behaviors at time $t$ are denoted by a pair of state and action $( s _ { d } ^ { t } , a _ { d } ^ { t } )$ and $( s _ { l } ^ { t } , a _ { l } ^ { t } )$ respectively. The history of their behaviors upon time $t$ is represented by trajectories $\\Gamma _ { d } ^ { \\acute { t } } = \\big \\{ ( s _ { d } ^ { \\tau } , a _ { d } ^ { \\tau } ) : \\tau = 1 , \\cdot \\cdot \\cdot , t \\big \\}$ and $\\Gamma _ { l } ^ { t } = \\{ ( s _ { l } ^ { \\tau } , a _ { l } ^ { \\tau } ) : \\tau = 1 , \\cdot \\cdot \\cdot , t \\}$ respectively. ", + "bbox": [ + 174, + 866, + 823, + 925 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our interactive agent modeling framework is illustrated in Figure 2, which consists of two parts: i) learner’s estimation of the demonstrator’s model and ii) the learner’s probing policy for a better understanding of the demonstrator’s model. ", + "bbox": [ + 176, + 103, + 823, + 145 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To estimate the demonstrator’s model, the learner maintains a behavior tracker, $\\mathcal { M } ( \\cdot )$ , to encode the observed trajectory of the demonstrator, which generates a latent vector, $m ^ { t } = \\mathcal { M } ( \\Gamma _ { d } ^ { t } )$ . This latent vector can be viewed as a simplified representation of the demonstrator’s mind upon time $t$ , hence the learner may use it to characterize the demonstrator’s policy, $\\pi _ { d } ( a _ { d } ^ { t } \\vert s _ { d } ^ { t } , m ^ { t - 1 } )$ , from which the learner may predict the demonstrator’s future action $\\hat { a } _ { d } ^ { t }$ . Note that for each demonstration, $m ^ { t }$ always starts from the same constant, ${ \\bf m } ^ { 0 } = { \\bf 0 }$ . This particular definition of the demonstrator’s policy may also be connected with the option framework in hierarchical RL (Sutton et al., 1999), where the behavior tracker serves as a global policy to update the temporal abstraction $m ^ { t }$ and consequently changes the local policy $\\pi _ { d }$ . ", + "bbox": [ + 173, + 150, + 825, + 276 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this work, we require a learner to interact with the environment and/or with the demonstrator instead of passively watching the demonstrations. We enable this by learning a probing policy for the learner, $\\pi _ { l } ( a _ { l } ^ { t } | \\dot { s _ { l } ^ { t } } , m ^ { t - 1 } )$ , where $m ^ { t - 1 }$ is from the current demonstration. The main purpose of the probing policy is to incite new behaviors of the demonstrator, thus we adopt a curiosity-driven reward to train this policy. Particularly, we define the reward function as ", + "bbox": [ + 173, + 281, + 825, + 349 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/270c26407b519468f6d4c4f996271ccacc00ce109eade27476175f74f1122f82.jpg", + "text": "$$\n\\begin{array} { r } { r ^ { t } = R ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } , m ^ { t } ) = | | m ^ { t } - m ^ { t - 1 } | | ^ { 2 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 356, + 647, + 375 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $m ^ { t }$ is the successive output of the behavior tracker after observing $( s _ { d } ^ { t } , a _ { d } ^ { t } )$ . ", + "bbox": [ + 176, + 381, + 710, + 396 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Finally, based on the probing policy, the learner can perform the probing as the rollout procedure outlined in Algorithm 1 (see Appendix A). ", + "bbox": [ + 173, + 401, + 821, + 430 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In summary, there are four key components in our probing-based interactive agent modeling: ", + "bbox": [ + 171, + 435, + 781, + 450 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• A behavior tracker $\\mathcal { M } ( \\Gamma _ { d } ^ { t } ; \\theta _ { M } )$ ; \n• The approximated demonstrator’s policy $\\pi _ { d } ( a _ { d } ^ { t } \\vert s _ { d } ^ { t } , m ^ { t - 1 } ; \\theta _ { d } )$ ; \n• A probing policy for the learner $\\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , m ^ { t - 1 } ; \\theta _ { l } )$ ; \n• A value function for the probing policy $V ( s _ { l } ^ { t } , m ^ { t - 1 } ; \\theta _ { V } )$ . ", + "bbox": [ + 215, + 460, + 640, + 539 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Please refer to Appendix C for the details of the network architecture. ", + "bbox": [ + 174, + 547, + 630, + 564 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 579, + 290, + 594 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As discussed above, we have two main learning objectives corresponding to the two parts in our model respectively: i) minimizing imitation error (i.e., cross-entropy loss for action prediction) and ii) maximizing accumulated probing reward (i.e., probing policy optimization). Consequently, our approach includes an imitation learning process for recovering demonstrator’s policy and a reinforcement learning process for optimizing the probing policy. These two processes are intertwined and influenced by each other: the IL process provides the behavior tracker guiding the probing policy while the RL process helps $\\mathrm { I L }$ to observe more diverse behaviors from the demonstrator, thus enabling an interactive learning scheme. Algorithm 2 in Appendix A summarizes the overall learning approach, where $N$ is the total number of training iterations. The optimization details for the two learning processes are introduced as follows. ", + "bbox": [ + 173, + 604, + 826, + 739 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2.1 IMITATION LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 755, + 383, + 768 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For $\\mathrm { I L }$ , we want to learn a good behavior tracker as well as the demonstrator’s policy. For this, we minimize a cross-entropy loss for predicting demonstrator’s actions: ", + "bbox": [ + 173, + 779, + 823, + 806 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/b59c9a697f4aa739947d7d41b2f555f240ff19973d0ea14bac94c86cbc450414.jpg", + "text": "$$\n\\mathcal { L } ( \\theta _ { M } , \\theta _ { d } ) = \\mathbb { E } \\left[ - \\log \\pi _ { d } ( a _ { d } ^ { t } | s _ { d } ^ { t } , m ^ { t - 1 } ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 361, + 813, + 635, + 833 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2.2 REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 176, + 844, + 429, + 859 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The goal of RL is to train a good probing policy that will maximize the change of behavior and/or discover new behaviors of the demonstrator to facilitate the imitation learning. Based on the reward $\\begin{array} { r } { \\mathbb { E } \\left[ \\sum _ { \\tau = 0 } ^ { \\infty } \\gamma ^ { \\tau } r ^ { t + \\tau } \\right] } \\end{array}$ n Eq. (, where $\\gamma$ this goal is equivalent to maximizing the accumulated reward, is the discounted factor. $J ( \\theta _ { l } ) =$ ", + "bbox": [ + 174, + 869, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For the policy optimization, we use Advantage Actor-Critic (A2C) (Mnih et al., 2016) to conduct on-policy training. The policy gradient is ", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/94dedab7ea514b04f933d7460cb599c35bc4b9c5d7f1fb734fbed77af46661d2.jpg", + "text": "$$\n\\nabla _ { \\theta _ { l } } J ( \\theta _ { l } ) = \\nabla _ { \\theta _ { l } } \\left[ \\log \\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , m ^ { t - 1 } ; \\theta _ { l } ) A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } ) + \\lambda \\mathcal { H } ( \\pi _ { l } ( \\cdot | s _ { l } ^ { t } , m ^ { t - 1 } ; \\theta _ { l } ) ) \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 220, + 137, + 776, + 157 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } )$ is the advantage estimation defined as $\\begin{array} { r } { A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } ) = \\sum _ { \\tau = 0 } ^ { \\infty } \\gamma ^ { \\tau } r ^ { t + \\tau } \\ - } \\end{array}$ $V ( s _ { l } ^ { t } , m ^ { t - 1 } )$ and $\\mathcal { H } ( \\cdot )$ is the entropy regularization weighted by the constant $\\lambda = 0 . 0 1$ for encouraging exploration. The value function is updated by the following gradient: ", + "bbox": [ + 173, + 165, + 826, + 209 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3e55479b0119c52ec1c9bebaff1d83382c9c087a98b80f79ba2c1efc6e84f2ff.jpg", + "text": "$$\n\\nabla _ { \\theta _ { V } } \\frac { 1 } { 2 } \\left( \\sum _ { \\tau = 0 } ^ { \\infty } \\gamma ^ { \\tau } r ^ { t + \\tau } - V ( s _ { l } ^ { t } , m ^ { t - 1 } ; \\theta _ { V } ) \\right) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 351, + 215, + 647, + 261 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that when we update the probing policy and the value function, the behavior tracker is fixed (i.e., no back propagation through the dashed path in Figure 2). Thus $\\theta _ { M }$ will only be updated by the IL loss in Eq. (2). This is to ensure that the change of $\\mathrm { \\Sigma } _ { m } \\bar { t }$ is only caused by the change in policy or in behaviors, and not by the change of the parameters of the mind model, $\\theta _ { M }$ . ", + "bbox": [ + 173, + 275, + 825, + 329 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 348, + 326, + 364 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To evaluate our approach, we introduce four tasks as shown in Figure 3, including three grid world tasks (passing through obstacles, maze navigation, construction) and an algorithmic problem (sorting). ", + "bbox": [ + 174, + 378, + 825, + 407 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In order to test the generalization ability of the learned agent model, we adopt a strict training procedure, where only one particular environment and task design is given during training. Specially, for grid world tasks, we fix the environment layout and/or item placement in each demonstration, whereas for the sorting task, we use the exact same input array throughout the training. At testing time, we randomize the task settings to an extent to create novel environments/inputs that have never been seen during training. We provide the specific settings in Section 4.1. ", + "bbox": [ + 174, + 414, + 825, + 494 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We implement rule-based policies for the demonstrator i) by searching the best plan from the initial state to the goal state for the grid world tasks or ii) by the bubble sort algorithm for sorting. When there is no possible path to reach the goal (e.g., blocked by the learner), the demonstrator will stop until a viable path appears. ", + "bbox": [ + 174, + 501, + 825, + 556 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 TASKS ", + "text_level": 1, + "bbox": [ + 174, + 571, + 261, + 585 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/3b9d1a1dd367e5002b7172940644e658adf68fc36a8b0e845a272a5d536c86d3.jpg", + "image_caption": [ + "Figure 3: Illustration of the evaluated tasks. " + ], + "image_footnote": [], + "bbox": [ + 205, + 601, + 794, + 688 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Passing. In this task modified from Baker et al. (2009), the demonstrator is initially located at the bottom-right corner and is trying to pass through the closest gap to get into the upper part of the room. The demonstrator can take 5 actions including moving in four directions and stopping, whereas the learner can move in four directions, stop, and also pickup or put down a wall block. The training environment is shown in Figure 3, where the gap is always located at the left end of the wall in the middle. In testing cases, we randomly place the location of the gap and the initial position of the demonstrator. ", + "bbox": [ + 173, + 728, + 825, + 823 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Maze Navigation. Inspired by similar tasks in recent literature (Andreas et al., 2017), we place a few door blocks and tools (a key and a hammer) in a four-room maze, where the key can be used to open the yellow door but has no effect on the blue door, which must be broken by the hammer. The demonstrator is trying to go from the top-right room to the top-left room. The two agents share the same action space including moving in four directions, picking up an item, and putting down an item. Also, they can only carry one item at a time. In the training setting, the initial positions of both agents and the door blocks are fixed as shown in Figure 3, whereas the tools may be randomly placed at only a few locations. The rules in this environment are in fact fairly complex compared to other grid world tasks in previous work, where multiple sub-goals such as getting the tools, getting the door blocks, placing the door blocks, and walking through the doors are involved. ", + "bbox": [ + 173, + 829, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 145 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Construction. We adapt the stacking tasks in Shu et al. (2018) into a grid world, where the demonstrator has a latent goal invisible to the learner, which is to construct a new block by putting two blocks with a specific color combination together. Three items are present in a room and they are assigned with different colors randomly. In each episode, the demonstrator is randomly assigned with a goal (i.e., a pair of colors). It then seeks the needed blocks and puts one of them beside the other one. In order to predict the demonstrator’s actions precisely, the learner must infer the correct goal first, which requires a sophisticated and dynamic agent modeling. Both agents share the same action space as in Maze Navigation. In training, there are no obstacles in the room. To increase the difficulty of goal inference, in testing scenarios, we randomly place a few wall blocks as obstacles around the colored blocks. ", + "bbox": [ + 174, + 150, + 825, + 285 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Sorting. Compared to a grid world, algorithmic problems are less visually informative and entail more abstract reasoning. For this, we design a sorting task where an array with certain length is given at the beginning. In our experiments, we set the length to be 10 and restrain the size of numbers in the array to be 4 bits (i.e., from 0 to 15). The demonstrator is able to perform a bubble sort algorithm to rearrange the input array in an ascending order. Its action at each step is to select a pair of numbers to swap. For every 5 steps done by the demonstrator, the learner can select a number and flipping one of the bit of that number. Both agents can choose to do nothing for a step. During training, we only provide one constant array so that the sorting always starts from the same initial array. This is a very challenging setting as only 10 out of 16 possible numbers are present in the training example and the fixed ordering may also easily cause overfitting. For testing, we generate random arrays as inputs. ", + "bbox": [ + 173, + 291, + 825, + 425 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For more details about the task settings, please refer to Appendix D. ", + "bbox": [ + 174, + 433, + 619, + 446 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 GENERALIZATION IN UNSEEN TASK SETTINGS ", + "text_level": 1, + "bbox": [ + 174, + 463, + 539, + 477 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/1871cb7eec22fea8ea3cee03fbce6bab4a89e5ce0e304bc11a26ace233854a63.jpg", + "image_caption": [ + "Figure 4: Action prediction accuracies in novel testing settings over numbers of training iterations. " + ], + "image_footnote": [], + "bbox": [ + 173, + 492, + 823, + 631 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/013eb5aa25e3450135a48626efaa1b683544a12479c08644269d9f5f88ec7693.jpg", + "image_caption": [ + "Figure 5: Action prediction accuracies in novel testing settings over numbers of training iterations with $10 \\%$ random actions. " + ], + "image_footnote": [], + "bbox": [ + 254, + 678, + 743, + 813 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "One of the main goals of learning to probe is to learn a good agent model that can be generalized to unseen scenarios. To evaluate how accurate our agent model is for approximating the true mind of the demonstrator, we may test the accuracy of predicting the demonstrator’s actions using the learned $\\pi _ { d }$ and behavior tracker $\\mathcal { M } ( \\cdot )$ in testing task settings unseen by the learner during training. A high ", + "bbox": [ + 174, + 869, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "prediction accuracy in unseen settings will indicate good generalization of the learned agent model. \nTo eliminate the effects from probing, we remove the learner from the environment during testing. ", + "bbox": [ + 173, + 103, + 823, + 131 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We compare our model with six baselines: i) ToMnet in Rabinowitz et al. (2018), which learns the demonstrator’s model by only observing the given demonstrations without interactions, ii) our model without training the probing policy using RL (i.e., the leaner always takes random actions), iii), ours without the attention-based fusion (concatenating state feature and $m ^ { t - 1 }$ instead), iv) using two LSTMs for the estimated demonstrator’s policy and the probing policy respectively (Figure 16), v) using count-based bonus as reward (Strehl & Littman, 2008; Bellemare et al., 2016; Tang et al., 2017), and vi) using cross-entropy loss for action prediction as reward (i.e., exploration by self-supervised prediction in Pathak et al. (2017)). To ensure fair comparison, the training settings and the testing settings are shared by all methods. We provide more details of the baselines in Appendix E. ", + "bbox": [ + 174, + 137, + 825, + 258 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 4 shows the predication accuracy in testing settings of the three approaches based on the models from different training iterations. It is clear that with more iterations, our probing policy can greatly help increase the accuracy by discovering new behaviors, and consequently yields much higher testing accuracy than the baselines do. The results of “ours w/o fusion” and “2-LSTM” baselines further demonstrates the importance of our attention-based fusion layer and the use of a separate behavior tracker. By randomizing $10 \\%$ of demonstrator’s actions (Figure 5), we show that the probing policy can also handle stochastic and sub-optimal policies. It can be clearly seen from the results that the performance of the two baselines based on different curiosity rewards is clearly inferior to ours, which demonstrates the advantage of defining the behavioral change as the intrinsic reward for the purpose of agent modeling. We have also evaluated the robustness of our approach by showing the standard deviation from multiple runs as shown in Figure 9, which demonstrates a reasonably low variance across multiple runs. ", + "bbox": [ + 174, + 265, + 825, + 425 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We demonstrate the effect of dimensionality of the latent vector $m ^ { t }$ (i.e., the complexity of the agent model) in Figure 4. In simple environments, our approach still outperforms the baselines even when the dimensionality is decreased from 8 to 2 or 4. In more complex tasks like Sorting, a higher dimension is necessary for the agent modeling. ", + "bbox": [ + 176, + 431, + 825, + 486 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 EVOLUTION OF LEARNED PROBING STRATEGY ", + "text_level": 1, + "bbox": [ + 174, + 502, + 540, + 516 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As training progresses, we observe that our probing policy is able to progressively discover new behaviors through interactions that are adapted to the demonstrator’s policy. For instance, in Maze Navigation, we find that the learner first learns to place one door, then gradually learns to place two doors at the appropriate moments to force the demonstrator to go back and forth to get the needed tools for opening the doors. Finally, the probing policy will even blocks the demonstrator for a while before it goes through the last door. Due to the space limit, we show this in the demo video. ", + "bbox": [ + 174, + 527, + 825, + 608 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We also provide more analysis and visualization of the probing behavior and the resulting latent vectors in Appendix B.2 and Appendix B.3. ", + "bbox": [ + 176, + 614, + 821, + 642 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.4 EMERGENCE OF OBSTRUCTIVE BEHAVIORS FROM PROBING ", + "text_level": 1, + "bbox": [ + 176, + 657, + 629, + 671 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Although we never explicitly set an adversarial goal for the learner, we do observe a natural emergence of obstructive behaviors caused by the probing, which can be quantitatively measured by the success rate of the demonstrator within a time limit as shown in Figure 6. This phenomenon is aligned with common sense that the optimal probing policy to discover new behaviors of the demonstrator should constantly force the demonstrator to change its plan, which will eventually delay the time when the demonstrator finishes the task. Because of the reward defined in Eq. (1), the probing policy learned from RL is also maximizing the accumulated behavioral change of the demonstrator just like the common sense. This further justifies our simple yet effective reward design. ", + "bbox": [ + 174, + 683, + 825, + 790 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/f79f849d4628693a408ffe9f369a47a5064b5907c7894cb647ef97d04aad2d23.jpg", + "image_caption": [ + "Figure 6: The average success rate of the demonstrator within the given time limit. " + ], + "image_footnote": [], + "bbox": [ + 200, + 801, + 789, + 896 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/22c56a09143e8dc063fbd0fed96feb71578073b1b84e64124f4c3a82addce74e.jpg", + "image_caption": [ + "Figure 7: The learning curves of the collaborative task (reward is rescaled). " + ], + "image_footnote": [], + "bbox": [ + 254, + 104, + 470, + 217 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/b2076ef0ebd4994b1be4f0abf9861f2baf22ffd6fcc9e875892dea55ad60f109.jpg", + "image_caption": [ + "Figure 8: The learning curves of the competitive task (reward is rescaled). " + ], + "image_footnote": [], + "bbox": [ + 526, + 106, + 741, + 215 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.5 APPLICATION 1: DISTILLING OPTIMAL PLANS TO A POLICY NET ", + "bbox": [ + 174, + 285, + 666, + 299 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/025ab377f9282f75bdab2cdf094ee9670458eb994c66192879d47898c2ec9c32.jpg", + "table_caption": [ + "Table 1: Success rates using the learned demonstrator’s policy in unseen tasking settings. Unless specified, the evaluated policies were learned from demonstrations without random actions. " + ], + "table_footnote": [], + "table_body": "
MethodPassingMaze NavigationConstructionSorting
Ours0.710.600.480.82
Ours (from sub-optimal demo)0.680.510.47N/A
Ours w/o RL0.130.310.230.80
Ours w/o fusion0.110.190.290.10
Ours (2-LSTM)0.170.020.170
Passive (ToMnet)0.1100.120
Count-based0.2200.310.39
Self-supervised0.2300.360.56
", + "bbox": [ + 199, + 363, + 797, + 500 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Optimal planning sometimes requires a long computational time. For acceleration, it is common to distill the optimal plans to a policy net (Lazaric et al., 2010; Guo et al., 2014). However, the distilled policy net may not generalize well in new scenarios if the training settings are not diverse enough. Thus, the nature of our approach makes it suitable for improving the generalization without manually designing a large number of diverse settings. ", + "bbox": [ + 174, + 522, + 825, + 589 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For this, we evaluate the success rates when the learner directly uses $\\pi _ { d }$ (with an 8-dim latent vector) to perform tasks in testing settings without finetuning. The results summarized in Table 1 are consistent with the findings based on action predictions. We have also tested the success rate of the policy learned from sub-optimal demonstrations with $10 \\%$ random actions. Its performance is comparable to the one learned from perfect demonstrations by our approach. It also outperforms the baselines trained from optimal demonstrations. We didn’t test the randomized demonstrations for Sorting as it is unnecessary to randomize bubble sort algorithm. Note that since the learner is unaware of the goal in Construction, we let the learner take over the task after the first block has been picked up. ", + "bbox": [ + 173, + 595, + 825, + 717 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.6 APPLICATION 2: COLLABORATION ", + "text_level": 1, + "bbox": [ + 176, + 741, + 454, + 755 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To test whether the improved agent modeling by learning a probing policy can facilitate multi-agent collaboration, we modify the Construction task to be a collaborative task, where the learner is trying to help the demonstrator (fixed policy) to finish the task. For every step the demonstrator takes, the learner will get a $- 0 . 0 5$ penalty. When the goal is reached, it will be given a reward of 1. This setting is difficult for training a collaborative policy since the demonstrator is capable of finishing the task by itself. In order to help finish the task faster, the learner must infer the true goal of the demonstrator quickly and shares part of the labor accordingly. ", + "bbox": [ + 173, + 768, + 825, + 863 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We fix the behavior tracker module, $\\mathcal { M } ( \\cdot )$ , trained from our interactive agent modeling and retrain the learner’s policy $\\pi _ { l }$ using the task reward defined above. For comparison, we implement two baselines: i) retraining $\\pi _ { l }$ based on $\\mathcal { M }$ learned from passive agent modeling (i.e., ToMnet) and ii) training a policy without agent modeling, i.e., $\\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , s _ { d } ^ { t } )$ . ", + "bbox": [ + 174, + 869, + 825, + 925 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 7 demonstrates the learning curves, where the reward is rescaled so that the theoretical maximum reward from a perfect policy is 1. From the curves, we may see that the policy trained with our interactively learned agent model significantly outperforms both baselines. ", + "bbox": [ + 174, + 103, + 823, + 145 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.7 APPLICATION 3: COMPETITION ", + "text_level": 1, + "bbox": [ + 176, + 161, + 431, + 175 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Similar to Section 4.6, we design a competitive task based on the Construction task, where the learner gets a 0.05 reward for every step and a -1.0 penalty if the opponent achieves its goal. We adopt the same training procedure as in Section 4.6, and also rescale the reward according to the maximum reward. As Figure 8 shows, the mind model learned by our approach improves the learning efficiency and the converged reward by a large margin. ", + "bbox": [ + 174, + 186, + 825, + 253 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 273, + 328, + 290 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work, we have proposed a novel agent modeling approach, i.e., probing-based interactive agent modeling. The core idea is to learn a probing policy using only a curiosity-driven reward, which is able to discover new behaviors of the target agent. We achieve this by incorporating two learning processes $\\mathrm { I L }$ and RL) together. We are able to validate our approach in four distinct tasks. The results show that by learning a probing policy, the learner in our approach can build a more accurate agent model of the demonstrator. Thanks to this interactively learned agent model, the learner is able to i) approximate the demonstrator’s policy more accurately in unseen settings compared to passive agent modeling, ii) efficiently learn a good collaborative policy to help the demonstrator, and iii) develop an adversarial policy to compete with the demonstrator. ", + "bbox": [ + 174, + 304, + 825, + 425 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In the future, we can extend this framework to simultaneous agent modeling and world modeling with a more complex mind representation. ", + "bbox": [ + 174, + 431, + 821, + 459 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 479, + 285, + 494 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Maruan Al-Shedivat, Trapit Bansal, Yuri Burda, Ilya Sutskever, Igor Mordatch, and Pieter Abbeel. Continuous adaptation via meta-learning in nonstationary and competitive environments. 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", + "bbox": [ + 174, + 646, + 826, + 688 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A PSEUDO CODE OF OUR ALGORITHMS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 524, + 118 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Algorithm 1 Rollout $ { T _ { \\mathrm { m a x } } } )$ ", + "text_level": 1, + "bbox": [ + 174, + 137, + 354, + 151 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Input: Maximum steps $T _ { \\mathrm { m a x } }$ \nOutput: Episode length $T$ , trajectories $\\Gamma _ { d } ^ { T }$ and $\\Gamma _ { l } ^ { T }$ , and the latent vector sequence $M$ \n1: Initialize the environment \n2: $\\Gamma _ { d } ^ { 0 } \\gets \\emptyset , \\Gamma _ { l } ^ { 0 } \\gets \\emptyset , M \\gets \\emptyset , m ^ { 0 } \\gets \\mathbf { 0 } , t \\gets 0$ \n3: repeat \n4: $t \\gets t + 1$ \n5: Observe $s _ { d } ^ { t }$ and $a _ { d } ^ { t }$ from the demonstrator \n6: Observe $s _ { l } ^ { t }$ from the environment \n7: Sample and execute the learner’s action $a _ { l } ^ { t } \\sim \\pi _ { l } ( s _ { l } ^ { t } , m ^ { t - 1 } )$ \n8: $\\Gamma _ { d } ^ { t } \\Gamma _ { d } ^ { t - 1 } \\cup \\{ ( s _ { d } ^ { t } , a _ { d } ^ { t } ) \\} , \\Gamma _ { l } ^ { t } \\Gamma _ { l } ^ { t - 1 } \\cup \\{ ( s _ { l } ^ { t } , a _ { l } ^ { t } ) \\}$ \n9: $m ^ { t } \\gets \\mathcal { M } ( \\Gamma _ { d } ^ { t } ) , M \\gets M \\cup \\{ m ^ { t } \\}$ ", + "bbox": [ + 174, + 156, + 681, + 297 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Algorithm 2 Learning Algorithm ", + "text_level": 1, + "bbox": [ + 174, + 342, + 395, + 356 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "1: Initialize parameters $\\Theta = \\left. \\theta _ { M } , \\theta _ { b } , \\theta _ { l } , \\theta _ { V } \\right.$ \n2: Set $T _ { \\mathrm { m a x } }$ (the maximum steps in an episode) and $N$ (the number of training iterations) \n3: $i \\gets 1$ \n4: repeat \n5: $T , \\Gamma _ { d } ^ { T } , \\Gamma _ { l } ^ { T } , M \\gets \\mathrm { R o l l o u t } ( T _ { \\mathrm { m a x } } )$ \n6: IL: Update $\\theta _ { M }$ and $\\theta _ { d }$ based on Eq. (2) using $\\Gamma _ { d } ^ { T }$ \n7: RL: Update $\\theta _ { l }$ and $\\theta _ { V }$ based on Eq. (3 and Eq. (4) respectively using $\\Gamma _ { l } ^ { T }$ , and $M$ \n8: $i \\gets i + 1$ \n9: until $i = N$ ", + "bbox": [ + 179, + 361, + 707, + 478 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B MORE RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 506, + 339, + 522 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B.1 ROBUSTNESS EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 536, + 405, + 551 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We show the mean and standard deviation of the prediction accuracy from 5 runs by our full model in Figure 9 for Maze Navigation to validate the robustness of our approach. ", + "bbox": [ + 174, + 561, + 825, + 590 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/d3766e4783ca2f1226bf81c173ed6690f9a07ff438e980d834c86b081f958f61.jpg", + "image_caption": [ + "10: until $t = T _ { \\operatorname* { m a x } }$ or the task is finished 11: $T \\gets t$ ", + "Figure 9: Mean and standard deviation of multiple runs in Maze Navigation. " + ], + "image_footnote": [], + "bbox": [ + 377, + 607, + 604, + 729 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B.2 VISUALIZATION OF LATENT VECTORS ", + "text_level": 1, + "bbox": [ + 176, + 775, + 482, + 789 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Figure 10 visualizes the latent vectors obtained from demonstrations with probing and without probing, where the latent vectors were computed by the same behavior tracker in both cases. This provides empirical evidences that by finding new latent vectors, we are able to discover new demonstrations with probing. ", + "bbox": [ + 173, + 801, + 826, + 854 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B.3 VISUALIZATION OF THE CHANGE IN $m ^ { t }$ AND THE CHANGE IN POLICY ", + "text_level": 1, + "bbox": [ + 174, + 869, + 700, + 885 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "To show that the change in $m ^ { t }$ indeed indicates the change in policy, we compute the correlation of $| | m ^ { t } - m ^ { t - 1 } | | ^ { 2 }$ and $\\tilde { K L } ( \\pi _ { d } ( \\cdot | s ^ { t + 1 } , m ^ { t } ) | | \\pi _ { d } ( \\cdot | s ^ { t + 1 } , m ^ { t - 1 } ) )$ (i.e., how different the policy conditioned on the new latent vector $m ^ { t }$ is compared to the one with the old latent vector $m ^ { t - 1 }$ ). Figure 11 demonstrates the correlation between the change in $m ^ { t }$ and the corresponding change in policy in testing settings. The high correlation validates our hypothesis that the distance between consecutive latent vectors $m ^ { t }$ and $\\bar { m } ^ { t - 1 }$ reflects the policy change of the demonstrator. ", + "bbox": [ + 174, + 896, + 828, + 925 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/f82b7a583fb0ccf3c5a6ef501adc2d8413d68eb07d6414f87e572db80aaacff7.jpg", + "image_caption": [ + "Figure 10: t-SNE embedding of $m ^ { t }$ . " + ], + "image_footnote": [], + "bbox": [ + 178, + 99, + 823, + 194 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/069435882d499904bb6f9b71b6a7dc219cc94a8b05cbac2d28d7728bc6f34cb1.jpg", + "image_caption": [ + "Figure 11: Correlation between the change in $m ^ { t }$ and the change in policy in testing settings $\\dot { \\boldsymbol { r } }$ is Pearson correlation coefficient). " + ], + "image_footnote": [], + "bbox": [ + 174, + 238, + 823, + 321 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 393, + 825, + 448 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C NETWORK ARCHITECTURE OF OUR MODEL ", + "text_level": 1, + "bbox": [ + 174, + 473, + 573, + 488 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "State Encoder. The input of the state encoder is a multi-channel tensor. For the grid world case, the input dimension is $1 1 \\bar { \\times } 1 1 \\times ( N _ { \\mathrm { b l o c k s } } + 1 )$ , where $1 1 \\times 1 1$ is the size of the grid world, $N _ { \\mathrm { i t e m s } }$ is the number of types of blocks, and the additional channel is to show the position of the corresponding agent, i.e., the position of the demonstrator for $s _ { d } ^ { t }$ or the position of the learner for $s _ { l } ^ { t }$ . The other agent is treated as an obstacle and its position is encoded into the channel corresponded to the wall block. In the case of Sorting task, the input dimension is $1 0 \\times 1 \\times 4$ , representing 10 numbers in an array where the size of each number is 4 bits. The state encoder has one convolutional layer which consists of 32 filters with kernel size of $1 \\times 1$ and stride of 1. ", + "bbox": [ + 173, + 506, + 825, + 614 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Behavior Tracker. Assuming the action space is $A$ , we combine the state input and the action input by augmenting the state input with $A$ channels, each of which corresponds to an action. We set the channel of the observed action to be all ones and set the remaining $A - 1$ channels to be zeros. This combined state and action input is then fed into a convolutional layer with 32 filters (the kernel size is $1 \\times 1$ and the stride is 1). The output is flatten into a vector and passed through two fully connected (FC) layers (all have 128 dimensions). The resulting 128-dim vector serves as the input of an LSTM with 128 hidden units. Finally, an FC layer takes in the hidden state from the LSTM and outputs the latent vector $m ^ { t }$ as the mind representation. ", + "bbox": [ + 173, + 621, + 825, + 728 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/9601707c6b27d9000ede144af1654db03552ded43028f6a05face04cd1b1a543.jpg", + "image_caption": [ + "Figure 12: The attention-based fusion module. " + ], + "image_footnote": [], + "bbox": [ + 369, + 746, + 604, + 896 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Fusion. As shown in Figure 12, we design our fusion module using an attention based mechanism similar to the one introduced by Chaplot et al. (2017), where the latent vector $m ^ { t - 1 }$ is fed into an FC layer outputting an $N$ -dim attention vector (each element is from 0 to 1) corresponding to the $N$ feature maps from the state encoder (here $N = 3 2$ ). Formally, we have an attention vector $h = \\sigma ( m ^ { t - 1 } ) \\in \\dot { \\mathbb { R } } ^ { 3 2 }$ , where $\\sigma ( \\cdot )$ is an FC layer with sigmoid activation. $h$ is spatially expanded to a $H \\times W \\times 3 2$ tensor, $\\pmb { H } ( h ) \\in \\mathbb { R } ^ { H \\times W \\times 3 2 }$ , where the elements in $k$ -th channel correspond to the $k$ -th element in $h$ . We then reweight each feature maps using the attention vector, which becomes the fusion output. I.e., $f ( \\phi ( s ^ { t } ) , m ^ { t - \\tilde { 1 } } ) = \\phi ( s ^ { t } ) \\odot H ( \\sigma \\tilde { ( } m ^ { t - 1 } ) \\mathbf { \\tilde { ) } }$ , where $\\phi ( s ^ { t } )$ are the feature maps from the state encoder, $f ( \\cdot )$ is the fusion layer, and $\\odot$ is element-wise product. ", + "bbox": [ + 173, + 103, + 825, + 227 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Policy. The input of this module is the flattened output from the fusion module, and is fed to an LSTM with 128 hidden units followed by an FC layer with softmax activation. The resulting output is an action distribution representing the policy (either $\\pi _ { d }$ or $\\pi _ { l }$ ). For Sorting task, we slightly modify the output to fit the problem. We decompose the demonstrator’s policy as ${ \\pi } _ { d } ( a _ { d } ^ { t } \\vert s _ { d } ^ { t } , m ^ { t - 1 } ) ~ = ~ { \\pi } _ { d } ^ { ( 1 ) } ( a _ { d } ^ { \\bar { t } , 1 } \\vert s _ { d } ^ { t } , m ^ { t - 1 } ) { \\pi } _ { d } ^ { ( 2 ) } ( a _ { d } ^ { t , 2 } \\vert s _ { d } ^ { t } , m ^ { t - 1 } )$ , where $a _ { d } ^ { t , 1 }$ and $a _ { d } ^ { t , 2 }$ are the indices of the numbers the demonstrator chooses to swap. For the learner’s policy, it is decomposed as $\\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , m ^ { t - 1 } ) = \\pi _ { l } ^ { \\mathrm { i d } } ( a _ { l } ^ { t , 1 } | s _ { l } ^ { t } , m ^ { t - 1 } ) \\pi _ { l } ^ { \\mathrm { b i t } } ( a _ { l } ^ { t , 2 } | s _ { l } ^ { t } , m ^ { t - 1 } )$ instead, where $a _ { l } ^ { t , 1 }$ indicates the number that the learner selects to change and $a _ { l } ^ { t , 2 }$ is the bit of that number that needs to be flipped. When $a _ { d } ^ { t , 1 }$ or $a _ { l } ^ { t , 1 }$ is larger than the length of the array, it means that the demonstrator or the learner is choosing to do nothing respectively. ", + "bbox": [ + 173, + 232, + 825, + 381 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Value. We also have a value net designed for training the learner’s policy using A2C (i.e., $V ( s _ { l } ^ { t } , m ^ { t - 1 } ) )$ , which takes in the hidden state from the LSTM in the learner’s policy module and outputs a scalar value after an FC layer. ", + "bbox": [ + 174, + 386, + 826, + 428 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The network is trained with RMSProp (Tieleman & Hinto, 2012) using a learning rate of 0.001. \nDuring training, $\\epsilon$ -greedy is applied to the rollout, where the $\\epsilon$ gradually decreases from 0.1 to 0.01. ", + "bbox": [ + 176, + 433, + 823, + 462 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D TASK SETTINGS ", + "text_level": 1, + "bbox": [ + 176, + 479, + 344, + 497 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We assume full observations of the world state for both agents in all tasks but the internal state of an agent (e.g., goals) is unobservable to another agent. The discounted factor is set to be $\\Gamma = 0 . 9 5$ . ", + "bbox": [ + 174, + 511, + 823, + 539 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For the demonstrator in the grid world tasks, we implemented search based path planning and used simple heuristics to perform branch and bound for acceleration. In particular, the state for the search algorithm in Passing is the map status, whereas the state in Maze Navigation and Construction is the combination of map status and the demonstrator’s inventory. ", + "bbox": [ + 174, + 545, + 825, + 599 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D.1 PASSING ", + "text_level": 1, + "bbox": [ + 174, + 616, + 279, + 631 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/104fb1162f260d597db6e2b8b643e42e84d9eb60187fab5116d2e2aba9e01f85.jpg", + "image_caption": [ + "Figure 13: The training setting and examples of testing settings for Passing. " + ], + "image_footnote": [], + "bbox": [ + 236, + 643, + 759, + 746 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Figure 13 shows the training setting where the locations of the gap and the staring point of the demonstrator are fixed, and the examples of testing settings where the placement of the gap and the initial position of the demonstrator is randomized. We terminate a training episode if the demonstrator has not passed the obstacle after 15 steps. ", + "bbox": [ + 173, + 786, + 825, + 840 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D.2 MAZE NAVIGATION ", + "text_level": 1, + "bbox": [ + 174, + 857, + 354, + 871 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The training setting in Maze Navigation is designed as shown in Figure 14a, where the placement of the tools is restrained in the purple region, and the positions of the demonstrator’s starting point and the doors are fixed. For testing, we randomly put one or two doors to fill the gaps; the demonstrator ", + "bbox": [ + 174, + 882, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/7e7f142e00277a702495ff544dfc962f5c601cf88c03a1f6483f2f64ce64d982.jpg", + "image_caption": [ + "Figure 14: The training setting and examples of testing settings for Maze Navigation. " + ], + "image_footnote": [], + "bbox": [ + 238, + 98, + 759, + 224 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "and the tools can be randomly placed in the purple region. The demonstrator is always guaranteed to be able to find a path from its starting point to the destination (the top-left room). A training episode has a time limit of 60 steps. ", + "bbox": [ + 174, + 276, + 825, + 319 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D.3 CONSTRUCTION ", + "text_level": 1, + "bbox": [ + 174, + 335, + 331, + 351 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/16b7b3aa92dee0ccd1525db6154d8b34c50807af6f08c7694f6469a3309c6d55.jpg", + "image_caption": [ + "Figure 15: The training setting and examples of testing settings for Construction. " + ], + "image_footnote": [], + "bbox": [ + 236, + 366, + 759, + 467 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The room layout in training setting is fixed and shown in Figure 15a. In testing settings, we randomly put six wall blocks around the three colored blocks to create obstacles. Figure 15b displays a few examples of testing scenarios. Note that in both training and testing, we allow randomized coloring as long as the goal can be achieved. ", + "bbox": [ + 174, + 510, + 823, + 564 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For each episode, we assign a random goal (a pair of colors) for the demonstrator. The maximum episode length is 30 steps during training. ", + "bbox": [ + 176, + 570, + 823, + 598 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D.4 SORTING ", + "text_level": 1, + "bbox": [ + 174, + 616, + 282, + 630 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Algorithm 3 Modified Bubble Sort ", + "text_level": 1, + "bbox": [ + 174, + 648, + 406, + 664 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Input: Initial array $X = [ x _ { 1 } , x _ { 2 } , \\cdot \\cdot \\cdot , x _ { n } ]$ , where $_ n$ is the length. \nOutput: Sorted array \n1: Last position $i \\gets 0$ \n2: Steps $t \\gets 0$ \n3: while $X$ is not in an ascending order do \n4: $c \\gets 0$ \n5: while $c < n - 1$ do \n6: if $x _ { i } > x _ { i + 1 }$ then \n7: Swap $x _ { i }$ and $x _ { i + 1 }$ , i.e., the demonstrator’s $t$ -th action is $( i , i + 1 )$ \n8: $t \\gets \\bar { t } + 1$ \n9: break \n10: end if \n11: $\\begin{array} { l } { c c + 1 } \\\\ { i ( i + 1 ) \\% ( n - 1 ) } \\end{array}$ \n12: \n13: end while \n14: end while ", + "bbox": [ + 174, + 667, + 648, + 863 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In training, there is only one sequence as the initial state, i.e., [2, 0, 5, 12, 14, 10, 3, 11, 9, 7]. The testing settings include 100 randomly generated initial sequences. Training episodes have a 30-step time limit. ", + "bbox": [ + 174, + 882, + 825, + 922 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Since the learner may change certain numbers during the process of sorting, the original bubble sort may fail to finish the sorting successfully since it will not look back at the sorted part of the array. To address this, we modify the original bubble sort algorithm so that it will continue to sort the sequence until it is in an ascending order. Algorithm 3 outlines how the demonstrator swaps the numbers. ", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E DETAILS OF BASELINES ", + "text_level": 1, + "bbox": [ + 176, + 178, + 408, + 194 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E.1 REWARD FUNCTIONS IN BASELINES ", + "text_level": 1, + "bbox": [ + 176, + 208, + 467, + 223 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We define the reward functions used for baselines, count-based reward and self-supervised prediction here. ", + "bbox": [ + 173, + 233, + 825, + 262 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Count-based reward: ", + "text_level": 1, + "bbox": [ + 174, + 268, + 323, + 282 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/050cbea808e73c012f9caec3a2ab7290b761ffdca5b30ab847c0b97247f62dae.jpg", + "text": "$$\nr ^ { t } = R ( s _ { d } ^ { t } ) = \\frac { \\beta } { \\sqrt { N ( s _ { d } ^ { t } ) } } ,\n$$", + "text_format": "latex", + "bbox": [ + 415, + 277, + 583, + 314 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\beta$ is a constant (we set $\\beta = 1$ , which gives the best results in our experiments), and $N ( s _ { d } ^ { t } )$ is the counts of state visitation. In our experiments, the counting can be efficiently implemented by hashing. This reward encourages the learner to push the demonstrator to new states in order to incite new demonstrations. ", + "bbox": [ + 173, + 316, + 825, + 371 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Self-supervised prediction: ", + "text_level": 1, + "bbox": [ + 174, + 378, + 361, + 392 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/149ec5b46e4be798ebbdfe6cbadc40e61d530a6a4ef523c79444671eb4c45609.jpg", + "text": "$$\n\\begin{array} { r } { r ^ { t } = R ( s _ { d } ^ { t } , m ^ { t - 1 } , a _ { d } ^ { t } ) = - \\log \\pi _ { d } ( a _ { d } ^ { t } | s _ { d } ^ { t } , m ^ { t - 1 } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 398, + 658, + 417 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $a _ { d } ^ { t }$ is the ground-truth action from the demonstrator. This reward essentially measures action prediction loss of the estimated demonstrator’s policy, which is designed to encourage the learner to find new scenarios where the previously learned demonstrator’s policy becomes less accurate. ", + "bbox": [ + 174, + 424, + 825, + 465 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E.2 NETWORK ARCHITECTURE OF THE 2-LSTM BASELINE ", + "bbox": [ + 173, + 481, + 601, + 496 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/4d94346a5d39928d4bc7f89cf3ad6b330ebd1c289bd68248bddb18577350c585.jpg", + "image_caption": [ + "Figure 16: The network architecture of the 2-LSTM baseline. " + ], + "image_footnote": [], + "bbox": [ + 334, + 508, + 663, + 688 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Figure 16 illustrates the network architecture of the 2-LSTM baseline, where two LSTMs all have 128 hidden units. ", + "bbox": [ + 168, + 727, + 825, + 756 + ], + "page_idx": 15 + } +] \ No newline at end of file diff --git a/parse/train/SJl98sR5tX/SJl98sR5tX_middle.json b/parse/train/SJl98sR5tX/SJl98sR5tX_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..39506492e10bf0a635ee3c6f7962eb4f992f5547 --- /dev/null +++ b/parse/train/SJl98sR5tX/SJl98sR5tX_middle.json @@ -0,0 +1,43030 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 79, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 98 + ], + "score": 1.0, + "content": "INTERACTIVE AGENT MODELING BY LEARNING TO", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 160, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 160, + 117 + ], + "score": 1.0, + "content": "PROBE", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 244, + 156 + ], + "lines": [ + { + "bbox": [ + 113, + 135, + 201, + 146 + ], + "spans": [ + { + "bbox": [ + 113, + 135, + 201, + 146 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 145, + 245, + 158 + ], + "spans": [ + { + "bbox": [ + 112, + 145, + 245, + 158 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 185, + 333, + 197 + ], + "lines": [ + { + "bbox": [ + 276, + 183, + 336, + 199 + ], + "spans": [ + { + "bbox": [ + 276, + 183, + 336, + 199 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 209, + 469, + 410 + ], + "lines": [ + { + "bbox": [ + 141, + 209, + 469, + 223 + ], + "spans": [ + { + "bbox": [ + 141, + 209, + 469, + 223 + ], + "score": 1.0, + "content": "The ability of modeling the other agents, such as understanding their intentions", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 220, + 470, + 232 + ], + "spans": [ + { + "bbox": [ + 141, + 220, + 470, + 232 + ], + "score": 1.0, + "content": "and skills, is essential to an agent’s interactions with other agents. Conventional", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 230, + 471, + 243 + ], + "spans": [ + { + "bbox": [ + 141, + 230, + 471, + 243 + ], + "score": 1.0, + "content": "agent modeling relies on passive observation from demonstrations. In this work,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 242, + 469, + 253 + ], + "spans": [ + { + "bbox": [ + 141, + 242, + 469, + 253 + ], + "score": 1.0, + "content": "we propose an interactive agent modeling scheme enabled by encouraging an", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 252, + 470, + 264 + ], + "spans": [ + { + "bbox": [ + 141, + 252, + 470, + 264 + ], + "score": 1.0, + "content": "agent to learn to probe. In particular, the probing agent (i.e., a learner) learns", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 263, + 470, + 275 + ], + "spans": [ + { + "bbox": [ + 141, + 263, + 470, + 275 + ], + "score": 1.0, + "content": "to interact with the environment and with a target agent (i.e., a demonstrator) to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 272, + 470, + 286 + ], + "spans": [ + { + "bbox": [ + 141, + 272, + 470, + 286 + ], + "score": 1.0, + "content": "maximize the change in the observed behaviors of that agent. 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We have validated", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 336, + 469, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 336, + 469, + 348 + ], + "score": 1.0, + "content": "our approach in four different tasks. 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A video demo is available at https://www.dropbox.com/", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 399, + 381, + 411 + ], + "spans": [ + { + "bbox": [ + 142, + 399, + 363, + 411 + ], + "score": 1.0, + "content": "s/8mz6rd3349tso67/Probing_Demo.mov?dl", + "type": "text" + }, + { + "bbox": [ + 364, + 400, + 376, + 409 + ], + "score": 0.64, + "content": "= 0", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 399, + 381, + 411 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 109, + 431, + 206, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "An accurate understanding of other agents is essential to many multi-agent problems, such as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "collaboration, competition, and learning from an expert agent. Humans achieve this not only by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "passively observing others’ behaviors, but also by actively probing others including interacting with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 507, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 507, + 500 + ], + "score": 1.0, + "content": "them or changing the environment and conditions so that they can understand others’ intentions,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "skills, and capabilities better. 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Different from common task-oriented policy training, the learning", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "of our probing policy is purely driven by the motivation of maximizing the knowledge about the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "score": 1.0, + "content": "target agent’s model. We show a simple case in Figure 1 to illustrate this idea, where the learner and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "the demonstrator are initially located in the upper part and the lower part of the room respectively.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "The true policy of the demonstrator is trying to go to the upper part by finding the shortest path.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "However, since the room layout is fixed, the learner may overfit the only path observed from the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "score": 1.0, + "content": "demonstrator. By actively creating new gaps, the learner is able to discover various paths, which will", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 641, + 417, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 417, + 653 + ], + "score": 1.0, + "content": "greatly improve the accuracy of the approximated model of the demonstrator.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 657, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 670 + ], + "score": 1.0, + "content": "We consider the following setting for probing-based interactive agent modeling. In an environment,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "there are two general types of agents: i) a demonstrator who possesses certain skills for a single", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "task or multiple tasks, and ii) a learner who has no prior knowledge of the environment and the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 690, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 700 + ], + "score": 1.0, + "content": "demonstrator’s skills. The purpose of the learner is to efficiently and thoroughly learn all of the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "demonstrator’s skills and goals by not only passively watching the demonstrations but also actively", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "interacting with the environment and/or the demonstrator. 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Conventional", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 230, + 471, + 243 + ], + "spans": [ + { + "bbox": [ + 141, + 230, + 471, + 243 + ], + "score": 1.0, + "content": "agent modeling relies on passive observation from demonstrations. In this work,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 242, + 469, + 253 + ], + "spans": [ + { + "bbox": [ + 141, + 242, + 469, + 253 + ], + "score": 1.0, + "content": "we propose an interactive agent modeling scheme enabled by encouraging an", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 252, + 470, + 264 + ], + "spans": [ + { + "bbox": [ + 141, + 252, + 470, + 264 + ], + "score": 1.0, + "content": "agent to learn to probe. In particular, the probing agent (i.e., a learner) learns", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 263, + 470, + 275 + ], + "spans": [ + { + "bbox": [ + 141, + 263, + 470, + 275 + ], + "score": 1.0, + "content": "to interact with the environment and with a target agent (i.e., a demonstrator) to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 272, + 470, + 286 + ], + "spans": [ + { + "bbox": [ + 141, + 272, + 470, + 286 + ], + "score": 1.0, + "content": "maximize the change in the observed behaviors of that agent. Through probing,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 283, + 470, + 295 + ], + "spans": [ + { + "bbox": [ + 141, + 283, + 470, + 295 + ], + "score": 1.0, + "content": "rich behaviors can be observed and are used for enhancing the agent modeling to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 294, + 470, + 306 + ], + "spans": [ + { + "bbox": [ + 141, + 294, + 470, + 306 + ], + "score": 1.0, + "content": "learn a more accurate mind model of the target agent. Our framework consists of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 304, + 470, + 317 + ], + "spans": [ + { + "bbox": [ + 141, + 304, + 470, + 317 + ], + "score": 1.0, + "content": "two learning processes: i) imitation learning for an approximated agent model and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 315, + 470, + 327 + ], + "spans": [ + { + "bbox": [ + 141, + 315, + 470, + 327 + ], + "score": 1.0, + "content": "ii) pure curiosity-driven reinforcement learning for an efficient probing policy to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 325, + 469, + 337 + ], + "spans": [ + { + "bbox": [ + 141, + 325, + 469, + 337 + ], + "score": 1.0, + "content": "discover new behaviors that otherwise can not be observed. We have validated", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 336, + 469, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 336, + 469, + 348 + ], + "score": 1.0, + "content": "our approach in four different tasks. The experimental results suggest that the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 347, + 469, + 358 + ], + "spans": [ + { + "bbox": [ + 141, + 347, + 469, + 358 + ], + "score": 1.0, + "content": "agent model learned by our approach i) generalizes better in novel scenarios than", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 357, + 470, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 470, + 370 + ], + "score": 1.0, + "content": "the ones learned by passive observation, random probing, and other curiosity-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 368, + 469, + 379 + ], + "spans": [ + { + "bbox": [ + 142, + 368, + 469, + 379 + ], + "score": 1.0, + "content": "driven approaches do, and ii) can be used for enhancing performance in multiple", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 378, + 470, + 390 + ], + "spans": [ + { + "bbox": [ + 141, + 378, + 470, + 390 + ], + "score": 1.0, + "content": "applications including distilling optimal planning to a policy net, collaboration,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 389, + 470, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 389, + 470, + 401 + ], + "score": 1.0, + "content": "and competition. A video demo is available at https://www.dropbox.com/", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 142, + 399, + 381, + 411 + ], + "spans": [ + { + "bbox": [ + 142, + 399, + 363, + 411 + ], + "score": 1.0, + "content": "s/8mz6rd3349tso67/Probing_Demo.mov?dl", + "type": "text" + }, + { + "bbox": [ + 364, + 400, + 376, + 409 + ], + "score": 0.64, + "content": "= 0", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 399, + 381, + 411 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 14, + "bbox_fs": [ + 141, + 209, + 471, + 411 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 431, + 206, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "An accurate understanding of other agents is essential to many multi-agent problems, such as", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "collaboration, competition, and learning from an expert agent. Humans achieve this not only by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "passively observing others’ behaviors, but also by actively probing others including interacting with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 507, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 507, + 500 + ], + "score": 1.0, + "content": "them or changing the environment and conditions so that they can understand others’ intentions,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "skills, and capabilities better. For instance, when working with a colleague for the first time, one may", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "intentionally create diverse situations where the true intention and skill set of that colleague can be", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 519, + 361, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 361, + 531 + ], + "score": 1.0, + "content": "clearly revealed, which in turn helps improve the collaboration.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 456, + 507, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "Inspired by this observation, in this work, we try to enable a probing agent (i.e., a learner) to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "automatically learn a good policy for probing in a way that helps it discover new behaviors of a target", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "agent (i.e., a demonstrator) and thus learn a better model of the target agent that is generalizable to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 566, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 581 + ], + "score": 1.0, + "content": "unseen environments or settings. Different from common task-oriented policy training, the learning", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "of our probing policy is purely driven by the motivation of maximizing the knowledge about the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 601 + ], + "score": 1.0, + "content": "target agent’s model. We show a simple case in Figure 1 to illustrate this idea, where the learner and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "the demonstrator are initially located in the upper part and the lower part of the room respectively.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "The true policy of the demonstrator is trying to go to the upper part by finding the shortest path.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "However, since the room layout is fixed, the learner may overfit the only path observed from the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "score": 1.0, + "content": "demonstrator. By actively creating new gaps, the learner is able to discover various paths, which will", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 641, + 417, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 417, + 653 + ], + "score": 1.0, + "content": "greatly improve the accuracy of the approximated model of the demonstrator.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 535, + 506, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 657, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 670 + ], + "score": 1.0, + "content": "We consider the following setting for probing-based interactive agent modeling. In an environment,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "there are two general types of agents: i) a demonstrator who possesses certain skills for a single", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "task or multiple tasks, and ii) a learner who has no prior knowledge of the environment and the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 690, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 505, + 700 + ], + "score": 1.0, + "content": "demonstrator’s skills. The purpose of the learner is to efficiently and thoroughly learn all of the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "demonstrator’s skills and goals by not only passively watching the demonstrations but also actively", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "interacting with the environment and/or the demonstrator. This learning process entails both imitation", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "learning (IL) for modeling the demonstrator’s skills and goals, and reinforcement learning (RL)", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 657, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 165, + 79, + 445, + 147 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 165, + 79, + 445, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 79, + 445, + 147 + ], + "spans": [ + { + "bbox": [ + 165, + 79, + 445, + 147 + ], + "score": 0.964, + "type": "image", + "image_path": "3c1afb04daa88192307507702776c69b03f1854302bfa148df1e924a35747df7.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 165, + 79, + 445, + 101.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 165, + 101.66666666666667, + 445, + 124.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 165, + 124.33333333333334, + 445, + 147.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 156, + 505, + 199 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 156, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 505, + 168 + ], + "score": 1.0, + "content": "Figure 1: Illustration of our probing-based interactive agent modeling. Here, the demonstrator tries", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 166, + 504, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 504, + 178 + ], + "score": 1.0, + "content": "to go from the bottom-right corner to the upper part of the room. The passive learner (left) only", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "observes one path in the fixed environment while the probing learner (right) removes a wall block to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 187, + 405, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 405, + 201 + ], + "score": 1.0, + "content": "create a new gap so that the demonstrator will change its path accordingly.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 219, + 503, + 252 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "for optimizing probing policy to diversify the task settings and the demonstrations to facilitate the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "imitation learning. Note that we assume that the demonstrator will always truthfully reveal its skills", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 241, + 235, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 235, + 253 + ], + "score": 1.0, + "content": "and intentions in any scenarios.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "A key idea in our approach is to use task independent RL training purely driven by a curiosity reward.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "score": 1.0, + "content": "For this, we represent the demonstrator’s mind by i) a latent vector to encode and track an agent’s", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "intention and belief, and ii) a policy conditioned on the latent vector and the agent’s observed state for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "action prediction (i.e., the agent’s skills). By introducing this latent vector, we are able to characterize", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "an agent’s policy by a low dimensional representation, and to reflect the change of policy by the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "score": 1.0, + "content": "change of this latent vector. Since the goal of probing policy is to cause the demonstrator to change", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "score": 1.0, + "content": "its policy so that the learner may observe diverse demonstrations, it is natural to apply the change in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 331, + 418, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 418, + 344 + ], + "score": 1.0, + "content": "the latent mind representation as the curiosity-driven incentive for the learner.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "We evaluate our approach on four tasks in different domains (grid worlds and algorithmic problems).", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 358, + 507, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 507, + 371 + ], + "score": 1.0, + "content": "The experimental results indicate that our probing-based interactive agent modeling framework can:", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "i) efficiently model the demonstrator’s mind that is generalizable in unseen scenarios, and ii) can", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "be applied to several applications including distilling optimal plans to a policy net by automatically", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "diversifying task settings, and improving multi-agent collaboration as well as competition using the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 193, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 193, + 413 + ], + "score": 1.0, + "content": "learned agent model.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 108, + 427, + 211, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 213, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 213, + 442 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "In multi-agent reinforcement learning (MARL), agent modeling or opponent modeling plays an", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "essential role as the ability of understanding other agent’s goals and predicting their actions can", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "greatly facilitate both collaborative and competitive purposes (Busoniu et al., 2008; Albrecht &", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 481, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 498 + ], + "score": 1.0, + "content": "Stone, 2018). Previous work has attempted to achieve this by task-oriented learning for maximizing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 492, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 506, + 508 + ], + "score": 1.0, + "content": "a specified collaborative or competitive reward in the given tasks. For instance, inspired by game", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "theory, there have been approaches aiming at finding Nash equilibira in multi-agent games, where", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "agents’ models are represented by their utilities (Littman, 1994; Hu & Wellman, 2003) and strategies", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "(Claus & Boutilier, 1998; Tesauro, 2004; Powers & Shoham, 2005; Heinrich et al., 2015; Heinrich &", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "Silver, 2016; Lanctot et al., 2017). Recently, some deep RL methods have incorporated simple agent", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 545, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 104, + 545, + 506, + 560 + ], + "score": 1.0, + "content": "modeling into Q-learning (Lowe et al., 2017) or policy updates (Foerster et al., 2018). Auxiliary", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "tasks like explicitly predicting other agents’ goals have also been applied to MARL (Mordatch &", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 567, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 506, + 582 + ], + "score": 1.0, + "content": "Abbeel, 2018). In previous work, the agents’ incentive of modeling other agents comes from reaching", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "a common goal or conflicting goals. In contrast, we never define a task-specific reward for the learner", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "since the goal of our probing-based interactive agent modeling is not to reach a predefined goal but", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 597, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 613 + ], + "score": 1.0, + "content": "rather to learn a good mind model of the demonstrator which can be generalized to unseen settings", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 609, + 454, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 454, + 622 + ], + "score": 1.0, + "content": "and transferred to multi-agent tasks afterwards when a task-dependent reward is given.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 625, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "score": 1.0, + "content": "Our work is greatly inspired by Theory of Mind (ToM) (Premack & Woodruff, 1978), which is a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "score": 1.0, + "content": "general and powerful framework to model an agent’s mind and use the mind model to better explain or", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "score": 1.0, + "content": "predict the agent’s behaviors. Baker et al. (2009) has proposed a Bayesian formulation to incorporate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "an agent’s desires, intentions, and belief about the world into the agent’s policy in order to predict", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "the agent’s goals and actions via inverse planning. Rabinowitz et al. (2018) adopts a simpler mind", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "representation (i.e., a latent vector) learned by a neural net (ToMnet). In our work, the learner is also", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "trying to learn the policy of the demonstrator with a simple mind modeling. However, instead of only", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "serving as a passive observer, we encourage the learner to probe so that it will learn to interact with", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "the environment and with the demonstrator to quickly and continuously discover new behaviors of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 378, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 378, + 733 + ], + "score": 1.0, + "content": "the demonstrator, which in turn helps learning a better mind model.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 45.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 165, + 79, + 445, + 147 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 165, + 79, + 445, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 79, + 445, + 147 + ], + "spans": [ + { + "bbox": [ + 165, + 79, + 445, + 147 + ], + "score": 0.964, + "type": "image", + "image_path": "3c1afb04daa88192307507702776c69b03f1854302bfa148df1e924a35747df7.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 165, + 79, + 445, + 101.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 165, + 101.66666666666667, + 445, + 124.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 165, + 124.33333333333334, + 445, + 147.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 156, + 505, + 199 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 156, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 505, + 168 + ], + "score": 1.0, + "content": "Figure 1: Illustration of our probing-based interactive agent modeling. Here, the demonstrator tries", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 166, + 504, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 504, + 178 + ], + "score": 1.0, + "content": "to go from the bottom-right corner to the upper part of the room. 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Note that we assume that the demonstrator will always truthfully reveal its skills", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 241, + 235, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 235, + 253 + ], + "score": 1.0, + "content": "and intentions in any scenarios.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 219, + 505, + 253 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 257, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 271 + ], + "score": 1.0, + "content": "A key idea in our approach is to use task independent RL training purely driven by a curiosity reward.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "score": 1.0, + "content": "For this, we represent the demonstrator’s mind by i) a latent vector to encode and track an agent’s", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "intention and belief, and ii) a policy conditioned on the latent vector and the agent’s observed state for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "action prediction (i.e., the agent’s skills). By introducing this latent vector, we are able to characterize", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "an agent’s policy by a low dimensional representation, and to reflect the change of policy by the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 324 + ], + "score": 1.0, + "content": "change of this latent vector. Since the goal of probing policy is to cause the demonstrator to change", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 334 + ], + "score": 1.0, + "content": "its policy so that the learner may observe diverse demonstrations, it is natural to apply the change in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 331, + 418, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 418, + 344 + ], + "score": 1.0, + "content": "the latent mind representation as the curiosity-driven incentive for the learner.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 257, + 506, + 344 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 348, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "We evaluate our approach on four tasks in different domains (grid worlds and algorithmic problems).", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 358, + 507, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 507, + 371 + ], + "score": 1.0, + "content": "The experimental results indicate that our probing-based interactive agent modeling framework can:", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "i) efficiently model the demonstrator’s mind that is generalizable in unseen scenarios, and ii) can", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "be applied to several applications including distilling optimal plans to a policy net by automatically", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "diversifying task settings, and improving multi-agent collaboration as well as competition using the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 401, + 193, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 193, + 413 + ], + "score": 1.0, + "content": "learned agent model.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 347, + 507, + 413 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 427, + 211, + 439 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 213, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 213, + 442 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "In multi-agent reinforcement learning (MARL), agent modeling or opponent modeling plays an", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 475 + ], + "score": 1.0, + "content": "essential role as the ability of understanding other agent’s goals and predicting their actions can", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "greatly facilitate both collaborative and competitive purposes (Busoniu et al., 2008; Albrecht &", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 481, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 498 + ], + "score": 1.0, + "content": "Stone, 2018). Previous work has attempted to achieve this by task-oriented learning for maximizing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 492, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 506, + 508 + ], + "score": 1.0, + "content": "a specified collaborative or competitive reward in the given tasks. For instance, inspired by game", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "theory, there have been approaches aiming at finding Nash equilibira in multi-agent games, where", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 528 + ], + "score": 1.0, + "content": "agents’ models are represented by their utilities (Littman, 1994; Hu & Wellman, 2003) and strategies", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "(Claus & Boutilier, 1998; Tesauro, 2004; Powers & Shoham, 2005; Heinrich et al., 2015; Heinrich &", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "Silver, 2016; Lanctot et al., 2017). Recently, some deep RL methods have incorporated simple agent", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 545, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 104, + 545, + 506, + 560 + ], + "score": 1.0, + "content": "modeling into Q-learning (Lowe et al., 2017) or policy updates (Foerster et al., 2018). Auxiliary", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 506, + 570 + ], + "score": 1.0, + "content": "tasks like explicitly predicting other agents’ goals have also been applied to MARL (Mordatch &", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 567, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 506, + 582 + ], + "score": 1.0, + "content": "Abbeel, 2018). In previous work, the agents’ incentive of modeling other agents comes from reaching", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "a common goal or conflicting goals. In contrast, we never define a task-specific reward for the learner", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "since the goal of our probing-based interactive agent modeling is not to reach a predefined goal but", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 597, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 613 + ], + "score": 1.0, + "content": "rather to learn a good mind model of the demonstrator which can be generalized to unseen settings", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 609, + 454, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 454, + 622 + ], + "score": 1.0, + "content": "and transferred to multi-agent tasks afterwards when a task-dependent reward is given.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 451, + 506, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 625, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "score": 1.0, + "content": "Our work is greatly inspired by Theory of Mind (ToM) (Premack & Woodruff, 1978), which is a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "score": 1.0, + "content": "general and powerful framework to model an agent’s mind and use the mind model to better explain or", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "score": 1.0, + "content": "predict the agent’s behaviors. Baker et al. (2009) has proposed a Bayesian formulation to incorporate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "an agent’s desires, intentions, and belief about the world into the agent’s policy in order to predict", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "the agent’s goals and actions via inverse planning. Rabinowitz et al. (2018) adopts a simpler mind", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "representation (i.e., a latent vector) learned by a neural net (ToMnet). In our work, the learner is also", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 702 + ], + "score": 1.0, + "content": "trying to learn the policy of the demonstrator with a simple mind modeling. However, instead of only", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "serving as a passive observer, we encourage the learner to probe so that it will learn to interact with", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "the environment and with the demonstrator to quickly and continuously discover new behaviors of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 378, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 378, + 733 + ], + "score": 1.0, + "content": "the demonstrator, which in turn helps learning a better mind model.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 625, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 206, + 80, + 404, + 243 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 206, + 80, + 404, + 243 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 206, + 80, + 404, + 243 + ], + "spans": [ + { + "bbox": [ + 206, + 80, + 404, + 243 + ], + "score": 0.971, + "type": "image", + "image_path": "84aa49337f3c7c08fcf23511a43c0078a3fcfdc51c45b78d41146d298740de5e.jpg" + } + ] + } + ], + "index": 5.5, + "virtual_lines": [ + { + "bbox": [ + 206, + 80, + 404, + 93.58333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 206, + 93.58333333333333, + 404, + 107.16666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 206, + 107.16666666666666, + 404, + 120.74999999999999 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 206, + 120.74999999999999, + 404, + 134.33333333333331 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 206, + 134.33333333333331, + 404, + 147.91666666666666 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 206, + 147.91666666666666, + 404, + 161.5 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 206, + 161.5, + 404, + 175.08333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 206, + 175.08333333333334, + 404, + 188.66666666666669 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 206, + 188.66666666666669, + 404, + 202.25000000000003 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 206, + 202.25000000000003, + 404, + 215.83333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 206, + 215.83333333333337, + 404, + 229.4166666666667 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 206, + 229.4166666666667, + 404, + 243.00000000000006 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 250, + 504, + 282 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 250, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 261 + ], + "score": 1.0, + "content": "Figure 2: An overview of our model. Architecture details are in Appendix C. Note that the modules", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "score": 1.0, + "content": "do not share weights, and the dashed line indicates that it is a feed forward only path (no back", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 337, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 337, + 284 + ], + "score": 1.0, + "content": "propagation through this path to update the mind model).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + } + ], + "index": 9.25 + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "Our task-independent reward is related to the curiosity-driven reward applied to an RL agent for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 316, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 328 + ], + "score": 1.0, + "content": "encouraging exploration (Strehl & Littman, 2008; Bellemare et al., 2016; Tang et al., 2017; Pathak", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "et al., 2017). Our probing framework differs from this in two ways: i) instead of exploring the world", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 349 + ], + "score": 1.0, + "content": "states, we encourage the learner to discover new behaviors of the demonstrator to learn a better", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "model of its mind; ii) the curiosity-driven rewards in previous work only serve as auxiliary rewards", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "for achieving specific goals, whereas in our case, the sole motivation of our learner agent is from", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "curiosity, and we demonstrate that this type of pure curiosity-driven learning can actually yield rich", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 379, + 263, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 263, + 393 + ], + "score": 1.0, + "content": "behaviors and general agent modeling.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "There is certain similarity between active learning and our learning to prob mechanism. As Yang &", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 506, + 419 + ], + "score": 1.0, + "content": "Shafto (2017) shows, active learning is more effective than optimal teaching when the learner and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "teacher are not conceptually aligned, which is exactly the scenario in our problem setting (the learner", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "does do not share any conceptual common ground with the demonstrator at the beginning). However,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 438, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 449 + ], + "score": 1.0, + "content": "active learning typically addresses problems such as classification (Tong & Koller, 2001; Kapoor", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "et al., 2007) by generating queries to an oracle to get additional ground-truth supervision. There are", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 471 + ], + "score": 1.0, + "content": "have been work on active imitation learning (Shon et al., 2007; nd Geoffrey J. Gordon & Bagnell,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 468, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 483 + ], + "score": 1.0, + "content": "2011; Judah et al., 2012) and active inverse reinforcement learning (Lopes et al., 2009) utilizing the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 480, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 506, + 492 + ], + "score": 1.0, + "content": "similar concept, where the learner asks quires at certain states to a human oracle for guidance on what", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "actions to take at those states. In contrast, our work goes beyond the scope of the existing work on", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "active learning – we aim at training a learner agent to directly interact with the environment and with", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "the target agent in order to automatically diversify the task settings and learn a better agent model", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "without any task-dependent training objectives so that the learned agent models can be applied to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 533, + 342, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 342, + 545 + ], + "score": 1.0, + "content": "improve the learner’s performance in various applications.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 549, + 505, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "Lastly, our task-independent learning objective can also be connected with meta-learning (Wang et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 558, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 573 + ], + "score": 1.0, + "content": "2016; Finn et al., 2017), which is to learn a meta strategy that can conduct efficient multi-task learning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 570, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 582 + ], + "score": 1.0, + "content": "(Maclaurin et al., 2015; Duan et al., 2017; Hariharan & Girshick, 2017; Wichrowska et al., 2017;", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 593 + ], + "score": 1.0, + "content": "Yu et al., 2018; Baker et al., 2017) or adapt an agent’s policy to its opponent’s policy (Al-Shedivat", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 590, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 604 + ], + "score": 1.0, + "content": "et al., 2018) in a competitive setting. 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Note that the modules", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "score": 1.0, + "content": "do not share weights, and the dashed line indicates that it is a feed forward only path (no back", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 337, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 337, + 284 + ], + "score": 1.0, + "content": "propagation through this path to update the mind model).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + } + ], + "index": 9.25 + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "Our task-independent reward is related to the curiosity-driven reward applied to an RL agent for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 316, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 328 + ], + "score": 1.0, + "content": "encouraging exploration (Strehl & Littman, 2008; Bellemare et al., 2016; Tang et al., 2017; Pathak", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "et al., 2017). 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The history of their behaviors upon time", + "type": "text" + }, + { + "bbox": [ + 374, + 710, + 379, + 719 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 707, + 506, + 723 + ], + "score": 1.0, + "content": "is represented by trajectories", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 720, + 431, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 234, + 733 + ], + "score": 0.91, + "content": "\\Gamma _ { d } ^ { \\acute { t } } = \\big \\{ ( s _ { d } ^ { \\tau } , a _ { d } ^ { \\tau } ) : \\tau = 1 , \\cdot \\cdot \\cdot , t \\big \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 720, + 252, + 734 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 720, + 377, + 733 + ], + "score": 0.92, + "content": "\\Gamma _ { l } ^ { t } = \\{ ( s _ { l } ^ { \\tau } , a _ { l } ^ { \\tau } ) : \\tau = 1 , \\cdot \\cdot \\cdot , t \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 720, + 431, + 734 + ], + "score": 1.0, + "content": "respectively.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 686, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Our interactive agent modeling framework is illustrated in Figure 2, which consists of two parts:", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "i) learner’s estimation of the demonstrator’s model and ii) the learner’s probing policy for a better", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 282, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 282, + 115 + ], + "score": 1.0, + "content": "understanding of the demonstrator’s model.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 119, + 505, + 219 + ], + "lines": [ + { + "bbox": [ + 106, + 119, + 504, + 131 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 422, + 131 + ], + "score": 1.0, + "content": "To estimate the demonstrator’s model, the learner maintains a behavior tracker,", + "type": "text" + }, + { + "bbox": [ + 422, + 120, + 446, + 131 + ], + "score": 0.88, + "content": "\\mathcal { M } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 119, + 504, + 131 + ], + "score": 1.0, + "content": ", to encode the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 130, + 506, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 398, + 143 + ], + "score": 1.0, + "content": "observed trajectory of the demonstrator, which generates a latent vector,", + "type": "text" + }, + { + "bbox": [ + 399, + 131, + 455, + 143 + ], + "score": 0.91, + "content": "m ^ { t } = \\mathcal { M } ( \\Gamma _ { d } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 130, + 506, + 143 + ], + "score": 1.0, + "content": ". This latent", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 504, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 457, + 153 + ], + "score": 1.0, + "content": "vector can be viewed as a simplified representation of the demonstrator’s mind upon time", + "type": "text" + }, + { + "bbox": [ + 457, + 143, + 461, + 151 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 141, + 504, + 153 + ], + "score": 1.0, + "content": ", hence the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 103, + 149, + 507, + 168 + ], + "spans": [ + { + "bbox": [ + 103, + 149, + 342, + 168 + ], + "score": 1.0, + "content": "learner may use it to characterize the demonstrator’s policy,", + "type": "text" + }, + { + "bbox": [ + 342, + 151, + 410, + 164 + ], + "score": 0.92, + "content": "\\pi _ { d } ( a _ { d } ^ { t } \\vert s _ { d } ^ { t } , m ^ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 149, + 507, + 168 + ], + "score": 1.0, + "content": ", from which the learner", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 163, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 285, + 177 + ], + "score": 1.0, + "content": "may predict the demonstrator’s future action", + "type": "text" + }, + { + "bbox": [ + 286, + 163, + 297, + 175 + ], + "score": 0.89, + "content": "\\hat { a } _ { d } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 163, + 437, + 177 + ], + "score": 1.0, + "content": ". Note that for each demonstration,", + "type": "text" + }, + { + "bbox": [ + 437, + 163, + 450, + 174 + ], + "score": 0.88, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 163, + 506, + 177 + ], + "score": 1.0, + "content": "always starts", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 203, + 188 + ], + "score": 1.0, + "content": "from the same constant,", + "type": "text" + }, + { + "bbox": [ + 204, + 175, + 237, + 186 + ], + "score": 0.9, + "content": "{ \\bf m } ^ { 0 } = { \\bf 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 174, + 505, + 188 + ], + "score": 1.0, + "content": ". This particular definition of the demonstrator’s policy may also be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "connected with the option framework in hierarchical RL (Sutton et al., 1999), where the behavior", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 195, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 371, + 209 + ], + "score": 1.0, + "content": "tracker serves as a global policy to update the temporal abstraction", + "type": "text" + }, + { + "bbox": [ + 371, + 196, + 384, + 206 + ], + "score": 0.88, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 195, + 505, + 209 + ], + "score": 1.0, + "content": "and consequently changes the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 205, + 171, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 155, + 221 + ], + "score": 1.0, + "content": "local policy", + "type": "text" + }, + { + "bbox": [ + 156, + 209, + 167, + 218 + ], + "score": 0.84, + "content": "\\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 205, + 171, + 221 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 223, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "In this work, we require a learner to interact with the environment and/or with the demonstrator", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "score": 1.0, + "content": "instead of passively watching the demonstrations. We enable this by learning a probing policy for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 243, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 155, + 258 + ], + "score": 1.0, + "content": "the learner,", + "type": "text" + }, + { + "bbox": [ + 155, + 244, + 218, + 257 + ], + "score": 0.92, + "content": "\\pi _ { l } ( a _ { l } ^ { t } | \\dot { s _ { l } ^ { t } } , m ^ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 243, + 250, + 258 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 250, + 244, + 274, + 255 + ], + "score": 0.91, + "content": "m ^ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 243, + 506, + 258 + ], + "score": 1.0, + "content": "is from the current demonstration. The main purpose of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "the probing policy is to incite new behaviors of the demonstrator, thus we adopt a curiosity-driven", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 265, + 397, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 397, + 278 + ], + "score": 1.0, + "content": "reward to train this policy. Particularly, we define the reward function as", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 282, + 396, + 297 + ], + "lines": [ + { + "bbox": [ + 214, + 282, + 396, + 297 + ], + "spans": [ + { + "bbox": [ + 214, + 282, + 396, + 297 + ], + "score": 0.92, + "content": "\\begin{array} { r } { r ^ { t } = R ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } , m ^ { t } ) = | | m ^ { t } - m ^ { t - 1 } | | ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "270c26407b519468f6d4c4f996271ccacc00ce109eade27476175f74f1122f82.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 214, + 282, + 396, + 297 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 302, + 435, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 435, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 133, + 316 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 302, + 146, + 312 + ], + "score": 0.86, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 299, + 399, + 316 + ], + "score": 1.0, + "content": "is the successive output of the behavior tracker after observing", + "type": "text" + }, + { + "bbox": [ + 399, + 301, + 431, + 315 + ], + "score": 0.92, + "content": "( s _ { d } ^ { t } , a _ { d } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 299, + 435, + 316 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 503, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "Finally, based on the probing policy, the learner can perform the probing as the rollout procedure", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 329, + 280, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 280, + 342 + ], + "score": 1.0, + "content": "outlined in Algorithm 1 (see Appendix A).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 105, + 345, + 478, + 357 + ], + "lines": [ + { + "bbox": [ + 104, + 343, + 479, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 479, + 361 + ], + "score": 1.0, + "content": "In summary, there are four key components in our probing-based interactive agent modeling:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 132, + 365, + 392, + 427 + ], + "lines": [ + { + "bbox": [ + 132, + 366, + 273, + 379 + ], + "spans": [ + { + "bbox": [ + 132, + 366, + 219, + 379 + ], + "score": 1.0, + "content": "• A behavior tracker", + "type": "text" + }, + { + "bbox": [ + 219, + 366, + 269, + 379 + ], + "score": 0.92, + "content": "\\mathcal { M } ( \\Gamma _ { d } ^ { t } ; \\theta _ { M } )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 366, + 273, + 379 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 132, + 380, + 393, + 396 + ], + "spans": [ + { + "bbox": [ + 132, + 380, + 306, + 396 + ], + "score": 1.0, + "content": "• The approximated demonstrator’s policy", + "type": "text" + }, + { + "bbox": [ + 307, + 381, + 389, + 395 + ], + "score": 0.91, + "content": "\\pi _ { d } ( a _ { d } ^ { t } \\vert s _ { d } ^ { t } , m ^ { t - 1 } ; \\theta _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 380, + 393, + 396 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 132, + 397, + 352, + 411 + ], + "spans": [ + { + "bbox": [ + 132, + 397, + 271, + 411 + ], + "score": 1.0, + "content": "• A probing policy for the learner", + "type": "text" + }, + { + "bbox": [ + 272, + 397, + 348, + 411 + ], + "score": 0.92, + "content": "\\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , m ^ { t - 1 } ; 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Consequently,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "our approach includes an imitation learning process for recovering demonstrator’s policy and a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "reinforcement learning process for optimizing the probing policy. These two processes are intertwined", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "and influenced by each other: the IL process provides the behavior tracker guiding the probing policy", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 542, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 213, + 557 + ], + "score": 1.0, + "content": "while the RL process helps", + "type": "text" + }, + { + "bbox": [ + 213, + 544, + 224, + 554 + ], + "score": 0.25, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 542, + 506, + 557 + ], + "score": 1.0, + "content": "to observe more diverse behaviors from the demonstrator, thus enabling", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "an interactive learning scheme. 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This latent", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 504, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 457, + 153 + ], + "score": 1.0, + "content": "vector can be viewed as a simplified representation of the demonstrator’s mind upon time", + "type": "text" + }, + { + "bbox": [ + 457, + 143, + 461, + 151 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 141, + 504, + 153 + ], + "score": 1.0, + "content": ", hence the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 103, + 149, + 507, + 168 + ], + "spans": [ + { + "bbox": [ + 103, + 149, + 342, + 168 + ], + "score": 1.0, + "content": "learner may use it to characterize the demonstrator’s policy,", + "type": "text" + }, + { + "bbox": [ + 342, + 151, + 410, + 164 + ], + "score": 0.92, + "content": "\\pi _ { d } ( a _ { d } ^ { t } \\vert s _ { d } ^ { t } , m ^ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 149, + 507, + 168 + ], + "score": 1.0, + "content": ", from which the learner", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 163, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 285, + 177 + ], + "score": 1.0, + "content": "may predict the demonstrator’s future action", + "type": "text" + }, + { + "bbox": [ + 286, + 163, + 297, + 175 + ], + "score": 0.89, + "content": "\\hat { a } _ { d } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 163, + 437, + 177 + ], + "score": 1.0, + "content": ". Note that for each demonstration,", + "type": "text" + }, + { + "bbox": [ + 437, + 163, + 450, + 174 + ], + "score": 0.88, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 163, + 506, + 177 + ], + "score": 1.0, + "content": "always starts", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 174, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 203, + 188 + ], + "score": 1.0, + "content": "from the same constant,", + "type": "text" + }, + { + "bbox": [ + 204, + 175, + 237, + 186 + ], + "score": 0.9, + "content": "{ \\bf m } ^ { 0 } = { \\bf 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 174, + 505, + 188 + ], + "score": 1.0, + "content": ". This particular definition of the demonstrator’s policy may also be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "connected with the option framework in hierarchical RL (Sutton et al., 1999), where the behavior", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 195, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 371, + 209 + ], + "score": 1.0, + "content": "tracker serves as a global policy to update the temporal abstraction", + "type": "text" + }, + { + "bbox": [ + 371, + 196, + 384, + 206 + ], + "score": 0.88, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 195, + 505, + 209 + ], + "score": 1.0, + "content": "and consequently changes the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 205, + 171, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 155, + 221 + ], + "score": 1.0, + "content": "local policy", + "type": "text" + }, + { + "bbox": [ + 156, + 209, + 167, + 218 + ], + "score": 0.84, + "content": "\\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 205, + 171, + 221 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7, + "bbox_fs": [ + 103, + 119, + 507, + 221 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 223, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "In this work, we require a learner to interact with the environment and/or with the demonstrator", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 246 + ], + "score": 1.0, + "content": "instead of passively watching the demonstrations. We enable this by learning a probing policy for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 243, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 155, + 258 + ], + "score": 1.0, + "content": "the learner,", + "type": "text" + }, + { + "bbox": [ + 155, + 244, + 218, + 257 + ], + "score": 0.92, + "content": "\\pi _ { l } ( a _ { l } ^ { t } | \\dot { s _ { l } ^ { t } } , m ^ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 243, + 250, + 258 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 250, + 244, + 274, + 255 + ], + "score": 0.91, + "content": "m ^ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 243, + 506, + 258 + ], + "score": 1.0, + "content": "is from the current demonstration. The main purpose of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 268 + ], + "score": 1.0, + "content": "the probing policy is to incite new behaviors of the demonstrator, thus we adopt a curiosity-driven", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 265, + 397, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 397, + 278 + ], + "score": 1.0, + "content": "reward to train this policy. Particularly, we define the reward function as", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 223, + 506, + 278 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 282, + 396, + 297 + ], + "lines": [ + { + "bbox": [ + 214, + 282, + 396, + 297 + ], + "spans": [ + { + "bbox": [ + 214, + 282, + 396, + 297 + ], + "score": 0.92, + "content": "\\begin{array} { r } { r ^ { t } = R ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } , m ^ { t } ) = | | m ^ { t } - m ^ { t - 1 } | | ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "270c26407b519468f6d4c4f996271ccacc00ce109eade27476175f74f1122f82.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 214, + 282, + 396, + 297 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 302, + 435, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 299, + 435, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 133, + 316 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 302, + 146, + 312 + ], + "score": 0.86, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 299, + 399, + 316 + ], + "score": 1.0, + "content": "is the successive output of the behavior tracker after observing", + "type": "text" + }, + { + "bbox": [ + 399, + 301, + 431, + 315 + ], + "score": 0.92, + "content": "( s _ { d } ^ { t } , a _ { d } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 299, + 435, + 316 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 299, + 435, + 316 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 503, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "Finally, based on the probing policy, the learner can perform the probing as the rollout procedure", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 329, + 280, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 280, + 342 + ], + "score": 1.0, + "content": "outlined in Algorithm 1 (see Appendix A).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 106, + 318, + 505, + 342 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 345, + 478, + 357 + ], + "lines": [ + { + "bbox": [ + 104, + 343, + 479, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 479, + 361 + ], + "score": 1.0, + "content": "In summary, there are four key components in our probing-based interactive agent modeling:", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 343, + 479, + 361 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 365, + 392, + 427 + ], + "lines": [ + { + "bbox": [ + 132, + 366, + 273, + 379 + ], + "spans": [ + { + "bbox": [ + 132, + 366, + 219, + 379 + ], + "score": 1.0, + "content": "• A behavior tracker", + "type": "text" + }, + { + "bbox": [ + 219, + 366, + 269, + 379 + ], + "score": 0.92, + "content": "\\mathcal { M } ( \\Gamma _ { d } ^ { t } ; 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Consequently,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 512, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 506, + 524 + ], + "score": 1.0, + "content": "our approach includes an imitation learning process for recovering demonstrator’s policy and a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "reinforcement learning process for optimizing the probing policy. These two processes are intertwined", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "and influenced by each other: the IL process provides the behavior tracker guiding the probing policy", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 542, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 213, + 557 + ], + "score": 1.0, + "content": "while the RL process helps", + "type": "text" + }, + { + "bbox": [ + 213, + 544, + 224, + 554 + ], + "score": 0.25, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 542, + 506, + 557 + ], + "score": 1.0, + "content": "to observe more diverse behaviors from the demonstrator, thus enabling", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "an interactive learning scheme. Algorithm 2 in Appendix A summarizes the overall learning approach,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 563, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 134, + 578 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 565, + 145, + 574 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 563, + 506, + 578 + ], + "score": 1.0, + "content": "is the total number of training iterations. The optimization details for the two learning", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 575, + 253, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 253, + 586 + ], + "score": 1.0, + "content": "processes are introduced as follows.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 480, + 506, + 586 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 598, + 235, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 597, + 235, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 235, + 611 + ], + "score": 1.0, + "content": "3.2.1 IMITATION LEARNING", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 504, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 123, + 631 + ], + "score": 1.0, + "content": "For", + "type": "text" + }, + { + "bbox": [ + 123, + 617, + 133, + 627 + ], + "score": 0.33, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 615, + 505, + 631 + ], + "score": 1.0, + "content": ", we want to learn a good behavior tracker as well as the demonstrator’s policy. For this, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 628, + 380, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 380, + 640 + ], + "score": 1.0, + "content": "minimize a cross-entropy loss for predicting demonstrator’s actions:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 615, + 505, + 640 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 644, + 389, + 660 + ], + "lines": [ + { + "bbox": [ + 221, + 644, + 389, + 660 + ], + "spans": [ + { + "bbox": [ + 221, + 644, + 389, + 660 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\theta _ { M } , \\theta _ { d } ) = \\mathbb { E } \\left[ - \\log \\pi _ { d } ( a _ { d } ^ { t } | s _ { d } ^ { t } , m ^ { t - 1 } ) \\right] .", + "type": "interline_equation", + "image_path": "b59c9a697f4aa739947d7d41b2f555f240ff19973d0ea14bac94c86cbc450414.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 221, + 644, + 389, + 660 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 669, + 263, + 681 + ], + "lines": [ + { + "bbox": [ + 105, + 669, + 264, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 264, + 682 + ], + "score": 1.0, + "content": "3.2.2 REINFORCEMENT LEARNING", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "The goal of RL is to train a good probing policy that will maximize the change of behavior and/or", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 505, + 711 + ], + "score": 1.0, + "content": "discover new behaviors of the demonstrator to facilitate the imitation learning. Based on the reward", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 710, + 505, + 739 + ], + "spans": [ + { + "bbox": [ + 107, + 720, + 178, + 734 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathbb { E } \\left[ \\sum _ { \\tau = 0 } ^ { \\infty } \\gamma ^ { \\tau } r ^ { t + \\tau } \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 710, + 208, + 739 + ], + "score": 1.0, + "content": "n Eq. 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The policy gradient is", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 109, + 475, + 125 + ], + "lines": [ + { + "bbox": [ + 135, + 109, + 475, + 125 + ], + "spans": [ + { + "bbox": [ + 135, + 109, + 475, + 125 + ], + "score": 0.9, + "content": "\\nabla _ { \\theta _ { l } } J ( \\theta _ { l } ) = \\nabla _ { \\theta _ { l } } \\left[ \\log \\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , m ^ { t - 1 } ; \\theta _ { l } ) A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } ) + \\lambda \\mathcal { H } ( \\pi _ { l } ( \\cdot | s _ { l } ^ { t } , m ^ { t - 1 } ; \\theta _ { l } ) ) \\right] ,", + "type": "interline_equation", + "image_path": "94dedab7ea514b04f933d7460cb599c35bc4b9c5d7f1fb734fbed77af46661d2.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 135, + 109, + 475, + 125 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 506, + 166 + ], + "lines": [ + { + "bbox": [ + 102, + 128, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 102, + 128, + 134, + 148 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 131, + 198, + 144 + ], + "score": 0.92, + "content": "A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 128, + 358, + 148 + ], + "score": 1.0, + "content": "is the advantage estimation defined as", + "type": "text" + }, + { + "bbox": [ + 358, + 131, + 506, + 144 + ], + "score": 0.92, + "content": "\\begin{array} { r } { A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } ) = \\sum _ { \\tau = 0 } ^ { \\infty } \\gamma ^ { \\tau } r ^ { t + \\tau } \\ - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 141, + 507, + 158 + ], + "spans": [ + { + "bbox": [ + 107, + 144, + 158, + 156 + ], + "score": 0.91, + "content": "V ( s _ { l } ^ { t } , m ^ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 141, + 176, + 158 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 177, + 144, + 196, + 156 + ], + "score": 0.89, + "content": "\\mathcal { H } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 141, + 411, + 158 + ], + "score": 1.0, + "content": "is the entropy regularization weighted by the constant", + "type": "text" + }, + { + "bbox": [ + 412, + 145, + 450, + 154 + ], + "score": 0.88, + "content": "\\lambda = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 141, + 507, + 158 + ], + "score": 1.0, + "content": "for encourag-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 154, + 400, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 400, + 167 + ], + "score": 1.0, + "content": "ing exploration. The value function is updated by the following gradient:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 171, + 396, + 207 + ], + "lines": [ + { + "bbox": [ + 215, + 171, + 396, + 207 + ], + "spans": [ + { + "bbox": [ + 215, + 171, + 396, + 207 + ], + "score": 0.93, + "content": "\\nabla _ { \\theta _ { V } } \\frac { 1 } { 2 } \\left( \\sum _ { \\tau = 0 } ^ { \\infty } \\gamma ^ { \\tau } r ^ { t + \\tau } - V ( s _ { l } ^ { t } , m ^ { t - 1 } ; \\theta _ { V } ) \\right) ^ { 2 } .", + "type": "interline_equation", + "image_path": "3e55479b0119c52ec1c9bebaff1d83382c9c087a98b80f79ba2c1efc6e84f2ff.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 215, + 171, + 396, + 189.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 215, + 189.0, + 396, + 207.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 218, + 505, + 261 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "score": 1.0, + "content": "Note that when we update the probing policy and the value function, the behavior tracker is fixed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 379, + 240 + ], + "score": 1.0, + "content": "(i.e., no back propagation through the dashed path in Figure 2). Thus", + "type": "text" + }, + { + "bbox": [ + 380, + 229, + 394, + 239 + ], + "score": 0.9, + "content": "\\theta _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 228, + 505, + 240 + ], + "score": 1.0, + "content": "will only be updated by the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 318, + 251 + ], + "score": 1.0, + "content": "IL loss in Eq. (2). This is to ensure that the change of", + "type": "text" + }, + { + "bbox": [ + 318, + 239, + 331, + 249 + ], + "score": 0.88, + "content": "\\mathrm { \\Sigma } _ { m } \\bar { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 239, + 505, + 251 + ], + "score": 1.0, + "content": "is only caused by the change in policy or in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 248, + 409, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 390, + 263 + ], + "score": 1.0, + "content": "behaviors, and not by the change of the parameters of the mind model,", + "type": "text" + }, + { + "bbox": [ + 390, + 250, + 404, + 260 + ], + "score": 0.91, + "content": "\\theta _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 248, + 409, + 263 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 276, + 200, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 201, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 201, + 291 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 300, + 505, + 323 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 504, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 504, + 313 + ], + "score": 1.0, + "content": "To evaluate our approach, we introduce four tasks as shown in Figure 3, including three grid world", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "tasks (passing through obstacles, maze navigation, construction) and an algorithmic problem (sorting).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 328, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 342 + ], + "score": 1.0, + "content": "In order to test the generalization ability of the learned agent model, we adopt a strict training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 339, + 507, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 507, + 352 + ], + "score": 1.0, + "content": "procedure, where only one particular environment and task design is given during training. Specially,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "for grid world tasks, we fix the environment layout and/or item placement in each demonstration,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 358, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 374 + ], + "score": 1.0, + "content": "whereas for the sorting task, we use the exact same input array throughout the training. At testing", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "time, we randomize the task settings to an extent to create novel environments/inputs that have never", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 381, + 403, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 403, + 393 + ], + "score": 1.0, + "content": "been seen during training. We provide the specific settings in Section 4.1.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "We implement rule-based policies for the demonstrator i) by searching the best plan from the initial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "state to the goal state for the grid world tasks or ii) by the bubble sort algorithm for sorting. When", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 418, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 432 + ], + "score": 1.0, + "content": "there is no possible path to reach the goal (e.g., blocked by the learner), the demonstrator will stop", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 429, + 216, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 216, + 442 + ], + "score": 1.0, + "content": "until a viable path appears.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 453, + 160, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 161, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 161, + 466 + ], + "score": 1.0, + "content": "4.1 TASKS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "image", + "bbox": [ + 126, + 476, + 486, + 545 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 126, + 476, + 486, + 545 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 476, + 486, + 545 + ], + "spans": [ + { + "bbox": [ + 126, + 476, + 486, + 545 + ], + "score": 0.969, + "type": "image", + "image_path": "3b9d1a1dd367e5002b7172940644e658adf68fc36a8b0e845a272a5d536c86d3.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 126, + 476, + 486, + 499.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 126, + 499.0, + 486, + 522.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 126, + 522.0, + 486, + 545.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 219, + 552, + 393, + 564 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 216, + 551, + 393, + 564 + ], + "spans": [ + { + "bbox": [ + 216, + 551, + 393, + 564 + ], + "score": 1.0, + "content": "Figure 3: Illustration of the evaluated tasks.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + } + ], + "index": 28.0 + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "Passing. In this task modified from Baker et al. (2009), the demonstrator is initially located at the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 587, + 507, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 507, + 601 + ], + "score": 1.0, + "content": "bottom-right corner and is trying to pass through the closest gap to get into the upper part of the room.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "The demonstrator can take 5 actions including moving in four directions and stopping, whereas the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "score": 1.0, + "content": "learner can move in four directions, stop, and also pickup or put down a wall block. The training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "environment is shown in Figure 3, where the gap is always located at the left end of the wall in the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 630, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 643 + ], + "score": 1.0, + "content": "middle. In testing cases, we randomly place the location of the gap and the initial position of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 641, + 164, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 164, + 653 + ], + "score": 1.0, + "content": "demonstrator.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 657, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "Maze Navigation. Inspired by similar tasks in recent literature (Andreas et al., 2017), we place a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "few door blocks and tools (a key and a hammer) in a four-room maze, where the key can be used to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "open the yellow door but has no effect on the blue door, which must be broken by the hammer. The", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "demonstrator is trying to go from the top-right room to the top-left room. The two agents share the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "same action space including moving in four directions, picking up an item, and putting down an item.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 711, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 722 + ], + "score": 1.0, + "content": "Also, they can only carry one item at a time. In the training setting, the initial positions of both agents", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "and the door blocks are fixed as shown in Figure 3, whereas the tools may be randomly placed at only", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "For the policy optimization, we use Advantage Actor-Critic (A2C) (Mnih et al., 2016) to conduct", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 273, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 273, + 106 + ], + "score": 1.0, + "content": "on-policy training. The policy gradient is", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 109, + 475, + 125 + ], + "lines": [ + { + "bbox": [ + 135, + 109, + 475, + 125 + ], + "spans": [ + { + "bbox": [ + 135, + 109, + 475, + 125 + ], + "score": 0.9, + "content": "\\nabla _ { \\theta _ { l } } J ( \\theta _ { l } ) = \\nabla _ { \\theta _ { l } } \\left[ \\log \\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , m ^ { t - 1 } ; \\theta _ { l } ) A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } ) + \\lambda \\mathcal { H } ( \\pi _ { l } ( \\cdot | s _ { l } ^ { t } , m ^ { t - 1 } ; \\theta _ { l } ) ) \\right] ,", + "type": "interline_equation", + "image_path": "94dedab7ea514b04f933d7460cb599c35bc4b9c5d7f1fb734fbed77af46661d2.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 135, + 109, + 475, + 125 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 506, + 166 + ], + "lines": [ + { + "bbox": [ + 102, + 128, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 102, + 128, + 134, + 148 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 131, + 198, + 144 + ], + "score": 0.92, + "content": "A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 128, + 358, + 148 + ], + "score": 1.0, + "content": "is the advantage estimation defined as", + "type": "text" + }, + { + "bbox": [ + 358, + 131, + 506, + 144 + ], + "score": 0.92, + "content": "\\begin{array} { r } { A ( s _ { l } ^ { t } , m ^ { t - 1 } , a _ { l } ^ { t } ) = \\sum _ { \\tau = 0 } ^ { \\infty } \\gamma ^ { \\tau } r ^ { t + \\tau } \\ - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 141, + 507, + 158 + ], + "spans": [ + { + "bbox": [ + 107, + 144, + 158, + 156 + ], + "score": 0.91, + "content": "V ( s _ { l } ^ { t } , m ^ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 141, + 176, + 158 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 177, + 144, + 196, + 156 + ], + "score": 0.89, + "content": "\\mathcal { H } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 141, + 411, + 158 + ], + "score": 1.0, + "content": "is the entropy regularization weighted by the constant", + "type": "text" + }, + { + "bbox": [ + 412, + 145, + 450, + 154 + ], + "score": 0.88, + "content": "\\lambda = 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 141, + 507, + 158 + ], + "score": 1.0, + "content": "for encourag-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 154, + 400, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 400, + 167 + ], + "score": 1.0, + "content": "ing exploration. The value function is updated by the following gradient:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 102, + 128, + 507, + 167 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 215, + 171, + 396, + 207 + ], + "lines": [ + { + "bbox": [ + 215, + 171, + 396, + 207 + ], + "spans": [ + { + "bbox": [ + 215, + 171, + 396, + 207 + ], + "score": 0.93, + "content": "\\nabla _ { \\theta _ { V } } \\frac { 1 } { 2 } \\left( \\sum _ { \\tau = 0 } ^ { \\infty } \\gamma ^ { \\tau } r ^ { t + \\tau } - V ( s _ { l } ^ { t } , m ^ { t - 1 } ; \\theta _ { V } ) \\right) ^ { 2 } .", + "type": "interline_equation", + "image_path": "3e55479b0119c52ec1c9bebaff1d83382c9c087a98b80f79ba2c1efc6e84f2ff.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 215, + 171, + 396, + 189.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 215, + 189.0, + 396, + 207.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 218, + 505, + 261 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 506, + 231 + ], + "score": 1.0, + "content": "Note that when we update the probing policy and the value function, the behavior tracker is fixed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 379, + 240 + ], + "score": 1.0, + "content": "(i.e., no back propagation through the dashed path in Figure 2). Thus", + "type": "text" + }, + { + "bbox": [ + 380, + 229, + 394, + 239 + ], + "score": 0.9, + "content": "\\theta _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 228, + 505, + 240 + ], + "score": 1.0, + "content": "will only be updated by the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 318, + 251 + ], + "score": 1.0, + "content": "IL loss in Eq. (2). This is to ensure that the change of", + "type": "text" + }, + { + "bbox": [ + 318, + 239, + 331, + 249 + ], + "score": 0.88, + "content": "\\mathrm { \\Sigma } _ { m } \\bar { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 239, + 505, + 251 + ], + "score": 1.0, + "content": "is only caused by the change in policy or in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 248, + 409, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 390, + 263 + ], + "score": 1.0, + "content": "behaviors, and not by the change of the parameters of the mind model,", + "type": "text" + }, + { + "bbox": [ + 390, + 250, + 404, + 260 + ], + "score": 0.91, + "content": "\\theta _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 248, + 409, + 263 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 217, + 506, + 263 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 276, + 200, + 289 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 201, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 201, + 291 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 300, + 505, + 323 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 504, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 504, + 313 + ], + "score": 1.0, + "content": "To evaluate our approach, we introduce four tasks as shown in Figure 3, including three grid world", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 311, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 506, + 325 + ], + "score": 1.0, + "content": "tasks (passing through obstacles, maze navigation, construction) and an algorithmic problem (sorting).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 106, + 301, + 506, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 328, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 342 + ], + "score": 1.0, + "content": "In order to test the generalization ability of the learned agent model, we adopt a strict training", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 339, + 507, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 507, + 352 + ], + "score": 1.0, + "content": "procedure, where only one particular environment and task design is given during training. Specially,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "for grid world tasks, we fix the environment layout and/or item placement in each demonstration,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 358, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 374 + ], + "score": 1.0, + "content": "whereas for the sorting task, we use the exact same input array throughout the training. At testing", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "time, we randomize the task settings to an extent to create novel environments/inputs that have never", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 381, + 403, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 403, + 393 + ], + "score": 1.0, + "content": "been seen during training. We provide the specific settings in Section 4.1.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 326, + 507, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 409 + ], + "score": 1.0, + "content": "We implement rule-based policies for the demonstrator i) by searching the best plan from the initial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "state to the goal state for the grid world tasks or ii) by the bubble sort algorithm for sorting. When", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 418, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 432 + ], + "score": 1.0, + "content": "there is no possible path to reach the goal (e.g., blocked by the learner), the demonstrator will stop", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 429, + 216, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 216, + 442 + ], + "score": 1.0, + "content": "until a viable path appears.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 397, + 506, + 442 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 453, + 160, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 161, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 161, + 466 + ], + "score": 1.0, + "content": "4.1 TASKS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "image", + "bbox": [ + 126, + 476, + 486, + 545 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 126, + 476, + 486, + 545 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 476, + 486, + 545 + ], + "spans": [ + { + "bbox": [ + 126, + 476, + 486, + 545 + ], + "score": 0.969, + "type": "image", + "image_path": "3b9d1a1dd367e5002b7172940644e658adf68fc36a8b0e845a272a5d536c86d3.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 126, + 476, + 486, + 499.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 126, + 499.0, + 486, + 522.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 126, + 522.0, + 486, + 545.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 219, + 552, + 393, + 564 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 216, + 551, + 393, + 564 + ], + "spans": [ + { + "bbox": [ + 216, + 551, + 393, + 564 + ], + "score": 1.0, + "content": "Figure 3: Illustration of the evaluated tasks.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + } + ], + "index": 28.0 + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "Passing. In this task modified from Baker et al. (2009), the demonstrator is initially located at the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 587, + 507, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 507, + 601 + ], + "score": 1.0, + "content": "bottom-right corner and is trying to pass through the closest gap to get into the upper part of the room.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "The demonstrator can take 5 actions including moving in four directions and stopping, whereas the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "score": 1.0, + "content": "learner can move in four directions, stop, and also pickup or put down a wall block. The training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "environment is shown in Figure 3, where the gap is always located at the left end of the wall in the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 630, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 643 + ], + "score": 1.0, + "content": "middle. In testing cases, we randomly place the location of the gap and the initial position of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 641, + 164, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 164, + 653 + ], + "score": 1.0, + "content": "demonstrator.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 578, + 507, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 657, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 506, + 669 + ], + "score": 1.0, + "content": "Maze Navigation. Inspired by similar tasks in recent literature (Andreas et al., 2017), we place a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "few door blocks and tools (a key and a hammer) in a four-room maze, where the key can be used to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "open the yellow door but has no effect on the blue door, which must be broken by the hammer. The", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "demonstrator is trying to go from the top-right room to the top-left room. The two agents share the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "same action space including moving in four directions, picking up an item, and putting down an item.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 711, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 722 + ], + "score": 1.0, + "content": "Also, they can only carry one item at a time. In the training setting, the initial positions of both agents", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "and the door blocks are fixed as shown in Figure 3, whereas the tools may be randomly placed at only", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "a few locations. The rules in this environment are in fact fairly complex compared to other grid world", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 105 + ], + "score": 1.0, + "content": "tasks in previous work, where multiple sub-goals such as getting the tools, getting the door blocks,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 383, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 383, + 116 + ], + "score": 1.0, + "content": "placing the door blocks, and walking through the doors are involved.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 657, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "a few locations. The rules in this environment are in fact fairly complex compared to other grid world", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 105 + ], + "score": 1.0, + "content": "tasks in previous work, where multiple sub-goals such as getting the tools, getting the door blocks,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 383, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 383, + 116 + ], + "score": 1.0, + "content": "placing the door blocks, and walking through the doors are involved.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 119, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 119, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 505, + 133 + ], + "score": 1.0, + "content": "Construction. We adapt the stacking tasks in Shu et al. (2018) into a grid world, where the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 129, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 505, + 145 + ], + "score": 1.0, + "content": "demonstrator has a latent goal invisible to the learner, which is to construct a new block by putting", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 505, + 154 + ], + "score": 1.0, + "content": "two blocks with a specific color combination together. Three items are present in a room and they", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "score": 1.0, + "content": "are assigned with different colors randomly. In each episode, the demonstrator is randomly assigned", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 505, + 173 + ], + "score": 1.0, + "content": "with a goal (i.e., a pair of colors). It then seeks the needed blocks and puts one of them beside the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "score": 1.0, + "content": "other one. In order to predict the demonstrator’s actions precisely, the learner must infer the correct", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "goal first, which requires a sophisticated and dynamic agent modeling. Both agents share the same", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "action space as in Maze Navigation. In training, there are no obstacles in the room. To increase the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "difficulty of goal inference, in testing scenarios, we randomly place a few wall blocks as obstacles", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 215, + 214, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 214, + 226 + ], + "score": 1.0, + "content": "around the colored blocks.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "Sorting. Compared to a grid world, algorithmic problems are less visually informative and entail", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "more abstract reasoning. For this, we design a sorting task where an array with certain length is given", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "score": 1.0, + "content": "at the beginning. In our experiments, we set the length to be 10 and restrain the size of numbers in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "the array to be 4 bits (i.e., from 0 to 15). The demonstrator is able to perform a bubble sort algorithm", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "to rearrange the input array in an ascending order. Its action at each step is to select a pair of numbers", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "to swap. For every 5 steps done by the demonstrator, the learner can select a number and flipping one", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "of the bit of that number. Both agents can choose to do nothing for a step. During training, we only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "provide one constant array so that the sorting always starts from the same initial array. This is a very", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "challenging setting as only 10 out of 16 possible numbers are present in the training example and the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 325, + 498, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 498, + 339 + ], + "score": 1.0, + "content": "fixed ordering may also easily cause overfitting. For testing, we generate random arrays as inputs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 343, + 379, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 341, + 380, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 380, + 356 + ], + "score": 1.0, + "content": "For more details about the task settings, please refer to Appendix D.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 107, + 367, + 330, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 331, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 331, + 380 + ], + "score": 1.0, + "content": "4.2 GENERALIZATION IN UNSEEN TASK SETTINGS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "image", + "bbox": [ + 106, + 390, + 504, + 500 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 390, + 504, + 500 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 390, + 504, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 504, + 500 + ], + "score": 0.971, + "type": "image", + "image_path": "1871cb7eec22fea8ea3cee03fbce6bab4a89e5ce0e304bc11a26ace233854a63.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 106, + 390, + 504, + 426.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 106, + 426.6666666666667, + 504, + 463.33333333333337 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 106, + 463.33333333333337, + 504, + 500.00000000000006 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 108, + 508, + 502, + 520 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 507, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 504, + 522 + ], + "score": 1.0, + "content": "Figure 4: Action prediction accuracies in novel testing settings over numbers of training iterations.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + } + ], + "index": 27.0 + }, + { + "type": "image", + "bbox": [ + 156, + 537, + 455, + 644 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 156, + 537, + 455, + 644 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 156, + 537, + 455, + 644 + ], + "spans": [ + { + "bbox": [ + 156, + 537, + 455, + 644 + ], + "score": 0.968, + "type": "image", + "image_path": "013eb5aa25e3450135a48626efaa1b683544a12479c08644269d9f5f88ec7693.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 156, + 537, + 455, + 572.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 156, + 572.6666666666666, + 455, + 608.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 156, + 608.3333333333333, + 455, + 643.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 653, + 505, + 675 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "Figure 5: Action prediction accuracies in novel testing settings over numbers of training iterations", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 663, + 212, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 126, + 675 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 664, + 146, + 674 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 663, + 212, + 675 + ], + "score": 1.0, + "content": "random actions.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + } + ], + "index": 31.25 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 703 + ], + "score": 1.0, + "content": "One of the main goals of learning to probe is to learn a good agent model that can be generalized to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "unseen scenarios. To evaluate how accurate our agent model is for approximating the true mind of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "the demonstrator, we may test the accuracy of predicting the demonstrator’s actions using the learned", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 722, + 118, + 732 + ], + "score": 0.85, + "content": "\\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 720, + 204, + 734 + ], + "score": 1.0, + "content": "and behavior tracker", + "type": "text" + }, + { + "bbox": [ + 205, + 721, + 228, + 732 + ], + "score": 0.91, + "content": "\\mathcal { M } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "in testing task settings unseen by the learner during training. A high", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 115 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 116 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 119, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 119, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 505, + 133 + ], + "score": 1.0, + "content": "Construction. We adapt the stacking tasks in Shu et al. (2018) into a grid world, where the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 129, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 505, + 145 + ], + "score": 1.0, + "content": "demonstrator has a latent goal invisible to the learner, which is to construct a new block by putting", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 505, + 154 + ], + "score": 1.0, + "content": "two blocks with a specific color combination together. Three items are present in a room and they", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "score": 1.0, + "content": "are assigned with different colors randomly. In each episode, the demonstrator is randomly assigned", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 505, + 173 + ], + "score": 1.0, + "content": "with a goal (i.e., a pair of colors). It then seeks the needed blocks and puts one of them beside the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "score": 1.0, + "content": "other one. In order to predict the demonstrator’s actions precisely, the learner must infer the correct", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "goal first, which requires a sophisticated and dynamic agent modeling. Both agents share the same", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "action space as in Maze Navigation. In training, there are no obstacles in the room. To increase the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "difficulty of goal inference, in testing scenarios, we randomly place a few wall blocks as obstacles", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 215, + 214, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 214, + 226 + ], + "score": 1.0, + "content": "around the colored blocks.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 119, + 506, + 226 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "Sorting. Compared to a grid world, algorithmic problems are less visually informative and entail", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "more abstract reasoning. For this, we design a sorting task where an array with certain length is given", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "score": 1.0, + "content": "at the beginning. In our experiments, we set the length to be 10 and restrain the size of numbers in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "the array to be 4 bits (i.e., from 0 to 15). The demonstrator is able to perform a bubble sort algorithm", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "to rearrange the input array in an ascending order. Its action at each step is to select a pair of numbers", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "to swap. For every 5 steps done by the demonstrator, the learner can select a number and flipping one", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "of the bit of that number. Both agents can choose to do nothing for a step. During training, we only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "provide one constant array so that the sorting always starts from the same initial array. This is a very", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "challenging setting as only 10 out of 16 possible numbers are present in the training example and the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 325, + 498, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 498, + 339 + ], + "score": 1.0, + "content": "fixed ordering may also easily cause overfitting. For testing, we generate random arrays as inputs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 230, + 505, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 343, + 379, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 341, + 380, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 380, + 356 + ], + "score": 1.0, + "content": "For more details about the task settings, please refer to Appendix D.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 341, + 380, + 356 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 367, + 330, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 331, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 331, + 380 + ], + "score": 1.0, + "content": "4.2 GENERALIZATION IN UNSEEN TASK SETTINGS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "image", + "bbox": [ + 106, + 390, + 504, + 500 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 390, + 504, + 500 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 390, + 504, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 504, + 500 + ], + "score": 0.971, + "type": "image", + "image_path": "1871cb7eec22fea8ea3cee03fbce6bab4a89e5ce0e304bc11a26ace233854a63.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 106, + 390, + 504, + 426.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 106, + 426.6666666666667, + 504, + 463.33333333333337 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 106, + 463.33333333333337, + 504, + 500.00000000000006 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 108, + 508, + 502, + 520 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 507, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 504, + 522 + ], + "score": 1.0, + "content": "Figure 4: Action prediction accuracies in novel testing settings over numbers of training iterations.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + } + ], + "index": 27.0 + }, + { + "type": "image", + "bbox": [ + 156, + 537, + 455, + 644 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 156, + 537, + 455, + 644 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 156, + 537, + 455, + 644 + ], + "spans": [ + { + "bbox": [ + 156, + 537, + 455, + 644 + ], + "score": 0.968, + "type": "image", + "image_path": "013eb5aa25e3450135a48626efaa1b683544a12479c08644269d9f5f88ec7693.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 156, + 537, + 455, + 572.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 156, + 572.6666666666666, + 455, + 608.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 156, + 608.3333333333333, + 455, + 643.9999999999999 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 653, + 505, + 675 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "Figure 5: Action prediction accuracies in novel testing settings over numbers of training iterations", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 663, + 212, + 675 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 126, + 675 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 664, + 146, + 674 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 663, + 212, + 675 + ], + "score": 1.0, + "content": "random actions.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + } + ], + "index": 31.25 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 703 + ], + "score": 1.0, + "content": "One of the main goals of learning to probe is to learn a good agent model that can be generalized to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "unseen scenarios. To evaluate how accurate our agent model is for approximating the true mind of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "the demonstrator, we may test the accuracy of predicting the demonstrator’s actions using the learned", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 722, + 118, + 732 + ], + "score": 0.85, + "content": "\\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 720, + 204, + 734 + ], + "score": 1.0, + "content": "and behavior tracker", + "type": "text" + }, + { + "bbox": [ + 205, + 721, + 228, + 732 + ], + "score": 0.91, + "content": "\\mathcal { M } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "in testing task settings unseen by the learner during training. A high", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 106, + 689, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 104 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "prediction accuracy in unseen settings will indicate good generalization of the learned agent model.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 500, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 500, + 107 + ], + "score": 1.0, + "content": "To eliminate the effects from probing, we remove the learner from the environment during testing.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 109, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 121 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 121 + ], + "score": 1.0, + "content": "We compare our model with six baselines: i) ToMnet in Rabinowitz et al. (2018), which learns the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 120, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 120, + 506, + 133 + ], + "score": 1.0, + "content": "demonstrator’s model by only observing the given demonstrations without interactions, ii) our model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 505, + 142 + ], + "score": 1.0, + "content": "without training the probing policy using RL (i.e., the leaner always takes random actions), iii), ours", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 139, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 384, + 155 + ], + "score": 1.0, + "content": "without the attention-based fusion (concatenating state feature and", + "type": "text" + }, + { + "bbox": [ + 384, + 141, + 407, + 151 + ], + "score": 0.91, + "content": "m ^ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 139, + 506, + 155 + ], + "score": 1.0, + "content": "instead), iv) using two", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 151, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 164 + ], + "score": 1.0, + "content": "LSTMs for the estimated demonstrator’s policy and the probing policy respectively (Figure 16), v)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 162, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 506, + 175 + ], + "score": 1.0, + "content": "using count-based bonus as reward (Strehl & Littman, 2008; Bellemare et al., 2016; Tang et al., 2017),", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 506, + 186 + ], + "score": 1.0, + "content": "and vi) using cross-entropy loss for action prediction as reward (i.e., exploration by self-supervised", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 181, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 506, + 198 + ], + "score": 1.0, + "content": "prediction in Pathak et al. (2017)). To ensure fair comparison, the training settings and the testing", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 194, + 474, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 474, + 207 + ], + "score": 1.0, + "content": "settings are shared by all methods. We provide more details of the baselines in Appendix E.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 210, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "Figure 4 shows the predication accuracy in testing settings of the three approaches based on the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 234 + ], + "score": 1.0, + "content": "models from different training iterations. It is clear that with more iterations, our probing policy can", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "greatly help increase the accuracy by discovering new behaviors, and consequently yields much higher", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "testing accuracy than the baselines do. The results of “ours w/o fusion” and “2-LSTM” baselines", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "further demonstrates the importance of our attention-based fusion layer and the use of a separate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 240, + 276 + ], + "score": 1.0, + "content": "behavior tracker. By randomizing", + "type": "text" + }, + { + "bbox": [ + 240, + 263, + 259, + 273 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "of demonstrator’s actions (Figure 5), we show that the probing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "policy can also handle stochastic and sub-optimal policies. It can be clearly seen from the results that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 284, + 507, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 507, + 297 + ], + "score": 1.0, + "content": "the performance of the two baselines based on different curiosity rewards is clearly inferior to ours,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "which demonstrates the advantage of defining the behavioral change as the intrinsic reward for the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 104, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "purpose of agent modeling. We have also evaluated the robustness of our approach by showing the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 314, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 329 + ], + "score": 1.0, + "content": "standard deviation from multiple runs as shown in Figure 9, which demonstrates a reasonably low", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 326, + 227, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 227, + 338 + ], + "score": 1.0, + "content": "variance across multiple runs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 108, + 342, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 360, + 355 + ], + "score": 1.0, + "content": "We demonstrate the effect of dimensionality of the latent vector", + "type": "text" + }, + { + "bbox": [ + 360, + 342, + 373, + 353 + ], + "score": 0.88, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "(i.e., the complexity of the agent", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "model) in Figure 4. In simple environments, our approach still outperforms the baselines even when", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "the dimensionality is decreased from 8 to 2 or 4. In more complex tasks like Sorting, a higher", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 372, + 297, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 297, + 388 + ], + "score": 1.0, + "content": "dimension is necessary for the agent modeling.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 107, + 398, + 331, + 409 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 332, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 332, + 411 + ], + "score": 1.0, + "content": "4.3 EVOLUTION OF LEARNED PROBING STRATEGY", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "As training progresses, we observe that our probing policy is able to progressively discover new", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "behaviors through interactions that are adapted to the demonstrator’s policy. For instance, in Maze", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "score": 1.0, + "content": "Navigation, we find that the learner first learns to place one door, then gradually learns to place two", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "doors at the appropriate moments to force the demonstrator to go back and forth to get the needed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "tools for opening the doors. Finally, the probing policy will even blocks the demonstrator for a while", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 470, + 474, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 474, + 483 + ], + "score": 1.0, + "content": "before it goes through the last door. Due to the space limit, we show this in the demo video.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 108, + 487, + 503, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "We also provide more analysis and visualization of the probing behavior and the resulting latent", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 498, + 284, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 284, + 509 + ], + "score": 1.0, + "content": "vectors in Appendix B.2 and Appendix B.3.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 108, + 521, + 385, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 386, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 386, + 533 + ], + "score": 1.0, + "content": "4.4 EMERGENCE OF OBSTRUCTIVE BEHAVIORS FROM PROBING", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "Although we never explicitly set an adversarial goal for the learner, we do observe a natural emergence", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "of obstructive behaviors caused by the probing, which can be quantitatively measured by the success", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "rate of the demonstrator within a time limit as shown in Figure 6. This phenomenon is aligned with", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "common sense that the optimal probing policy to discover new behaviors of the demonstrator should", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "constantly force the demonstrator to change its plan, which will eventually delay the time when the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "demonstrator finishes the task. Because of the reward defined in Eq. (1), the probing policy learned", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "from RL is also maximizing the accumulated behavioral change of the demonstrator just like the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 614, + 411, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 614, + 411, + 629 + ], + "score": 1.0, + "content": "common sense. 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(2017)). To ensure fair comparison, the training settings and the testing", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 194, + 474, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 474, + 207 + ], + "score": 1.0, + "content": "settings are shared by all methods. We provide more details of the baselines in Appendix E.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 110, + 506, + 207 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 210, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "Figure 4 shows the predication accuracy in testing settings of the three approaches based on the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 234 + ], + "score": 1.0, + "content": "models from different training iterations. It is clear that with more iterations, our probing policy can", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "greatly help increase the accuracy by discovering new behaviors, and consequently yields much higher", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "testing accuracy than the baselines do. The results of “ours w/o fusion” and “2-LSTM” baselines", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "further demonstrates the importance of our attention-based fusion layer and the use of a separate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 240, + 276 + ], + "score": 1.0, + "content": "behavior tracker. By randomizing", + "type": "text" + }, + { + "bbox": [ + 240, + 263, + 259, + 273 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "of demonstrator’s actions (Figure 5), we show that the probing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "policy can also handle stochastic and sub-optimal policies. It can be clearly seen from the results that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 284, + 507, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 507, + 297 + ], + "score": 1.0, + "content": "the performance of the two baselines based on different curiosity rewards is clearly inferior to ours,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "which demonstrates the advantage of defining the behavioral change as the intrinsic reward for the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 104, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "purpose of agent modeling. We have also evaluated the robustness of our approach by showing the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 314, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 329 + ], + "score": 1.0, + "content": "standard deviation from multiple runs as shown in Figure 9, which demonstrates a reasonably low", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 326, + 227, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 227, + 338 + ], + "score": 1.0, + "content": "variance across multiple runs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 210, + 507, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 342, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 342, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 360, + 355 + ], + "score": 1.0, + "content": "We demonstrate the effect of dimensionality of the latent vector", + "type": "text" + }, + { + "bbox": [ + 360, + 342, + 373, + 353 + ], + "score": 0.88, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 342, + 505, + 355 + ], + "score": 1.0, + "content": "(i.e., the complexity of the agent", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "model) in Figure 4. In simple environments, our approach still outperforms the baselines even when", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "the dimensionality is decreased from 8 to 2 or 4. In more complex tasks like Sorting, a higher", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 372, + 297, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 297, + 388 + ], + "score": 1.0, + "content": "dimension is necessary for the agent modeling.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 342, + 505, + 388 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 398, + 331, + 409 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 332, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 332, + 411 + ], + "score": 1.0, + "content": "4.3 EVOLUTION OF LEARNED PROBING STRATEGY", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "As training progresses, we observe that our probing policy is able to progressively discover new", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "behaviors through interactions that are adapted to the demonstrator’s policy. For instance, in Maze", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "score": 1.0, + "content": "Navigation, we find that the learner first learns to place one door, then gradually learns to place two", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "doors at the appropriate moments to force the demonstrator to go back and forth to get the needed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "tools for opening the doors. Finally, the probing policy will even blocks the demonstrator for a while", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 470, + 474, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 474, + 483 + ], + "score": 1.0, + "content": "before it goes through the last door. Due to the space limit, we show this in the demo video.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 418, + 505, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 487, + 503, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 500 + ], + "score": 1.0, + "content": "We also provide more analysis and visualization of the probing behavior and the resulting latent", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 498, + 284, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 284, + 509 + ], + "score": 1.0, + "content": "vectors in Appendix B.2 and Appendix B.3.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 486, + 505, + 509 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 521, + 385, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 521, + 386, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 386, + 533 + ], + "score": 1.0, + "content": "4.4 EMERGENCE OF OBSTRUCTIVE BEHAVIORS FROM PROBING", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "Although we never explicitly set an adversarial goal for the learner, we do observe a natural emergence", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "of obstructive behaviors caused by the probing, which can be quantitatively measured by the success", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "rate of the demonstrator within a time limit as shown in Figure 6. This phenomenon is aligned with", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "common sense that the optimal probing policy to discover new behaviors of the demonstrator should", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "constantly force the demonstrator to change its plan, which will eventually delay the time when the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "demonstrator finishes the task. Because of the reward defined in Eq. (1), the probing policy learned", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "from RL is also maximizing the accumulated behavioral change of the demonstrator just like the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 614, + 411, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 614, + 411, + 629 + ], + "score": 1.0, + "content": "common sense. 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Unless", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 267, + 474, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 474, + 280 + ], + "score": 1.0, + "content": "specified, the evaluated policies were learned from demonstrations without random actions.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "table_body", + "bbox": [ + 122, + 288, + 488, + 396 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 288, + 488, + 396 + ], + "spans": [ + { + "bbox": [ + 122, + 288, + 488, + 396 + ], + "score": 0.985, + "html": "
MethodPassingMaze NavigationConstructionSorting
Ours0.710.600.480.82
Ours (from sub-optimal demo)0.680.510.47N/A
Ours w/o RL0.130.310.230.80
Ours w/o fusion0.110.190.290.10
Ours (2-LSTM)0.170.020.170
Passive (ToMnet)0.1100.120
Count-based0.2200.310.39
Self-supervised0.2300.360.56
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For acceleration, it is common to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "distill the optimal plans to a policy net (Lazaric et al., 2010; Guo et al., 2014). However, the distilled", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "policy net may not generalize well in new scenarios if the training settings are not diverse enough.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 444, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 459 + ], + "score": 1.0, + "content": "Thus, the nature of our approach makes it suitable for improving the generalization without manually", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 455, + 287, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 287, + 469 + ], + "score": 1.0, + "content": "designing a large number of diverse settings.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 505, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 473, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 399, + 484 + ], + "score": 1.0, + "content": "For this, we evaluate the success rates when the learner directly uses", + "type": "text" + }, + { + "bbox": [ + 400, + 474, + 412, + 483 + ], + "score": 0.85, + "content": "\\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 473, + 505, + 484 + ], + "score": 1.0, + "content": "(with an 8-dim latent", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "vector) to perform tasks in testing settings without finetuning. 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MethodPassingMaze NavigationConstructionSorting
Ours0.710.600.480.82
Ours (from sub-optimal demo)0.680.510.47N/A
Ours w/o RL0.130.310.230.80
Ours w/o fusion0.110.190.290.10
Ours (2-LSTM)0.170.020.170
Passive (ToMnet)0.1100.120
Count-based0.2200.310.39
Self-supervised0.2300.360.56
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From the curves, we may see that the policy trained with", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 423, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 423, + 116 + ], + "score": 1.0, + "content": "our interactively learned agent model significantly outperforms both baselines.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 108, + 128, + 264, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 127, + 267, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 267, + 141 + ], + "score": 1.0, + "content": "4.7 APPLICATION 3: COMPETITION", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 148, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 106, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 506, + 161 + ], + "score": 1.0, + "content": "Similar to Section 4.6, we design a competitive task based on the Construction task, where the learner", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 172 + ], + "score": 1.0, + "content": "gets a 0.05 reward for every step and a -1.0 penalty if the opponent achieves its goal. We adopt the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "same training procedure as in Section 4.6, and also rescale the reward according to the maximum", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 179, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 505, + 194 + ], + "score": 1.0, + "content": "reward. As Figure 8 shows, the mind model learned by our approach improves the learning efficiency", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 190, + 285, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 285, + 203 + ], + "score": 1.0, + "content": "and the converged reward by a large margin.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 217, + 201, + 230 + ], + "lines": [ + { + "bbox": [ + 104, + 214, + 203, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 203, + 233 + ], + "score": 1.0, + "content": "5 CONCLUSIONS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 241, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "In this work, we have proposed a novel agent modeling approach, i.e., probing-based interactive agent", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "modeling. The core idea is to learn a probing policy using only a curiosity-driven reward, which is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 261, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 276 + ], + "score": 1.0, + "content": "able to discover new behaviors of the target agent. We achieve this by incorporating two learning", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 151, + 286 + ], + "score": 1.0, + "content": "processes", + "type": "text" + }, + { + "bbox": [ + 152, + 274, + 163, + 284 + ], + "score": 0.28, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "and RL) together. We are able to validate our approach in four distinct tasks. The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "score": 1.0, + "content": "results show that by learning a probing policy, the learner in our approach can build a more accurate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "agent model of the demonstrator. Thanks to this interactively learned agent model, the learner is able", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "to i) approximate the demonstrator’s policy more accurately in unseen settings compared to passive", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "agent modeling, ii) efficiently learn a good collaborative policy to help the demonstrator, and iii)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 326, + 363, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 363, + 339 + ], + "score": 1.0, + "content": "develop an adversarial policy to compete with the demonstrator.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 503, + 364 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 356 + ], + "score": 1.0, + "content": "In the future, we can extend this framework to simultaneous agent modeling and world modeling", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 277, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 277, + 365 + ], + "score": 1.0, + "content": "with a more complex mind representation.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 107, + 380, + 175, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 176, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 176, + 394 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 105, + 397, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 506, + 410 + ], + "score": 1.0, + "content": "Maruan Al-Shedivat, Trapit Bansal, Yuri Burda, Ilya Sutskever, Igor Mordatch, and Pieter Abbeel.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 116, + 409, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 116, + 409, + 506, + 422 + ], + "score": 1.0, + "content": "Continuous adaptation via meta-learning in nonstationary and competitive environments. 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We adopt the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "same training procedure as in Section 4.6, and also rescale the reward according to the maximum", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 179, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 505, + 194 + ], + "score": 1.0, + "content": "reward. As Figure 8 shows, the mind model learned by our approach improves the learning efficiency", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 190, + 285, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 285, + 203 + ], + "score": 1.0, + "content": "and the converged reward by a large margin.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 147, + 506, + 203 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 217, + 201, + 230 + ], + "lines": [ + { + "bbox": [ + 104, + 214, + 203, + 233 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 203, + 233 + ], + "score": 1.0, + "content": "5 CONCLUSIONS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 241, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "In this work, we have proposed a novel agent modeling approach, i.e., probing-based interactive agent", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "modeling. The core idea is to learn a probing policy using only a curiosity-driven reward, which is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 261, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 276 + ], + "score": 1.0, + "content": "able to discover new behaviors of the target agent. We achieve this by incorporating two learning", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 273, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 151, + 286 + ], + "score": 1.0, + "content": "processes", + "type": "text" + }, + { + "bbox": [ + 152, + 274, + 163, + 284 + ], + "score": 0.28, + "content": "\\mathrm { I L }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 273, + 506, + 286 + ], + "score": 1.0, + "content": "and RL) together. We are able to validate our approach in four distinct tasks. The", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 298 + ], + "score": 1.0, + "content": "results show that by learning a probing policy, the learner in our approach can build a more accurate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "agent model of the demonstrator. Thanks to this interactively learned agent model, the learner is able", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "to i) approximate the demonstrator’s policy more accurately in unseen settings compared to passive", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "agent modeling, ii) efficiently learn a good collaborative policy to help the demonstrator, and iii)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 326, + 363, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 363, + 339 + ], + "score": 1.0, + "content": "develop an adversarial policy to compete with the demonstrator.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 241, + 506, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 503, + 364 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 356 + ], + "score": 1.0, + "content": "In the future, we can extend this framework to simultaneous agent modeling and world modeling", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 277, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 277, + 365 + ], + "score": 1.0, + "content": "with a more complex mind representation.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 340, + 505, + 365 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 380, + 175, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 176, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 176, + 394 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "list", + "bbox": [ + 105, + 397, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 506, + 410 + ], + "score": 1.0, + "content": "Maruan Al-Shedivat, Trapit Bansal, Yuri Burda, Ilya Sutskever, Igor Mordatch, and Pieter Abbeel.", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 116, + 409, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 116, + 409, + 506, + 422 + ], + "score": 1.0, + "content": "Continuous adaptation via meta-learning in nonstationary and competitive environments. 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Finally, an FC layer takes in the hidden state from the LSTM and outputs the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 565, + 282, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 158, + 578 + ], + "score": 1.0, + "content": "latent vector", + "type": "text" + }, + { + "bbox": [ + 158, + 565, + 172, + 576 + ], + "score": 0.89, + "content": "m ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 565, + 282, + 578 + ], + "score": 1.0, + "content": "as the mind representation.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 491, + 506, + 578 + ] + }, + { + "type": "image", + "bbox": [ + 226, + 591, + 370, + 710 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 226, + 591, + 370, + 710 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 226, + 591, + 370, + 710 + ], + "spans": [ + { + "bbox": [ + 226, + 591, + 370, + 710 + ], + "score": 0.953, + "type": "image", + "image_path": "9601707c6b27d9000ede144af1654db03552ded43028f6a05face04cd1b1a543.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 226, + 591, + 370, + 650.5 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 226, + 650.5, + 370, + 710.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 212, + 717, + 399, + 729 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 210, + 717, + 399, + 730 + ], + "spans": [ + { + "bbox": [ + 210, + 717, + 399, + 730 + ], + "score": 1.0, + "content": "Figure 12: The attention-based fusion module.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + } + ], + "index": 31.25 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Fusion. As shown in Figure 12, we design our fusion module using an attention based mechanism", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 434, + 105 + ], + "score": 1.0, + "content": "similar to the one introduced by Chaplot et al. (2017), where the latent vector", + "type": "text" + }, + { + "bbox": [ + 434, + 92, + 458, + 103 + ], + "score": 0.91, + "content": "m ^ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 92, + 505, + 105 + ], + "score": 1.0, + "content": "is fed into", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 214, + 117 + ], + "score": 1.0, + "content": "an FC layer outputting an", + "type": "text" + }, + { + "bbox": [ + 214, + 104, + 225, + 114 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "-dim attention vector (each element is from 0 to 1) corresponding to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 122, + 127 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 122, + 115, + 132, + 124 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 113, + 306, + 127 + ], + "score": 1.0, + "content": "feature maps from the state encoder (here", + "type": "text" + }, + { + "bbox": [ + 306, + 115, + 341, + 124 + ], + "score": 0.88, + "content": "N = 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "). Formally, we have an attention vector", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 507, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 190, + 136 + ], + "score": 0.92, + "content": "h = \\sigma ( m ^ { t - 1 } ) \\in \\dot { \\mathbb { R } } ^ { 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 122, + 222, + 139 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 222, + 125, + 239, + 136 + ], + "score": 0.89, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 122, + 399, + 139 + ], + "score": 1.0, + "content": "is an FC layer with sigmoid activation.", + "type": "text" + }, + { + "bbox": [ + 399, + 125, + 407, + 135 + ], + "score": 0.74, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 122, + 507, + 139 + ], + "score": 1.0, + "content": "is spatially expanded to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 133, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 104, + 133, + 114, + 150 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 114, + 137, + 170, + 147 + ], + "score": 0.89, + "content": "H \\times W \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 133, + 200, + 150 + ], + "score": 1.0, + "content": "tensor,", + "type": "text" + }, + { + "bbox": [ + 201, + 136, + 282, + 148 + ], + "score": 0.92, + "content": "\\pmb { H } ( h ) \\in \\mathbb { R } ^ { H \\times W \\times 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 133, + 378, + 150 + ], + "score": 1.0, + "content": ", where the elements in", + "type": "text" + }, + { + "bbox": [ + 378, + 137, + 385, + 147 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 133, + 506, + 150 + ], + "score": 1.0, + "content": "-th channel correspond to the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 146, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 107, + 147, + 113, + 157 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 146, + 168, + 159 + ], + "score": 1.0, + "content": "-th element in", + "type": "text" + }, + { + "bbox": [ + 169, + 147, + 176, + 157 + ], + "score": 0.78, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 146, + 505, + 159 + ], + "score": 1.0, + "content": ". We then reweight each feature maps using the attention vector, which becomes the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 182, + 171 + ], + "score": 1.0, + "content": "fusion output. I.e.,", + "type": "text" + }, + { + "bbox": [ + 182, + 156, + 347, + 169 + ], + "score": 0.93, + "content": "f ( \\phi ( s ^ { t } ) , m ^ { t - \\tilde { 1 } } ) = \\phi ( s ^ { t } ) \\odot H ( \\sigma \\tilde { ( } m ^ { t - 1 } ) \\mathbf { \\tilde { ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 155, + 377, + 171 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 378, + 157, + 400, + 169 + ], + "score": 0.93, + "content": "\\phi ( s ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 155, + 506, + 171 + ], + "score": 1.0, + "content": "are the feature maps from", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 167, + 401, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 178, + 181 + ], + "score": 1.0, + "content": "the state encoder,", + "type": "text" + }, + { + "bbox": [ + 178, + 169, + 196, + 180 + ], + "score": 0.9, + "content": "f ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 167, + 289, + 181 + ], + "score": 1.0, + "content": "is the fusion layer, and", + "type": "text" + }, + { + "bbox": [ + 290, + 170, + 299, + 178 + ], + "score": 0.83, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 167, + 401, + 181 + ], + "score": 1.0, + "content": "is element-wise product.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 184, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 184, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 506, + 196 + ], + "score": 1.0, + "content": "Policy. The input of this module is the flattened output from the fusion module, and is fed to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 195, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 506, + 207 + ], + "score": 1.0, + "content": "an LSTM with 128 hidden units followed by an FC layer with softmax activation. The result-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 385, + 218 + ], + "score": 1.0, + "content": "ing output is an action distribution representing the policy (either", + "type": "text" + }, + { + "bbox": [ + 385, + 207, + 397, + 217 + ], + "score": 0.85, + "content": "\\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 205, + 411, + 218 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 412, + 207, + 422, + 217 + ], + "score": 0.8, + "content": "\\pi _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "). For Sorting task,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "score": 1.0, + "content": "we slightly modify the output to fit the problem. We decompose the demonstrator’s policy as", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 222, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 351, + 242 + ], + "score": 0.9, + "content": "{ \\pi } _ { d } ( a _ { d } ^ { t } \\vert s _ { d } ^ { t } , m ^ { t - 1 } ) ~ = ~ { \\pi } _ { d } ^ { ( 1 ) } ( a _ { d } ^ { \\bar { t } , 1 } \\vert s _ { d } ^ { t } , m ^ { t - 1 } ) { \\pi } _ { d } ^ { ( 2 ) } ( a _ { d } ^ { t , 2 } \\vert s _ { d } ^ { t } , m ^ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 222, + 384, + 248 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 384, + 227, + 401, + 242 + ], + "score": 0.91, + "content": "a _ { d } ^ { t , 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 222, + 422, + 248 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 422, + 227, + 439, + 242 + ], + "score": 0.91, + "content": "a _ { d } ^ { t , 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 222, + 505, + 248 + ], + "score": 1.0, + "content": "are the indices", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "of the numbers the demonstrator chooses to swap. For the learner’s policy, it is decomposed as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 246, + 509, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 333, + 266 + ], + "score": 0.89, + "content": "\\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , m ^ { t - 1 } ) = \\pi _ { l } ^ { \\mathrm { i d } } ( a _ { l } ^ { t , 1 } | s _ { l } ^ { t } , m ^ { t - 1 } ) \\pi _ { l } ^ { \\mathrm { b i t } } ( a _ { l } ^ { t , 2 } | s _ { l } ^ { t } , m ^ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 246, + 398, + 272 + ], + "score": 1.0, + "content": "instead, where", + "type": "text" + }, + { + "bbox": [ + 398, + 251, + 415, + 265 + ], + "score": 0.92, + "content": "a _ { l } ^ { t , 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 246, + 509, + 272 + ], + "score": 1.0, + "content": "indicates the number", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 103, + 260, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 103, + 260, + 252, + 282 + ], + "score": 1.0, + "content": "that the learner selects to change and", + "type": "text" + }, + { + "bbox": [ + 253, + 264, + 268, + 279 + ], + "score": 0.9, + "content": "a _ { l } ^ { t , 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 260, + 487, + 282 + ], + "score": 1.0, + "content": "is the bit of that number that needs to be flipped. When", + "type": "text" + }, + { + "bbox": [ + 487, + 264, + 504, + 279 + ], + "score": 0.9, + "content": "a _ { d } ^ { t , 1 }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 102, + 274, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 102, + 274, + 117, + 294 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 117, + 277, + 133, + 291 + ], + "score": 0.93, + "content": "a _ { l } ^ { t , 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 274, + 507, + 294 + ], + "score": 1.0, + "content": "is larger than the length of the array, it means that the demonstrator or the learner is choosing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 289, + 214, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 214, + 303 + ], + "score": 1.0, + "content": "to do nothing respectively.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 306, + 506, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "Value. We also have a value net designed for training the learner’s policy using A2C (i.e.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 315, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 107, + 316, + 161, + 329 + ], + "score": 0.91, + "content": "V ( s _ { l } ^ { t } , m ^ { t - 1 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 315, + 506, + 330 + ], + "score": 1.0, + "content": ", which takes in the hidden state from the LSTM in the learner’s policy module and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 327, + 267, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 267, + 340 + ], + "score": 1.0, + "content": "outputs a scalar value after an FC layer.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 343, + 504, + 366 + ], + "lines": [ + { + "bbox": [ + 107, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 107, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "The network is trained with RMSProp (Tieleman & Hinto, 2012) using a learning rate of 0.001.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 173, + 367 + ], + "score": 1.0, + "content": "During training,", + "type": "text" + }, + { + "bbox": [ + 173, + 356, + 178, + 365 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 354, + 349, + 367 + ], + "score": 1.0, + "content": "-greedy is applied to the rollout, where the", + "type": "text" + }, + { + "bbox": [ + 349, + 356, + 355, + 364 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "gradually decreases from 0.1 to 0.01.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 108, + 380, + 211, + 394 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 213, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 213, + 396 + ], + "score": 1.0, + "content": "D TASK SETTINGS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 504, + 427 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "We assume full observations of the world state for both agents in all tasks but the internal state of an", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 416, + 491, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 448, + 428 + ], + "score": 1.0, + "content": "agent (e.g., goals) is unobservable to another agent. The discounted factor is set to be", + "type": "text" + }, + { + "bbox": [ + 448, + 416, + 487, + 426 + ], + "score": 0.88, + "content": "\\Gamma = 0 . 9 5", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 416, + 491, + 428 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "For the demonstrator in the grid world tasks, we implemented search based path planning and used", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "simple heuristics to perform branch and bound for acceleration. In particular, the state for the search", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "algorithm in Passing is the map status, whereas the state in Maze Navigation and Construction is the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 463, + 350, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 350, + 478 + ], + "score": 1.0, + "content": "combination of map status and the demonstrator’s inventory.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 107, + 488, + 171, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 172, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 172, + 501 + ], + "score": 1.0, + "content": "D.1 PASSING", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "image", + "bbox": [ + 145, + 510, + 465, + 591 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 510, + 465, + 591 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 510, + 465, + 591 + ], + "spans": [ + { + "bbox": [ + 145, + 510, + 465, + 591 + ], + "score": 0.97, + "type": "image", + "image_path": "104fb1162f260d597db6e2b8b643e42e84d9eb60187fab5116d2e2aba9e01f85.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 145, + 510, + 465, + 537.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 145, + 537.0, + 465, + 564.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 145, + 564.0, + 465, + 591.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 153, + 598, + 456, + 610 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 152, + 596, + 458, + 613 + ], + "spans": [ + { + "bbox": [ + 152, + 596, + 458, + 613 + ], + "score": 1.0, + "content": "Figure 13: The training setting and examples of testing settings for Passing.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + } + ], + "index": 34.0 + }, + { + "type": "text", + "bbox": [ + 106, + 623, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "Figure 13 shows the training setting where the locations of the gap and the staring point of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "demonstrator are fixed, and the examples of testing settings where the placement of the gap and the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "initial position of the demonstrator is randomized. We terminate a training episode if the demonstrator", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 655, + 274, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 274, + 667 + ], + "score": 1.0, + "content": "has not passed the obstacle after 15 steps.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "title", + "bbox": [ + 107, + 679, + 217, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 678, + 219, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 219, + 692 + ], + "score": 1.0, + "content": "D.2 MAZE NAVIGATION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "The training setting in Maze Navigation is designed as shown in Figure 14a, where the placement of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "the tools is restrained in the purple region, and the positions of the demonstrator’s starting point and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "the doors are fixed. For testing, we randomly put one or two doors to fill the gaps; the demonstrator", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "Fusion. As shown in Figure 12, we design our fusion module using an attention based mechanism", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 434, + 105 + ], + "score": 1.0, + "content": "similar to the one introduced by Chaplot et al. 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Formally, we have an attention vector", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 122, + 507, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 124, + 190, + 136 + ], + "score": 0.92, + "content": "h = \\sigma ( m ^ { t - 1 } ) \\in \\dot { \\mathbb { R } } ^ { 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 122, + 222, + 139 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 222, + 125, + 239, + 136 + ], + "score": 0.89, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 122, + 399, + 139 + ], + "score": 1.0, + "content": "is an FC layer with sigmoid activation.", + "type": "text" + }, + { + "bbox": [ + 399, + 125, + 407, + 135 + ], + "score": 0.74, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 122, + 507, + 139 + ], + "score": 1.0, + "content": "is spatially expanded to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 133, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 104, + 133, + 114, + 150 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 114, + 137, + 170, + 147 + ], + "score": 0.89, + "content": "H \\times W \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 133, + 200, + 150 + ], + "score": 1.0, + "content": "tensor,", + "type": "text" + }, + { + "bbox": [ + 201, + 136, + 282, + 148 + ], + "score": 0.92, + "content": "\\pmb { H } ( h ) \\in \\mathbb { R } ^ { H \\times W \\times 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 133, + 378, + 150 + ], + "score": 1.0, + "content": ", where the elements in", + "type": "text" + }, + { + "bbox": [ + 378, + 137, + 385, + 147 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 133, + 506, + 150 + ], + "score": 1.0, + "content": "-th channel correspond to the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 146, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 107, + 147, + 113, + 157 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 146, + 168, + 159 + ], + "score": 1.0, + "content": "-th element in", + "type": "text" + }, + { + "bbox": [ + 169, + 147, + 176, + 157 + ], + "score": 0.78, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 146, + 505, + 159 + ], + "score": 1.0, + "content": ". We then reweight each feature maps using the attention vector, which becomes the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 155, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 182, + 171 + ], + "score": 1.0, + "content": "fusion output. I.e.,", + "type": "text" + }, + { + "bbox": [ + 182, + 156, + 347, + 169 + ], + "score": 0.93, + "content": "f ( \\phi ( s ^ { t } ) , m ^ { t - \\tilde { 1 } } ) = \\phi ( s ^ { t } ) \\odot H ( \\sigma \\tilde { ( } m ^ { t - 1 } ) \\mathbf { \\tilde { ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 155, + 377, + 171 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 378, + 157, + 400, + 169 + ], + "score": 0.93, + "content": "\\phi ( s ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 155, + 506, + 171 + ], + "score": 1.0, + "content": "are the feature maps from", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 167, + 401, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 178, + 181 + ], + "score": 1.0, + "content": "the state encoder,", + "type": "text" + }, + { + "bbox": [ + 178, + 169, + 196, + 180 + ], + "score": 0.9, + "content": "f ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 167, + 289, + 181 + ], + "score": 1.0, + "content": "is the fusion layer, and", + "type": "text" + }, + { + "bbox": [ + 290, + 170, + 299, + 178 + ], + "score": 0.83, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 167, + 401, + 181 + ], + "score": 1.0, + "content": "is element-wise product.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 81, + 507, + 181 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 184, + 505, + 302 + ], + "lines": [ + { + "bbox": [ + 106, + 184, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 184, + 506, + 196 + ], + "score": 1.0, + "content": "Policy. The input of this module is the flattened output from the fusion module, and is fed to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 195, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 506, + 207 + ], + "score": 1.0, + "content": "an LSTM with 128 hidden units followed by an FC layer with softmax activation. The result-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 385, + 218 + ], + "score": 1.0, + "content": "ing output is an action distribution representing the policy (either", + "type": "text" + }, + { + "bbox": [ + 385, + 207, + 397, + 217 + ], + "score": 0.85, + "content": "\\pi _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 205, + 411, + 218 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 412, + 207, + 422, + 217 + ], + "score": 0.8, + "content": "\\pi _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "). For Sorting task,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 506, + 228 + ], + "score": 1.0, + "content": "we slightly modify the output to fit the problem. We decompose the demonstrator’s policy as", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 222, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 351, + 242 + ], + "score": 0.9, + "content": "{ \\pi } _ { d } ( a _ { d } ^ { t } \\vert s _ { d } ^ { t } , m ^ { t - 1 } ) ~ = ~ { \\pi } _ { d } ^ { ( 1 ) } ( a _ { d } ^ { \\bar { t } , 1 } \\vert s _ { d } ^ { t } , m ^ { t - 1 } ) { \\pi } _ { d } ^ { ( 2 ) } ( a _ { d } ^ { t , 2 } \\vert s _ { d } ^ { t } , m ^ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 222, + 384, + 248 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 384, + 227, + 401, + 242 + ], + "score": 0.91, + "content": "a _ { d } ^ { t , 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 222, + 422, + 248 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 422, + 227, + 439, + 242 + ], + "score": 0.91, + "content": "a _ { d } ^ { t , 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 222, + 505, + 248 + ], + "score": 1.0, + "content": "are the indices", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "of the numbers the demonstrator chooses to swap. For the learner’s policy, it is decomposed as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 246, + 509, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 333, + 266 + ], + "score": 0.89, + "content": "\\pi _ { l } ( a _ { l } ^ { t } | s _ { l } ^ { t } , m ^ { t - 1 } ) = \\pi _ { l } ^ { \\mathrm { i d } } ( a _ { l } ^ { t , 1 } | s _ { l } ^ { t } , m ^ { t - 1 } ) \\pi _ { l } ^ { \\mathrm { b i t } } ( a _ { l } ^ { t , 2 } | s _ { l } ^ { t } , m ^ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 246, + 398, + 272 + ], + "score": 1.0, + "content": "instead, where", + "type": "text" + }, + { + "bbox": [ + 398, + 251, + 415, + 265 + ], + "score": 0.92, + "content": "a _ { l } ^ { t , 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 246, + 509, + 272 + ], + "score": 1.0, + "content": "indicates the number", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 103, + 260, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 103, + 260, + 252, + 282 + ], + "score": 1.0, + "content": "that the learner selects to change and", + "type": "text" + }, + { + "bbox": [ + 253, + 264, + 268, + 279 + ], + "score": 0.9, + "content": "a _ { l } ^ { t , 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 260, + 487, + 282 + ], + "score": 1.0, + "content": "is the bit of that number that needs to be flipped. When", + "type": "text" + }, + { + "bbox": [ + 487, + 264, + 504, + 279 + ], + "score": 0.9, + "content": "a _ { d } ^ { t , 1 }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 102, + 274, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 102, + 274, + 117, + 294 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 117, + 277, + 133, + 291 + ], + "score": 0.93, + "content": "a _ { l } ^ { t , 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 274, + 507, + 294 + ], + "score": 1.0, + "content": "is larger than the length of the array, it means that the demonstrator or the learner is choosing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 289, + 214, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 214, + 303 + ], + "score": 1.0, + "content": "to do nothing respectively.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 13.5, + "bbox_fs": [ + 102, + 184, + 509, + 303 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 306, + 506, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "Value. We also have a value net designed for training the learner’s policy using A2C (i.e.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 315, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 107, + 316, + 161, + 329 + ], + "score": 0.91, + "content": "V ( s _ { l } ^ { t } , m ^ { t - 1 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 315, + 506, + 330 + ], + "score": 1.0, + "content": ", which takes in the hidden state from the LSTM in the learner’s policy module and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 327, + 267, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 267, + 340 + ], + "score": 1.0, + "content": "outputs a scalar value after an FC layer.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 305, + 506, + 340 + ] + }, + { + "type": "list", + "bbox": [ + 108, + 343, + 504, + 366 + ], + "lines": [ + { + "bbox": [ + 107, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 107, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "The network is trained with RMSProp (Tieleman & Hinto, 2012) using a learning rate of 0.001.", + "type": "text" + } + ], + "index": 22, + "is_list_end_line": true + }, + { + "bbox": [ + 106, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 173, + 367 + ], + "score": 1.0, + "content": "During training,", + "type": "text" + }, + { + "bbox": [ + 173, + 356, + 178, + 365 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 354, + 349, + 367 + ], + "score": 1.0, + "content": "-greedy is applied to the rollout, where the", + "type": "text" + }, + { + "bbox": [ + 349, + 356, + 355, + 364 + ], + "score": 0.66, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "gradually decreases from 0.1 to 0.01.", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 22.5, + "bbox_fs": [ + 106, + 344, + 506, + 367 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 380, + 211, + 394 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 213, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 213, + 396 + ], + "score": 1.0, + "content": "D TASK SETTINGS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 504, + 427 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "We assume full observations of the world state for both agents in all tasks but the internal state of an", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 416, + 491, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 448, + 428 + ], + "score": 1.0, + "content": "agent (e.g., goals) is unobservable to another agent. The discounted factor is set to be", + "type": "text" + }, + { + "bbox": [ + 448, + 416, + 487, + 426 + ], + "score": 0.88, + "content": "\\Gamma = 0 . 9 5", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 416, + 491, + 428 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 406, + 505, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 505, + 445 + ], + "score": 1.0, + "content": "For the demonstrator in the grid world tasks, we implemented search based path planning and used", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "simple heuristics to perform branch and bound for acceleration. 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The demonstrator is always guaranteed to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "be able to find a path from its starting point to the destination (the top-left room). A training episode", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 240, + 219, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 219, + 254 + ], + "score": 1.0, + "content": "has a time limit of 60 steps.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 266, + 203, + 278 + ], + "lines": [ + { + "bbox": [ + 105, + 265, + 204, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 204, + 280 + ], + "score": 1.0, + "content": "D.3 CONSTRUCTION", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "image", + "bbox": [ + 145, + 290, + 465, + 370 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 145, + 290, + 465, + 370 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 145, + 290, + 465, + 370 + ], + "spans": [ + { + "bbox": [ + 145, + 290, + 465, + 370 + ], + "score": 0.967, + "type": "image", + "image_path": "16b7b3aa92dee0ccd1525db6154d8b34c50807af6f08c7694f6469a3309c6d55.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 145, + 290, + 465, + 316.6666666666667 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 145, + 316.6666666666667, + 465, + 343.33333333333337 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 145, + 343.33333333333337, + 465, + 370.00000000000006 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 142, + 378, + 467, + 390 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 142, + 377, + 468, + 391 + ], + "spans": [ + { + "bbox": [ + 142, + 377, + 468, + 391 + ], + "score": 1.0, + "content": "Figure 15: The training setting and examples of testing settings for Construction.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + } + ], + "index": 10.0 + }, + { + "type": "text", + "bbox": [ + 107, + 404, + 504, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "The room layout in training setting is fixed and shown in Figure 15a. In testing settings, we randomly", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "score": 1.0, + "content": "put six wall blocks around the three colored blocks to create obstacles. Figure 15b displays a few", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 423, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 439 + ], + "score": 1.0, + "content": "examples of testing scenarios. 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The", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "testing settings include 100 randomly generated initial sequences. 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MethodPassingMaze NavigationConstructionSorting
Ours0.710.600.480.82
Ours (from sub-optimal demo)0.680.510.47N/A
Ours w/o RL0.130.310.230.80
Ours w/o fusion0.110.190.290.10
Ours (2-LSTM)0.170.020.170
Passive (ToMnet)0.1100.120
Count-based0.2200.310.39
Self-supervised0.2300.360.56
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The nonlinearity of the generator implies", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 254, + 469, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 469, + 267 + ], + "score": 1.0, + "content": "that the latent space gives a distorted view of the input space. Under mild condi-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 266, + 469, + 278 + ], + "spans": [ + { + "bbox": [ + 141, + 266, + 469, + 278 + ], + "score": 1.0, + "content": "tions, we show that this distortion can be characterized by a stochastic Rieman-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 277, + 469, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 469, + 289 + ], + "score": 1.0, + "content": "nian metric, and we demonstrate that distances and interpolants are significantly", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 469, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 300 + ], + "score": 1.0, + "content": "improved under this metric. This in turn improves probability distributions, sam-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 470, + 311 + ], + "score": 1.0, + "content": "pling algorithms and clustering in the latent space. Our geometric analysis further", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 470, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 470, + 322 + ], + "score": 1.0, + "content": "reveals that current generators provide poor variance estimates and we propose a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 470, + 333 + ], + "score": 1.0, + "content": "new generator architecture with vastly improved variance estimates. Results are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 331, + 470, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 331, + 470, + 344 + ], + "score": 1.0, + "content": "demonstrated on convolutional and fully connected variational autoencoders, but", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 343, + 405, + 354 + ], + "spans": [ + { + "bbox": [ + 142, + 343, + 405, + 354 + ], + "score": 1.0, + "content": "the formalism easily generalizes to other deep generative models.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11.5, + "bbox_fs": [ + 141, + 222, + 470, + 354 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 374, + 206, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 208, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 208, + 390 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 108, + 398, + 504, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "Deep generative models (Goodfellow et al., 2014; Kingma & Welling, 2014; Rezende et al., 2014)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 410, + 504, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 283, + 422 + ], + "score": 1.0, + "content": "model the data distribution of observations", + "type": "text" + }, + { + "bbox": [ + 284, + 410, + 314, + 420 + ], + "score": 0.9, + "content": "\\mathbf { x } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 410, + 474, + 422 + ], + "score": 1.0, + "content": "through corresponding latent variables", + "type": "text" + }, + { + "bbox": [ + 474, + 410, + 504, + 420 + ], + "score": 0.87, + "content": "\\mathbf { z } \\in { \\mathcal { Z } }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 421, + 309, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 249, + 433 + ], + "score": 1.0, + "content": "and a stochastic generator function", + "type": "text" + }, + { + "bbox": [ + 249, + 421, + 296, + 433 + ], + "score": 0.92, + "content": "f : { \\mathcal { Z } } \\to { \\mathcal { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 421, + 309, + 433 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 398, + 506, + 433 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 283, + 436, + 327, + 450 + ], + "lines": [ + { + "bbox": [ + 283, + 436, + 327, + 450 + ], + "spans": [ + { + "bbox": [ + 283, + 436, + 327, + 450 + ], + "score": 0.91, + "content": "\\mathbf { x } = f ( \\mathbf { z } ) .", + "type": "interline_equation", + "image_path": "8ffae7edd90cf1ce2bdd1598e9938a0a3c3398024b8da8f67874582e30856c26.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 283, + 436, + 327, + 450 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 504, + 509 + ], + "lines": [ + { + "bbox": [ + 107, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 107, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "Using reasonably low-dimensional latent variables and highly flexible generator functions allows", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "these models to efficiently represent a useful distribution over the underlying data manifold. These", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "approaches have recently attracted a lot of attention, as deep neural networks are suitable generators", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 506, + 500 + ], + "score": 1.0, + "content": "which lead to the impressive performance of current variational autoencoders (VAEs) (Kingma &", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 496, + 455, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 455, + 510 + ], + "score": 1.0, + "content": "Welling, 2014) and generative adversarial networks (GANs) (Goodfellow et al., 2014).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 454, + 506, + 510 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 514, + 505, + 625 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "Consider the left panel of Fig. 1, which shows the latent representations of digits 0 and 1 from", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 524, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 539 + ], + "score": 1.0, + "content": "MNIST under a VAE. Three latent points are highlighted: one point (A) far away from the class", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "score": 1.0, + "content": "boundary, and two points (B, C) near the boundary, but on opposite sides. Points B and C near the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "boundary seem to be very close to each other, while the third is far away from the others. Intuitively,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "we would hope that points from the same class (A and B) are closer to each other than to members of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "other classes (C), but this is seemingly not the case. In this paper, we argue this seemed conclusion", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 451, + 594 + ], + "score": 1.0, + "content": "is incorrect and only due to a misinterpretation of the latent space — in fact points", + "type": "text" + }, + { + "bbox": [ + 451, + 581, + 460, + 591 + ], + "score": 0.27, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 580, + 479, + 594 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 480, + 580, + 489, + 591 + ], + "score": 0.55, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 580, + 506, + 594 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 225, + 604 + ], + "score": 1.0, + "content": "closer to each other than to", + "type": "text" + }, + { + "bbox": [ + 225, + 591, + 234, + 601 + ], + "score": 0.4, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "in the latent representation. Correcting this misinterpretation not", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 601, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 617 + ], + "score": 1.0, + "content": "only improves our understanding of generative models, but also improves interpolations, clusterings,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 613, + 423, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 423, + 626 + ], + "score": 1.0, + "content": "latent probability distributions, sampling algorithms, interpretability and more.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 514, + 506, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 630, + 504, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "In general, latent space distances lack physical units (making them difficult to interpret) and are sen-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "sitive to specifics of the underlying neural nets. It is therefore more robust to consider infinitesimal", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 652, + 506, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 333, + 665 + ], + "score": 1.0, + "content": "distances along the data manifold in the input space. Let", + "type": "text" + }, + { + "bbox": [ + 333, + 654, + 339, + 662 + ], + "score": 0.61, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 652, + 435, + 665 + ], + "score": 1.0, + "content": "be a latent point and let", + "type": "text" + }, + { + "bbox": [ + 436, + 653, + 455, + 663 + ], + "score": 0.91, + "content": "\\Delta \\mathbf { z } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 652, + 473, + 665 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 473, + 652, + 492, + 663 + ], + "score": 0.91, + "content": "\\Delta { \\bf z } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 652, + 506, + 665 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 662, + 336, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 336, + 676 + ], + "score": 1.0, + "content": "infinitesimals, then we can compute the squared distance", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 630, + 506, + 676 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 678, + 480, + 706 + ], + "lines": [ + { + "bbox": [ + 118, + 678, + 480, + 706 + ], + "spans": [ + { + "bbox": [ + 118, + 678, + 480, + 706 + ], + "score": 0.89, + "content": "\\left\\| f ( \\mathbf { z } + \\Delta \\mathbf { z } _ { 1 } ) - f ( \\mathbf { z } + \\Delta \\mathbf { z } _ { 2 } ) \\right\\| _ { 2 } ^ { 2 } = ( \\Delta \\mathbf { z } _ { 1 } - \\Delta \\mathbf { z } _ { 2 } ) ^ { \\mathsf { T } } \\left( \\mathbf { J } _ { \\mathbf { z } } ^ { \\mathsf { T } } \\mathbf { J } _ { \\mathbf { z } } \\right) ( \\Delta \\mathbf { z } _ { 1 } - \\Delta \\mathbf { z } _ { 2 } ) , \\quad \\mathbf { J } _ { \\mathbf { z } } = \\frac { \\partial f } { \\partial \\mathbf { z } } \\bigg | _ { \\mathbf { z } = \\mathbf { z } } ,", + "type": "interline_equation", + "image_path": "018fd672f671aba8f592e71d6b6c8f0101995cb63495024b7f27da927440cc5e.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 118, + 678, + 480, + 687.3333333333334 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 118, + 687.3333333333334, + 480, + 696.6666666666667 + ], + "spans": [], + "index": 43 + }, + { + "bbox": [ + 118, + 696.6666666666667, + 480, + 706.0000000000001 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 416, + 722 + ], + "score": 1.0, + "content": "using Taylor’s Theorem. This implies that the natural distance function in", + "type": "text" + }, + { + "bbox": [ + 416, + 710, + 426, + 720 + ], + "score": 0.81, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "changes locally as", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "it is governed by the local Jacobian. Mathematically, the latent space should not then be seen", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "as a linear Euclidean space, but rather as a curved space. The right panel of Fig. 1 provides an", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 278, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 290 + ], + "score": 1.0, + "content": "example of the implications of this curvature. The figure shows synthetic data from two classes,", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "score": 1.0, + "content": "and the corresponding latent representation of the data. The background color of the latent space", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 169, + 313 + ], + "score": 1.0, + "content": "corresponds to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 169, + 299, + 223, + 313 + ], + "score": 0.93, + "content": "\\mathrm { \\sqrt { d e t } } ( \\mathbf { J _ { z } ^ { \\mathsf { T } } } \\mathbf { J _ { z } } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 223, + 300, + 506, + 313 + ], + "score": 1.0, + "content": ", which can be seen as a measure of the local distortion of the latent", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "space. We interpolate two points from the same class by walking along the connecting straight", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "line (red); in the right panel, we show points along this straight line which have been mapped by", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "the generator to the input space. Since the generator defines a surface in the input space, we can", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "alternatively seek the shortest curve along this surface that connects the two points; this is perhaps", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "score": 1.0, + "content": "the most natural choice of interpolant. We show this shortest curve in green. From the center panel", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "it is evident that the natural interpolant is rather different from the straight line. This is due to the", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 377, + 380, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 380, + 391 + ], + "score": 1.0, + "content": "distortion of the latent space, which is the topic of the present paper.", + "type": "text", + "cross_page": true + } + ], + "index": 16 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 78, + 506, + 198 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 78, + 506, + 198 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 78, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 506, + 198 + ], + "score": 0.968, + "type": "image", + "image_path": "4f6ab73930c0c558bca7cac815f1ab4cf01c5f356a658bc9c3f933e9ccea2d5e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 78, + 506, + 118.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 118.0, + 506, + 158.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 158.0, + 506, + 198.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 505, + 241 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "Figure 1: Left: An example of how latent space distances do not reflect actual data distances. Right:", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "Shortest paths on the surface spanned by the generator do not correspond to straight lines in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 230, + 315, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 315, + 242 + ], + "score": 1.0, + "content": "latent space, as is assumed by the Euclidean metric.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "as a linear Euclidean space, but rather as a curved space. The right panel of Fig. 1 provides an", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 278, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 290 + ], + "score": 1.0, + "content": "example of the implications of this curvature. The figure shows synthetic data from two classes,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 301 + ], + "score": 1.0, + "content": "and the corresponding latent representation of the data. The background color of the latent space", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 169, + 313 + ], + "score": 1.0, + "content": "corresponds to", + "type": "text" + }, + { + "bbox": [ + 169, + 299, + 223, + 313 + ], + "score": 0.93, + "content": "\\mathrm { \\sqrt { d e t } } ( \\mathbf { J _ { z } ^ { \\mathsf { T } } } \\mathbf { J _ { z } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 300, + 506, + 313 + ], + "score": 1.0, + "content": ", which can be seen as a measure of the local distortion of the latent", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "space. We interpolate two points from the same class by walking along the connecting straight", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "line (red); in the right panel, we show points along this straight line which have been mapped by", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 347 + ], + "score": 1.0, + "content": "the generator to the input space. Since the generator defines a surface in the input space, we can", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "alternatively seek the shortest curve along this surface that connects the two points; this is perhaps", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "score": 1.0, + "content": "the most natural choice of interpolant. We show this shortest curve in green. From the center panel", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "it is evident that the natural interpolant is rather different from the straight line. This is due to the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 377, + 380, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 380, + 391 + ], + "score": 1.0, + "content": "distortion of the latent space, which is the topic of the present paper.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 494 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "Outline. In Sec. 2 we briefly present the VAE as a representative instance of generative models.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 429 + ], + "score": 1.0, + "content": "In Sec. 3 we connect generative models with their underlying geometry, and in Sec. 4 we argue that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "a stochastic Riemannian metric is naturally induced in the latent space by the generator. This metric", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "enables us to compute length-minimizing curves and corresponding distances. This analysis, how-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "ever, reveals that the traditional variance approximations in VAEs are rather poor and misleading;", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "we propose a solution in Sec. 4.1. In Sec. 5 we demonstrate how the resulting view of the latent", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "space improves latent interpolations, gives rise to more meaningful latent distributions, clusterings", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 482, + 486, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 486, + 495 + ], + "score": 1.0, + "content": "and more. We discuss related work in Sec. 6 and conclude the paper with an outlook in Sec. 7.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 106, + 514, + 459, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 460, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 460, + 528 + ], + "score": 1.0, + "content": "2 THE VARIATIONAL AUTOENCODERS ACTING AS THE GENERATOR", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 504, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "The variational autoencoder (VAE) proposed by Kingma & Welling (2014) is a simple yet powerful", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "generative model which consists of two parts: (1) an inference network or recognition network", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 418, + 576 + ], + "score": 1.0, + "content": "(encoder) learns the latent representation (codes) of the data in the input space", + "type": "text" + }, + { + "bbox": [ + 419, + 563, + 456, + 574 + ], + "score": 0.92, + "content": "\\chi = \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 564, + 506, + 576 + ], + "score": 1.0, + "content": "; and (2) the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 574, + 494, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 456, + 587 + ], + "score": 1.0, + "content": "generator (decoder) learns how to reconstruct the data from these latent space codes in", + "type": "text" + }, + { + "bbox": [ + 456, + 574, + 489, + 585 + ], + "score": 0.9, + "content": "\\mathcal { Z } = \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 574, + 494, + 587 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 389, + 605 + ], + "score": 1.0, + "content": "Formally, a prior distribution is defined for the latent representations", + "type": "text" + }, + { + "bbox": [ + 389, + 591, + 460, + 604 + ], + "score": 0.92, + "content": "p ( \\mathbf { z } ) = \\mathcal { N } ( \\mathbf { 0 } , \\mathbb { I } _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 591, + 506, + 605 + ], + "score": 1.0, + "content": ", and there", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 217, + 615 + ], + "score": 1.0, + "content": "exists a mapping function", + "type": "text" + }, + { + "bbox": [ + 217, + 603, + 280, + 613 + ], + "score": 0.9, + "content": "\\mu _ { \\theta } : \\mathcal { Z } \\to \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 602, + 394, + 615 + ], + "score": 1.0, + "content": "that generates a surface in", + "type": "text" + }, + { + "bbox": [ + 395, + 604, + 404, + 612 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 602, + 506, + 615 + ], + "score": 1.0, + "content": ". Moreover, we assume", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 611, + 507, + 628 + ], + "spans": [ + { + "bbox": [ + 104, + 611, + 194, + 628 + ], + "score": 1.0, + "content": "that another function", + "type": "text" + }, + { + "bbox": [ + 195, + 614, + 261, + 626 + ], + "score": 0.9, + "content": "\\sigma _ { \\theta } : \\mathcal { Z } \\to \\mathbb { R } _ { + } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 611, + 507, + 628 + ], + "score": 1.0, + "content": "captures the error (or uncertainty) between the actual data", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 624, + 504, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 155, + 637 + ], + "score": 1.0, + "content": "observation", + "type": "text" + }, + { + "bbox": [ + 156, + 625, + 187, + 635 + ], + "score": 0.9, + "content": "\\mathbf { x } \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 624, + 291, + 637 + ], + "score": 1.0, + "content": "and its reconstruction as", + "type": "text" + }, + { + "bbox": [ + 291, + 624, + 383, + 636 + ], + "score": 0.92, + "content": "\\mathbf { x } = \\pmb { \\mu } _ { \\boldsymbol { \\theta } } ( \\mathbf { z } ) + \\pmb { \\sigma } _ { \\boldsymbol { \\theta } } \\odot \\pmb { \\epsilon } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 624, + 414, + 637 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 415, + 624, + 475, + 636 + ], + "score": 0.92, + "content": "\\epsilon \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbb { I } _ { D } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 624, + 495, + 637 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 495, + 625, + 504, + 635 + ], + "score": 0.81, + "content": "\\odot", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 634, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 453, + 648 + ], + "score": 1.0, + "content": "is the Hadamard (element-wise) product. Then the likelihood is naturally defined as", + "type": "text" + }, + { + "bbox": [ + 453, + 636, + 505, + 648 + ], + "score": 0.91, + "content": "p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } \\mid \\mathbf { z } ) =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 645, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 206, + 658 + ], + "score": 0.64, + "content": "\\mathcal { N } ( \\mathbf { x } \\mid \\pmb { \\mu } _ { \\boldsymbol { \\theta } } ( \\mathbf { z } ) , \\mathbb { I } _ { D } \\pmb { \\sigma } _ { \\boldsymbol { \\theta } } ^ { 2 } ( \\mathbf { z } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 645, + 303, + 659 + ], + "score": 1.0, + "content": ". 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Right:", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "score": 1.0, + "content": "Shortest paths on the surface spanned by the generator do not correspond to straight lines in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 230, + 315, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 315, + 242 + ], + "score": 1.0, + "content": "latent space, as is assumed by the Euclidean metric.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 505, + 389 + ], + "lines": [], + "index": 11, + "bbox_fs": [ + 105, + 266, + 506, + 391 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 494 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "Outline. In Sec. 2 we briefly present the VAE as a representative instance of generative models.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 417, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 429 + ], + "score": 1.0, + "content": "In Sec. 3 we connect generative models with their underlying geometry, and in Sec. 4 we argue that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "a stochastic Riemannian metric is naturally induced in the latent space by the generator. This metric", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "enables us to compute length-minimizing curves and corresponding distances. This analysis, how-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "ever, reveals that the traditional variance approximations in VAEs are rather poor and misleading;", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "we propose a solution in Sec. 4.1. 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We discuss related work in Sec. 6 and conclude the paper with an outlook in Sec. 7.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 405, + 506, + 495 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 514, + 459, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 460, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 460, + 528 + ], + "score": 1.0, + "content": "2 THE VARIATIONAL AUTOENCODERS ACTING AS THE GENERATOR", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 504, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "The variational autoencoder (VAE) proposed by Kingma & Welling (2014) is a simple yet powerful", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "generative model which consists of two parts: (1) an inference network or recognition network", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 418, + 576 + ], + "score": 1.0, + "content": "(encoder) learns the latent representation (codes) of the data in the input space", + "type": "text" + }, + { + "bbox": [ + 419, + 563, + 456, + 574 + ], + "score": 0.92, + "content": "\\chi = \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 564, + 506, + 576 + ], + "score": 1.0, + "content": "; and (2) the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 574, + 494, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 456, + 587 + ], + "score": 1.0, + "content": "generator (decoder) learns how to reconstruct the data from these latent space codes in", + "type": "text" + }, + { + "bbox": [ + 456, + 574, + 489, + 585 + ], + "score": 0.9, + "content": "\\mathcal { Z } = \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 574, + 494, + 587 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 541, + 506, + 587 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 389, + 605 + ], + "score": 1.0, + "content": "Formally, a prior distribution is defined for the latent representations", + "type": "text" + }, + { + "bbox": [ + 389, + 591, + 460, + 604 + ], + "score": 0.92, + "content": "p ( \\mathbf { z } ) = \\mathcal { N } ( \\mathbf { 0 } , \\mathbb { I } _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 591, + 506, + 605 + ], + "score": 1.0, + "content": ", and there", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 217, + 615 + ], + "score": 1.0, + "content": "exists a mapping function", + "type": "text" + }, + { + "bbox": [ + 217, + 603, + 280, + 613 + ], + "score": 0.9, + "content": "\\mu _ { \\theta } : \\mathcal { Z } \\to \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 602, + 394, + 615 + ], + "score": 1.0, + "content": "that generates a surface in", + "type": "text" + }, + { + "bbox": [ + 395, + 604, + 404, + 612 + ], + "score": 0.82, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 602, + 506, + 615 + ], + "score": 1.0, + "content": ". 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A Riemannian metric", + "type": "text" + }, + { + "bbox": [ + 252, + 551, + 319, + 563 + ], + "score": 0.91, + "content": "\\mathbf { M } : \\mathcal { Z } \\mathbb { R } ^ { d \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 549, + 506, + 566 + ], + "score": 1.0, + "content": "is a smooth function that assigns a symmetric", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 564, + 273, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 261, + 576 + ], + "score": 1.0, + "content": "positive definite matrix to any point in", + "type": "text" + }, + { + "bbox": [ + 261, + 564, + 270, + 573 + ], + "score": 0.8, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 564, + 273, + 576 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 305, + 596 + ], + "score": 1.0, + "content": "It should be clear that if the generator function", + "type": "text" + }, + { + "bbox": [ + 305, + 584, + 312, + 595 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 581, + 430, + 596 + ], + "score": 1.0, + "content": "is sufficiently smooth, then", + "type": "text" + }, + { + "bbox": [ + 430, + 583, + 448, + 596 + ], + "score": 0.9, + "content": "{ { \\bf { M } } _ { \\gamma } }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 581, + 506, + 596 + ], + "score": 1.0, + "content": "in Eq. 5 is a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 594, + 188, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 188, + 606 + ], + "score": 1.0, + "content": "Riemannian metric.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "When defining distances across a given surface, it is meaningful to seek the shortest curve connecting", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 621, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 636 + ], + "score": 1.0, + "content": "two points. 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This implies that the length of a curve", + "type": "text" + }, + { + "bbox": [ + 453, + 470, + 464, + 480 + ], + "score": 0.88, + "content": "\\gamma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 467, + 505, + 481 + ], + "score": 1.0, + "content": "along the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 479, + 446, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 446, + 492 + ], + "score": 1.0, + "content": "surface can be computed directly in the latent space using the (locally defined) norm", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 106, + 467, + 505, + 492 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 493, + 429, + 515 + ], + "lines": [ + { + "bbox": [ + 181, + 493, + 429, + 515 + ], + "spans": [ + { + "bbox": [ + 181, + 493, + 429, + 515 + ], + "score": 0.92, + "content": "\\| \\mathbf { J } _ { \\gamma } \\dot { \\gamma } \\| _ { 2 } = \\sqrt { ( \\mathbf { J } _ { \\gamma } \\dot { \\gamma } ) ^ { \\intercal } ( \\mathbf { J } _ { \\gamma } \\dot { \\gamma } ) } = \\sqrt { \\dot { \\gamma } ^ { \\intercal } ( \\mathbf { J } _ { \\gamma } ^ { \\intercal } \\mathbf { J } _ { \\gamma } ) \\dot { \\gamma } } = \\sqrt { \\dot { \\gamma } ^ { \\intercal } \\mathbf { M } _ { \\gamma } \\dot { \\gamma } } .", + "type": "interline_equation", + "image_path": "b0dc3530177e3341805b36e0b25729b5a4eb3075b25b6e81ceffd4fe656d7c2e.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 181, + 493, + 429, + 515 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 516, + 502, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 131, + 529 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 516, + 186, + 530 + ], + "score": 0.93, + "content": "{ \\bf M } _ { \\gamma } = { \\bf J } _ { \\gamma } ^ { \\top } { \\bf J } _ { \\gamma }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "is a symmetric positive definite matrix, which acts akin to a local Mahalanobis", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 107, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "distance measure. This gives rise to the definition of a Riemannian metric, which represents a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 540, + 281, + 551 + ], + "spans": [ + { + "bbox": [ + 107, + 540, + 281, + 551 + ], + "score": 1.0, + "content": "smoothly changing inner product structure.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 516, + 505, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 503, + 575 + ], + "lines": [ + { + "bbox": [ + 104, + 549, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 251, + 566 + ], + "score": 1.0, + "content": "Definition 1. A Riemannian metric", + "type": "text" + }, + { + "bbox": [ + 252, + 551, + 319, + 563 + ], + "score": 0.91, + "content": "\\mathbf { M } : \\mathcal { Z } \\mathbb { R } ^ { d \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 549, + 506, + 566 + ], + "score": 1.0, + "content": "is a smooth function that assigns a symmetric", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 564, + 273, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 261, + 576 + ], + "score": 1.0, + "content": "positive definite matrix to any point in", + "type": "text" + }, + { + "bbox": [ + 261, + 564, + 270, + 573 + ], + "score": 0.8, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 564, + 273, + 576 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5, + "bbox_fs": [ + 104, + 549, + 506, + 576 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 305, + 596 + ], + "score": 1.0, + "content": "It should be clear that if the generator function", + "type": "text" + }, + { + "bbox": [ + 305, + 584, + 312, + 595 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 581, + 430, + 596 + ], + "score": 1.0, + "content": "is sufficiently smooth, then", + "type": "text" + }, + { + "bbox": [ + 430, + 583, + 448, + 596 + ], + "score": 0.9, + "content": "{ { \\bf { M } } _ { \\gamma } }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 581, + 506, + 596 + ], + "score": 1.0, + "content": "in Eq. 5 is a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 594, + 188, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 188, + 606 + ], + "score": 1.0, + "content": "Riemannian metric.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 581, + 506, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "When defining distances across a given surface, it is meaningful to seek the shortest curve connecting", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 621, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 636 + ], + "score": 1.0, + "content": "two points. Then a distance can be defined as the length of this curve. The shortest curve connecting", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 633, + 269, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 133, + 644 + ], + "score": 1.0, + "content": "points", + "type": "text" + }, + { + "bbox": [ + 133, + 635, + 144, + 644 + ], + "score": 0.86, + "content": "\\mathbf { z } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 633, + 162, + 644 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 634, + 173, + 644 + ], + "score": 0.87, + "content": "\\mathbf { z } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 633, + 269, + 644 + ], + "score": 1.0, + "content": "is by (trivial) definition", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 610, + 506, + 644 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 647, + 429, + 670 + ], + "lines": [ + { + "bbox": [ + 182, + 647, + 429, + 670 + ], + "spans": [ + { + "bbox": [ + 182, + 647, + 429, + 670 + ], + "score": 0.9, + "content": "\\boldsymbol \\gamma _ { t } ^ { ( \\mathrm { s h o r t e s t } ) } = \\underset { \\boldsymbol \\gamma _ { t } } { \\mathrm { a r g m i n L e n g t h } } [ f ( \\boldsymbol \\gamma _ { t } ) ] , \\qquad \\boldsymbol \\gamma _ { 0 } = \\mathbf z _ { 0 } , \\boldsymbol \\gamma _ { 1 } = \\mathbf z _ { 1 } .", + "type": "interline_equation", + "image_path": "c6c87e3ff4dca39c5a2ac584843c5f35f74c2708b716f8da661d9cc33e6fde58.jpg" + } + ] + } + ], + "index": 52, + "virtual_lines": [ + { + "bbox": [ + 182, + 647, + 429, + 670 + ], + "spans": [], + "index": 52 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 672, + 507, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "A classic result of differential geometry (do Carmo, 1992) is that solutions to this optimization", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 683, + 425, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 425, + 696 + ], + "score": 1.0, + "content": "problem satisfy the following system of ordinary differential equations (ODEs)", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 53.5, + "bbox_fs": [ + 105, + 672, + 505, + 696 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 697, + 451, + 731 + ], + "lines": [ + { + "bbox": [ + 158, + 697, + 451, + 731 + ], + "spans": [ + { + "bbox": [ + 158, + 697, + 451, + 731 + ], + "score": 0.94, + "content": "\\ddot { \\gamma } _ { t } = - \\frac { 1 } { 2 } \\mathbf { M } _ { \\gamma _ { t } } ^ { - 1 } \\left[ 2 \\big ( \\mathbb { I } _ { d } \\otimes \\dot { \\gamma } _ { t } ^ { \\intercal } \\big ) \\frac { \\partial \\mathrm { v e c } \\big [ \\mathbf { M } _ { \\gamma _ { t } } \\big ] } { \\partial \\gamma _ { t } } \\dot { \\gamma } _ { t } - \\frac { \\partial \\mathrm { v e c } \\big [ \\mathbf { M } _ { \\gamma _ { t } } \\big ] ^ { \\intercal } } { \\partial \\gamma _ { t } } ( \\dot { \\gamma } _ { t } \\otimes \\dot { \\gamma } _ { t } ) \\right] ,", + "type": "interline_equation", + "image_path": "92d967178546532454e8e29aa71c57b216ef3688ad7afcc29dadc022f87384f2.jpg" + } + ] + } + ], + "index": 56, + "virtual_lines": [ + { + "bbox": [ + 158, + 697, + 451, + 708.3333333333334 + ], + "spans": [], + "index": 55 + }, + { + "bbox": [ + 158, + 708.3333333333334, + 451, + 719.6666666666667 + ], + "spans": [], + "index": 56 + }, + { + "bbox": [ + 158, + 719.6666666666667, + 451, + 731.0000000000001 + ], + "spans": [], + "index": 57 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 365, + 95 + ], + "score": 1.0, + "content": "where vec[·] stacks the columns of a matrix into a vector and", + "type": "text" + }, + { + "bbox": [ + 366, + 84, + 375, + 93 + ], + "score": 0.8, + "content": "\\otimes", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "is the Kronecker product. For", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "completeness, we provide a derivation of this result in Appendix A. Shortest curves can then be", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 474, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 387, + 117 + ], + "score": 1.0, + "content": "computed by solving the ODEs numerically; our implementation uses", + "type": "text" + }, + { + "bbox": [ + 387, + 105, + 419, + 116 + ], + "score": 0.32, + "content": "_ { \\mathrm { b v p } 5 \\mathrm { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 105, + 474, + 117 + ], + "score": 1.0, + "content": "from Matlab.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 107, + 132, + 369, + 145 + ], + "lines": [ + { + "bbox": [ + 105, + 130, + 369, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 369, + 147 + ], + "score": 1.0, + "content": "4 THE GEOMETRY OF STOCHASTIC GENERATORS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 157, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 157, + 505, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 366, + 170 + ], + "score": 1.0, + "content": "In the previous section, we considered deterministic generators", + "type": "text" + }, + { + "bbox": [ + 367, + 158, + 374, + 169 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 157, + 505, + 170 + ], + "score": 1.0, + "content": "to provide relevant background", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 169, + 465, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 465, + 180 + ], + "score": 1.0, + "content": "information. We now extend these results to the stochastic case; in particular we consider", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 186, + 459, + 201 + ], + "lines": [ + { + "bbox": [ + 152, + 186, + 459, + 201 + ], + "spans": [ + { + "bbox": [ + 152, + 186, + 459, + 201 + ], + "score": 0.88, + "content": "f ( \\mathbf { z } ) = \\pmb { \\mu } ( \\mathbf { z } ) + \\pmb { \\sigma } ( \\mathbf { z } ) \\odot \\epsilon , \\qquad \\pmb { \\mu } : \\mathscr { Z } \\pmb { \\chi } , ~ \\pmb { \\sigma } : \\mathscr { Z } \\mathbb { R } _ { + } ^ { D } , ~ \\epsilon \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbb { I } _ { D } ) .", + "type": "interline_equation", + "image_path": "4bbcc4852b2a197fa832ac95f26c5f0558f78c7c90e484248f1c32be9784c0f2.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 152, + 186, + 459, + 201 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 205, + 505, + 228 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 446, + 219 + ], + "score": 1.0, + "content": "This is the generator driving VAEs and related models. For our purposes, we will call", + "type": "text" + }, + { + "bbox": [ + 447, + 206, + 465, + 218 + ], + "score": 0.91, + "content": "\\mu ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 204, + 505, + 219 + ], + "score": 1.0, + "content": "the mean", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 216, + 273, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 158, + 229 + ], + "score": 1.0, + "content": "function and", + "type": "text" + }, + { + "bbox": [ + 159, + 216, + 182, + 229 + ], + "score": 0.92, + "content": "\\sigma ^ { 2 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 216, + 273, + 229 + ], + "score": 1.0, + "content": "the variance function.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "Following the discussion from the previous section, it is natural to consider the Riemannian metric", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 107, + 245, + 161, + 257 + ], + "score": 0.92, + "content": "\\mathbf { M _ { z } } = \\mathbf { J _ { z } ^ { \\intercal } } \\mathbf { J _ { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "in the latent space. Since the generator is now stochastic, this metric also becomes", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 256, + 463, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 463, + 268 + ], + "score": 1.0, + "content": "stochastic, which complicates analysis. The following results, however, simplify matters.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 504, + 294 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "Theorem 1. If the stochastic generator in Eq. 8 has mean and variance functions that are at least", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 282, + 316, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 316, + 294 + ], + "score": 1.0, + "content": "twice differentiable, then the expected metric equals", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 298, + 424, + 321 + ], + "lines": [ + { + "bbox": [ + 186, + 298, + 424, + 321 + ], + "spans": [ + { + "bbox": [ + 186, + 298, + 424, + 321 + ], + "score": 0.92, + "content": "\\overline { { \\mathbf { M } } } _ { \\mathbf { z } } = \\mathbb { E } _ { p ( \\boldsymbol { \\epsilon } ) } [ \\mathbf { M } _ { \\mathbf { z } } ] = \\left( \\mathbf { J } _ { \\mathbf { z } } ^ { ( \\boldsymbol { \\mu } ) } \\right) ^ { \\intercal } \\left( \\mathbf { J } _ { \\mathbf { z } } ^ { ( \\boldsymbol { \\mu } ) } \\right) + \\left( \\mathbf { J } _ { \\mathbf { z } } ^ { ( \\boldsymbol { \\sigma } ) } \\right) ^ { \\intercal } \\left( \\mathbf { J } _ { \\mathbf { z } } ^ { ( \\boldsymbol { \\sigma } ) } \\right) ,", + "type": "interline_equation", + "image_path": "ce69d035222cc7f0bc3d5c07cfa0ab021927ac0473ac326fc95f5f0c88378ac9.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 186, + 298, + 424, + 321 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 366, + 342 + ], + "lines": [ + { + "bbox": [ + 103, + 322, + 364, + 345 + ], + "spans": [ + { + "bbox": [ + 103, + 322, + 133, + 345 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 326, + 152, + 340 + ], + "score": 0.91, + "content": "\\mathbf { J _ { z } ^ { ( \\mu ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 322, + 171, + 345 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 171, + 326, + 190, + 340 + ], + "score": 0.9, + "content": "\\mathbf { J } _ { \\mathbf { z } } ^ { ( \\sigma ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 322, + 308, + 345 + ], + "score": 1.0, + "content": "are the Jacobian matrices of", + "type": "text" + }, + { + "bbox": [ + 308, + 329, + 326, + 342 + ], + "score": 0.91, + "content": "\\mu ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 322, + 345, + 345 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 345, + 329, + 364, + 342 + ], + "score": 0.91, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 203, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 205, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 205, + 363 + ], + "score": 1.0, + "content": "Proof. See Appendix B.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 215, + 378 + ], + "score": 1.0, + "content": "Remark 1. By Definition", + "type": "text" + }, + { + "bbox": [ + 216, + 366, + 221, + 375 + ], + "score": 0.28, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 365, + 505, + 378 + ], + "score": 1.0, + "content": ", the metric tensor must change smoothly, which implies that the Ja-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "cobians must be smooth functions as well. This is easily ensured with activation functions for the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 386, + 444, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 207, + 401 + ], + "score": 1.0, + "content": "neural networks that are", + "type": "text" + }, + { + "bbox": [ + 207, + 387, + 218, + 398 + ], + "score": 0.84, + "content": "\\mathcal { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 386, + 315, + 401 + ], + "score": 1.0, + "content": "differentiable, e.g. tanh", + "type": "text" + }, + { + "bbox": [ + 315, + 388, + 326, + 399 + ], + "score": 0.27, + "content": "( \\cdot ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 386, + 444, + 401 + ], + "score": 1.0, + "content": ", sigmoid(·), and softplus(·).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 502, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 401, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 418, + 415 + ], + "score": 1.0, + "content": "Theorem 2 (Due to Tosi et al. (2014)). The variance of the metric under the", + "type": "text" + }, + { + "bbox": [ + 418, + 403, + 430, + 414 + ], + "score": 0.87, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 401, + 504, + 415 + ], + "score": 1.0, + "content": "measure vanishes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 413, + 388, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 285, + 426 + ], + "score": 1.0, + "content": "when the data dimension goes to infinity, i.e.", + "type": "text" + }, + { + "bbox": [ + 285, + 414, + 384, + 425 + ], + "score": 0.74, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { D \\to \\infty } \\mathrm { V a r } \\left( \\mathbf { M } _ { \\mathbf { z } } \\right) = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 413, + 388, + 426 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 436, + 375, + 480 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 376, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 349, + 448 + ], + "score": 1.0, + "content": "Theorem 2 suggests that the (deterministic) expected metric", + "type": "text" + }, + { + "bbox": [ + 349, + 434, + 366, + 447 + ], + "score": 0.89, + "content": "\\overline { { \\mathbf { M } } } _ { \\mathbf { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 435, + 376, + 448 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 447, + 376, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 376, + 458 + ], + "score": 1.0, + "content": "a good approximation to the underlying stochastic metric when the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 376, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 376, + 470 + ], + "score": 1.0, + "content": "data dimension is large. We make this approximation, which allows", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 469, + 308, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 308, + 481 + ], + "score": 1.0, + "content": "us to apply the theory of deterministic generators.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 375, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 376, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 376, + 498 + ], + "score": 1.0, + "content": "This expected metric has a particularly appealing form, where the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 497, + 376, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 376, + 508 + ], + "score": 1.0, + "content": "two terms capture the distortion of the mean and the variance func-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 102, + 504, + 375, + 527 + ], + "spans": [ + { + "bbox": [ + 102, + 504, + 317, + 527 + ], + "score": 1.0, + "content": "tions respectively. In particular, the variance term", + "type": "text" + }, + { + "bbox": [ + 317, + 507, + 375, + 522 + ], + "score": 0.92, + "content": "( \\mathbf { J _ { z } ^ { ( \\sigma ) } } ) ^ { \\mathsf { T } } ( \\mathbf { J _ { z } ^ { ( \\sigma ) } } )", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 522, + 376, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 376, + 533 + ], + "score": 1.0, + "content": "will be large in regions of the latent space, where the generator has", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 532, + 376, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 376, + 544 + ], + "score": 1.0, + "content": "large variance. This implies that induced distances will be large in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 543, + 376, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 376, + 555 + ], + "score": 1.0, + "content": "regions of the latent space where the generator is highly uncertain,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 554, + 376, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 376, + 566 + ], + "score": 1.0, + "content": "such that shortest paths will tend to avoid these regions. 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Fig. 3. It is worth stressing, that no learning", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "is needed to compute this metric: it only consists of terms that can be derived directly from the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 588, + 149, + 599 + ], + "spans": [ + { + "bbox": [ + 104, + 588, + 149, + 599 + ], + "score": 1.0, + "content": "generator.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + }, + { + "type": "title", + "bbox": [ + 106, + 612, + 473, + 623 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 474, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 474, + 624 + ], + "score": 1.0, + "content": "4.1 ENSURING PROPER GEOMETRY THROUGH MEANINGFUL VARIANCE FUNCTIONS", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "Theorem 1 informs us about how the geometry of the generative model depends on both the mean", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "and the variance of the generator. Assuming successful training of the generator, we can expect to", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "have good estimates of the geometry in regions near the data. But what happens in regions further", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "away from the data? In general, the mean function cannot be expected to give useful extrapolations", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "to such regions, so it is reasonable to require that the generator has high variance in regions that are", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "not near the data. In practice, the neural net used to represent the variance function is only trained", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "in regions where data is available, which implies that variance estimates are extrapolated to regions", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "with no data. As neural nets tend to extrapolate poorly, practical variance estimates tend to be", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 720, + 269, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 269, + 734 + ], + "score": 1.0, + "content": "arbitrarily poor in regions without data.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 365, + 95 + ], + "score": 1.0, + "content": "where vec[·] stacks the columns of a matrix into a vector and", + "type": "text" + }, + { + "bbox": [ + 366, + 84, + 375, + 93 + ], + "score": 0.8, + "content": "\\otimes", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "is the Kronecker product. For", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "completeness, we provide a derivation of this result in Appendix A. Shortest curves can then be", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 474, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 387, + 117 + ], + "score": 1.0, + "content": "computed by solving the ODEs numerically; our implementation uses", + "type": "text" + }, + { + "bbox": [ + 387, + 105, + 419, + 116 + ], + "score": 0.32, + "content": "_ { \\mathrm { b v p } 5 \\mathrm { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 105, + 474, + 117 + ], + "score": 1.0, + "content": "from Matlab.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 117 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 132, + 369, + 145 + ], + "lines": [ + { + "bbox": [ + 105, + 130, + 369, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 369, + 147 + ], + "score": 1.0, + "content": "4 THE GEOMETRY OF STOCHASTIC GENERATORS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 157, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 106, + 157, + 505, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 366, + 170 + ], + "score": 1.0, + "content": "In the previous section, we considered deterministic generators", + "type": "text" + }, + { + "bbox": [ + 367, + 158, + 374, + 169 + ], + "score": 0.87, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 157, + 505, + 170 + ], + "score": 1.0, + "content": "to provide relevant background", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 169, + 465, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 465, + 180 + ], + "score": 1.0, + "content": "information. We now extend these results to the stochastic case; in particular we consider", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 106, + 157, + 505, + 180 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 186, + 459, + 201 + ], + "lines": [ + { + "bbox": [ + 152, + 186, + 459, + 201 + ], + "spans": [ + { + "bbox": [ + 152, + 186, + 459, + 201 + ], + "score": 0.88, + "content": "f ( \\mathbf { z } ) = \\pmb { \\mu } ( \\mathbf { z } ) + \\pmb { \\sigma } ( \\mathbf { z } ) \\odot \\epsilon , \\qquad \\pmb { \\mu } : \\mathscr { Z } \\pmb { \\chi } , ~ \\pmb { \\sigma } : \\mathscr { Z } \\mathbb { R } _ { + } ^ { D } , ~ \\epsilon \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbb { I } _ { D } ) .", + "type": "interline_equation", + "image_path": "4bbcc4852b2a197fa832ac95f26c5f0558f78c7c90e484248f1c32be9784c0f2.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 152, + 186, + 459, + 201 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 205, + 505, + 228 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 446, + 219 + ], + "score": 1.0, + "content": "This is the generator driving VAEs and related models. For our purposes, we will call", + "type": "text" + }, + { + "bbox": [ + 447, + 206, + 465, + 218 + ], + "score": 0.91, + "content": "\\mu ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 204, + 505, + 219 + ], + "score": 1.0, + "content": "the mean", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 216, + 273, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 158, + 229 + ], + "score": 1.0, + "content": "function and", + "type": "text" + }, + { + "bbox": [ + 159, + 216, + 182, + 229 + ], + "score": 0.92, + "content": "\\sigma ^ { 2 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 216, + 273, + 229 + ], + "score": 1.0, + "content": "the variance function.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 204, + 505, + 229 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 234, + 505, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 505, + 246 + ], + "score": 1.0, + "content": "Following the discussion from the previous section, it is natural to consider the Riemannian metric", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 107, + 245, + 161, + 257 + ], + "score": 0.92, + "content": "\\mathbf { M _ { z } } = \\mathbf { J _ { z } ^ { \\intercal } } \\mathbf { J _ { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "in the latent space. Since the generator is now stochastic, this metric also becomes", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 256, + 463, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 256, + 463, + 268 + ], + "score": 1.0, + "content": "stochastic, which complicates analysis. The following results, however, simplify matters.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 234, + 505, + 268 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 504, + 294 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "Theorem 1. If the stochastic generator in Eq. 8 has mean and variance functions that are at least", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 282, + 316, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 316, + 294 + ], + "score": 1.0, + "content": "twice differentiable, then the expected metric equals", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 106, + 270, + 505, + 294 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 298, + 424, + 321 + ], + "lines": [ + { + "bbox": [ + 186, + 298, + 424, + 321 + ], + "spans": [ + { + "bbox": [ + 186, + 298, + 424, + 321 + ], + "score": 0.92, + "content": "\\overline { { \\mathbf { M } } } _ { \\mathbf { z } } = \\mathbb { E } _ { p ( \\boldsymbol { \\epsilon } ) } [ \\mathbf { M } _ { \\mathbf { z } } ] = \\left( \\mathbf { J } _ { \\mathbf { z } } ^ { ( \\boldsymbol { \\mu } ) } \\right) ^ { \\intercal } \\left( \\mathbf { J } _ { \\mathbf { z } } ^ { ( \\boldsymbol { \\mu } ) } \\right) + \\left( \\mathbf { J } _ { \\mathbf { z } } ^ { ( \\boldsymbol { \\sigma } ) } \\right) ^ { \\intercal } \\left( \\mathbf { J } _ { \\mathbf { z } } ^ { ( \\boldsymbol { \\sigma } ) } \\right) ,", + "type": "interline_equation", + "image_path": "ce69d035222cc7f0bc3d5c07cfa0ab021927ac0473ac326fc95f5f0c88378ac9.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 186, + 298, + 424, + 321 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 326, + 366, + 342 + ], + "lines": [ + { + "bbox": [ + 103, + 322, + 364, + 345 + ], + "spans": [ + { + "bbox": [ + 103, + 322, + 133, + 345 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 326, + 152, + 340 + ], + "score": 0.91, + "content": "\\mathbf { J _ { z } ^ { ( \\mu ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 322, + 171, + 345 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 171, + 326, + 190, + 340 + ], + "score": 0.9, + "content": "\\mathbf { J } _ { \\mathbf { z } } ^ { ( \\sigma ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 322, + 308, + 345 + ], + "score": 1.0, + "content": "are the Jacobian matrices of", + "type": "text" + }, + { + "bbox": [ + 308, + 329, + 326, + 342 + ], + "score": 0.91, + "content": "\\mu ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 322, + 345, + 345 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 345, + 329, + 364, + 342 + ], + "score": 0.91, + "content": "\\sigma ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 103, + 322, + 364, + 345 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 350, + 203, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 205, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 205, + 363 + ], + "score": 1.0, + "content": "Proof. See Appendix B.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 349, + 205, + 363 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 505, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 215, + 378 + ], + "score": 1.0, + "content": "Remark 1. By Definition", + "type": "text" + }, + { + "bbox": [ + 216, + 366, + 221, + 375 + ], + "score": 0.28, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 365, + 505, + 378 + ], + "score": 1.0, + "content": ", the metric tensor must change smoothly, which implies that the Ja-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "cobians must be smooth functions as well. This is easily ensured with activation functions for the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 386, + 444, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 207, + 401 + ], + "score": 1.0, + "content": "neural networks that are", + "type": "text" + }, + { + "bbox": [ + 207, + 387, + 218, + 398 + ], + "score": 0.84, + "content": "\\mathcal { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 386, + 315, + 401 + ], + "score": 1.0, + "content": "differentiable, e.g. tanh", + "type": "text" + }, + { + "bbox": [ + 315, + 388, + 326, + 399 + ], + "score": 0.27, + "content": "( \\cdot ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 386, + 444, + 401 + ], + "score": 1.0, + "content": ", sigmoid(·), and softplus(·).", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 365, + 505, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 502, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 401, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 418, + 415 + ], + "score": 1.0, + "content": "Theorem 2 (Due to Tosi et al. (2014)). The variance of the metric under the", + "type": "text" + }, + { + "bbox": [ + 418, + 403, + 430, + 414 + ], + "score": 0.87, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 401, + 504, + 415 + ], + "score": 1.0, + "content": "measure vanishes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 413, + 388, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 285, + 426 + ], + "score": 1.0, + "content": "when the data dimension goes to infinity, i.e.", + "type": "text" + }, + { + "bbox": [ + 285, + 414, + 384, + 425 + ], + "score": 0.74, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { D \\to \\infty } \\mathrm { V a r } \\left( \\mathbf { M } _ { \\mathbf { z } } \\right) = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 413, + 388, + 426 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 401, + 504, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 436, + 375, + 480 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 376, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 349, + 448 + ], + "score": 1.0, + "content": "Theorem 2 suggests that the (deterministic) expected metric", + "type": "text" + }, + { + "bbox": [ + 349, + 434, + 366, + 447 + ], + "score": 0.89, + "content": "\\overline { { \\mathbf { M } } } _ { \\mathbf { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 435, + 376, + 448 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 447, + 376, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 376, + 458 + ], + "score": 1.0, + "content": "a good approximation to the underlying stochastic metric when the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 457, + 376, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 376, + 470 + ], + "score": 1.0, + "content": "data dimension is large. We make this approximation, which allows", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 469, + 308, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 308, + 481 + ], + "score": 1.0, + "content": "us to apply the theory of deterministic generators.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 434, + 376, + 481 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 375, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 376, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 376, + 498 + ], + "score": 1.0, + "content": "This expected metric has a particularly appealing form, where the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 497, + 376, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 376, + 508 + ], + "score": 1.0, + "content": "two terms capture the distortion of the mean and the variance func-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 102, + 504, + 375, + 527 + ], + "spans": [ + { + "bbox": [ + 102, + 504, + 317, + 527 + ], + "score": 1.0, + "content": "tions respectively. In particular, the variance term", + "type": "text" + }, + { + "bbox": [ + 317, + 507, + 375, + 522 + ], + "score": 0.92, + "content": "( \\mathbf { J _ { z } ^ { ( \\sigma ) } } ) ^ { \\mathsf { T } } ( \\mathbf { J _ { z } ^ { ( \\sigma ) } } )", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 522, + 376, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 376, + 533 + ], + "score": 1.0, + "content": "will be large in regions of the latent space, where the generator has", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 532, + 376, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 376, + 544 + ], + "score": 1.0, + "content": "large variance. This implies that induced distances will be large in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 543, + 376, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 376, + 555 + ], + "score": 1.0, + "content": "regions of the latent space where the generator is highly uncertain,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 554, + 376, + 566 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 376, + 566 + ], + "score": 1.0, + "content": "such that shortest paths will tend to avoid these regions. These paths", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 563, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 578 + ], + "score": 1.0, + "content": "will then tend to follow the data in the latent space, c.f. Fig. 3. 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Assuming successful training of the generator, we can expect to", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "have good estimates of the geometry in regions near the data. But what happens in regions further", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "away from the data? 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The first two panels show the data and its corresponding latent", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "representations (here both input and latent dimensions are 2 to ease illustration). The third panel", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "shows the variance function under a standard architecture, deep multilayer perceptron with softplus", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "score": 1.0, + "content": "nonlinearity for the output layer. It is evident that variance estimates in regions without data are", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "score": 1.0, + "content": "not representative of either uncertainty or error of the generative process; sometimes variance is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "high, sometimes it is low. From a probabilistic modeling point-of-view, this is disheartening. An", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 278, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 289 + ], + "score": 1.0, + "content": "informal survey of publicly available VAE implementations also reveals that it is common to enforce", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 289, + 368, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 368, + 301 + ], + "score": 1.0, + "content": "a constant unit variance everywhere; this is further disheartening.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "For our purposes, we need well-behaved variance functions to ensure a well-behaved geometry,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "but reasonable variance estimates are of general use. Here, as a general strategy, we propose to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 327, + 504, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 504, + 338 + ], + "score": 1.0, + "content": "model the inverse variance with a network that extrapolates towards zero. This at least ensures that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 335, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 437, + 356 + ], + "score": 1.0, + "content": "variances are large in regions without data. Specifically, we model the precision as", + "type": "text" + }, + { + "bbox": [ + 438, + 337, + 501, + 354 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\beta _ { \\psi } ( \\mathbf { z } ) = \\frac { 1 } { \\pmb { \\sigma } _ { \\psi } ^ { 2 } ( \\mathbf { z } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 335, + 505, + 356 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "where all operations are element-wise. Then, we model this precision with a radial basis function", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 363, + 387, + 377 + ], + "spans": [ + { + "bbox": [ + 107, + 364, + 132, + 375 + ], + "score": 0.75, + "content": "( R B F )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 363, + 387, + 377 + ], + "score": 1.0, + "content": "neural network (Que & Belkin, 2016). Formally this is written", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 376, + 468, + 398 + ], + "lines": [ + { + "bbox": [ + 143, + 376, + 468, + 398 + ], + "spans": [ + { + "bbox": [ + 143, + 376, + 468, + 398 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\beta _ { \\psi } ( { \\bf z } ) = { \\bf W } { \\bf v } ( { \\bf z } ) + \\boldsymbol { \\zeta } , \\quad \\mathrm { w i t h } \\quad v _ { k } ( { \\bf z } ) = \\exp \\left( - \\lambda _ { k } \\left. { \\bf z } - { \\bf c } _ { k } \\right. _ { 2 } ^ { 2 } \\right) , k = 1 , \\dots , K , } \\end{array}", + "type": "interline_equation", + "image_path": "3497e224dfd202107a9c1a9fe0bc09cdc511c58d74865d312b5888457867c2e1.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 143, + 376, + 468, + 398 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 398, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 102, + 393, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 102, + 393, + 133, + 420 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 401, + 142, + 412 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 393, + 218, + 420 + ], + "score": 1.0, + "content": "are all parameters,", + "type": "text" + }, + { + "bbox": [ + 219, + 399, + 271, + 413 + ], + "score": 0.93, + "content": "\\mathbf { W } \\in \\mathbb { R } _ { > 0 } ^ { D \\times K }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 393, + 402, + 420 + ], + "score": 1.0, + "content": "are the positive weights of the n", + "type": "text" + }, + { + "bbox": [ + 414, + 393, + 506, + 420 + ], + "score": 1.0, + "content": "ork (positivity ensures", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 411, + 413, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 192, + 424 + ], + "score": 1.0, + "content": "a positive precision),", + "type": "text" + }, + { + "bbox": [ + 205, + 411, + 223, + 424 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 223, + 412, + 236, + 423 + ], + "score": 0.87, + "content": "\\lambda _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 411, + 402, + 424 + ], + "score": 1.0, + "content": "are the centers and the bandwidth of the", + "type": "text" + }, + { + "bbox": [ + 402, + 412, + 413, + 421 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 124, + 435 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 423, + 154, + 434 + ], + "score": 0.91, + "content": "\\zeta \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "is a vector of positive constants to prevent division by zero. It is easy to see that with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 434, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 504, + 446 + ], + "score": 1.0, + "content": "this approach the variance of the generator increases with the distance to the centers. The right-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "most panel of Fig. 4 shows an estimated variance function, which indeed has the desired property", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "that variance is large outside the data support. Further, note the increased variance between the two", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "clusters, which captures that even interpolating between clusters comes with a level of uncertainty. In", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "Appendix C we also demonstrate that this simple variance model improves the marginal likelihood", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 198, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 126, + 500 + ], + "score": 0.9, + "content": "p ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 488, + 198, + 501 + ], + "score": 1.0, + "content": "on held-out data.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 364, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 364, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 364, + 517 + ], + "score": 1.0, + "content": "Training the variance network amounts to fitting the RBF net-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 516, + 364, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 364, + 528 + ], + "score": 1.0, + "content": "work. 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The first two panels show the data and its corresponding latent", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "representations (here both input and latent dimensions are 2 to ease illustration). The third panel", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "shows the variance function under a standard architecture, deep multilayer perceptron with softplus", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 506, + 257 + ], + "score": 1.0, + "content": "nonlinearity for the output layer. It is evident that variance estimates in regions without data are", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 267 + ], + "score": 1.0, + "content": "not representative of either uncertainty or error of the generative process; sometimes variance is", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "high, sometimes it is low. From a probabilistic modeling point-of-view, this is disheartening. An", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 278, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 289 + ], + "score": 1.0, + "content": "informal survey of publicly available VAE implementations also reveals that it is common to enforce", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 289, + 368, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 368, + 301 + ], + "score": 1.0, + "content": "a constant unit variance everywhere; this is further disheartening.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 212, + 506, + 301 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 304, + 505, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "For our purposes, we need well-behaved variance functions to ensure a well-behaved geometry,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "but reasonable variance estimates are of general use. Here, as a general strategy, we propose to", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 327, + 504, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 504, + 338 + ], + "score": 1.0, + "content": "model the inverse variance with a network that extrapolates towards zero. This at least ensures that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 335, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 437, + 356 + ], + "score": 1.0, + "content": "variances are large in regions without data. Specifically, we model the precision as", + "type": "text" + }, + { + "bbox": [ + 438, + 337, + 501, + 354 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\beta _ { \\psi } ( \\mathbf { z } ) = \\frac { 1 } { \\pmb { \\sigma } _ { \\psi } ^ { 2 } ( \\mathbf { z } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 335, + 505, + 356 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "where all operations are element-wise. Then, we model this precision with a radial basis function", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 363, + 387, + 377 + ], + "spans": [ + { + "bbox": [ + 107, + 364, + 132, + 375 + ], + "score": 0.75, + "content": "( R B F )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 363, + 387, + 377 + ], + "score": 1.0, + "content": "neural network (Que & Belkin, 2016). Formally this is written", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 304, + 506, + 377 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 143, + 376, + 468, + 398 + ], + "lines": [ + { + "bbox": [ + 143, + 376, + 468, + 398 + ], + "spans": [ + { + "bbox": [ + 143, + 376, + 468, + 398 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\beta _ { \\psi } ( { \\bf z } ) = { \\bf W } { \\bf v } ( { \\bf z } ) + \\boldsymbol { \\zeta } , \\quad \\mathrm { w i t h } \\quad v _ { k } ( { \\bf z } ) = \\exp \\left( - \\lambda _ { k } \\left. { \\bf z } - { \\bf c } _ { k } \\right. _ { 2 } ^ { 2 } \\right) , k = 1 , \\dots , K , } \\end{array}", + "type": "interline_equation", + "image_path": "3497e224dfd202107a9c1a9fe0bc09cdc511c58d74865d312b5888457867c2e1.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 143, + 376, + 468, + 398 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 398, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 102, + 393, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 102, + 393, + 133, + 420 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 401, + 142, + 412 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 393, + 218, + 420 + ], + "score": 1.0, + "content": "are all parameters,", + "type": "text" + }, + { + "bbox": [ + 219, + 399, + 271, + 413 + ], + "score": 0.93, + "content": "\\mathbf { W } \\in \\mathbb { R } _ { > 0 } ^ { D \\times K }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 393, + 402, + 420 + ], + "score": 1.0, + "content": "are the positive weights of the n", + "type": "text" + }, + { + "bbox": [ + 414, + 393, + 506, + 420 + ], + "score": 1.0, + "content": "ork (positivity ensures", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 411, + 413, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 192, + 424 + ], + "score": 1.0, + "content": "a positive precision),", + "type": "text" + }, + { + "bbox": [ + 205, + 411, + 223, + 424 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 223, + 412, + 236, + 423 + ], + "score": 0.87, + "content": "\\lambda _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 411, + 402, + 424 + ], + "score": 1.0, + "content": "are the centers and the bandwidth of the", + "type": "text" + }, + { + "bbox": [ + 402, + 412, + 413, + 421 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 124, + 435 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 423, + 154, + 434 + ], + "score": 0.91, + "content": "\\zeta \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "is a vector of positive constants to prevent division by zero. It is easy to see that with", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 434, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 504, + 446 + ], + "score": 1.0, + "content": "this approach the variance of the generator increases with the distance to the centers. The right-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "most panel of Fig. 4 shows an estimated variance function, which indeed has the desired property", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "that variance is large outside the data support. Further, note the increased variance between the two", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "clusters, which captures that even interpolating between clusters comes with a level of uncertainty. In", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "Appendix C we also demonstrate that this simple variance model improves the marginal likelihood", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 198, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 126, + 500 + ], + "score": 0.9, + "content": "p ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 488, + 198, + 501 + ], + "score": 1.0, + "content": "on held-out data.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24, + "bbox_fs": [ + 102, + 393, + 506, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 364, + 560 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 364, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 364, + 517 + ], + "score": 1.0, + "content": "Training the variance network amounts to fitting the RBF net-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 516, + 364, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 364, + 528 + ], + "score": 1.0, + "content": "work. Assuming we have already trained the inference network", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 527, + 365, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 316, + 539 + ], + "score": 1.0, + "content": "(Sec. 2), we can encode the training data, and use", + "type": "text" + }, + { + "bbox": [ + 316, + 527, + 323, + 537 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 527, + 365, + 539 + ], + "score": 1.0, + "content": "-means to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 538, + 364, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 364, + 549 + ], + "score": 1.0, + "content": "estimate the RBF centers. 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Figure 5 shows this volume measure for both standard", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 709, + 367, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 367, + 722 + ], + "score": 1.0, + "content": "variance functions as well as our proposed RBF model. We see", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 721, + 299, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 299, + 732 + ], + "score": 1.0, + "content": "the trend of the data, unlike the standard model.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 662, + 367, + 732 + ] + }, + { + "type": "image", + "bbox": [ + 372, + 493, + 499, + 673 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 372, + 493, + 499, + 673 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 372, + 493, + 499, + 673 + ], + "spans": [ + { + "bbox": [ + 372, + 493, + 499, + 673 + ], + "score": 0.962, + "type": "image", + "image_path": "c7075f8077cba97a8ced072d15552c951119f42cbb37ef44a5700aff838d8091.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 372, + 493, + 499, + 673 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 371, + 675, + 504, + 708 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 370, + 672, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 370, + 672, + 505, + 688 + ], + "score": 1.0, + "content": "Figure 5: Comparison of (log)", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 371, + 686, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 371, + 686, + 505, + 698 + ], + "score": 1.0, + "content": "measures of standard (top) and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 371, + 696, + 491, + 709 + ], + "spans": [ + { + "bbox": [ + 371, + 696, + 491, + 709 + ], + "score": 1.0, + "content": "proposed (bottom) variances.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50 + } + ], + "index": 46.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 234, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 235, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 235, + 96 + ], + "score": 1.0, + "content": "5 EMPIRICAL RESULTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "We demonstrate the usefulness of the geometric view of the latent space with several experiments.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "Model and implementation details can be found in Appendix D. In all experiments we first train a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 127, + 310, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 310, + 139 + ], + "score": 1.0, + "content": "VAE and then use the induced Riemannian metric.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 107, + 152, + 243, + 163 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 244, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 244, + 165 + ], + "score": 1.0, + "content": "5.1 MEANINGFUL DISTANCES", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 282, + 250 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 282, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 282, + 185 + ], + "score": 1.0, + "content": "First we seek to quantify if the induced Rie-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 183, + 282, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 282, + 196 + ], + "score": 1.0, + "content": "mannian distance in the latent space is more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 194, + 282, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 282, + 206 + ], + "score": 1.0, + "content": "useful than the usual Euclidean distance.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 206, + 282, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 213, + 217 + ], + "score": 1.0, + "content": "For this we perform basic", + "type": "text" + }, + { + "bbox": [ + 213, + 206, + 220, + 216 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 206, + 282, + 217 + ], + "score": 1.0, + "content": "-means cluster-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 217, + 283, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 283, + 228 + ], + "score": 1.0, + "content": "ing under the two metrics. We construct 3", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 227, + 282, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 282, + 239 + ], + "score": 1.0, + "content": "sets of MNIST digits, using 1000 random", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 238, + 282, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 282, + 250 + ], + "score": 1.0, + "content": "samples for each digit. We train a VAE for", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + }, + { + "type": "table", + "bbox": [ + 294, + 170, + 500, + 227 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 294, + 170, + 500, + 227 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 294, + 170, + 500, + 227 + ], + "spans": [ + { + "bbox": [ + 294, + 170, + 500, + 227 + ], + "score": 0.976, + "html": "
DigitsLinearRiemannian
{0,1,2}77.57(±0.87)%94.28(±1.14)%
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One example result is shown in Fig. 6. Here it is evident that, since the latent points", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 465, + 285 + ], + "score": 1.0, + "content": "roughly follow a unit Gaussian, there is little structure to be discovered by the Euclidean", + "type": "text" + }, + { + "bbox": [ + 466, + 272, + 473, + 281 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "-means,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 281, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 296 + ], + "score": 1.0, + "content": "and consequently it performs poorly. The Riemannian clustering is remarkably accurate. Summary", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 293, + 504, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 423, + 306 + ], + "score": 1.0, + "content": "statistics across all subsets are provided in Table 1, which shows the established", + "type": "text" + }, + { + "bbox": [ + 423, + 294, + 432, + 303 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 293, + 504, + 306 + ], + "score": 1.0, + "content": "-measure for clus-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "tering accuracy. Again, the Riemannian metric significantly improves clustering. This implies that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 315, + 436, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 436, + 328 + ], + "score": 1.0, + "content": "the underlying Riemannian distance is more useful than its Euclidean counterpart.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "image", + "bbox": [ + 112, + 335, + 498, + 443 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 335, + 498, + 443 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 335, + 498, + 443 + ], + "spans": [ + { + "bbox": [ + 112, + 335, + 498, + 443 + ], + "score": 0.971, + "type": "image", + "image_path": "d806b849a937f63ce95cada43c48d104ccea6c002aa6302cf87a7bb9d7b7e0eb.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 112, + 335, + 498, + 371.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 112, + 371.0, + 498, + 407.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 112, + 407.0, + 498, + 443.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 450, + 504, + 472 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 450, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 200, + 462 + ], + "score": 1.0, + "content": "Figure 6: The result of", + "type": "text" + }, + { + "bbox": [ + 201, + 451, + 208, + 460 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 450, + 506, + 462 + ], + "score": 1.0, + "content": "-means comparing the distance measures. For the decision boundaries we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 461, + 209, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 209, + 473 + ], + "score": 1.0, + "content": "used 7-NN classification.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + } + ], + "index": 26.25 + }, + { + "type": "title", + "bbox": [ + 107, + 492, + 206, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 207, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 207, + 504 + ], + "score": 1.0, + "content": "5.2 INTERPOLATIONS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 512, + 506, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "Next, we investigate whether the Riemannian metric gives more meaningful interpolations. First,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "score": 1.0, + "content": "we train a VAE for the digits 0 and 1 from MNIST. The upper left panel of Fig. 7 shows the latent", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "space with the Riemannian measure as background color, together with two interpolations. Images", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "generated by both Riemannian and Euclidean interpolations are shown in the bottom of Fig. 7. The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "Euclidean interpolations seem to have a very abrupt change when transitioning from one class to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "another. The Riemannian interpolant gives smoother changes in the generated images. 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Figure 8 shows the corresponding latent", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "space and some sample interpolations. As before, we see more smooth changes in generated images", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 623, + 311, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 311, + 634 + ], + "score": 1.0, + "content": "when we take the Riemannian metric into account.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 107, + 646, + 297, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 646, + 299, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 299, + 659 + ], + "score": 1.0, + "content": "5.3 LATENT PROBABILITY DISTRIBUTIONS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 667, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 504, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 504, + 680 + ], + "score": 1.0, + "content": "We have seen strong indications that the Riemannian metric gives a more meaningful view of the la-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "tent space, which may also improve probability distributions in the latent space. A relevant candidate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 688, + 460, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 460, + 702 + ], + "score": 1.0, + "content": "distribution is the locally adaptive normal distribution (LAND) (Arvanitidis et al., 2016)", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 703, + 397, + 731 + ], + "lines": [ + { + "bbox": [ + 213, + 703, + 397, + 731 + ], + "spans": [ + { + "bbox": [ + 213, + 703, + 397, + 731 + ], + "score": 0.94, + "content": "\\mathrm { L A N D } ( { \\bf z } \\mid { \\pmb \\mu } , { \\pmb \\Sigma } ) \\propto \\exp \\left( - \\frac { 1 } { 2 } \\mathrm { d i s t } _ { { \\pmb \\Sigma } } ^ { 2 } ( { \\bf z } , { \\pmb \\mu } ) \\right) ,", + "type": "interline_equation", + "image_path": "99f94b62df4a0b633745d11f671ac4c38fbacca968ef6193dfc25ed86c3a2e62.jpg" + } + ] + } + ], + "index": 45.5, + "virtual_lines": [ + { + "bbox": [ + 213, + 703, + 397, + 717.0 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 213, + 717.0, + 397, + 731.0 + ], + "spans": [], + "index": 46 + } + ] + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 234, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 235, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 235, + 96 + ], + "score": 1.0, + "content": "5 EMPIRICAL RESULTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "We demonstrate the usefulness of the geometric view of the latent space with several experiments.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "Model and implementation details can be found in Appendix D. In all experiments we first train a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 127, + 310, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 310, + 139 + ], + "score": 1.0, + "content": "VAE and then use the induced Riemannian metric.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 105, + 506, + 139 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 152, + 243, + 163 + ], + "lines": [ + { + "bbox": [ + 105, + 150, + 244, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 244, + 165 + ], + "score": 1.0, + "content": "5.1 MEANINGFUL DISTANCES", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 282, + 250 + ], + "lines": [ + { + "bbox": [ + 106, + 172, + 282, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 282, + 185 + ], + "score": 1.0, + "content": "First we seek to quantify if the induced Rie-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 183, + 282, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 282, + 196 + ], + "score": 1.0, + "content": "mannian distance in the latent space is more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 194, + 282, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 282, + 206 + ], + "score": 1.0, + "content": "useful than the usual Euclidean distance.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 206, + 282, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 213, + 217 + ], + "score": 1.0, + "content": "For this we perform basic", + "type": "text" + }, + { + "bbox": [ + 213, + 206, + 220, + 216 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 206, + 282, + 217 + ], + "score": 1.0, + "content": "-means cluster-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 217, + 283, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 283, + 228 + ], + "score": 1.0, + "content": "ing under the two metrics. 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One example result is shown in Fig. 6. Here it is evident that, since the latent points", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 465, + 285 + ], + "score": 1.0, + "content": "roughly follow a unit Gaussian, there is little structure to be discovered by the Euclidean", + "type": "text" + }, + { + "bbox": [ + 466, + 272, + 473, + 281 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "-means,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 281, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 296 + ], + "score": 1.0, + "content": "and consequently it performs poorly. The Riemannian clustering is remarkably accurate. Summary", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 293, + 504, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 423, + 306 + ], + "score": 1.0, + "content": "statistics across all subsets are provided in Table 1, which shows the established", + "type": "text" + }, + { + "bbox": [ + 423, + 294, + 432, + 303 + ], + "score": 0.84, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 293, + 504, + 306 + ], + "score": 1.0, + "content": "-measure for clus-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "tering accuracy. Again, the Riemannian metric significantly improves clustering. This implies that", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 315, + 436, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 436, + 328 + ], + "score": 1.0, + "content": "the underlying Riemannian distance is more useful than its Euclidean counterpart.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 249, + 506, + 328 + ] + }, + { + "type": "image", + "bbox": [ + 112, + 335, + 498, + 443 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 335, + 498, + 443 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 335, + 498, + 443 + ], + "spans": [ + { + "bbox": [ + 112, + 335, + 498, + 443 + ], + "score": 0.971, + "type": "image", + "image_path": "d806b849a937f63ce95cada43c48d104ccea6c002aa6302cf87a7bb9d7b7e0eb.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 112, + 335, + 498, + 371.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 112, + 371.0, + 498, + 407.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 112, + 407.0, + 498, + 443.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 450, + 504, + 472 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 450, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 200, + 462 + ], + "score": 1.0, + "content": "Figure 6: The result of", + "type": "text" + }, + { + "bbox": [ + 201, + 451, + 208, + 460 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 450, + 506, + 462 + ], + "score": 1.0, + "content": "-means comparing the distance measures. For the decision boundaries we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 461, + 209, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 209, + 473 + ], + "score": 1.0, + "content": "used 7-NN classification.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + } + ], + "index": 26.25 + }, + { + "type": "title", + "bbox": [ + 107, + 492, + 206, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 207, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 207, + 504 + ], + "score": 1.0, + "content": "5.2 INTERPOLATIONS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 512, + 506, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "Next, we investigate whether the Riemannian metric gives more meaningful interpolations. First,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 535 + ], + "score": 1.0, + "content": "we train a VAE for the digits 0 and 1 from MNIST. The upper left panel of Fig. 7 shows the latent", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "space with the Riemannian measure as background color, together with two interpolations. Images", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "generated by both Riemannian and Euclidean interpolations are shown in the bottom of Fig. 7. The", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "Euclidean interpolations seem to have a very abrupt change when transitioning from one class to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "another. The Riemannian interpolant gives smoother changes in the generated images. The top-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "right panel of the figure shows the auto-correlation of images along the interpolants; again we see", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "a very abrupt change in the Euclidean interpolant, while the Riemannian is significantly smoother.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "We also train a convolutional VAE on frames from a video. Figure 8 shows the corresponding latent", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "space and some sample interpolations. As before, we see more smooth changes in generated images", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 623, + 311, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 311, + 634 + ], + "score": 1.0, + "content": "when we take the Riemannian metric into account.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 512, + 506, + 634 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 646, + 297, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 646, + 299, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 299, + 659 + ], + "score": 1.0, + "content": "5.3 LATENT PROBABILITY DISTRIBUTIONS", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 667, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 504, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 504, + 680 + ], + "score": 1.0, + "content": "We have seen strong indications that the Riemannian metric gives a more meaningful view of the la-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "tent space, which may also improve probability distributions in the latent space. A relevant candidate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 688, + 460, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 460, + 702 + ], + "score": 1.0, + "content": "distribution is the locally adaptive normal distribution (LAND) (Arvanitidis et al., 2016)", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 106, + 667, + 505, + 702 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 703, + 397, + 731 + ], + "lines": [ + { + "bbox": [ + 213, + 703, + 397, + 731 + ], + "spans": [ + { + "bbox": [ + 213, + 703, + 397, + 731 + ], + "score": 0.94, + "content": "\\mathrm { L A N D } ( { \\bf z } \\mid { \\pmb \\mu } , { \\pmb \\Sigma } ) \\propto \\exp \\left( - \\frac { 1 } { 2 } \\mathrm { d i s t } _ { { \\pmb \\Sigma } } ^ { 2 } ( { \\bf z } , { \\pmb \\mu } ) \\right) ,", + "type": "interline_equation", + "image_path": "99f94b62df4a0b633745d11f671ac4c38fbacca968ef6193dfc25ed86c3a2e62.jpg" + } + ] + } + ], + "index": 45.5, + "virtual_lines": [ + { + "bbox": [ + 213, + 703, + 397, + 717.0 + ], + "spans": [], + "index": 45 + }, + { + "bbox": [ + 213, + 717.0, + 397, + 731.0 + ], + "spans": [], + "index": 46 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 80, + 503, + 363 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 80, + 503, + 363 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 80, + 503, + 363 + ], + "spans": [ + { + "bbox": [ + 108, + 80, + 503, + 363 + ], + "score": 0.975, + "type": "image", + "image_path": "7c6c7604b2beef99c6958e92dc2f972869e6161897c59d460e8cbd4c5540d244.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 80, + 503, + 174.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 174.33333333333331, + 503, + 268.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 268.66666666666663, + 503, + 362.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 372, + 505, + 406 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "Figure 7: Left: the latent space with example interpolants. Right: auto-correlations of Riemannian", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "(top) and Euclidean (bottom) samples along the curves of the left panel. Bottom: decoded images", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 394, + 396, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 396, + 407 + ], + "score": 1.0, + "content": "along Euclidean (top rows) and Riemannian (bottom rows) interpolants.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "image", + "bbox": [ + 109, + 419, + 503, + 501 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 419, + 503, + 501 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 109, + 419, + 503, + 501 + ], + "spans": [ + { + "bbox": [ + 109, + 419, + 503, + 501 + ], + "score": 0.969, + "type": "image", + "image_path": "1a1c6b5de149bfc16c6c0898140b356140a1b13ca85c6283cfaa6b15b9031579.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 109, + 419, + 503, + 446.3333333333333 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 109, + 446.3333333333333, + 503, + 473.66666666666663 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 109, + 473.66666666666663, + 503, + 500.99999999999994 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 510, + 504, + 533 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 509, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 524 + ], + "score": 1.0, + "content": "Figure 8: Left: the latent space and geodesic interpolants. Right: samples comparing Euclidean (top", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 522, + 476, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 476, + 533 + ], + "score": 1.0, + "content": "row) with Riemannian (bottom row) interpolation. Corresponding videos can be found here.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 8.25 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 133, + 567 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 556, + 158, + 566 + ], + "score": 0.82, + "content": "\\mathrm { d i s t } _ { \\Sigma }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "is the Riemannian extension of Mahalanobis distance. We fit a mixture of two LANDs", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "to the MNIST data from Sec. 5.2 alongside a mixture of Euclidean normal distributions. The first", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 578, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 589 + ], + "score": 1.0, + "content": "column of Fig. 9 shows the density functions of the two mixture models. Only the Riemannian model", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "reveals the underlying clusters. We then sample 40 points from each component of these generative", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 598, + 504, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 504, + 612 + ], + "score": 1.0, + "content": "models1 (center column of the figure). We see that the Riemannian model generates high-quality", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "samples, whereas the Euclidean model generates several samples in regions where the generator", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "is not trained and therefore produces blurry images. Finally, the right column of Fig. 9 shows all", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 631, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 646 + ], + "score": 1.0, + "content": "pairwise distances between the latent points under both Riemannian and Euclidean distances. Again,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 644, + 387, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 387, + 655 + ], + "score": 1.0, + "content": "we see that the geometric view clearly reveals the underlying clusters.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 668, + 312, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 312, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 312, + 681 + ], + "score": 1.0, + "content": "5.4 RANDOM WALK ON THE DATA MANIFOLD", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 109, + 689, + 504, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "Finally, we consider random walks over the data manifold, which is a common tool for exploring la-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "tent spaces. To avoid the walk drifting outside the data support, practical implementations artificially", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 113, + 721, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 718, + 504, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 718, + 504, + 734 + ], + "score": 1.0, + "content": "1We do not follow common practice and sort samples by their likelihood, as this hides low-quality samples.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 80, + 503, + 363 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 80, + 503, + 363 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 80, + 503, + 363 + ], + "spans": [ + { + "bbox": [ + 108, + 80, + 503, + 363 + ], + "score": 0.975, + "type": "image", + "image_path": "7c6c7604b2beef99c6958e92dc2f972869e6161897c59d460e8cbd4c5540d244.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 80, + 503, + 174.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 174.33333333333331, + 503, + 268.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 268.66666666666663, + 503, + 362.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 372, + 505, + 406 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "Figure 7: Left: the latent space with example interpolants. 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Right: samples comparing Euclidean (top", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 522, + 476, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 476, + 533 + ], + "score": 1.0, + "content": "row) with Riemannian (bottom row) interpolation. Corresponding videos can be found here.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + } + ], + "index": 8.25 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 654 + ], + "lines": [ + { + "bbox": [ + 106, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 133, + 567 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 556, + 158, + 566 + ], + "score": 0.82, + "content": "\\mathrm { d i s t } _ { \\Sigma }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "is the Riemannian extension of Mahalanobis distance. 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Finally, the right column of Fig. 9 shows all", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 631, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 646 + ], + "score": 1.0, + "content": "pairwise distances between the latent points under both Riemannian and Euclidean distances. Again,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 644, + 387, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 387, + 655 + ], + "score": 1.0, + "content": "we see that the geometric view clearly reveals the underlying clusters.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 555, + 505, + 655 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 668, + 312, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 312, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 312, + 681 + ], + "score": 1.0, + "content": "5.4 RANDOM WALK ON THE DATA MANIFOLD", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 109, + 689, + 504, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "Finally, we consider random walks over the data manifold, which is a common tool for exploring la-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "tent spaces. To avoid the walk drifting outside the data support, practical implementations artificially", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 241, + 487 + ], + "score": 1.0, + "content": "restrict the walk to stay inside the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 241, + 473, + 274, + 486 + ], + "score": 0.93, + "content": "[ - 1 , 1 ] ^ { d }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 275, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "hypercube. Here, we consider unrestricted Brownian mo-", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "tion under both the Euclidean and Riemannian metric. 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This is explained in the left panel of Fig. 10, which shows", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "that the variance term in the Riemannian metric creates a “wall” around the data, which the random", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 541, + 450, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 450, + 552 + ], + "score": 1.0, + "content": "walk will only rarely cross. 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This unsupervised learning category attracted a lot of attention, especially,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 115, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 130 + ], + "score": 1.0, + "content": "due to the advances on the deep neural networks. We have considered VAEs (Kingma & Welling,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 140 + ], + "score": 1.0, + "content": "2014; Rezende et al., 2014), but the ideas extend to similar related models. These include exten-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "sions that provide more flexible approximate posteriors (Rezende & Mohamed, 2015; Kingma et al.,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 162 + ], + "score": 1.0, + "content": "2016). GANs (Goodfellow et al., 2014) also fall in this category, as these models have an explicit", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "generator. While the inference network is not a necessary component in the GAN model, it has been", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "shown that incorporating it improves overall performance (Donahue et al., 2017; Dumoulin et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "2017). The same thoughts hold for approaches that transform the latent space through a sequence of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 194, + 259, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 259, + 205 + ], + "score": 1.0, + "content": "bijective functions (Dinh et al., 2017)", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 230 + ], + "score": 1.0, + "content": "Geometry in neural networks. Bengio et al. (2013) discuss the importance of geometry in neural", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "networks as a tool to understand local generalization. For instance, the Jacobian matrix is a measure", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 238, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 505, + 253 + ], + "score": 1.0, + "content": "of smoothness for a function that interpolates a surface to the given data. This is exactly the implica-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "tion in (Rifai et al., 2011), where the norm of the Jacobian acts as a regularizer for the deterministic", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 492, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 492, + 273 + ], + "score": 1.0, + "content": "autoencoder. Recently, Kumar et al. (2017) used the Jacobian to inject invariances in a classifier.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 298 + ], + "score": 1.0, + "content": "Riemannian Geometry. Like the present paper, Tosi et al. (2014) derive a suitable Riemannian", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "metric in Gaussian process (GP) latent variable models (Lawrence, 2005), but the computational", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "complexity of GPs causes practical concerns. Unlike works that explicitly learn a Riemannian metric", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "(Hauberg et al., 2012; Peltonen et al., 2004), our metric is fully derived from the generator and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 329, + 336, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 336, + 341 + ], + "score": 1.0, + "content": "requires no extra learning once the generator is available.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 356, + 336, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 338, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 338, + 371 + ], + "score": 1.0, + "content": "7 DISCUSSION AND FURTHER EXTENSIONS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 381, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "The geometric interpretation of representation learning is that the latent space is a compressed and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "flattened version of the data manifold. We show that the actual geometry of the data manifold can", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 403, + 259, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 259, + 416 + ], + "score": 1.0, + "content": "be more complex than it first appears.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "Here we have initiated the study of proper geometries for generative models. We showed that the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "latent space not only provides a low-dimensional representation of the data manifold, but at the same", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "time, can reveal the underlying geometrical structure. We proposed a new variance network for the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "generator, which provides meaningful uncertainty estimates while regularizing the geometry. The", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "new detailed understanding of the geometry provides us with more relevant distance measures, as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 475, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 237, + 487 + ], + "score": 1.0, + "content": "demonstrated by the fact that a", + "type": "text" + }, + { + "bbox": [ + 237, + 475, + 244, + 485 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 475, + 506, + 487 + ], + "score": 1.0, + "content": "-means clustering, on these distances, is better aligned with the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "ground truth label structure than a clustering based on conventional Euclidean distances. We also", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "score": 1.0, + "content": "found that the new distance measure produces smoother interpolation, and when training Rieman-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "nian “LAND” mixture models based on the new geometry, the components aligned much better with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 519, + 504, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 504, + 532 + ], + "score": 1.0, + "content": "the ground truth group structure. Finally, inspired by the recent interest in sequence generation by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 530, + 504, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 504, + 542 + ], + "score": 1.0, + "content": "random walks in latent space, we found that geometrically informed random walks stayed on the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 541, + 434, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 434, + 552 + ], + "score": 1.0, + "content": "manifold for much longer runs than sequences based on Euclidean random walks.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 504, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "The presented analysis easily extends to sophisticated generative models, where the latent space will", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "be potentially endowed with more flexible nonlinear structures. This directly implies particularly", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 594 + ], + "score": 1.0, + "content": "interesting geometrical models. An obvious question is: can the geometry of the latent space play", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 590, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 604 + ], + "score": 1.0, + "content": "a role while we learn the generative model? Either way, we believe that this geometric perspective", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "provides a new way of thinking and further interpreting the generative models, while at the same", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 612, + 448, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 448, + 626 + ], + "score": 1.0, + "content": "time it encourages development of new nonlinear models in the representation space.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5 + }, + { + "type": "title", + "bbox": [ + 108, + 637, + 200, + 647 + ], + "lines": [ + { + "bbox": [ + 107, + 637, + 200, + 648 + ], + "spans": [ + { + "bbox": [ + 107, + 637, + 200, + 648 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 710 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "LKH is supported by Innovation Fund Denmark / the Danish Center for Big Data Analytics Driven", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "Innovation. SH was supported by a research grant (15334) from VILLUM FONDEN. This project", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "has received funding from the European Research Council (ERC) under the European Union’s Hori-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 359, + 702 + ], + "score": 1.0, + "content": "zon 2020 research and innovation programme (grant agreement", + "type": "text" + }, + { + "bbox": [ + 360, + 689, + 370, + 699 + ], + "score": 0.41, + "content": "\\boldsymbol { \\mathrm { n ^ { \\circ } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "757360). We gratefully acknowl-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 470, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 470, + 712 + ], + "score": 1.0, + "content": "edge the support of the NVIDIA Corporation with the donation of the used Titan Xp GPU.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 211, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 213, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 213, + 95 + ], + "score": 1.0, + "content": "6 RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "Generative models. This unsupervised learning category attracted a lot of attention, especially,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 115, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 130 + ], + "score": 1.0, + "content": "due to the advances on the deep neural networks. We have considered VAEs (Kingma & Welling,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 128, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 505, + 140 + ], + "score": 1.0, + "content": "2014; Rezende et al., 2014), but the ideas extend to similar related models. These include exten-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "sions that provide more flexible approximate posteriors (Rezende & Mohamed, 2015; Kingma et al.,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 162 + ], + "score": 1.0, + "content": "2016). GANs (Goodfellow et al., 2014) also fall in this category, as these models have an explicit", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 505, + 173 + ], + "score": 1.0, + "content": "generator. While the inference network is not a necessary component in the GAN model, it has been", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 184 + ], + "score": 1.0, + "content": "shown that incorporating it improves overall performance (Donahue et al., 2017; Dumoulin et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 195 + ], + "score": 1.0, + "content": "2017). The same thoughts hold for approaches that transform the latent space through a sequence of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 194, + 259, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 259, + 205 + ], + "score": 1.0, + "content": "bijective functions (Dinh et al., 2017)", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 105, + 506, + 205 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 230 + ], + "score": 1.0, + "content": "Geometry in neural networks. Bengio et al. (2013) discuss the importance of geometry in neural", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "networks as a tool to understand local generalization. For instance, the Jacobian matrix is a measure", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 238, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 505, + 253 + ], + "score": 1.0, + "content": "of smoothness for a function that interpolates a surface to the given data. This is exactly the implica-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "tion in (Rifai et al., 2011), where the norm of the Jacobian acts as a regularizer for the deterministic", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 492, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 492, + 273 + ], + "score": 1.0, + "content": "autoencoder. Recently, Kumar et al. (2017) used the Jacobian to inject invariances in a classifier.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 216, + 505, + 273 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 285, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 298 + ], + "score": 1.0, + "content": "Riemannian Geometry. Like the present paper, Tosi et al. (2014) derive a suitable Riemannian", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "metric in Gaussian process (GP) latent variable models (Lawrence, 2005), but the computational", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "complexity of GPs causes practical concerns. Unlike works that explicitly learn a Riemannian metric", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "(Hauberg et al., 2012; Peltonen et al., 2004), our metric is fully derived from the generator and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 329, + 336, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 336, + 341 + ], + "score": 1.0, + "content": "requires no extra learning once the generator is available.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 284, + 505, + 341 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 356, + 336, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 338, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 338, + 371 + ], + "score": 1.0, + "content": "7 DISCUSSION AND FURTHER EXTENSIONS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 108, + 381, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 394 + ], + "score": 1.0, + "content": "The geometric interpretation of representation learning is that the latent space is a compressed and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "flattened version of the data manifold. We show that the actual geometry of the data manifold can", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 403, + 259, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 259, + 416 + ], + "score": 1.0, + "content": "be more complex than it first appears.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 381, + 505, + 416 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 420, + 505, + 551 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "Here we have initiated the study of proper geometries for generative models. We showed that the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "latent space not only provides a low-dimensional representation of the data manifold, but at the same", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "time, can reveal the underlying geometrical structure. We proposed a new variance network for the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "generator, which provides meaningful uncertainty estimates while regularizing the geometry. The", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 477 + ], + "score": 1.0, + "content": "new detailed understanding of the geometry provides us with more relevant distance measures, as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 475, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 237, + 487 + ], + "score": 1.0, + "content": "demonstrated by the fact that a", + "type": "text" + }, + { + "bbox": [ + 237, + 475, + 244, + 485 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 475, + 506, + 487 + ], + "score": 1.0, + "content": "-means clustering, on these distances, is better aligned with the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "ground truth label structure than a clustering based on conventional Euclidean distances. We also", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 510 + ], + "score": 1.0, + "content": "found that the new distance measure produces smoother interpolation, and when training Rieman-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "nian “LAND” mixture models based on the new geometry, the components aligned much better with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 519, + 504, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 504, + 532 + ], + "score": 1.0, + "content": "the ground truth group structure. Finally, inspired by the recent interest in sequence generation by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 530, + 504, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 504, + 542 + ], + "score": 1.0, + "content": "random walks in latent space, we found that geometrically informed random walks stayed on the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 541, + 434, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 434, + 552 + ], + "score": 1.0, + "content": "manifold for much longer runs than sequences based on Euclidean random walks.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 420, + 506, + 552 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 504, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "The presented analysis easily extends to sophisticated generative models, where the latent space will", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "be potentially endowed with more flexible nonlinear structures. This directly implies particularly", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 578, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 594 + ], + "score": 1.0, + "content": "interesting geometrical models. An obvious question is: can the geometry of the latent space play", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 590, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 604 + ], + "score": 1.0, + "content": "a role while we learn the generative model? Either way, we believe that this geometric perspective", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "provides a new way of thinking and further interpreting the generative models, while at the same", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 612, + 448, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 448, + 626 + ], + "score": 1.0, + "content": "time it encourages development of new nonlinear models in the representation space.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 557, + 506, + 626 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 637, + 200, + 647 + ], + "lines": [ + { + "bbox": [ + 107, + 637, + 200, + 648 + ], + "spans": [ + { + "bbox": [ + 107, + 637, + 200, + 648 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 710 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "LKH is supported by Innovation Fund Denmark / the Danish Center for Big Data Analytics Driven", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "Innovation. SH was supported by a research grant (15334) from VILLUM FONDEN. This project", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "has received funding from the European Research Council (ERC) under the European Union’s Hori-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 359, + 702 + ], + "score": 1.0, + "content": "zon 2020 research and innovation programme (grant agreement", + "type": "text" + }, + { + "bbox": [ + 360, + 689, + 370, + 699 + ], + "score": 0.41, + "content": "\\boldsymbol { \\mathrm { n ^ { \\circ } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "757360). 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208, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 348, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 349, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 349, + 96 + ], + "score": 1.0, + "content": "Proof. As we introduced in Eq. 8 the stochastic generator is", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 99, + 459, + 114 + ], + "lines": [ + { + "bbox": [ + 151, + 99, + 459, + 114 + ], + "spans": [ + { + "bbox": [ + 151, + 99, + 459, + 114 + ], + "score": 0.87, + "content": "f ( \\mathbf { z } ) = \\pmb { \\mu } ( \\mathbf { z } ) + \\pmb { \\sigma } ( \\mathbf { z } ) \\odot \\epsilon , \\qquad \\pmb { \\mu } : \\mathscr { Z } \\pmb { \\chi } , ~ \\pmb { \\sigma } : \\mathscr { Z } \\mathbb { R } _ { + } ^ { D } , ~ \\epsilon \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbb { I } _ { D } ) .", + "type": "interline_equation", + "image_path": "90430b51e3fba3f5f00b313b2805212223ae00b8d4230e8ec2acd0c1bfbe2cab.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 151, + 99, + 459, + 114 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 118, + 353, + 131 + 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The other term", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 483, + 440, + 594 + ], + "lines": [ + { + "bbox": [ + 156, + 483, + 440, + 594 + ], + "spans": [ + { + "bbox": [ + 156, + 483, + 440, + 594 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\mathbb { E } _ { p ( \\epsilon ) } [ \\mathbf { B } ^ { \\mathsf { T } } \\mathbf { B } ] = \\mathbb { E } _ { p ( \\epsilon ) } \\left( \\left[ \\begin{array} { c } { \\epsilon ^ { \\mathsf { T } } \\mathbf { S } _ { 1 } } \\\\ { \\epsilon ^ { \\mathsf { T } } \\mathbf { S } _ { 2 } } \\\\ { \\vdots } \\\\ { \\epsilon ^ { \\mathsf { T } } \\mathbf { S } _ { d } } \\end{array} \\right] _ { d \\times D } \\quad [ \\mathbf { S } _ { 1 } \\epsilon , \\mathbf { S } _ { 2 } \\epsilon , \\cdots , \\mathbf { S } _ { d } \\epsilon ] \\right) } \\\\ & { \\quad \\quad \\quad \\quad = \\mathbb { E } _ { p ( \\epsilon ) } \\left( \\left[ \\begin{array} { c 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As we introduced in Eq. 8 the stochastic generator is", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 349, + 96 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 99, + 459, + 114 + ], + "lines": [ + { + "bbox": [ + 151, + 99, + 459, + 114 + ], + "spans": [ + { + "bbox": [ + 151, + 99, + 459, + 114 + ], + "score": 0.87, + "content": "f ( \\mathbf { z } ) = \\pmb { \\mu } ( \\mathbf { z } ) + \\pmb { \\sigma } ( \\mathbf { z } ) \\odot \\epsilon , \\qquad \\pmb { \\mu } : \\mathscr { Z } \\pmb { \\chi } , ~ \\pmb { \\sigma } : \\mathscr { Z } \\mathbb { R } _ { + } ^ { D } , ~ \\epsilon \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbb { I } _ { D } ) .", + "type": "interline_equation", + "image_path": "90430b51e3fba3f5f00b313b2805212223ae00b8d4230e8ec2acd0c1bfbe2cab.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 151, + 99, + 459, + 114 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": 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So the expectation of the induced Riemannian", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 212, + 286, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 286, + 225 + ], + "score": 1.0, + "content": "metric in the latent space by the generator is", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 228, + 382, + 251 + ], + "lines": [ + { + "bbox": [ + 228, + 228, + 382, + 251 + ], + "spans": [ + { + "bbox": [ + 228, + 228, + 382, + 251 + ], + "score": 0.93, + "content": "\\bar { \\mathbf { M _ { z } } } = \\left( \\mathbf { J _ { z } ^ { ( \\mu ) } } \\right) ^ { \\mathsf { T } } \\mathbf { J _ { z } ^ { ( \\mu ) } } + \\left( \\mathbf { J _ { z } ^ { ( \\sigma ) } } \\right) ^ { \\mathsf { T } } \\mathbf { J _ { z } ^ { ( \\sigma ) } }", + "type": "interline_equation", + "image_path": "9bc76b185cc7440cddbfecc6482d114be433ed6e502e4d358679c315aa83dba4.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 228, + 228, + 382, + 251 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 216, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 254, + 217, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 217, + 267 + ], + "score": 1.0, + "content": "which concludes the proof.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 106, + 281, + 432, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 434, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 434, + 297 + ], + "score": 1.0, + "content": "C INFLUENCE OF VARIANCE ON THE MARGINAL LIKELIHOOD", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 306, + 506, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 362, + 320 + ], + "score": 1.0, + "content": "We trained a VAE on the digits 0 and 1 of the MNIST scaled to", + "type": "text" + }, + { + "bbox": [ + 362, + 307, + 390, + 319 + ], + "score": 0.88, + "content": "[ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 305, + 506, + 320 + ], + "score": 1.0, + "content": ". We randomly split the data", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 116, + 330 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 318, + 136, + 329 + ], + "score": 0.89, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 317, + 187, + 330 + ], + "score": 1.0, + "content": "training and", + "type": "text" + }, + { + "bbox": [ + 187, + 318, + 207, + 329 + ], + "score": 0.89, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "test data, ensuring balanced classes. First, we only trained the encoder and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "the mean function of the decoder. 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Encoder/DecoderLayer 1Layer 2Layer 3
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Fur-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "ther, the mean and the variance functions share the weights of the first layer. The input space", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 483, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 162, + 497 + ], + "score": 1.0, + "content": "dimension is", + "type": "text" + }, + { + "bbox": [ + 162, + 484, + 203, + 494 + ], + "score": 0.9, + "content": "D = 7 8 4", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 483, + 393, + 497 + ], + "score": 1.0, + "content": ". 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Each mixture component is defined by the generator through the likelihood", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 250, + 635 + ], + "score": 0.92, + "content": "p ( \\mathbf { x } | \\mathbf { z } ) = \\mathcal { N } \\left( \\mathbf { x } \\mid \\pmb { \\mu } _ { \\theta } ( \\mathbf { z } ) , \\mathbb { I } _ { D } \\pmb { \\sigma } _ { \\theta } ^ { 2 } ( \\mathbf { z } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 621, + 506, + 634 + ], + "score": 1.0, + "content": ". Considering the variance term, the standard neural network", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "approach is trained on the given data points and the corresponding latent codes. Unfortunately,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "its behavior is arbitrary in regions where there are not any encoded data. On the other hand our", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "proposed model assigns large variance to these regions, while on the regions where we have latent", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "codes its behavior will be approximately the same with the standard neural network. 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The RBF variance model ensures that mixture", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "components in these regions have high variance, whereas the standard architecture assign arbitrary", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 268, + 722 + ], + "score": 1.0, + "content": "variance. 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So the expectation of the induced Riemannian", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 212, + 286, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 286, + 225 + ], + "score": 1.0, + "content": "metric in the latent space by the generator is", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5, + "bbox_fs": [ + 102, + 192, + 509, + 225 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 228, + 382, + 251 + ], + "lines": [ + { + "bbox": [ + 228, + 228, + 382, + 251 + ], + "spans": [ + { + "bbox": [ + 228, + 228, + 382, + 251 + ], + "score": 0.93, + "content": "\\bar { \\mathbf { M _ { z } } } = \\left( \\mathbf { J _ { z } ^ { ( \\mu ) } } \\right) ^ { \\mathsf { T } } \\mathbf { J _ { z } ^ { ( \\mu ) } } + \\left( \\mathbf { J _ { z } ^ { ( \\sigma ) } } \\right) ^ { \\mathsf { T } } \\mathbf { J _ { z } ^ { ( \\sigma ) } }", + "type": "interline_equation", + "image_path": "9bc76b185cc7440cddbfecc6482d114be433ed6e502e4d358679c315aa83dba4.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 228, + 228, + 382, + 251 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 254, + 216, + 266 + ], + "lines": [ + { + "bbox": [ + 106, + 254, + 217, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 217, + 267 + ], + "score": 1.0, + "content": "which concludes the proof.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 254, + 217, + 267 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 281, + 432, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 434, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 434, + 297 + ], + "score": 1.0, + "content": "C INFLUENCE OF VARIANCE ON THE MARGINAL LIKELIHOOD", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 306, + 506, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 362, + 320 + ], + "score": 1.0, + "content": "We trained a VAE on the digits 0 and 1 of the MNIST scaled to", + "type": "text" + }, + { + "bbox": [ + 362, + 307, + 390, + 319 + ], + "score": 0.88, + "content": "[ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 305, + 506, + 320 + ], + "score": 1.0, + "content": ". We randomly split the data", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 116, + 330 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 318, + 136, + 329 + ], + "score": 0.89, + "content": "9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 317, + 187, + 330 + ], + "score": 1.0, + "content": "training and", + "type": "text" + }, + { + "bbox": [ + 187, + 318, + 207, + 329 + ], + "score": 0.89, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "test data, ensuring balanced classes. First, we only trained the encoder and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "the mean function of the decoder. Then, keeping these fixed, we trained two variance functions: one", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 353 + ], + "score": 1.0, + "content": "based on standard deep neural network architecture, and the other using our proposed RBF model.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 350, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 363 + ], + "score": 1.0, + "content": "Clearly, we have two generators with the same mean function, but different variance functions.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "Below we present the architectures for the standard neural networks. 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Encoder/DecoderLayer 1Layer 2Layer 3
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Fur-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 472, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 505, + 485 + ], + "score": 1.0, + "content": "ther, the mean and the variance functions share the weights of the first layer. The input space", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 483, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 162, + 497 + ], + "score": 1.0, + "content": "dimension is", + "type": "text" + }, + { + "bbox": [ + 162, + 484, + 203, + 494 + ], + "score": 0.9, + "content": "D = 7 8 4", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 483, + 393, + 497 + ], + "score": 1.0, + "content": ". Then, we computed the marginal likelihood", + "type": "text" + }, + { + "bbox": [ + 394, + 484, + 414, + 496 + ], + "score": 0.92, + "content": "p ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 483, + 505, + 497 + ], + "score": 1.0, + "content": "of the test data using", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 494, + 174, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 174, + 506 + ], + "score": 1.0, + "content": "Monte Carlo as:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 462, + 505, + 506 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 525, + 420, + 555 + ], + "lines": [ + { + "bbox": [ + 190, + 525, + 420, + 555 + ], + "spans": [ + { + "bbox": [ + 190, + 525, + 420, + 555 + ], + "score": 0.93, + "content": "p ( \\mathbf { x } ) = \\int _ { \\mathbb { Z } } p ( \\mathbf { x } | \\mathbf { z } ) p ( \\mathbf { z } ) \\mathrm { d } \\mathbf { z } \\simeq \\frac { 1 } { S } \\sum _ { s = 1 } p ( \\mathbf { x } | \\mathbf { z } _ { s } ) , \\quad \\mathbf { z } _ { s } \\sim p ( \\mathbf { z } )", + "type": "interline_equation", + "image_path": "1a757a08df1275929556d32b4313e786bc65a950a73b21d9b12328aa9c5c5997.jpg" + } + ] + } + ], + "index": 26.5, + "virtual_lines": [ + { + "bbox": [ + 190, + 525, + 420, + 540.0 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 190, + 540.0, + 420, + 555.0 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 559, + 503, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 130, + 571 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 131, + 559, + 177, + 570 + ], + "score": 0.88, + "content": "S = 1 0 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "samples. 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The marginal", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "likelihood under the Monte Carlo estimation is, essentially, a large Gaussian mixture model with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 608, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 608, + 168, + 624 + ], + "score": 1.0, + "content": "equal weights", + "type": "text" + }, + { + "bbox": [ + 168, + 609, + 176, + 622 + ], + "score": 0.87, + "content": "\\textstyle { \\frac { 1 } { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 608, + 506, + 624 + ], + "score": 1.0, + "content": ". Each mixture component is defined by the generator through the likelihood", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 250, + 635 + ], + "score": 0.92, + "content": "p ( \\mathbf { x } | \\mathbf { z } ) = \\mathcal { N } \\left( \\mathbf { x } \\mid \\pmb { \\mu } _ { \\theta } ( \\mathbf { z } ) , \\mathbb { I } _ { D } \\pmb { \\sigma } _ { \\theta } ^ { 2 } ( \\mathbf { z } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 621, + 506, + 634 + ], + "score": 1.0, + "content": ". Considering the variance term, the standard neural network", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "approach is trained on the given data points and the corresponding latent codes. 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Encoder/DecoderLayer 1Layer 2Layer 3
64,(tanh)32,(tanh)d,(linear)
64, (tanh)32, (tanh)d,(softplus)
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EncoderLayer 1 (Conv)Layer 2 (Conv)Layer 3 (MLP)Layer 4 (MLP)
山32,3,2,(tanh)32,3,2,(tanh)1024, (tanh)d,(linear)
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DecoderrLayer1(RBF)Layer 2 (Deconv）Layer 3 (Conv)
βW ∈ R(D/2)×641,3,2 (linear)3,3,1 (linear)
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Encoder/DecoderLayer 1Layer 2Layer 3
64,(tanh)32,(tanh)d,(linear)
64, (tanh)32, (tanh)d,(softplus)
μ32,(tanh)64,(tanh)D,(sigmoid)
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EncoderLayer 1 (Conv)Layer 2 (Conv)Layer 3 (MLP)Layer 4 (MLP)
山32,3,2,(tanh)32,3,2,(tanh)1024, (tanh)d,(linear)
o32,3,2,(tanh)32,3,2,(tanh)1024, (tanh)d,(softplus)
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DecoderrLayer1(RBF)Layer 2 (Deconv）Layer 3 (Conv)
βW ∈ R(D/2)×641,3,2 (linear)3,3,1 (linear)
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sha256:1ceb631997fa5d3c34039256f340692ddb18ed5d77764c26254adbae70b64d71 +size 78924 diff --git a/parse/train/ryfMLoCqtQ/ryfMLoCqtQ.md b/parse/train/ryfMLoCqtQ/ryfMLoCqtQ.md new file mode 100644 index 0000000000000000000000000000000000000000..d045899c557197d0640e243ad4347abf8b6860cf --- /dev/null +++ b/parse/train/ryfMLoCqtQ/ryfMLoCqtQ.md @@ -0,0 +1,476 @@ +# AN ANALYTIC THEORY OF GENERALIZATION DYNAMICS AND TRANSFER LEARNING IN DEEP LINEAR NETWORKS + +Andrew K. Lampinen Department of Psychology Stanford University lampinen@stanford.edu + +Surya Ganguli +Department of Applied Physics +Stanford University +and +Google Brain +sganguli@stanford.edu + +# ABSTRACT + +Much attention has been devoted recently to the generalization puzzle in deep learning: large, deep networks can generalize well, but existing theories bounding generalization error are exceedingly loose, and thus cannot explain this striking performance. Furthermore, a major hope is that knowledge may transfer across tasks, so that multi-task learning can improve generalization on individual tasks. However we lack analytic theories that can quantitatively predict how the degree of knowledge transfer depends on the relationship between the tasks. We develop an analytic theory of the nonlinear dynamics of generalization in deep linear networks, both within and across tasks. In particular, our theory provides analytic solutions to the training and testing error of deep networks as a function of training time, number of examples, network size and initialization, and the task structure and SNR. Our theory reveals that deep networks progressively learn the most important task structure first, so that generalization error at the early stopping time primarily depends on task structure and is independent of network size. This suggests any tight bound on generalization error must take into account task structure, and explains observations about real data being learned faster than random data. Intriguingly our theory also reveals the existence of a learning algorithm that proveably out-performs neural network training through gradient descent. Finally, for transfer learning, our theory reveals that knowledge transfer depends sensitively, but computably, on the SNRs and input feature alignments of pairs of tasks. + +# 1 INTRODUCTION + +Many deep learning practitioners closely monitor both training and test errors, hoping to achieve both a small training error and a small generalization error, or gap between testing and training errors. Training is usually stopped early, before overfitting sets in and increases the test error. This procedure often results in large networks that generalize well on structured tasks, raising an important generalization puzzle (Zhang et al., 2016): many existing theories that upper bound generalization error (Bartlett & Mendelson, 2002; Neyshabur et al., 2015; Dziugaite & Roy, 2017; Golowich et al., 2017; Neyshabur et al., 2017; Bartlett et al., 2017; Arora et al., 2018, e.g) in terms of various measures of network complexity yield very loose bounds. Therefore they cannot explain the impressive generalization capabilities of deep nets. + +In the absence of any such tight and computable theory of deep network generalization error, we develop an analytic theory of generalization error for deep linear networks. Such networks exhibit highly nonlinear learning dynamics (Saxe et al., 2013a;b) including many prominent phenomena like learning plateaus, saddle points, and sudden drops in training error. Moreover, theory developed for the learning dynamics of deep linear networks directly inspired better initialization schemes for nonlinear networks (Schoenholz et al., 2016; Pennington et al., 2017; 2018). Here we show that deep linear networks also provide a good theoretical model for generalization dynamics. In particular we develop an analytic theory for both the training and test error of a deep linear network as a function of training time, number of training examples, network architecture, initialization, and task structure and SNR. Our theory matches simulations and reveals that deep networks with small weight initialization learn the most important aspects of a task first. Thus the optimal test error at the early stopping time depends largely on task structure and SNR, and not on network architecture, as long as the architecture is expressive enough to attain small training error. Thus our exact analysis of generalization dynamics reveals the important lesson that any theory that seeks to upper bound generalization error based only on network architecture, and not on task structure, is likely to yield exceedingly loose upper bounds. Intriguingly our theory also reveals a non-gradient-descent learning algorithm that proveably out-performs neural network training through gradient descent. + +We also apply our theory to multi-task learning, which enables knowledge transfer from one task to another, thereby further lowering generalization error (Dong et al., 2015; Rusu et al., 2015; Luong et al., 2016, e.g.). Moreover, knowledge transfer across tasks may be key to human generalization capabilities (Hansen et al., 2017; Lampinen et al., 2017). We provide an analytic theory for how much knowledge is transferred between pairs of tasks, and we find that it displays a sensitive but computable dependence on the relationship between pairs of tasks, in particular, their SNRs and feature space alignments. + +We note that a related prior work (Advani & Saxe, 2017) studied generalization in shallow and deep linear networks, but that work was limited to networks with a single output, thereby precluding the possibility of addressing the issue of transfer learning. Moreover, analyzing networks with a single output also precludes the possibility of addressing interesting tasks that require higher dimensional outputs, for example in language (Dong et al., 2015, e.g.), generative models (Goodfellow et al., 2014, e.g), and reinforcement learning (Mnih et al., 2015; Silver et al., 2016, e.g). + +# 2 THEORETICAL FRAMEWORK + +We work in a student-teacher scenario in which we consider an ensemble of low rank, noisy teacher networks that generate training data for a potentially more complex student network, and define the training and test errors whose dynamics we wish to understand. + +# 2.1 AN ENSEMBLE OF LOW-RANK NOISY TEACHERS + +We first consider an ensemble of 3-layer linear teacher networks with $\overline { { N } } _ { i }$ units in layer $i$ , and weight matrices $\overline { { \mathbf { W } } } ^ { 2 1 } \in \mathbb { R } ^ { \overline { { N _ { 2 } } } \times \overline { { N } } _ { 1 } }$ and $\overline { { \mathbf { W } } } ^ { 3 2 } \in \mathbb { R } ^ { \overline { { N _ { 3 } } } \times \overline { { N _ { 2 } } } }$ between the input to hidden, and hidden to output layers, respectively. The teacher network thus computes the composite map $\overline { { \mathbf { y } } } = \overline { { \mathbf { W } } } \mathbf { x }$ , where $\dot { \overline { { \mathbf { W } } } } \equiv \overline { { \mathbf { W } } } ^ { 3 \bar { 2 } } \overline { { \mathbf { W } } } ^ { 2 1 }$ . Of critical importance is the singular value decomposition (SVD) of $\overline { { \mathbf { W } } }$ : + +$$ +\overline { { \mathbf { W } } } = \overline { { \mathbf { U } } } \overline { { \mathbf { S } } } \overline { { \mathbf { V } } } ^ { T } = \sum _ { \alpha = 1 } ^ { \overline { { N _ { 2 } } } } \overline { { s } } ^ { \alpha } \overline { { \mathbf { u } } } ^ { \alpha } \overline { { \mathbf { v } } } ^ { \alpha T } , +$$ + +Where $\overline { { \mathbf { U } } } \in \mathbb { R } ^ { \overline { { N _ { 3 } } } \times \overline { { N } } _ { 2 } }$ and $\overline { { \mathbf { V } } } \in \mathbb { R } ^ { \overline { { N _ { 1 } } } \times \overline { { N } } _ { 2 } }$ are both matrices with orthonormal columns and $\overline { \mathbf { S } }$ is an $\overline { { N _ { 2 } } } \times \overline { { N _ { 2 } } }$ diagonal matrix. We construct a random teacher by picking $\overline { { \mathbf { U } } }$ and $\overline { { \mathbf { V } } }$ to be random matrices with orthonormal columns and choosing $O ( 1 )$ values for the diagonal elements of $\overline { { \mathbf { S } } }$ . We work in the limit $\overline { { N _ { 1 } } } , \overline { { N _ { 3 } } } \infty$ with an $O ( 1 )$ aspect ratio $\mathcal { A } = \overline { { N _ { 3 } } } / \overline { { N _ { 1 } } } \in ( 0 , 1 ]$ so that the teacher has fewer outputs than inputs. Also, we hold $\overline { { N } } _ { 2 } \sim O ( 1 )$ , so the teacher has a low, finite rank, and we study generalization performance as a function of the $\overline { { N } } _ { 2 }$ teacher singular values. + +We further assume the teacher generates noisy outputs from a set of $\overline { { N } } _ { 1 }$ orthonormal inputs: + +$$ +\hat { { \bf y } } ^ { \mu } = \overline { { { \bf W } } } \hat { \bf x } ^ { \mu } + { \bf z } ^ { \mu } \qquad \mathrm { f o r } \quad \mu = { \bf 1 } , \ldots , \overline { { { \bf N } } } _ { \bf 1 } . +$$ + +This training set yields important second-order training statistics that will guide student learning: + +$$ +\begin{array} { r } { \pmb { \Sigma } ^ { 1 1 } \equiv \overset { \overline { { \boldsymbol { N } } } _ { 1 } } { \mu = 1 } \hat { \mathbf { x } } ^ { \mu } \hat { \mathbf { x } } ^ { \mu T } = \mathbf { I } , \qquad \pmb { \Sigma } ^ { 3 1 } \equiv \overset { \overline { { \boldsymbol { N } } } _ { 1 } } { \mu = 1 } \hat { \mathbf { y } } ^ { \mu } \hat { \mathbf { x } } ^ { \mu T } = \overline { { \mathbf { W } } } + \mathbf { Z } \hat { \mathbf { X } } ^ { T } . } \end{array} +$$ + +Here the input covariance $\pmb { \Sigma } ^ { 1 1 }$ is assumed to be white (a common pre-processing step), the inputoutput covariance $\pmb { \Sigma } ^ { 3 1 }$ is simplified using (2), and $\mathbf { Z } \in \mathbb { R } ^ { \overline { { N } } _ { 3 } \times \overline { { N } } _ { 1 } }$ is the noise matrix, whose $\mu$ ’th column is $\mathbf { z } ^ { \mu }$ . Its matrix elements $z _ { i } ^ { \mu }$ are drawn iid. from a Gaussian with zero mean and variance $\sigma _ { z } ^ { 2 } / \overline { { N } } _ { 1 }$ . The noise scaling is chosen so the singular values of the teacher $\overline { { \mathbf { W } } }$ and the noise $\mathbf { Z }$ are both $O ( 1 )$ , leading to non-trivial generalization effects. As generalization performance will depend on the ratio of teacher singular values to the noise variance parameter $\sigma _ { z } ^ { 2 }$ , we simply set $\sigma _ { z } = 1$ in the following. Thus we can think of teacher singular values as signal to noise ratios (SNRs). + +Finally, we note that while we focus for ease of exposition in the main paper on the case of one hidden layer networks and a full orthonormal basis of $P = { \overline { { N _ { 1 } } } }$ training inputs in the main paper, neither of these assumptions are essential to our theory. Indeed in Section 3.4 and App. A we extend our theory to networks of arbitrary depth, and in App. G we extend our theory to the case of white inputs with $P \neq \overline { { N } } _ { 1 }$ , obtaining a good match between theory and experiment in both cases. + +# 2.2 STUDENT TRAINING AND TEST ERROR + +Now consider a student network with $N _ { i }$ units in each layer. We assume the first and last layers match the teacher (i.e. $N _ { 1 } = \overline { { N _ { 1 } } }$ and $N _ { 3 } = \overline { { N _ { 3 } } }$ ) but $N _ { 2 } \geq \overline { { N _ { 2 } } }$ , allowing the student to have more hidden units than the teacher. We also consider deeper students (see below and App. A). Now consider any student whose input-output map is given by $\mathbf { \dot { y } } = \mathbf { W } ^ { 3 2 } \mathbf { W } ^ { 2 1 } \equiv \mathbf { W } \mathbf { x }$ . Its training error on the teacher dataset in (2) and its test error over a distribution of new inputs are given by + +$$ +\varepsilon _ { \mathrm { t r a i n } } \equiv \frac { \sum _ { \mu = 1 } ^ { \overline { { N _ { 1 } } } } | | \mathbf { W } \hat { \mathbf { x } } ^ { \mu } - \hat { \mathbf { y } } ^ { \mu } | | _ { 2 } ^ { 2 } } { \sum _ { \mu = 1 } ^ { \overline { { N _ { 1 } } } } | | \hat { \mathbf { y } } ^ { \mu } | | _ { 2 } ^ { 2 } } , \varepsilon _ { \mathrm { t e s t } } \equiv \frac { \langle | | \mathbf { W } \overline { { \mathbf { x } } } - \overline { { \mathbf { y } } } | | _ { 2 } ^ { 2 } \rangle } { \langle | | \overline { { \mathbf { y } } } | | _ { 2 } ^ { 2 } \rangle } , +$$ + +respectively. Here $\hat { \mathbf { x } } ^ { \mu }$ and ${ \hat { \mathbf { y } } } ^ { \mu }$ are the noisy training set inputs and outputs in (2), whereas $\overline { { \mathbf { x } } }$ denotes a random test input drawn from zero mean Gaussian with identity covariance, $\overline { { \mathbf { y } } } ^ { \mu } = \overline { { \mathbf { W } } } \overline { { \mathbf { x } } } ^ { \mu }$ is noise free teacher output, and $\langle \cdot \rangle$ denotes an average w.r.t the distribution of the test input $\overline { { \mathbf { x } } }$ . Due to the orthonormality of the training and isotropy of the test inputs, both $\varepsilon _ { \mathrm { t r a i n } }$ and ${ \varepsilon } _ { \mathrm { t e s t } }$ can be expressed as + +$$ +\mathrm { ~ \xi ~ } _ { \mathrm { t r a i n } } = \frac { \mathrm { T } \mathbf { F } \mathbf { W } ^ { T } \mathbf { W } - 2 \mathrm { T r } \mathbf { W } ^ { T } { \boldsymbol { \Sigma } } ^ { 3 1 } + \mathrm { T r } { \boldsymbol { \Sigma } } ^ { 3 1 T } { \boldsymbol { \Sigma } } ^ { 3 1 } } { \mathrm { T r } \boldsymbol { \Sigma } ^ { 3 1 T } \boldsymbol { \Sigma } ^ { 3 1 } } , \mathrm { ~ \xi ~ } _ { \mathrm { f e s t } } = \frac { \mathrm { T r } \mathbf { W } ^ { T } \mathbf { W } - 2 \mathrm { T r } \mathbf { W } ^ { T } \overline { { \mathbf { W } } } + \mathrm { T r } \overline { { \mathbf { W } } } ^ { T } \overline { { \mathbf { W } } } } { \mathrm { T r } \overline { { \mathbf { W } } } ^ { T } \overline { { \mathbf { W } } } } . +$$ + +Both $\varepsilon _ { \mathrm { t r a i n } }$ and $\varepsilon _ { \mathrm { t e s t } }$ can be further expressed in terms of the student, training data and teacher SVDs, which we denote by $\mathbf { W } = \mathbf { U } \mathbf { S } \mathbf { V } ^ { T }$ , $\begin{array} { r } { \pmb { \Sigma } ^ { 3 1 } = \hat { \mathbf { U } } \hat { \mathbf { S } } \hat { \mathbf { V } } ^ { T } } \end{array}$ , and $\overline { { \mathbf { W } } } = \overline { { \mathbf { U } } } \overline { { \mathbf { S } } } \overline { { \mathbf { V } } } ^ { T }$ respectively. Specifically, + +$$ +\begin{array} { r l } & { \varepsilon _ { \mathrm { t r a i n } } = \left[ \overbrace { \sum _ { \beta = 1 } ^ { N _ { 3 } } } ^ { \overline { { S } } _ { 3 } } \hat { s } _ { \beta } ^ { 2 } \right] ^ { - 1 } \left[ \underset { \alpha = 1 } { \overset { N _ { 2 } } { \sum _ { \alpha } } } s _ { \alpha } ^ { 2 } + \underset { \beta = 1 } { \overset { \overline { { N } } _ { 3 } } { \sum _ { \beta } } } \hat { s } _ { \beta } ^ { 2 } - 2 \underset { \alpha = 1 } { \overset { N _ { 2 } } { \sum _ { \beta = 1 } } } \sum _ { \beta = 1 } ^ { \overline { { N _ { 3 } } } } s _ { \alpha } \hat { s } _ { \beta } \left( \mathbf { u } ^ { \alpha } \cdot \hat { \mathbf { u } } ^ { \beta } \right) \left( \mathbf { v } ^ { \alpha } \cdot \hat { \mathbf { v } } ^ { \beta } \right) \right] , } \\ & { \varepsilon _ { \mathrm { t e s t } } = \left[ \underset { \beta = 1 } { \overset { \overline { { N } } _ { 2 } } { \sum _ { \beta } } } \overline { { s } } _ { \beta } ^ { 2 } \right] ^ { - 1 } \left[ \underset { \alpha = 1 } { \overset { N _ { 2 } } { \sum _ { \alpha } } } s _ { \alpha } ^ { 2 } + \underset { \beta = 1 } { \overset { \overline { { N } } _ { 2 } } { \sum _ { \beta } } } \frac { 1 } { s _ { \beta } ^ { 2 } } - 2 \underset { \alpha = 1 } { \overset { N _ { 2 } } { \sum _ { \alpha } } } \sum _ { \beta = 1 } ^ { \overline { { N _ { 2 } } } } s _ { \alpha } \overline { { s } } _ { \beta } \left( \mathbf { u } ^ { \alpha } \cdot \overline { { \mathbf { u } } } ^ { \beta } \right) \left( \mathbf { v } ^ { \alpha } \cdot \overline { { \mathbf { v } } } ^ { \beta } \right) \right] . } \end{array} +$$ + +Thus as the student learns, its training and test error dynamics depends on the alignment of the time-evolving student singular modes $\{ s ^ { \alpha } , \mathbf { u } ^ { \alpha } , \mathbf { v } ^ { \alpha } \}$ with the fixed training data $\{ \hat { s } ^ { \alpha } , \hat { \mathbf { u } } ^ { \alpha } , \hat { \mathbf { v } } ^ { \alpha } \}$ and teacher $\{ \overline { { s } } ^ { \alpha } , \bar { \overline { { \mathbf { u } } } } ^ { \alpha } , \overline { { \mathbf { v } } } ^ { \alpha } \}$ singular modes respectively. + +# 3 SINGLE TASK GENERALIZATION DYNAMICS: THEORY AND EXPERIMENT + +Here we derive and numerically test analytic formulas for both the training and test errors of a student network as it learns from training data generated from a teacher network. We explore the dependence of these quantitites on the student network size, student initialization, teacher SNR, and training time. + +# 3.1 STUDENT TRAINING DYNAMICS AND TRAINING-ALIGNED (TA) NETWORKS + +We assume the student weights undergo batch gradient descent with learning rate $\lambda$ on the training error $\begin{array} { r } { \sum _ { \mu } | | \hat { \mathbf { y } } ^ { \mu } - \mathbf { W } ^ { 3 2 } \mathbf { W } ^ { 2 1 } \bar { \hat { \mathbf { x } } ^ { \mu } } | | _ { 2 } ^ { 2 } } \end{array}$ , which for small $\lambda$ is well approximated by the differential equations: + +![](images/df635d5402308d3a9a9f10644ccaa9227749fea9232affb61828b212aa929642.jpg) +Figure 1: Learning dynamics as a function of singular dimension strength. (a) shows how modes of different singular value are learned, (b) shows that there is a wave of learning that picks up singular dimensions with smaller and smaller singular values as $t \to \infty$ . + +$$ +\tau \frac { d } { d t } \mathbf { W } ^ { 2 1 } = \mathbf { W } ^ { 3 2 ^ { T } } \left( \boldsymbol { \Sigma } ^ { 3 1 } - \mathbf { W } ^ { 3 2 } \mathbf { W } ^ { 2 1 } \boldsymbol { \Sigma } ^ { 1 1 } \right) , \qquad \tau \frac { d } { d t } \mathbf { W } ^ { 3 2 } = \left( \boldsymbol { \Sigma } ^ { 3 1 } - \mathbf { W } ^ { 3 2 } \mathbf { W } ^ { 2 1 } \boldsymbol { \Sigma } ^ { 1 1 } \right) \mathbf { W } ^ { 2 1 ^ { T } } , +$$ + +(where $\tau \equiv 1 / \lambda$ ), which must be solved from an initial set of student weights at time $t = 0$ (Saxe et al., 2013a). We consider two classes of student initializations. The first initialization corresponds to a random student where the weights $\mathbf { W } ^ { 2 1 }$ and $\mathbf { W ^ { 3 2 } }$ are chosen such that the composite map $\mathbf { W } = \mathbf { W ^ { 3 2 } W ^ { 2 1 } }$ has an SVD $\mathbf { W } = \epsilon \mathbf { U } \mathbf { V } ^ { \mathbf { T } }$ , where $\mathbf { U }$ and $\mathbf { V }$ are random singular vector matrices and all student singular vala time dependent evolution $\epsilon$ te map undergoes. For white inputs, $\begin{array} { r } { \mathbf { W } ( t ) = \mathbf { U } ( t ) \mathbf { S } ( t ) \mathbf { V } ( t ) ^ { T } = \sum _ { \alpha = 1 } ^ { N _ { 2 } } \mathbf { s } _ { \alpha } ( t ) \mathbf { u } ^ { \alpha } ( t ) \mathbf { v } ^ { \alpha } ( t ) ^ { T } . } \end{array}$ as , , and so the time-dependent student singular modes $\{ s ^ { \alpha } ( t ) , \mathbf { u } ^ { \alpha } ( \mathbf { t } ) , \mathbf { v } ^ { ( } \mathbf { t } ) \}$ converge to the training data singular modes $\{ \hat { s } ^ { \alpha } , \hat { \mathbf { u } } ^ { \alpha } , \hat { \mathbf { v } } ^ { \alpha } \}$ . However, the explicit dynamics of the student singular modes can be difficult to obtain analytically from random initial conditions. + +Thus we also consider a special class of training aligned (TA) initial conditions in which $\mathbf { W ^ { 2 1 } }$ and $\mathbf { W ^ { 3 2 } }$ are chosen such that the composite map $\mathbf { W } = \mathbf { W ^ { 3 2 } W ^ { 2 1 } }$ has an SVD $\mathbf { W } = \epsilon \hat { \mathbf { U } } \hat { \mathbf { V } } ^ { \mathbf { T } }$ . That is, the TA network (henceforth referred to simply as the TA) has the same singular vectors as the training data covariance $\boldsymbol { \Sigma } ^ { 3 1 }$ , but has all singular values equal to $\epsilon$ . As shown in (Saxe et al., 2013a), as the TA learns according to (8), the singular vectors of its composite map W remain unchanged, while the singular values evolve as $s ^ { \alpha } ( t ) \stackrel { \textstyle - } { = } s ( t , { \hat { s } } ^ { \alpha } )$ , where the learning curve function $s ( t , { \hat { s } } )$ as well as its functional inverse $t ( s , { \hat { s } } )$ is given by + +$$ +s ( t , \hat { s } ) = \frac { \hat { s } e ^ { 2 \hat { s } t / \tau } } { e ^ { 2 \hat { s } t / \tau } - 1 + \hat { s } / \epsilon } , \qquad t ( s , \hat { s } ) = \frac { \tau } { 2 \hat { s } } \ln \frac { \hat { s } / \epsilon - 1 } { \hat { s } / s - 1 } . +$$ + +Here the function $s ( t , { \hat { s } } )$ describes analytically how each training set singular value $\hat { s }$ drives the dynamics of the corresponding TA singular value $s$ , and for notational simplicity, we have suppressed the dependence of $s ( t , { \hat { s } } )$ on $\tau$ and the initial condition $\epsilon$ . As shown in Fig. 1A, for each $\hat { s }$ , $s ( t , { \hat { s } } )$ is a sigmoidal learning curve that undergoes aat which it rises from its small initial value of haat sition around time to its asymptotic v $\begin{array} { r } { t / \tau = \frac { 1 } { 2 \hat { s } } \ln { \left( \hat { s } / \epsilon - \mathrm { 1 } \right) } } \end{array}$ $\epsilon$ $t = 0$ $\hat { s }$ $t / \tau \to \infty$ Alternatively, we can plot $s ( t , { \hat { s } } ) / { \hat { s } }$ as a function of $\hat { s }$ for different training times $t / \tau$ , as in Fig. 1B. This shows that TA learning corresponds to a singular mode detection wave which progressively sweeps from large to small singular values. At any given training time $t$ , training data modes with singular values $\hat { s } > t / \tau$ have been learned, while those with singular values $\hat { s } < t / \tau$ have not. + +While the TA is more sophisticated than the random student, since it already knows the singular vectors of the training data before learning, we will see that the analytic solution for the TA learning dynamics provides a good approximation to the student learning dynamics, not only for the training error, as shown in (Saxe et al., 2013a), but also for the generalization error as shown below. + +The results in this section assume a single hidden layer, but Saxe et al. (2013a) derived $t ( s , { \hat { s } } )$ for networks of arbitrary depth and we apply our theory to some deeper networks. The general differential equation and derivations for deeper networks can be found in Appendix A. + +# 3.2 HOW THE TEACHER IS BURIED IN THE TRAINING DATA: A RANDOM MATRIX ANALYSIS + +In the previous section, we reviewed an exact analytic solution for the composite map of a TA network, namely that its singular modes are related to those of the training data through the relation + +$$ +\begin{array} { r } { s _ { \alpha } ( t ) = s ( t , \hat { s } _ { \alpha } ) , \qquad { \bf u } ^ { \alpha } ( t ) = \hat { \bf u } ^ { \alpha } , \qquad { \bf v } ^ { \alpha } ( t ) = \hat { \bf v } ^ { \alpha } . } \end{array} +$$ + +However, computation of the generalization error through (5) then requires understanding how the teacher singular modes of $\overline { { \mathbf { W } } }$ are buried within the noisy training data singular modes of $\pmb { \Sigma } ^ { 3 1 }$ through the relation (3). Since the input matrix $\hat { \mathbf X }$ is orthonormal, $\pmb { \Sigma } ^ { 3 1 }$ is simply a perturbation of the low rank teacher $\overline { { \mathbf { W } } }$ by a high dimensional noise matrix $\mathbf { Z }$ . The relation between the singular modes of a low rank matrix and its noise perturbed version has been studied extensively in Benaych-Georges $\&$ Nadakuditi (2012), in the high dimensional limit we are working in, namely $\overline { { N _ { 1 } } } , \overline { { N _ { 3 } } } \overline { { } } \infty$ with the aspect ratio $\mathcal { A } = \overline { { N _ { 3 } } } / \overline { { N _ { 1 } } } \in \overline { { ( 0 , 1 ] } }$ , and $\overline { { N } } _ { 2 } \sim O ( 1 )$ . + +![](images/74aa5b143963ba7d2cf4b03e7444309ed0f7fd59b33a147d86b619f88f63f9e7.jpg) +Figure 2: The teacher’s signal through the noise. Theoretical vs. empirical (a) histogram of singular values of noisy teacher $\hat { s }$ . (b) $\hat { s }$ as a function of $\overline { { s } }$ . (c) alignment of noisy teacher and noiseless teacher singular vectors as a function of $\overline { { s } }$ . ${ \widetilde { N _ { 1 } } } = { \overline { { N _ { 3 } } } } = 1 0 0 .$ ) + +In this limit, the top $\overline { { N } } _ { 2 }$ singular values and vectors of $\pmb { \Sigma } ^ { 3 1 }$ converge to $\hat { s } ( \overline { { s } } _ { \alpha } )$ , where the transfer function from a teacher singular value $\overline { { s } }$ to a training data singular value $\hat { s }$ is given by the function + +$$ +{ \hat { s } } ( { \overline { { s } } } ) = { \left\{ \begin{array} { l l } { ( { \overline { { s } } } ) ^ { - 1 } { \sqrt { ( 1 + { \overline { { s } } } ^ { 2 } ) ( A + { \overline { { s } } } ^ { 2 } ) } } } & { { \mathrm { ~ i f ~ } } { \overline { { s } } } > A ^ { 1 / 4 } } \\ { 1 + { \sqrt { A } } } & { { \mathrm { ~ o t h e r w i s e . } } } \end{array} \right. } +$$ + +The associated top $\overline { { N } } _ { 2 }$ singular vectors of $\pmb { \Sigma } ^ { 3 1 }$ can also acquire a nontrivial overlap with the $\overline { { N } } _ { 2 }$ modes of the teacher through the relation $\left| \hat { \mathbf { u } } ^ { \alpha } \cdot \overline { { \mathbf { u } } } ^ { \alpha } \right| \left| \hat { \mathbf { v } } ^ { \alpha } \cdot \overline { { \mathbf { v } } } ^ { \alpha } \right| = \mathcal { O } ( \overline { { s } } _ { \alpha } )$ , where the singular vector overlap function is given by + +$$ +\begin{array} { r } { \mathcal { O } ( \overline { { s } } ) = \left\{ \begin{array} { l l } { \left[ 1 - \frac { \mathcal { A } ( 1 + \overline { { s } } ^ { 2 } ) } { \overline { { s } } ^ { 2 } ( A + \overline { { s } } ^ { 2 } ) } \right] ^ { 1 / 2 } \left[ 1 - \frac { ( \mathcal { A } + \overline { { s } } ^ { 2 } ) } { \overline { { s } } ^ { 2 } ( 1 + \overline { { s } } ^ { 2 } ) } \right] ^ { 1 / 2 } } & { \mathrm { i f } \overline { { s } } > \mathcal { A } ^ { 1 / 4 } } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \end{array} +$$ + +The rest of the $N _ { 3 } - { \overline { { N _ { 2 } } } }$ singular vectors of $\pmb { \Sigma } ^ { 3 1 }$ are orthogonal to the top $\overline { { N } } _ { 2 }$ ones, and their singular values are distributed according to the the Marchenko-Pastur (MP) distribution: + +$$ +P ( \hat { s } ) = \left\{ \begin{array} { l l } { \frac { \sqrt { 4 A - ( \hat { s } ^ { 2 } - ( 1 + A ) ) ^ { 2 } } } { \pi A \hat { s } } } & { \hat { s } \in [ 1 - \sqrt { \mathcal { A } } , 1 + \sqrt { \mathcal { A } } ] } \\ { 0 } & { \mathrm { o t h e r w i s e } . } \end{array} \right. +$$ + +Overall, these equations describe a singular vector phase transition in the training data, as illustrated in Fig. 2BC. For example in the case of no teacher, the training data is simply noise and the singular values of $\pmb { \Sigma } ^ { 3 1 }$ are distributed as an MP sea spread between $1 \pm { \sqrt { A } }$ . When one adds a teacher, how each teacher singular mode is imprinted on the training data depends crucially on the teacher singular value $\overline { { s } }$ , and the nature of this imprinting undergoes a phase transition at $\overline { { s } } = \mathcal { A } ^ { 1 / 4 }$ . For $\overline { { s } } \leq { \mathcal { A } } ^ { 1 / 4 }$ , the teacher mode SNR is too low and this mode is not imprinted in the noisy training data; the associated√ training data singular value $\hat { s }$ remains at the edge of the MP sea at $1 + { \sqrt { A } }$ , and the overlap $\mathcal { O } ( \overline { { s } } )$ between training and teacher singular vectors remains zero. + +However, when $\overline { { s } } > \mathcal { A } ^ { 1 / 4 }$ , this teacher mode is imprinted in the training data; there is an associated training data singular value $\hat { s }$ that pops out of the MP sea (Fig. 2AB). However, the training data singular value emerges at a position $\hat { s } > \overline { { s } }$ that is inflated by the noise, though the inflation effect decreases at larger $\overline { { s } }$ , with the ratio $\hat { s } / \overline { { s } }$ approaching the unity line as $\overline { { s } }$ becomes large (Fig. 2B). Similarly, the corresponding training data singular vectors acquire a non-trivial overlap with the teacher singular vectors when $\overline { { s } } > \mathcal { A } ^ { 1 / 4 }$ , and the alignment approaches unity as $\overline { { s } }$ increases (Fig. 2C). + +# 3.3 PUTTING IT ALL TOGETHER: AN ANALYTIC THEORY OF GENERALIZATION DYNAMICS + +Based on an analytic understanding of how the singular mode structure $\{ \overline { { s } } ^ { \alpha } , \overline { { \mathbf { u } } } ^ { \alpha } , \overline { { \mathbf { v } } } ^ { \alpha } \}$ of the teacher $\overline { { \mathbf { W } } }$ is imprinted in the modes $\{ \hat { s } ^ { \alpha } , \hat { \mathbf { u } } ^ { \alpha } , \hat { \mathbf { v } } ^ { \alpha } \}$ of the training data covariance $\pmb { \Sigma } ^ { 3 1 }$ through (11), (12) and (13), and in turn how this training data singular structure drives the time evolving singular modes of a + +![](images/314c03e7d659a75d3bce2929de3fdf61f6a46155c699119ff46ac7743510ffe3.jpg) +Figure 3: Match between theory and experiment for rank 1 (row 1, a-d) and rank 3 (row 2, e-h) teachers with single-hidden-layer students: (a-b, e-f) log train and test error, respectively, showing very close match between theory and experiment for TA, and close match for the random student. (c,g) comparing TA and randomly initialized students minimum generalization errors, showing almost perfect match. (d,h) comparing TA and randomly initialized students optimal stopping times, showing small lag due to alignment. ( $N _ { 1 } = 1 0 0$ , $N _ { 2 } = 5 0$ , $N _ { 3 } = 5 0 .$ ) + +TA network $\{ s ^ { \alpha } ( t ) , \hat { \mathbf { u } } ^ { \alpha } , \hat { \mathbf { v } } ^ { \alpha } \}$ of through (9), we can now derive analytic expressions for $\varepsilon _ { \mathrm { t r a i n } }$ and $\varepsilon _ { \mathrm { t e s t } }$ in (6) and (7), for a TA network. We will also show that these learning curves closely approximate those of a random student with time-evolving singular vectors $\{ \mathbf { u } ^ { \alpha } ( t ) , \mathbf { \bar { v } } ^ { \alpha } ( t ) \}$ , and match on several key aspects. First, inserting the TA dynamics in (10) into $\varepsilon _ { \mathrm { t r a i n } }$ in (6), we obtain + +$$ +\mathfrak { c } _ { \mathrm { r e a i n } } ( t ) = \left[ \sum _ { \alpha = 1 } ^ { \overline { { N } } _ { 3 } } \hat { s } _ { \alpha } ^ { 2 } \right] ^ { - 1 } \left[ ( N _ { 3 } - N _ { 2 } ) \langle \hat { s } ^ { 2 } \rangle _ { \mathcal { R } _ { o u t } } + ( N _ { 2 } - \overline { { N } } _ { 2 } ) \langle ( s ( \hat { s } , t ) - \hat { s } ) ^ { 2 } \rangle _ { \mathcal { R } _ { i n } } + \sum _ { \alpha = 1 } ^ { \overline { { N } } _ { 2 } } \left[ s _ { \alpha } ( t ) - \hat { s } _ { \alpha } \right] ^ { 2 } \right] +$$ + +Here, $s _ { \alpha } ( t ) = s ( \hat { s } _ { \alpha } , t )$ as defined in (9) are the TA singular values, and $\hat { s } _ { \alpha } = \hat { s } ( \overline { { s } } _ { \alpha } )$ as defined in (11) are the training data singular values associated with the teacher singular values $\overline { { s } } _ { \alpha }$ . Also $\langle \cdot \rangle _ { \mathcal { R } }$ denotes an average with respect to the MP distribution in (13) over a region $\mathcal { R }$ . Two distinct regions contribute to training error. First $\mathcal { R } _ { i n }$ contains those top $N _ { 2 } - \overline { { N } } _ { 2 }$ training data singular values that do not correspond to the $\overline { { N } } _ { 2 }$ singular values of the teacher but will be learned by a rank $N _ { 2 }$ student. Second, $\mathcal { R } _ { o u t }$ corresponds to the remaining $N _ { 3 } - N _ { 2 }$ lowest training data singular values√ that cannot be learned by a rank √ $N _ { 2 }$ student. In terms of the MP distribution, $\mathcal { R } _ { o u t } = [ 1 - \sqrt { \mathcal { A } } , f ]$ and $\mathcal { R } _ { i n } = [ f , 1 + \sqrt { \mathcal { A } } ]$ , where $f$ is the point at which the MP density has $1 - N _ { 2 } / N _ { 3 }$ of its mass to the left and $N _ { 2 } / N _ { 3 }$ of its mass to the right. In the simple case of a full rank student, $f = 1 - \sqrt { \mathcal { A } }$ , and one need only integrate over $\mathcal { R } _ { i n }$ which is the entire range. Equation (14) for $\varepsilon _ { \mathrm { t r a i n } }$ makes it manifest that it will go to zero for a full rank student as its singular values approach those of the training data. + +Of course the test error can behave very differently. Inserting the TA training dynamics in (10) into $\varepsilon _ { \mathrm { t e s t } }$ in (7), and using (11), (12) and (13) to relate training data to the teacher, we find + +$$ +\varepsilon _ { \mathrm { t e x t } } ( t ) = \left[ \sum _ { \alpha = 1 } ^ { \overline { { N } } _ { 2 } } \overline { { s } } _ { \alpha } ^ { 2 } \right] ^ { - 1 } \left[ ( N _ { 2 } - \overline { { N } } _ { 2 } ) \langle s ( \hat { s } , t ) ^ { 2 } \rangle _ { \mathcal { R } _ { i n } } + \sum _ { \alpha = 1 } ^ { \overline { { N } } _ { 2 } } \left[ ( s _ { \alpha } ( t ) - \overline { { s } } _ { \alpha } ) ^ { 2 } + 2 s _ { \alpha } ( t ) \overline { { s } } _ { \alpha } ( 1 - \mathcal { O } ( \overline { { s } } _ { \alpha } ) ) \right] \right] +$$ + +Together (14) and (15) constitute a complete theory of generalization dynamics in terms of the structure of the data distribution (i.e. the teacher rank $\overline { { N } } _ { 2 }$ , teacher SNRs $\left\{ \overline { { s } } _ { \alpha } \right\}$ , and the teacher aspect ratio $\mathcal { A } = \overline { { N } } _ { 3 } / \overline { { N } } _ { 1 } )$ , the architectural complexity of the student (i.e. its rank $N _ { 2 }$ , its number of layers $N _ { l }$ , and the norm $\epsilon$ of its initialization), and the training time $t$ . They yield considerable insight into the dynamics of good generalization early in learning and overfitting later, as we show below. + +![](images/be9716a9b1de318c5906d81cbfaaab3bf19f9183b7b3ce2f88ef2d6bf3ba7e80.jpg) +Figure 4: Our theory applies to deeper networks: match between theory and simulation for rank 1 (row 1, a-d) and rank 3 (row 2, e-h) teachers with $n _ { l } = 5$ students: (a-b, e-f) log train and test error, respectively, showing very close match between theory and experiment for TA. (c,g) comparing TA and randomly initialized students minimum generalization errors, showing almost perfect match. (d,h) comparing TA and randomly initialized students optimal stopping times, showing large lag due to slower alignment in deeper networks. $N _ { 1 } = 1 0 0$ , $N _ { 2 } = 5 0$ , $N _ { 3 } = 5 0 .$ .) + +# 3.4 NUMERICAL TESTS OF THE THEORY OF NEURAL NETWORK GENERALIZATION DYNAMICS + +Fig. 3 demonstrates an excellent match between the theory and simulations for the TA, and a close match for random students, for single-hidden-layer students and various teacher ranks $\overline { { N } } _ { 2 }$ . Intuitively, as time $t$ proceeds, learning corresponds to singular mode detection wave sweeping from large to small training data singular values (i.e. the wave in Fig. 1B sweeps across the training data spectrum in Fig 2A). Initially, strong singular values associated with large SNR teacher modes are learned and both $\varepsilon _ { \mathrm { t r a i n } }$ and $\varepsilon _ { \mathrm { t e s t } }$ drop. Fig. 3A-D are for a rank 1 teacher, and so in Fig 3AB we see a single sharp drop early on, if the teacher SNR is sufficiently high. By contrast, with a rank 3 teacher in Fig. 3E-H, there are several early drops as the three modes are picked up. However, as time progresses, the singular mode detection wave penetrates the MP sea, and the student picks up noise structure in the data, so $\varepsilon _ { \mathrm { t r a i n } }$ drops but $\varepsilon _ { \mathrm { t e s t } }$ rises, indicating the onset of overfitting. + +The main difference between the random student and TA learning curves is that the random student learning is slightly delayed relative to the TA, especially late in training. This is understandable because the TA already knows the singular vectors of the training data, while the random student must learn them. Nevertheless, two of the most important aspects of learning, namely the optimal early stopping time $t _ { \mathrm { g r a d i e n t } } ^ { \mathrm { o p t } } \equiv \mathrm { a r g m i n } _ { t } \varepsilon _ { \mathrm { t e s t } } ( t )$ and the minimal test error achieved at this time $\varepsilon _ { \mathrm { g r a d i e n t } } ^ { \mathrm { o p t } } \equiv$ $\mathrm { m i n } _ { t } \varepsilon _ { \mathrm { t e s t } } ( t )$ , match well between TA and random student, as shown in Fig. 3CD. At low teacher SNRs, the student takes a little longer to learn than the TA, but their optimal test errors match. + +Our theory can also be easily extended to describe the learning dynamics deeper networks. Saxe et al. (2013a) derived $t ( s , { \hat { s } } )$ for networks of arbitrary depth, so we only need to adjust this factor in our formulas, see App. A for details. In Fig. 4 we show that again there is an excellent match between TA networks and theory for student networks with $N _ { l } = 5$ layers (i.e. 3 hidden layers). Randomly-initialized networks show a much longer alignment lag for deeper networks (see App. B for details), but the curves are qualitatively similar and optimal stopping errors match. We also demonstrate extensions of our theory to different numbers of training examples (App. G). + +Importantly, many of the phenomena we observe in linear networks are qualitatively replicated in nonlinear networks (Fig. 5), suggesting that our theory may help guide understanding of the nonlinear case. In particular, features such as stage-like initial learning, followed by a plateau if SNR is high, and finally followed by overfitting, are replicated. However, there are some discrepancies, in particular nonlinear networks (especially deeper ones) begin overfitting earlier than linear networks. This is likely because a mode in a non-linear network can be co-opted by an orthogonal mode, while in a linear network it cannot. Thus noise modes are able to “stow away” on the strong signal modes once they are learned. However, overall learning patterns are similar, and we show below that many interesting phenomena in nonlinear networks are understandable in the linear case, such as the (non-)effects of overparameterization, the dynamics of memorization, and the benefits of transfer. + +![](images/4c96fe2cc1b69b79f16e01bb40a77f5c5b9805452e0fd502c4194097a289ca0c.jpg) +Figure 5: Train (first row, A-D) and test (second row, E-H) error for nonlinear networks (leaky relu at all hidden layers) with one hidden layer (first two columns) or three hidden layers (last two columns) trained on the tasks above, with a rank 1 teacher (first and third columns) or a rank 3 teacher (second and fourth columns). Note that many of the qualitative phenomena observed in linear networks, such as stage-like improvement in the errors, followed by a plateau, followed by overfitting, also appear in nonlinear networks. Compare the first column to Fig. 3AB, the second column to Fig. 3EF, the third to Fig. 4AB, and the fourth to Fig. 4EF. ( $N _ { 1 } = 1 0 0$ , $N _ { 2 } = 5 0$ , $N _ { 3 } = 5 0 .$ ) + +# .5 RANDOMIZED DATA VS. REAL DATA: A LEARNING TIME PUZZLE + +An intriguing observation that resurrected the generalization puzzle in deep learning was the observation by Zhang et al. (2016) that deep networks can memorize data with the labels randomly permuted. However, as Arpit et al. (2017) pointed out, the learning dynamics of training error for randomized labels can be slower than than for structured data. This phenomenon also arises in deep linear networks, and our theory yields an analytic explanation for why. We randomize data by choosing orthonormal inputs $\hat { \mathbf { x } } ^ { \mu }$ as in the structured case, but we choose the outputs ${ \hat { \mathbf { y } } } ^ { \mu }$ to be i.i.d. Gaussian with zero mean and the same diagonal variance as the structured training data generated by the teacher. For structured data generated by a low rank teacher with singular values $\overline { { s } } _ { \alpha }$ , the diagonal output variance is given by $\begin{array} { r } { \sigma _ { r } ^ { 2 } = \frac { 1 } { N _ { 3 } } \left[ \sum _ { i = \alpha } ^ { \overline { { N } } _ { 2 } } \bar { s } _ { \alpha } ^ { 2 } \right] + \frac { 1 } { \overline { { N } } _ { 1 } } \sigma _ { z } ^ { 2 } } \end{array}$ , where $\sigma _ { z }$ is the noise variance, as before. Since there is no relation between input and output, $\pmb { \Sigma } ^ { 3 1 }$ is now distributed as a MP distribution whose support is $[ ( \sigma _ { r } ( 1 - \sqrt { \mathcal { A } } ) , \sigma _ { r } ( 1 + \sqrt { \mathcal { A } } ) ]$ . Thus randomization essentially destroys the outlier signal singular values in $\pmb { \Sigma } ^ { 3 1 }$ reflecting the teacher, and distributes them across all randomized data modes, yielding this stretched MP distribution (compare 6A top and bottom). However, even on this stretched MP distribution, the right edge will be much smaller than the signal singular values, since the signal variance will be diluted by spreading it out over many more modes in the randomized data. Thus the randomized data will lead to slower initial training error drops relative to the structured data (Fig. 6B) since the singular mode detection wave encounters the first signal singular values in structured data earlier than it encounters the edge of the stretched MP sea in randomized data. + +![](images/214421209393a1f729207d74055cfc8b2447742fb08e8821164fae3a79d95a61.jpg) +Figure 6: Learning randomized data: Comparing (a) singular value distributions and (b) learning curves for data with a signal vs. random data that preserves basic statistics (mean, variance). Randomizing the data dilutes the signal singular values, spreading their variance out over many modes, hence randomly labelled data is learned more slowly. $N _ { 1 } = 1 0 0$ , $N _ { 2 } = 5 0$ , $N _ { 3 } = 5 0 .$ .) + +![](images/6f50ba8e93b7f0f0f5390f316c743f888d62c7d9566d1790b18925c63d2e3904.jpg) +Figure 7: Transfer setting– If two different tasks are combined, how well students of the combined teacher perform on each task depends on the alignment and SNRs of the teachers. + +![](images/0a5b9a9dbd5641d64b838b1df6081e9aedcd5fcf53c8a71c1f83e3bbc6d1722c.jpg) +Figure 8: Transfer benefit $\mathcal { T } ^ { A B } ( \overline { { s } } _ { A } , \overline { { s } } _ { B } , q )$ plotted at different values of ${ \overline { { s } } } _ { A }$ . (a) $\overline { { s } } _ { A } = 0 . 8 4 = \sqrt [ 4 ] { A }$ . Although this task is impossible to learn on its own, with support from another aligned task, especially one with high SNR, learning can occur. (b) ${ \overline { { s } } } _ { A } = 3$ . Tasks with modest signals will face interference from poorly aligned tasks, but benefits from well aligned tasks. These effects are amplified by SNR. (c) $\overline { { s } } _ { A } = 1 0 0$ . Tasks with very strong signals will show little effect from other tasks (note y-axis scales), but any impact will be negative unless the tasks are very well aligned. $N _ { 1 } = 1 0 0$ $\dot { \overline { { { N } } } } _ { 2 } ^ { A } = \overline { { { N } } } _ { 2 } ^ { B } = 1 .$ , $N _ { 2 } = 5 0$ , $N _ { 3 } = 5 0 .$ ) + +# 3.6 OUT-PERFORMING OPTIMAL EARLY STOPPING THROUGH A NON-GRADIENT ALGORITHM + +For the case of a rank 1 teacher, it is straightforward to derive a good analytic approximation to the important quantities $\varepsilon _ { \mathrm { g r a d i e n t } } ^ { \mathrm { o p t } }$ and topt $t _ { \mathrm { g r a d i e n t } } ^ { \mathrm { o p t } }$ . We assume the teacher SNR is beyond the phase transition point so its unique singular value $\overline { { s } } _ { 1 } > A ^ { 1 / 4 }$ , yielding a separation between the training data singular value $\hat { s _ { 1 } }$ in (11) and the edge of the MP sea. In this scenario, optimal early stopping will occur at a time before the detection wave in Fig. 1B penetrates the MP sea, so to minimize test error, we can neglect the first term in (15). Then optimizing the second term yields the optimal student singular value $s _ { 1 } = \overline { { s } } _ { 1 } \mathcal { O } ( \overline { { s } } _ { 1 } )$ . Inserting this value into (15) yields εoptgradient = 1 − O(s1)2, and inserting it into (9) yields $t _ { \mathrm { g r a d i e n t } } ^ { \mathrm { o p t } }$ . Thus the optimal generalization error with a rank 1 teacher is very simply related to the alignment of the top training data singular vectors with the teacher singular vectors, and it decreases as this alignment increases. In App. E, we show this match in the rank 1 case. + +With higher rank teachers, teacher modes with differen $\varepsilon _ { \mathrm { g r a d i e n t } } ^ { \mathrm { o p t } }$ and For must negotiate a more complex trade-off betweene, as the singular mode detection wave passes the top training data singular value, $s _ { 1 } ( t ) \to \hat { s } _ { 1 }$ which is greater than the optimal $s _ { 1 } = \overline { { s } } _ { 1 } \mathcal { O } ( \overline { { s } } _ { 1 } )$ for mode 1. Thus as learning progresses, the student overfits on the first mode but learns lower modes. However, this neural generalization dynamics suggests a superior non-gradient training algorithm that simply optimally sets each $s _ { \alpha }$ to $\overline { { s } } _ { \alpha } \mathcal { O } ( \overline { { s } } _ { \alpha } )$ in (15), yielding an optimal generalization error: + +$$ +\varepsilon _ { \mathrm { n o n - g r a d i e n t } } ^ { \mathrm { o p t } } = \left[ \sum _ { \alpha = 1 } ^ { \overline { { N } } _ { 2 } } \overline { { s } } _ { \alpha } ^ { 2 } \right] ^ { - 1 } \left[ \sum _ { \alpha = 1 } ^ { \overline { { N } } _ { 2 } } \overline { { s } } _ { \alpha } ^ { 2 } ( 1 - \mathcal { O } ( \overline { { s } } _ { \alpha } ) ^ { 2 } ) \right] . +$$ + +Standard gradient descent learning cannot achieve this low generalization error because it cannot independently adjust all student singular values. A simple algorithm that achieves $\varepsilon _ { \mathrm { n o n - g r a d i e n t } } ^ { \mathrm { o p t } }$ is as follows. From the training data covariance $\pmb { \Sigma } ^ { 3 1 }$ , extract the top singular values $\hat { s } _ { \alpha }$ that pop-out of the MP sea, use the functional inverse of (11) to compute $\overline { { s } } _ { a } ( \widehat { s } _ { \alpha } )$ , use (12) to compute the optimal $s _ { \alpha }$ , and then construct a matrix $\mathbf { W }$ with the same top singular vectors as $\pmb { \Sigma } ^ { 3 1 }$ , but with the outlier singular values shrunk from $\hat { s } _ { \alpha }$ to $s _ { \alpha }$ and the rest set to zero. This non-gradient singular value shrinkage algorithm provably outperforms neural network training with $\varepsilon _ { \mathrm { n o n - g r a d i e n t } } ^ { \mathrm { o p t } ^ { - } } \le \varepsilon _ { \mathrm { g r a d i e n t } } ^ { \mathrm { o p t } } .$ . + +# 4 A THEORY FOR THE TRANSFER OF KNOWLEDGE ACROSS MULTIPLE TASKS + +Consider two tasks $A$ and $B$ , described by $\overline { { N } } _ { 3 }$ by $\overline { { N } } _ { 1 }$ teacher maps $\overline { { \mathbf { W } } } ^ { A }$ and $\overline { { \mathbf { W } } } ^ { B }$ , of ranks $\overline { { N } } _ { 2 } ^ { A }$ and $\overline { { N } } _ { 2 } ^ { B }$ , respectively. Now two student networks can learn from the two teacher networks separately, each achieving optimal early stopping test errors composite teacher (and student) that concatenate $\varepsilon _ { A } ^ { \mathrm { { o p t } } }$ and hid $\varepsilon _ { B } ^ { \mathrm { { o p t } } }$ . Alternatively, one could construct a and output units, but shares the same $\overline { { N } } _ { 3 }$ ut units (Fig. 7). The composite student and teacher each have two heads, one for each tasneurons per head. Optimal early stopping on each head of the student yields test errors opt opt $\varepsilon _ { A B } ^ { \mathrm { o p t } }$ and . We define the transfer benefit that task B confers on task A to be $\mathcal { T } ^ { A B } \equiv \varepsilon _ { A } ^ { \mathrm { o p t } } - \varepsilon _ { A B } ^ { \mathrm { o p t } }$ A postive (negative) transfer benefit implies learning tasks A and B simultaneously yields a lower (higher) optimal test error on task A compared to just learning task A alone. + +A foundational question is how the transfer benefit $\scriptstyle { \mathcal { T } } ^ { A B }$ depends on the two tasks defined by the teachers $\overline { { \mathbf { W } } } ^ { A }$ and $\overline { { \mathbf { W } } } ^ { B }$ . To answer this, consider the SVDs of each teacher alone: $\begin{array} { r } { \overline { { \mathbf { W } } } ^ { A } = } \end{array}$ $\overline { { \mathbf { U } } } ^ { A } \overline { { \mathbf { S } } } ^ { A } \overline { { \mathbf { V } } } ^ { A ^ { T } }$ and $\overline { { \mathbf { W } } } ^ { B } = \overline { { \mathbf { U } } } ^ { B } \overline { { \mathbf { S } } } ^ { B } \overline { { \mathbf { V } } } ^ { B ^ { T } }$ . From the above, we know that $\varepsilon _ { A } ^ { \mathrm { { o p t } } }$ depends on $\overline { { \mathbf { W } } } ^ { A }$ only through $\overline { { \mathbf { S } } } ^ { A }$ . In App. D we show that the transfer benefit depends on both $\overline { { \mathbf { W } } } ^ { A }$ and $\overline { { \mathbf { W } } } ^ { B }$ only through $\overline { { \mathbf { S } } } ^ { A } , \overline { { \mathbf { S } } } ^ { B }$ , and the $\overline { { N } } _ { 2 } ^ { A }$ by $\overline { { N } } _ { 2 } ^ { B }$ similarity matrix $\mathbf { \overline { { Q } } } = \mathbf { \overline { { V } } } ^ { A ^ { T } } \mathbf { \overline { { V } } } ^ { B }$ . If we think of the columns of each $\overline { { \mathbf { V } } }$ as spanning a low dimensional feature space in $\overline { { N } } _ { 1 }$ dimensional input space that is important for each task, then $\overline { { \mathbf { Q } } }$ reflects the input feature subspace similarity matrix. Interestingly, the transfer benefit is independent of output singular vectors $\bar { \mathbf { U } } ^ { A }$ and $\overline { { \mathbf { U } } } ^ { B }$ . What matters for knowledge transfer in this setting are the relevant input features, not how you must respond to them. + +We describe the transfer benefit for the simple case of two rank one teachers. Then $\overline { { \mathbf { S } } } ^ { A } , \overline { { \mathbf { S } } } ^ { B }$ , and $\overline { { \mathbf { Q } } }$ are simply scalars $s _ { A } , s _ { B }$ and $q$ , and we explore the function $\mathcal { T } ^ { A B } ( \overline { { s } } _ { A } , \overline { { s } } _ { B } , q )$ in Fig. 5ABC, which reveals several interesting features. First, knowledge can be transferred from a high SNR task to a low SNR task (Fig. 5A) and the degree of transfer increases with task alignment $q$ . This can make it possible to capture signals from task $A$ which would otherwise sink into the MP sea by learning jointly with a related task, even if the tasks are only weakly aligned (Fig. 5A). However, if task $A$ already has a high SNR, task $B$ must be very well aligned to it for transfer to be beneficial – otherwise there will be interference. The degree of alignment required increases as the task $A$ SNR increases, but the quantity of benefit or interference decreases correspondingly (Fig. 5BC). In Appendix D we explain why our theory predicts these results. Furthermore, in Appendix F we demonstrate these phenomena are qualitatively recapitulated in nonlinear networks, which suggests that our theory may give insight into how to choose auxiliary tasks. + +# 5 DISCUSSION + +In summary, our analytic theory of generalization dynamics in deep linear networks reveals that many puzzling aspects of generalization in deep learning already arise in the simple linear setting, where the puzzles can be understood analytically. In particular, deep linear networks learn more important structure in data first, leading to generalization errors that depend on task structure much more than network size. Our theory explains why deep linear networks learn randomized data more slowly than structured data, and provides a non-gradient based learning method that out-performs gradient descent learning in the linear case. Finally, we provide an analytic theory of how knowledge is transferred from one task to another, demonstrating that the degree of alignment of input features important for each task, but not how one must respond to these features, is critical for facilitating knowledge transfer. We think these analytic results provide useful insight into the similar generalization and transfer phenomena observed in the nonlinear case. Among other things, we hope our work will motivate and enable: (1) the search for tighter upper bounds on generalization error that take into account task structure; (2) the design of non gradient based training algorithms that outperform gradient-based learning; and (3) the theory-driven selection of auxiliary tasks that maximize knowledge transfer. + +# REFERENCES + +Madhu S. Advani and Andrew M. Saxe. High-dimensional dynamics of generalization error in neural networks. arXiv, pp. 1–32, 2017. + +Sanjeev Arora, Rong Ge, Behnam Neyshabur, and Yi Zhang. Stronger generalization bounds for deep nets via a compression approach. arXiv preprint, pp. 1–39, 2018. URL http://arxiv. org/abs/1802.05296. + +Devansh Arpit, Stanisław Jastrze¸bski, Nicolas Ballas, David Krueger, Emmanuel Bengio, Maxinder S Kanwal, Tegan Maharaj, Asja Fischer, Aaron Courville, Yoshua Bengio, and Simon Lacoste-Julien. A Closer Look at Memorization in Deep Networks. arXiv preprint, 2017. ISSN 1938-7228. 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Understanding deep learning requires rethinking generalization. arXiv preprint, 2016. ISSN 10414347. doi: 10.1109/TKDE.2015.2507132. URL http://arxiv.org/abs/1611.03530. + +# A LEARNING DYNAMICS FOR DEEPER NETWORKS + +In the main text, we described the dynamics of how a single-hidden-layer network converges toward the training data singular modes $\{ \hat { s } ^ { \alpha } , \hat { \mathbf { u } } ^ { \alpha } , \hat { \mathbf { v } } ^ { \alpha } \}$ , which were originally derived in Saxe et al. (2013a). There it was also proven that for a network with $N _ { l }$ layers (i.e. $N _ { l } - 2$ hidden layers), the strength of the mode obeys the differential equation: + +$$ +\tau { \frac { d } { d t } } u = ( N _ { l } - 1 ) u ^ { 2 - 2 / ( N _ { l } - 1 ) } ( s - u ) +$$ + +This equation is separable and can be integrated for any integer number of layers. In particular, we consider the case of 5 layers (3 hidden), in which case: + +$$ +t ( s , \hat { s } ) = \frac { \tau } { 2 } \left[ \frac { \operatorname { t a n h } ^ { - 1 } \left( \sqrt { \frac { u } { \hat { s } } } \right) } { \hat { s } ^ { 3 / 2 } } - \frac { 1 } { \hat { s } \sqrt { u } } \right] _ { \epsilon } ^ { s } +$$ + +This expression cannot be analytically inverted to find $s ( t , { \hat { s } } )$ , so we numerically invert it where necessary. + +# B ALIGNMENT LAG IN RANDOMLY INITIALIZED NETWORKS + +As noted in the main text, the randomly-initialized networks behave quite similarly to the TA networks, except that the randomly-initialized networks show a lag due to the time it takes for the network’s modes to align with the data modes. In fig. 9 we explore this lag by plotting the alignment of the modes and the increase in the singular value for several randomly initialized networks. + +Notice that stronger modes align more quickly. Furthermore, the mode alignment is relatively independent – whether the teacher is rank 1 or rank 3, the alignment of the modes is similar for the mode of singular value 2. Most importantly, note how the deeper networks show substantially slower mode alignment, with alignment not completed until around when the singular value increases. This explains why deeper networks show a larger lag between randomly-initialized and TA networks – the alignment process is much slower for deeper networks. + +# C TRAIN AND TEST ERRORS AFTER A PROJECTION + +In the case of transfer learning, or more generally when we want to evaluate a network’s loss on a subset of its outputs, we need to use a slight generalization of the train and test error formulas given in the main text. Suppose we are interested in the train and test errors after applying a projection operator $\mathbf { P }$ : + +$$ +\varepsilon _ { \mathrm { t r a i n } } \equiv \frac { \sum _ { \mu = 1 } ^ { \overline { { N _ { 1 } } } } | | \mathbf { P } \mathbf { W } \hat { \mathbf { x } } ^ { \mu } - \mathbf { P } \hat { \mathbf { y } } ^ { \mu } | | _ { 2 } ^ { 2 } } { \sum _ { \mu = 1 } ^ { \overline { { N _ { 1 } } } } | | \mathbf { P } \hat { \mathbf { y } } ^ { \mu } | | _ { 2 } ^ { 2 } } , \varepsilon _ { \mathrm { t e s t } } \equiv \frac { \sum _ { \mu = 1 } ^ { \overline { { N _ { 1 } } } } | | \mathbf { P } \mathbf { W } \overline { { \mathbf { x } } } ^ { \mu } - \mathbf { P } \overline { { \mathbf { y } } } ^ { \mu } | | _ { 2 } ^ { 2 } } { \sum _ { \mu = 1 } ^ { \overline { { N _ { 1 } } } } | | \mathbf { P } \overline { { \mathbf { y } } } ^ { \mu } | | _ { 2 } ^ { 2 } } , +$$ + +respectively. As in the main text, we can rexpress these as: + +$$ +\varepsilon _ { \mathrm { t r a i n } } = \frac { \mathrm { T r } { \mathbf { W } } ^ { T } { \mathbf { P } } ^ { T } { \mathbf { P } } { \mathbf { W } } - 2 \mathrm { T r } { \mathbf { W } } ^ { T } { \mathbf { P } } ^ { T } { \mathbf { P } } \Sigma ^ { 3 1 } + \mathrm { T r } { \Sigma ^ { 3 1 } } ^ { T } { \mathbf { P } } ^ { T } { \mathbf { P } } \Sigma ^ { 3 1 } } { \mathrm { T r } \Sigma ^ { 3 1 } ^ { T } { \mathbf { P } } ^ { T } { \mathbf { P } } \Sigma ^ { 3 1 } } , +$$ + +$$ +\varepsilon _ { \mathrm { t e s t } } = \frac { \mathrm { T r } { \mathbf { W } } ^ { T } { \mathbf { P } } ^ { T } { \mathbf { P } } { \mathbf { W } } - 2 \mathrm { T r } { \mathbf { W } } ^ { T } { \mathbf { P } } ^ { T } { \mathbf { P } } \overline { { { \mathbf { W } } } } + \mathrm { T r } \overline { { { \mathbf { W } } } } ^ { T } { \mathbf { P } } ^ { T } { \mathbf { P } } \overline { { { \mathbf { W } } } } } { \mathrm { T r } \overline { { { \mathbf { W } } } } ^ { T } { \mathbf { P } } ^ { T } { \mathbf { P } } \overline { { { \mathbf { W } } } } } . +$$ + +Using the cyclic property of the trace, we can modify these to get: + +$$ +\varepsilon _ { \mathrm { t r a i n } } = \frac { \mathrm { T r } { \bf P } { \bf W } { \bf W } ^ { T } { \bf P } ^ { T } - 2 \mathrm { T r } { \bf P } \Sigma ^ { 3 1 } { \bf W } ^ { T } { \bf P } ^ { T } + \mathrm { T r } { \bf P } \Sigma ^ { 3 1 } \Sigma ^ { 3 1 ^ { T } } { \bf P } ^ { T } } { \mathrm { T r } { \bf P } \Sigma ^ { 3 1 } \Sigma ^ { 3 1 ^ { T } } { \bf P } ^ { T } } , +$$ + +$$ +\varepsilon _ { \mathrm { t e s t } } = \frac { \mathrm { T r } \mathbf { P } \mathbf { W } \mathbf { W } ^ { T } \mathbf { P } ^ { T } - 2 \mathrm { T r } \mathbf { P } \overline { { \mathbf { W } } } \mathbf { W } ^ { T } \mathbf { P } ^ { T } + \mathrm { T r } \mathbf { P } \overline { { \mathbf { W } } } \mathbf { W } ^ { T } \mathbf { P } ^ { T } } { \mathrm { T r } \mathbf { P } \overline { { \mathbf { W } } } \mathbf { W } ^ { T } \mathbf { P } ^ { T } } . +$$ + +As before, we express these in terms of the student, training data and teacher SVDs, $\mathbf { W } = \mathbf { U } \mathbf { S } \mathbf { V } ^ { T }$ $\begin{array} { r } { \pmb { \Sigma } ^ { 3 1 } = \hat { \mathbf { U } } \hat { \mathbf { S } } \hat { \mathbf { V } } ^ { T } } \end{array}$ , and $\overline { { \mathbf { W } } } = \overline { { \mathbf { U } } } \overline { { \mathbf { S } } } \overline { { \mathbf { V } } } ^ { T }$ respectively. Specifically, + +$$ +\begin{array} { r } { \operatorname { t r a i n } = \Big [ \sum _ { \beta = 1 } ^ { \mathbb { N } _ { 3 } } \hat { s } _ { \beta } ^ { 2 } \vert \vert { \mathbf { P } } \hat { \mathbf { a } } ^ { \alpha } \vert \vert _ { 2 } ^ { 2 } \Big ] ^ { - 1 } \Big [ \sum _ { \alpha = 1 } ^ { N _ { 2 } } s _ { \alpha } ^ { 2 } \vert \vert { \mathbf { P } } \mathbf { u } ^ { \alpha } \vert \vert _ { 2 } ^ { 2 } + \sum _ { \beta = 1 } ^ { \mathbb { N } _ { 3 } } \hat { s } _ { \beta } ^ { 2 } \vert \vert { \mathbf { P } } \hat { \mathbf { a } } ^ { \alpha } \vert \vert _ { 2 } ^ { 2 } - 2 \sum _ { \alpha = 1 } ^ { N _ { 2 } } \sum _ { \beta = 1 } ^ { \mathbb { N } _ { 3 } } s _ { \alpha } \hat { s } _ { \beta } \big ( { \mathbf { P } } \mathbf { u } ^ { \alpha } \cdot { \mathbf { P } } \hat { \mathbf { u } } ^ { \beta } \big ) \big ( { \mathbf { v } } ^ { \alpha } \cdot { \mathbf { \hat { v } } } ^ { \beta } \big ) \Big ] , } \end{array} +$$ + +$$ +\begin{array} { r } { \varepsilon _ { \mathrm { t e s t } } = \Big [ \sum _ { \beta = 1 } ^ { \mathbb { N } _ { 2 } } \bar { s } _ { \beta } ^ { 2 } \big | \big | { \bf P } \bar { \bf u } ^ { \alpha } \big | \big | _ { 2 } ^ { 2 } \Big ] ^ { - 1 } \Big [ \sum _ { \alpha = 1 } ^ { N _ { 2 } } s _ { \alpha } ^ { 2 } \big | \big | { \bf P } { \bf u } ^ { \alpha } \big | \big | _ { 2 } ^ { 2 } + \sum _ { \beta = 1 } ^ { \mathbb { N } _ { 2 } } \bar { s } _ { \beta } ^ { 2 } \big | \big | { \bf P } \bar { \bf u } ^ { \alpha } \big | \big | _ { 2 } ^ { 2 } - 2 \sum _ { \alpha = 1 } ^ { N _ { 2 } } \sum _ { \beta = 1 } ^ { \mathbb { N } _ { 2 } } s _ { \alpha } \bar { s } _ { \beta } \big ( { \bf P } { \bf u } ^ { \alpha } \cdot { \bf P } { \bf { \bar { u } } } ^ { \beta } \big ) \big ( { \bf v } ^ { \alpha } \cdot { \bf \bar { v } } ^ { \beta } \big ) \Big ] . } \end{array} +$$ + +# D TRANSFER LEARNING DERIVATIONS & DETAILS + +Thm 1 (Transfer theorem) The transfer benefit $\scriptstyle { \mathcal { T } } ^ { A B }$ : + +• Is unaffected by the $\overline { { \mathbf { U } } } ^ { A }$ and $\overline { { \mathbf { U } } } ^ { B }$ . +• Is completely determined by only $\sigma _ { z } ^ { 2 }$ $, \overline { { \mathbf { S } } } ^ { A } , \overline { { \mathbf { S } } } ^ { B }$ , and the $\overline { { N } } _ { 2 } ^ { A }$ by $\overline { { N } } _ { 2 } ^ { B }$ similarity matrix $\overline { { \mathbf { Q } } } =$ $\overline { { \mathbf { V } } } ^ { A T } \overline { { \mathbf { V } } } ^ { B }$ . + +![](images/f7d69e78559f4561bf6dde026948060bf03ae8cb1ead40ce9131543b8df1331b.jpg) +Figure 9: Alignment of randomly-initialized network modes to data modes and growth of singular values, plotted for 1 hidden layer (first two rows, a-d) and 3 hidden layers (last two rows, e-h), and for a rank 1 teacher (first and third rows, a & e), or a rank 3 teacher (second and fourth rows, b-d & f-h). The columns are the different modes, with respective singular values of 6, 4, and 2. $\sigma _ { z }$ was set to 1. The deeper networks show substantially slower mode alignment, with alignment not completed until around when the singular value increases. + +Proof: We define + +$$ +\begin{array} { l l } { { { \overline { { { \cal { \bf { U } } } } } } ^ { A B } = \begin{array} { c } { { { \overline { { { \cal { N } } } } } _ { 3 } } } \\ { { { \overline { { { \cal { N } } } } } _ { 3 } } } \\ { { { \overline { { { \cal { N } } } } } _ { 3 } } } \end{array} } } & { { { \overline { { { \cal { N } } } } } _ { 3 } ^ { A } \begin{array} { c } { { { \overline { { { \cal { N } } } } } _ { 2 } ^ { B } } } \\ { { \left| \begin{array} { c } { { { \bf { \overline { { { \bf { U } } } } } } ^ { A } } } \end{array} \right| } } } \\ { { { \overline { { { \bf { U } } } } } ^ { B } } } \end{array} } & { { { \overline { { { \bf { S } } } } } ^ { A B } = \begin{array} { c } { { { \overline { { { \cal { N } } } } } _ { 2 } ^ { A } } } \\ { { { \overline { { { \cal { N } } } } } _ { 2 } ^ { B } } } \end{array} } } \\ { { { \overline { { { \cal { N } } } } } _ { 2 } ^ { A B } = \begin{array} { c } { { { \overline { { { \cal { N } } } } } _ { 3 } ^ { A } } } \end{array} } } & { { { \overline { { { \bf { U } } } } } _ { 2 } ^ { B } } } \end{array} +$$ + +$$ +\mathbf { \overline { { W } } } ^ { A + B } = [ \frac { \mathbf { \overline { { U } } } ^ { A } } { \mathbf { 0 } } | \begin{array} { c } { \mathbf { 0 } } \\ { \mathbf { \overline { { U } } } ^ { B } } \end{array} ] [ \frac { \mathbf { \overline { { S } } } ^ { A } } { \mathbf { 0 } } | \begin{array} { c } { \mathbf { 0 } } \\ { \mathbf { \overline { { S } } } ^ { B } } \end{array} ] [ \frac { \mathbf { \overline { { V } } } ^ { A ^ { T } } } { \mathbf { \overline { { V } } } ^ { B ^ { T } } } ] +$$ + +Because of the 0 blocks in UAB, the vectors in blocks corresponding to task $A$ and task $B$ are completely orthogonal, so UAB remains orthonormal. Thus the relationship between the $\overline { { \mathbf { U } } } ^ { A }$ and $\overline { { \mathbf { U } } } ^ { A }$ is irrelevant to the transfer. (In our simulations we use arbitrary orthonormal matrices for $\overline { { \mathbf { U } } } ^ { A }$ and $\overline { { \mathbf { U } } } ^ { B }$ .) Therefore the transfer effects will be entirely driven by the relationship between the matrices $\overline { { \mathbf { V } } } ^ { A }$ and $\overline { { \mathbf { V } } } ^ { B }$ and the singular values. + +We define $\overline { { N } } _ { 2 } ^ { A }$ by $\overline { { N } } _ { 2 } ^ { B }$ similarity matrix $\mathbf { \overline { { Q } } } = \mathbf { \overline { { V } } } ^ { A ^ { T } } \mathbf { \overline { { V } } } ^ { B }$ . If we think of the columns of each $\overline { { \mathbf { V } } }$ as spanning a low dimensional feature space in $\overline { { N } } _ { 1 }$ dimensional input space that is important for each task, then $\overline { { \mathbf { Q } } }$ reflects the input feature subspace similarity matrix. We can now calculate the singular values of WA+ B. First, note that the input singular modes of WA+B a re eigenvectors of $\overline { { \mathbf { W } } } ^ { A + B ^ { T } } \overline { { \mathbf { W } } } ^ { A + B }$ B, and the associated singular values are square roots of the eigenvalues of WA+B. Now + +$$ +\begin{array} { r } { \overline { { \mathbf { W } } } ^ { A + B ^ { T } } \overline { { \mathbf { W } } } ^ { A + B } = \overline { { \mathbf { V } } } ^ { A B } \overline { { \mathbf { S } } } ^ { A B } \overline { { \mathbf { U } } } ^ { A B ^ { T } } \overline { { \mathbf { U } } } ^ { A B } \overline { { \mathbf { S } } } ^ { A B } \overline { { \mathbf { V } } } ^ { A B ^ { T } } = \overline { { \mathbf { V } } } ^ { A B } \overline { { \mathbf { S } } } ^ { A B ^ { 2 } } \overline { { \mathbf { V } } } ^ { A B ^ { T } } } \end{array} +$$ + +Now if $\vec { c }$ is an eigenvector of this matrix: + +$$ +\overline { { { \bf V } } } ^ { A B } \overline { { { \bf S } } } ^ { A B ^ { 2 } } \overline { { { \bf V } } } ^ { A B ^ { T } } \vec { c } = \lambda \vec { c } +$$ + +This implies that + +$$ +\mathbf { \overline { { { V } } } } ^ { A B ^ { T } } \mathbf { \overline { { { V } } } } ^ { A B } \mathbf { \overline { { { S } } } } ^ { A B ^ { 2 } } \mathbf { \overline { { { V } } } } ^ { A B ^ { T } } { \vec { c } } = \lambda \mathbf { \overline { { { V } } } } ^ { A B ^ { T } } { \vec { c } } +$$ + +Hence eigenvalues of VABSAB2V T are also eigenvalues of VABT V $\overline { { { \bf V } } } ^ { A B ^ { T } } \overline { { { \bf V } } } ^ { A B } \overline { { { \bf S } } } ^ { A B ^ { 2 } }$ , with the mapping between the eigenvectors given by $\overline { { \mathbf { V } } } ^ { A B }$ . Furthermore, this mapping must be a bijection for eigenvectors with non-zero eigenvalues, since the matrices have the same rank (the rank of $\overline { { \mathbf { V } } } ^ { A B }$ ). To see this, note that SAB 2 i s full rank. From this, it is clear that + +$$ +\mathrm { r a n k } \overline { { { \mathbf { V } } } } ^ { A B ^ { T } } \overline { { { \mathbf { V } } } } ^ { A B } \overline { { { \mathbf { S } } } } ^ { A B ^ { 2 } } = \mathrm { r a n k } \overline { { { \mathbf { V } } } } ^ { A B ^ { T } } \overline { { { \mathbf { V } } } } ^ { A B } = \mathrm { r a n k } \overline { { { \mathbf { V } } } } ^ { A B } . +$$ + +Furthermore, SAB2 is positive definite, so + +$$ +\operatorname { r a n k } { \overline { { \mathbf { V } } } } ^ { A B } { \overline { { \mathbf { S } } } } ^ { A B ^ { 2 } } { \overline { { \mathbf { V } } } } ^ { A B ^ { T } } = \operatorname { r a n k } { \overline { { \mathbf { V } } } } ^ { A B } . +$$ + +Now that we know the eigenvectors of these matrices are in bijection, note that: + +$$ +\begin{array}{c} \overline { { \mathbf { V } } } ^ { A B ^ { T } } \overline { { \mathbf { V } } } ^ { A B } \overline { { \mathbf { S } } } ^ { A B ^ { 2 } } = [ \frac { \overline { { \mathbf { V } } } ^ { A ^ { T } } } { \overline { { \mathbf { V } } } ^ { B ^ { T } } } ] [ \begin{array} { c } { \overline { { \mathbf { V } } } ^ { A } \begin{array} { c } { \overline { { \mathbf { V } } } ^ { B } } \end{array} } \end{array} ] [ \frac { \overline { { \mathbf { S } } } ^ { A ^ { 2 } } } { \mathbf { 0 } } \end{array} ] \frac { \mathbf { 0 } } { \overline { { \mathbf { S } } } ^ { B ^ { 2 } } } ] = [ \begin{array} { c c } { \mathbf { I } } & { \mathbf { Q } } \\ { \mathbf { Q } ^ { \mathbf { T } } } & { \mathbf { I } } \end{array} ] [ \frac { \overline { { \mathbf { S } } } ^ { A ^ { 2 } } } { \mathbf { 0 } } \begin{array} { c } { \mathbf { [ \begin{array} { c } { \mathbf { 0 } } \\ { \overline { { \mathbf { S } } } ^ { B ^ { 2 } } } \end{array} ] } \overline { { \mathbf { S } } } ^ { B ^ { 2 } } } \end{array} ] +$$ + +Because the output modes don’t matter (as noted above), the alignment between the eigenvectors of VABSAB2V ABT and $\overline { { \mathbf { V } } } ^ { A }$ , weighted by their respective eigenvalues, gives the transfer benefit. + +For any given tasks, the transfer benefit can be calculated using our theory. However, in certain special cases, we can give exact answers. For example, in the rank one case with equal singular values between the tasks $\mathit { \Pi } _ { \overline { { { s } } } _ { A } } ^ { \prime } = \mathit { \Pi } _ { \overline { { { s } } } _ { B } } ^ { \prime } = \mathit { \Pi } _ { \overline { { { s } } } } ^ { \prime }$ ), the matrix + +$$ +\left[ \begin{array} { l l } { \mathbf { I } } & { \mathbf { Q } } \\ { \mathbf { Q } ^ { \mathbf { T } } } & { \mathbf { I } } \end{array} \right] \left[ \frac { \mathbf { \overline { { S } } } ^ { A ^ { 2 } } \mathbf { \Phi } } { \mathbf { 0 } } \right] \mathbf { \overline { { S } } } ^ { B ^ { 2 } } \mathbf { \Phi } ] +$$ + +reduces to + +$$ +\left[ \begin{array} { l l } { 1 } & { q } \\ { q } & { 1 } \end{array} \right] \overline { { s } } ^ { 2 } +$$ + +with eigenvalues $s \sqrt { 1 \pm q }$ and eigenvectors + +$$ +\left[ \begin{array} { c c } { 1 } & { 1 } \\ { 1 } & { - 1 } \end{array} \right] +$$ + +Corresponding to the shared structure between the tasks and the differences between them. We note that the sign of the alignment $q$ is irrelevant as a special case of the fact (noted above) that any orthogonal transformation on the output modes does not affect transfer. + +# D.1 MISALIGNMENT AND INTERFERENCE + +Why is there interference between tasks which are not well aligned? In the rank one case, we are effectively changing the (input) singular dimensions of $\overline { { \mathbf { Y } } } _ { A }$ from $\overline { { \mathbf { V } } } ^ { A }$ to $\overline { { \mathbf { V } } } ^ { A B }$ . The two singular modes of VAB correspond to the shared structure between the tasks (weighted by the relative signal strengths), and the differences between them, respectively. Although we may be improving our estimates of the shared mode if $q > 0$ (by increasing its singular value relative to ${ \overline { { s } } } _ { A }$ ), we are actually decreasing its alignment with $\overline { { \mathbf { V } } } ^ { A }$ unless $q = 1$ . This misalignment is captured by the second mode of $\overline { { \mathbf { V } } } ^ { A B }$ , but the increase in the singular value of the first mode must come at the cost of a decrease in the singular value of the second mode. See Fig. 10 for a conceptual illustration of this. This means that the multi-task setting allows the distinctions between the tasks to sink towards the sea of noise, while pulling out the common structure. In other words, transferring knowledge from one task always comes at the cost of ignoring differences between the tasks. Furthermore, incorporating a task $B$ allows its noise to seep into the task $A$ signal. Together, these two effects help to explain why transfer can be sometimes beneficial but sometimes detrimental. + +![](images/2ff9eeb17e34a1c5c018c10339262e9b2643717551b0503ce8da75485a108ec8.jpg) +Figure 10: Conceptual cartoon of how $\mathcal { T } ^ { A B }$ , the transfer benefit (or cost) arises from alignment between the task’s input modes. + +# E NON-GRADIENT TRAINING ALGORITHM + +In Fig. 11 we show the match between the error achieved by training the student by gradient descent and the optimal stopping error predicted by the non-gradient shrinkage algorithm in the case of a rank-1 teacher. + +![](images/8fe699450a90e0a1da920afb212efc64c0b70cce8fa419ef6569d47e281026cc.jpg) +Figure 11: Match between optimal stopping error prediction from non-gradient training algorithm and empirical optimal stopping error for a rank-1 teacher. + +# F TRANSFER RESULTS GENERALIZE TO NON-LINEAR NETWORKS + +Since most deep learning practitioners do not train linear networks, it is important that our theoretical insights generalize beyond this simple case. In this section we show that the transfer patterns qualitatively generalize to non-linear networks. + +Here, we show results from teacher networks with $\overline { { N } } _ { 1 } = 1 0 0 \overline { { N } } _ { 3 } = 5 0$ , $\overline { { N } } _ { 2 } = 4$ (thus the task is higher rank) and leaky relu non-linearities at the hidden and output layers. We train a student with leaky relu units and $N _ { 2 } = N _ { 3 }$ to solve this task. Results qualitatively look quite similar to those in Fig 5. of the main text for rank one linear teachers, see below. Thus our insights into transfer may help to understand multi-task benefits in more complicated architectures. + +![](images/3aa21b341a46731773f7366f14f4c4df08b4e4ea25985a881fa5ca23d11fc270.jpg) +Figure 12: Transfer benefit $\mathcal { T } ^ { A B } ( \overline { { s } } _ { A } , \overline { { s } } _ { B } , q )$ for non-linear teachers and students, plotted at different values of ${ \overline { { s } } } _ { A }$ . (a) $\overline { { s } } _ { A } = 0 . 8 4 = \sqrt [ 4 ] { A }$ . With support from another aligned task, especially one with moderately higher SNR, performance on a low SNR task will improve. (b) ${ \overline { { s } } } _ { A } = 3$ . Tasks with modest signals will face interference from poorly aligned tasks, but benefits from well aligned tasks. These effects are amplified by SNR. (c) $\overline { { s } } _ { A } = 1 0 0$ . Tasks with very strong signals will show little effect from other tasks (note y-axis scale), but any impact will be negative unless the tasks are very well aligned. + +# G VARYING THE NUMBER OF TRAINING EXAMPLES + +In the main text, we focused on the test error dynamics in the case in which the number of examples equalled the number of inputs. Here we show how the formula for test error curves is modified as the number of training examples $P$ is varied. For simplicity, when $P \neq N _ { 1 }$ , we focus on the case of a full rank student with aspect ratio $A = 1$ (so that $N _ { 1 } = N _ { 2 } = N _ { 3 }$ ). The more general case of lower rank students with non-unity aspect ratios can be easily found from this case, but with some additional bookkeeping. + +As before, we assume the teacher generates noisy outputs from a set of $P$ inputs: + +$$ +\hat { \mathbf { y } } ^ { \mu } = \overline { { \mathbf { W } } } \hat { \mathbf { x } } ^ { \mu } + \mathbf { z } ^ { \mu } \qquad \mathrm { f o r } \quad \mu = 1 , \ldots , \mathbf { P } . +$$ + +![](images/cdfbc5d1756ec978413726647b732f78c395fed4934518082d5a82318c0a50f6.jpg) +Figure 13: The effects of varying the number of training examples $P$ . (a) Test error for a student learning from a rank-1 teacher with an SNR of 3, with different numbers of inputs. (b,c) Minimum generalization error plotted against $\sqrt { P / N _ { 1 } }$ and SNR · $\sqrt { P / N _ { 1 } }$ , respectively, at different SNRs. When $P \geq N _ { 1 }$ , the minimum generalization error is simply determined by $\mathrm { S N R } \sqrt { P / N _ { 1 } }$ , so all curves converge to a single asymptotic line in (c) as $P$ increases. When $P < N _ { 1 }$ , however, the curves for different SNRs separate because the projection and noise effects depend on initial SNR. (d) Optimal stopping error for gaussian vs. orthogonal inputs, showing a strong correlation. Thus our use of orthogonal inputs in the theory also yields insight into the more general case of approximately unit norm Gaussian inputs. (For all panels $N _ { 1 } = N _ { 2 } = N _ { 3 } = 1 0 0$ , ${ \overline { { N } } } _ { 2 } ^ { - } = 1 .$ ) + +This training set yields important second-order training statistics that will guide student learning: + +$$ +\begin{array} { r } { \pmb { \Sigma } ^ { 1 1 } \equiv \hat { \mathbf { X } } \hat { \mathbf { X } } ^ { T } \qquad \pmb { \Sigma } ^ { 3 1 } \equiv \hat { \mathbf { Y } } \hat { \mathbf { X } } ^ { T } = \overline { { \mathbf { W } } } \hat { \mathbf { X } } \hat { \mathbf { X } } ^ { T } + \mathbf { Z } \hat { \mathbf { X } } ^ { T } . } \end{array} +$$ + +Here $\hat { \mathbf { X } } , \hat { \mathbf { Y } }$ , and $\mathbf { Z }$ are each $\overline { { N } } _ { 1 }$ by $P$ , $\overline { { N } } _ { 3 }$ by $P$ , and $\overline { { N } } _ { 3 }$ by $P$ matrices respectively, whose $\mu ^ { \mathrm { i } }$ ’th columns are $\hat { \mathbf { x } } ^ { \mu } , \hat { \mathbf { y } } ^ { \mu }$ , and $\hat { \mathbf { z } } ^ { \mu }$ , respectively. $\dot { \mathbf { \Sigma } } ^ { 1 \mathrm { 1 } }$ is an $\overline { { N } } _ { 1 }$ by $\overline { { N } } _ { 1 }$ input correlation matrix, and $\pmb { \Sigma } ^ { 3 1 }$ is an $\overline { { N } } _ { 3 }$ by $\overline { { N } } _ { 1 }$ the input-output correlation matrix. We choose the matrix elements $z _ { i } ^ { \mu }$ of the noise matrix $\mathbf { Z }$ to be drawn iid from a Gaussian with zero mean and variance $\sigma _ { z } ^ { 2 } / \overline { { N } } _ { 1 }$ . The noise scaling is chosen so the singular values of the teacher $\overline { { \mathbf { W } } }$ and the noise $\mathbf { Z }$ are both $O ( 1 )$ , leading to non-trivial generalization effects. Furthermore, we chose training inputs to be close to unit-norm, and make the input covariance matrix $\pmb { \Sigma } ^ { 1 1 }$ as white as possible (whitening is a common pre-processing step for inputs). When $P > \overline { { N } } _ { 1 }$ , this can be done by choosing the rows of $\hat { \bf X }$ to be orthonormal and then scaling up by $\sqrt { P / \overline { { N } } _ { 1 } }$ , so the columns are approximately unit norm. Then $\Sigma ^ { 1 1 } = P / \overline { { N } } _ { 1 } \mathbf { I }$ is proportional to the identity. On the otherhand, if $P < \overline { { N } } _ { 1 }$ , we choose the columns of $\hat { \bf X }$ to be orthonormal, so that $\pmb { \Sigma } ^ { 1 1 } = \pmb { \mathcal { P } } ^ { | | }$ , where $\mathcal { P } ^ { | | }$ is a projection operator onto the $P$ dimensional column space of $\hat { \mathbf X }$ spanned by the input examples. Both these choices are intended to approximate the situation in which the columns of $\hat { \bf X }$ are chosen to be iid unit-norm vectors. Finally, as generalization performance will depend on the ratio of teacher singular values to the noise variance parameter $\sigma _ { z } ^ { 2 }$ we simply set $\sigma _ { z } = 1$ as in the main text. Thus, given the unit-norm inputs, we can think of the teacher singular values as signal to noise ratios (SNRs). We now examine how the dynamics of the test error evolves as we vary the number of training examples $P$ . We split our analyses into two distinct regimes: (1) the oversampled regime in which the data density $\bar { \mathcal { D } } \equiv P / N _ { 1 } > \mathrm { ~ \bar { ~ } { ~ 1 ~ } ~ }$ , and (2) the undersampled regime in which $\mathcal { D } < 1$ . + +# G.1 THE OVERSAMPLED REGIME + +The oversampled regime $\mathcal { D } > 1 \vert$ ) is relatively simple. First $\pmb { \Sigma } ^ { 1 1 }$ is scaled up by a factor of $\mathcal { D }$ . And in the input-output covariance matrix, ${ \boldsymbol { \Sigma } } ^ { 3 1 } = \overline { { \mathbf { W } } } \hat { \mathbf { X } } \hat { \mathbf { X } } ^ { T } + \mathbf { Z } \hat { \mathbf { X } } ^ { T }$ , the signal component, $\overline { { \mathbf { W } } } \hat { \mathbf { X } } \hat { \mathbf { X } } ^ { T }$ is scaled up by a factor of $\mathcal { D }$ while the noise component √ $\mathbf { Z } \hat { \mathbf { X } } ^ { T }$ has the same singular value spectrum as the $\mathcal { D } = 1$ case, up to an overall scaling by √ $\sqrt { \mathcal { D } }$ (since the rows of $\hat { \mathbf X }$ are orthogonal and all its√ singular values are equal to $\sqrt { \mathcal { D } }$ ). This leads to an increase in the effective SNR by a factor of $\sqrt { \mathcal { D } }$ . Thus overall, the test error curves for the case of $\mathcal { D } > 1$ can be simply obtained from the theory of the test error curves for $\mathcal { D } = 1$ through two modifications: (1) a boost in the SNR for the $\mathcal { D } = 1$ case by a multiplicative factor of $\sqrt { \mathcal { D } }$ , and (2) and an overall speed up in the learning time by a multiplicative factor of $\mathcal { D }$ . + +# G.2 THE UNDERSAMPLED REGIME + +For the undersampled regime $\mathcal { D } < 1 \dot { }$ ), we must account for the fact that the $P$ training inputs do not span the full $N _ { 1 }$ dimensional space of all inputs. Thus the projection operator $\mathcal { P } ^ { | | }$ onto the $P$ dimensional column space of $\hat { \mathbf X }$ plays a crucial role. Indeed the input-correlation $\pmb { \Sigma } ^ { 1 1 } = \pmb { \mathcal { P } } ^ { | | }$ . And ${ \boldsymbol { \Sigma } } ^ { 3 1 } = \overline { { \mathbf { W } } } \mathbf { \mathcal { P } } ^ { | | } + \mathbf { Z } \hat { \mathbf { X } } ^ { T }$ . This implies that the learning dynamics only transforms the composite student map W from the $P$ dimensional subspace spanned by the inputs to the $N _ { 3 }$ dimensional output space. In contrast, the student map from the $N _ { 1 } - P$ dimensional subspace orthogonal to the image of $\mathcal { P } ^ { | | }$ remains frozen. Tracing through the equations of the main paper and accounting for the projection operator $\mathcal { P } ^ { | | }$ , we find the effective aspect ratio for this undersampled learning problem (when $N _ { 3 } = N _ { 2 } = N _ { 1 } ;$ is no longer $\mathcal { A } = N _ { 3 } / \bar { N _ { 1 } }$ but rather $\mathcal { D } = P / N _ { 1 }$ . Furthermore, in the limit $\overline { { N } } _ { 3 } , \overline { { N } } _ { 1 } \infty$ while $\overline { { N } } _ { 2 }$ remains √ $O ( 1 )$ , the singular values of the signal component $\overline { { \mathbf { W } } } \mathcal { P } ^ { | | }$ of $\pmb { \Sigma } ^ { 3 1 }$ are attenuated by a factor of $\sqrt { \mathcal { D } }$ , making the associated singular vectors more susceptible to noise. Again tracing through the equations of the main paper, with all of these modifications, we find the final formula for test error curves in the undersampled measurement regime: + +$$ +\varepsilon _ { \mathsf { t e s t } } ( t ) = { \frac { \left[ ( N _ { 3 } - P ) \epsilon ^ { 2 } + ( P - { \overline { { N } } } _ { 2 } ) \langle s ( { \hat { s } } , t ) ^ { 2 } \rangle + \sum _ { \alpha = 1 } ^ { \overline { { N } } _ { 2 } } \left[ ( s _ { \alpha } ( t ) - { \overline { { s } } } _ { \alpha } ) ^ { 2 } + 2 s _ { \alpha } ( t ) { \overline { { s } } } _ { \alpha } ( 1 - { \mathcal { O } } ( { \sqrt { \overline { { D } } } } { \overline { { s } } } _ { \alpha } ) ) \right] \right] } { \left[ \sum _ { \alpha = 1 } ^ { \overline { { N } } _ { 2 } } { \overline { { s } } } _ { \alpha } ^ { 2 } \right] } } +$$ + +This equation has several modifications compared to the case $P = N _ { 1 }$ in (15). First the term in the numerator involving $N _ { 3 } - P$ reflects generalization error due to the $N _ { 3 } - P$ dimensional frozen subspace, and the initial weight variance $\epsilon ^ { 2 }$ contributes to this generalization error. The second term in the numerator involves all the $P - { \overline { { N } } } _ { 2 }$ training modes which cannot be correlated with the teacher, and the average $\langle \cdot \rangle$ is over a Marcenko-Pasteur distribution of singular values (see (13)) except with the aspect ratio $\mathcal { A }$ replaced by $\mathcal { D }$ . The third term accounts for learned correlations between the student and teacher. It involves the transformation from teacher singular values $\overline { { s } }$ to training data singular values $\hat { s }$ through the formula (11) except with the aspect ratio replacement √ $A \mathcal { D }$ , and the effective teacher singular value attenuation $\overline { { s } } \sqrt { D } \overline { { s } }$ . Similarly, the computation of the singular vector overlap is done through (12) also with the replacements $A \mathcal { D }$ and $\overline { { s } } \sqrt { D } \overline { { s } }$ . + +# G.3 COMPARISON OF THEORY AND EXPERIMENT FOR UNDER AND OVER SAMPLED MEASUREMENT REGIMES + +In Fig. 13, we show an excellent match between our theory and empirical simulations for varying values of $P$ , both in the oversampled and undersampled measurement regimes. There are a number of interesting features to note. First, although the minimum generalization error improves monotonically with $P$ , the asymptotic $t \to \infty$ ) generalization error does not, because of a frozen subspace (Advani & Saxe, 2017) of the modes that are not overfit when $P < N _ { 1 }$ , because the training data rank is $\le P$ . Second, when $P \geq N _ { 1 }$ , the minimum generalization error is simply determined by $\mathbf { S N R } \sqrt { P / N _ { 1 } }$ , so all curves converge to a single asymptotic line as $P$ increases. When $P < N _ { 1 }$ , however, the curves for different SNRs separate because the projection and noise effects depend on initial SNR. Finally, in Fig. 13D we show that approximately unit norm i.i.d. gaussian inputs yield similar results to the orthogonalized data matrices we employed in the theory, although the gaussian inputs do result in slightly higher optimal stopping error. + +# H LESS THAN FULL RANK STUDENTS + +Although we generally assumed students were full rank in the main text to simplify the calculations, our theory remains exact for TA networks of any rank. Furthermore, as shown in Fig. 14, the TA and random networks again show very similar optimal stopping generalization error, but with the optimal stopping time of the random networks lagging behind that of the TA networks. Furthermore, this lag increases as the rank of the random network decreases (because a low rank network will have less initial projection onto the random modes, there is is more alignment to be done). However, reducing the student rank does not change the optimal stopping error (as long as it is still greater than the teacher rank). + +![](images/6f1091544ee918b4ed7b6456f495e4dd5c10e618ea9cc7217b0d7c4d68a60fb8.jpg) +(c) Optimal generalization error vs. optimal(d) Optimal generalization error vs. optimal stopping time for stopping time for randomly initialized networksinitially aligned networks +Figure 14: Empirical verification that the simplifying assumptions of our theory are approximately valid in the regime we are considering at different student ranks. Initializations with random initial weights (random init.) and initializations with initial weight aligned to the noisy data SVD (aligned init.) are compared across varying student ranks. (a) The minimum generalization errors are almost identical between the different initializations and different student ranks. (b) The optimal stopping time in the randomly initialized networks consistently lags behind the aligned networks, because it takes time for the alignment to occur. This lag increases as the students rank decreases. (c) Randomly initialized networks of varying ranks obey qualitatively similar trends of increase in optimal stopping error and optimal stopping time as SNR decreases. (d) The theory predicts the aligned networks trends of increase in optimal stopping error and optimal stopping time with decreasing SNR almost perfectly. (All plots are made with a rank 1 teacher and $N _ { 1 } = N _ { 3 } = 1 0 0 \rangle$ ) \ No newline at end of file diff --git a/parse/train/ryfMLoCqtQ/ryfMLoCqtQ_content_list.json b/parse/train/ryfMLoCqtQ/ryfMLoCqtQ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..b68b6dd5f07da53c159b230dc934b5be3b76c787 --- /dev/null +++ b/parse/train/ryfMLoCqtQ/ryfMLoCqtQ_content_list.json @@ -0,0 +1,2352 @@ +[ + { + "type": "text", + "text": "AN ANALYTIC THEORY OF GENERALIZATION DYNAMICS AND TRANSFER LEARNING IN DEEP LINEAR NETWORKS ", + "text_level": 1, + "bbox": [ + 173, + 101, + 828, + 170 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Andrew K. Lampinen Department of Psychology Stanford University lampinen@stanford.edu ", + "bbox": [ + 184, + 195, + 392, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Surya Ganguli \nDepartment of Applied Physics \nStanford University \nand \nGoogle Brain \nsganguli@stanford.edu ", + "bbox": [ + 506, + 195, + 715, + 279 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 315, + 544, + 330 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Much attention has been devoted recently to the generalization puzzle in deep learning: large, deep networks can generalize well, but existing theories bounding generalization error are exceedingly loose, and thus cannot explain this striking performance. Furthermore, a major hope is that knowledge may transfer across tasks, so that multi-task learning can improve generalization on individual tasks. However we lack analytic theories that can quantitatively predict how the degree of knowledge transfer depends on the relationship between the tasks. We develop an analytic theory of the nonlinear dynamics of generalization in deep linear networks, both within and across tasks. In particular, our theory provides analytic solutions to the training and testing error of deep networks as a function of training time, number of examples, network size and initialization, and the task structure and SNR. Our theory reveals that deep networks progressively learn the most important task structure first, so that generalization error at the early stopping time primarily depends on task structure and is independent of network size. This suggests any tight bound on generalization error must take into account task structure, and explains observations about real data being learned faster than random data. Intriguingly our theory also reveals the existence of a learning algorithm that proveably out-performs neural network training through gradient descent. Finally, for transfer learning, our theory reveals that knowledge transfer depends sensitively, but computably, on the SNRs and input feature alignments of pairs of tasks. ", + "bbox": [ + 233, + 345, + 766, + 623 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 641, + 336, + 657 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Many deep learning practitioners closely monitor both training and test errors, hoping to achieve both a small training error and a small generalization error, or gap between testing and training errors. Training is usually stopped early, before overfitting sets in and increases the test error. This procedure often results in large networks that generalize well on structured tasks, raising an important generalization puzzle (Zhang et al., 2016): many existing theories that upper bound generalization error (Bartlett & Mendelson, 2002; Neyshabur et al., 2015; Dziugaite & Roy, 2017; Golowich et al., 2017; Neyshabur et al., 2017; Bartlett et al., 2017; Arora et al., 2018, e.g) in terms of various measures of network complexity yield very loose bounds. Therefore they cannot explain the impressive generalization capabilities of deep nets. ", + "bbox": [ + 174, + 666, + 825, + 791 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In the absence of any such tight and computable theory of deep network generalization error, we develop an analytic theory of generalization error for deep linear networks. Such networks exhibit highly nonlinear learning dynamics (Saxe et al., 2013a;b) including many prominent phenomena like learning plateaus, saddle points, and sudden drops in training error. Moreover, theory developed for the learning dynamics of deep linear networks directly inspired better initialization schemes for nonlinear networks (Schoenholz et al., 2016; Pennington et al., 2017; 2018). Here we show that deep linear networks also provide a good theoretical model for generalization dynamics. In particular we develop an analytic theory for both the training and test error of a deep linear network as a function of training time, number of training examples, network architecture, initialization, and task structure and SNR. Our theory matches simulations and reveals that deep networks with small weight initialization learn the most important aspects of a task first. Thus the optimal test error at the early stopping time depends largely on task structure and SNR, and not on network architecture, as long as the architecture is expressive enough to attain small training error. Thus our exact analysis of generalization dynamics reveals the important lesson that any theory that seeks to upper bound generalization error based only on network architecture, and not on task structure, is likely to yield exceedingly loose upper bounds. Intriguingly our theory also reveals a non-gradient-descent learning algorithm that proveably out-performs neural network training through gradient descent. ", + "bbox": [ + 174, + 799, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We also apply our theory to multi-task learning, which enables knowledge transfer from one task to another, thereby further lowering generalization error (Dong et al., 2015; Rusu et al., 2015; Luong et al., 2016, e.g.). Moreover, knowledge transfer across tasks may be key to human generalization capabilities (Hansen et al., 2017; Lampinen et al., 2017). We provide an analytic theory for how much knowledge is transferred between pairs of tasks, and we find that it displays a sensitive but computable dependence on the relationship between pairs of tasks, in particular, their SNRs and feature space alignments. ", + "bbox": [ + 174, + 222, + 825, + 319 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We note that a related prior work (Advani & Saxe, 2017) studied generalization in shallow and deep linear networks, but that work was limited to networks with a single output, thereby precluding the possibility of addressing the issue of transfer learning. Moreover, analyzing networks with a single output also precludes the possibility of addressing interesting tasks that require higher dimensional outputs, for example in language (Dong et al., 2015, e.g.), generative models (Goodfellow et al., 2014, e.g), and reinforcement learning (Mnih et al., 2015; Silver et al., 2016, e.g). ", + "bbox": [ + 174, + 325, + 825, + 411 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 THEORETICAL FRAMEWORK ", + "text_level": 1, + "bbox": [ + 176, + 430, + 441, + 446 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We work in a student-teacher scenario in which we consider an ensemble of low rank, noisy teacher networks that generate training data for a potentially more complex student network, and define the training and test errors whose dynamics we wish to understand. ", + "bbox": [ + 174, + 462, + 825, + 503 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 AN ENSEMBLE OF LOW-RANK NOISY TEACHERS ", + "text_level": 1, + "bbox": [ + 174, + 521, + 545, + 535 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We first consider an ensemble of 3-layer linear teacher networks with $\\overline { { N } } _ { i }$ units in layer $i$ , and weight matrices $\\overline { { \\mathbf { W } } } ^ { 2 1 } \\in \\mathbb { R } ^ { \\overline { { N _ { 2 } } } \\times \\overline { { N } } _ { 1 } }$ and $\\overline { { \\mathbf { W } } } ^ { 3 2 } \\in \\mathbb { R } ^ { \\overline { { N _ { 3 } } } \\times \\overline { { N _ { 2 } } } }$ between the input to hidden, and hidden to output layers, respectively. The teacher network thus computes the composite map $\\overline { { \\mathbf { y } } } = \\overline { { \\mathbf { W } } } \\mathbf { x }$ , where $\\dot { \\overline { { \\mathbf { W } } } } \\equiv \\overline { { \\mathbf { W } } } ^ { 3 \\bar { 2 } } \\overline { { \\mathbf { W } } } ^ { 2 1 }$ . Of critical importance is the singular value decomposition (SVD) of $\\overline { { \\mathbf { W } } }$ : ", + "bbox": [ + 173, + 545, + 825, + 612 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/18e92e9c21acd8fdfe61f3b0f2f83ca93e2158c5227d0fe9be82985d71af48cd.jpg", + "text": "$$\n\\overline { { \\mathbf { W } } } = \\overline { { \\mathbf { U } } } \\overline { { \\mathbf { S } } } \\overline { { \\mathbf { V } } } ^ { T } = \\sum _ { \\alpha = 1 } ^ { \\overline { { N _ { 2 } } } } \\overline { { s } } ^ { \\alpha } \\overline { { \\mathbf { u } } } ^ { \\alpha } \\overline { { \\mathbf { v } } } ^ { \\alpha T } ,\n$$", + "text_format": "latex", + "bbox": [ + 387, + 618, + 607, + 664 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Where $\\overline { { \\mathbf { U } } } \\in \\mathbb { R } ^ { \\overline { { N _ { 3 } } } \\times \\overline { { N } } _ { 2 } }$ and $\\overline { { \\mathbf { V } } } \\in \\mathbb { R } ^ { \\overline { { N _ { 1 } } } \\times \\overline { { N } } _ { 2 } }$ are both matrices with orthonormal columns and $\\overline { \\mathbf { S } }$ is an $\\overline { { N _ { 2 } } } \\times \\overline { { N _ { 2 } } }$ diagonal matrix. We construct a random teacher by picking $\\overline { { \\mathbf { U } } }$ and $\\overline { { \\mathbf { V } } }$ to be random matrices with orthonormal columns and choosing $O ( 1 )$ values for the diagonal elements of $\\overline { { \\mathbf { S } } }$ . We work in the limit $\\overline { { N _ { 1 } } } , \\overline { { N _ { 3 } } } \\infty$ with an $O ( 1 )$ aspect ratio $\\mathcal { A } = \\overline { { N _ { 3 } } } / \\overline { { N _ { 1 } } } \\in ( 0 , 1 ]$ so that the teacher has fewer outputs than inputs. Also, we hold $\\overline { { N } } _ { 2 } \\sim O ( 1 )$ , so the teacher has a low, finite rank, and we study generalization performance as a function of the $\\overline { { N } } _ { 2 }$ teacher singular values. ", + "bbox": [ + 173, + 670, + 825, + 765 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We further assume the teacher generates noisy outputs from a set of $\\overline { { N } } _ { 1 }$ orthonormal inputs: ", + "bbox": [ + 169, + 770, + 774, + 786 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/9e0840961c481665c78b2718bcdff3a35f1556ac3337518adc63d8550c81159a.jpg", + "text": "$$\n\\hat { { \\bf y } } ^ { \\mu } = \\overline { { { \\bf W } } } \\hat { \\bf x } ^ { \\mu } + { \\bf z } ^ { \\mu } \\qquad \\mathrm { f o r } \\quad \\mu = { \\bf 1 } , \\ldots , \\overline { { { \\bf N } } } _ { \\bf 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 352, + 791, + 647, + 810 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This training set yields important second-order training statistics that will guide student learning: ", + "bbox": [ + 169, + 816, + 807, + 832 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/f25ec0c2167a37977c208003ddeb0568aa5c5a202c1a5e030b90eba093b332e8.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { \\Sigma } ^ { 1 1 } \\equiv \\overset { \\overline { { \\boldsymbol { N } } } _ { 1 } } { \\mu = 1 } \\hat { \\mathbf { x } } ^ { \\mu } \\hat { \\mathbf { x } } ^ { \\mu T } = \\mathbf { I } , \\qquad \\pmb { \\Sigma } ^ { 3 1 } \\equiv \\overset { \\overline { { \\boldsymbol { N } } } _ { 1 } } { \\mu = 1 } \\hat { \\mathbf { y } } ^ { \\mu } \\hat { \\mathbf { x } } ^ { \\mu T } = \\overline { { \\mathbf { W } } } + \\mathbf { Z } \\hat { \\mathbf { X } } ^ { T } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 838, + 710, + 885 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Here the input covariance $\\pmb { \\Sigma } ^ { 1 1 }$ is assumed to be white (a common pre-processing step), the inputoutput covariance $\\pmb { \\Sigma } ^ { 3 1 }$ is simplified using (2), and $\\mathbf { Z } \\in \\mathbb { R } ^ { \\overline { { N } } _ { 3 } \\times \\overline { { N } } _ { 1 } }$ is the noise matrix, whose $\\mu$ ’th column is $\\mathbf { z } ^ { \\mu }$ . Its matrix elements $z _ { i } ^ { \\mu }$ are drawn iid. from a Gaussian with zero mean and variance $\\sigma _ { z } ^ { 2 } / \\overline { { N } } _ { 1 }$ . The noise scaling is chosen so the singular values of the teacher $\\overline { { \\mathbf { W } } }$ and the noise $\\mathbf { Z }$ are both $O ( 1 )$ , leading to non-trivial generalization effects. As generalization performance will depend on the ratio of teacher singular values to the noise variance parameter $\\sigma _ { z } ^ { 2 }$ , we simply set $\\sigma _ { z } = 1$ in the following. Thus we can think of teacher singular values as signal to noise ratios (SNRs). ", + "bbox": [ + 174, + 892, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 102, + 825, + 176 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Finally, we note that while we focus for ease of exposition in the main paper on the case of one hidden layer networks and a full orthonormal basis of $P = { \\overline { { N _ { 1 } } } }$ training inputs in the main paper, neither of these assumptions are essential to our theory. Indeed in Section 3.4 and App. A we extend our theory to networks of arbitrary depth, and in App. G we extend our theory to the case of white inputs with $P \\neq \\overline { { N } } _ { 1 }$ , obtaining a good match between theory and experiment in both cases. ", + "bbox": [ + 174, + 181, + 825, + 253 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 STUDENT TRAINING AND TEST ERROR ", + "text_level": 1, + "bbox": [ + 176, + 271, + 478, + 285 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Now consider a student network with $N _ { i }$ units in each layer. We assume the first and last layers match the teacher (i.e. $N _ { 1 } = \\overline { { N _ { 1 } } }$ and $N _ { 3 } = \\overline { { N _ { 3 } } }$ ) but $N _ { 2 } \\geq \\overline { { N _ { 2 } } }$ , allowing the student to have more hidden units than the teacher. We also consider deeper students (see below and App. A). Now consider any student whose input-output map is given by $\\mathbf { \\dot { y } } = \\mathbf { W } ^ { 3 2 } \\mathbf { W } ^ { 2 1 } \\equiv \\mathbf { W } \\mathbf { x }$ . Its training error on the teacher dataset in (2) and its test error over a distribution of new inputs are given by ", + "bbox": [ + 174, + 296, + 826, + 367 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/ec10a694331c27b34d51ae231b55eb44404ae9aa4b000936042e75c64c0f3aab.jpg", + "text": "$$\n\\varepsilon _ { \\mathrm { t r a i n } } \\equiv \\frac { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\mathbf { W } \\hat { \\mathbf { x } } ^ { \\mu } - \\hat { \\mathbf { y } } ^ { \\mu } | | _ { 2 } ^ { 2 } } { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\hat { \\mathbf { y } } ^ { \\mu } | | _ { 2 } ^ { 2 } } , \\varepsilon _ { \\mathrm { t e s t } } \\equiv \\frac { \\langle | | \\mathbf { W } \\overline { { \\mathbf { x } } } - \\overline { { \\mathbf { y } } } | | _ { 2 } ^ { 2 } \\rangle } { \\langle | | \\overline { { \\mathbf { y } } } | | _ { 2 } ^ { 2 } \\rangle } ,\n$$", + "text_format": "latex", + "bbox": [ + 281, + 375, + 715, + 422 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "respectively. Here $\\hat { \\mathbf { x } } ^ { \\mu }$ and ${ \\hat { \\mathbf { y } } } ^ { \\mu }$ are the noisy training set inputs and outputs in (2), whereas $\\overline { { \\mathbf { x } } }$ denotes a random test input drawn from zero mean Gaussian with identity covariance, $\\overline { { \\mathbf { y } } } ^ { \\mu } = \\overline { { \\mathbf { W } } } \\overline { { \\mathbf { x } } } ^ { \\mu }$ is noise free teacher output, and $\\langle \\cdot \\rangle$ denotes an average w.r.t the distribution of the test input $\\overline { { \\mathbf { x } } }$ . Due to the orthonormality of the training and isotropy of the test inputs, both $\\varepsilon _ { \\mathrm { t r a i n } }$ and ${ \\varepsilon } _ { \\mathrm { t e s t } }$ can be expressed as ", + "bbox": [ + 174, + 429, + 825, + 488 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2164005cd357741499916e1ea4597b4f17af0fa591d2c70c9b6722b832ef8c1d.jpg", + "text": "$$\n\\mathrm { ~ \\xi ~ } _ { \\mathrm { t r a i n } } = \\frac { \\mathrm { T } \\mathbf { F } \\mathbf { W } ^ { T } \\mathbf { W } - 2 \\mathrm { T r } \\mathbf { W } ^ { T } { \\boldsymbol { \\Sigma } } ^ { 3 1 } + \\mathrm { T r } { \\boldsymbol { \\Sigma } } ^ { 3 1 T } { \\boldsymbol { \\Sigma } } ^ { 3 1 } } { \\mathrm { T r } \\boldsymbol { \\Sigma } ^ { 3 1 T } \\boldsymbol { \\Sigma } ^ { 3 1 } } , \\mathrm { ~ \\xi ~ } _ { \\mathrm { f e s t } } = \\frac { \\mathrm { T r } \\mathbf { W } ^ { T } \\mathbf { W } - 2 \\mathrm { T r } \\mathbf { W } ^ { T } \\overline { { \\mathbf { W } } } + \\mathrm { T r } \\overline { { \\mathbf { W } } } ^ { T } \\overline { { \\mathbf { W } } } } { \\mathrm { T r } \\overline { { \\mathbf { W } } } ^ { T } \\overline { { \\mathbf { W } } } } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 494, + 825, + 535 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Both $\\varepsilon _ { \\mathrm { t r a i n } }$ and $\\varepsilon _ { \\mathrm { t e s t } }$ can be further expressed in terms of the student, training data and teacher SVDs, which we denote by $\\mathbf { W } = \\mathbf { U } \\mathbf { S } \\mathbf { V } ^ { T }$ , $\\begin{array} { r } { \\pmb { \\Sigma } ^ { 3 1 } = \\hat { \\mathbf { U } } \\hat { \\mathbf { S } } \\hat { \\mathbf { V } } ^ { T } } \\end{array}$ , and $\\overline { { \\mathbf { W } } } = \\overline { { \\mathbf { U } } } \\overline { { \\mathbf { S } } } \\overline { { \\mathbf { V } } } ^ { T }$ respectively. Specifically, ", + "bbox": [ + 174, + 545, + 831, + 579 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9924429ca3ea20bc11ef1ca8fe58cf05d95015678f3a03aceca1e14b494d6ac4.jpg", + "text": "$$\n\\begin{array} { r l } & { \\varepsilon _ { \\mathrm { t r a i n } } = \\left[ \\overbrace { \\sum _ { \\beta = 1 } ^ { N _ { 3 } } } ^ { \\overline { { S } } _ { 3 } } \\hat { s } _ { \\beta } ^ { 2 } \\right] ^ { - 1 } \\left[ \\underset { \\alpha = 1 } { \\overset { N _ { 2 } } { \\sum _ { \\alpha } } } s _ { \\alpha } ^ { 2 } + \\underset { \\beta = 1 } { \\overset { \\overline { { N } } _ { 3 } } { \\sum _ { \\beta } } } \\hat { s } _ { \\beta } ^ { 2 } - 2 \\underset { \\alpha = 1 } { \\overset { N _ { 2 } } { \\sum _ { \\beta = 1 } } } \\sum _ { \\beta = 1 } ^ { \\overline { { N _ { 3 } } } } s _ { \\alpha } \\hat { s } _ { \\beta } \\left( \\mathbf { u } ^ { \\alpha } \\cdot \\hat { \\mathbf { u } } ^ { \\beta } \\right) \\left( \\mathbf { v } ^ { \\alpha } \\cdot \\hat { \\mathbf { v } } ^ { \\beta } \\right) \\right] , } \\\\ & { \\varepsilon _ { \\mathrm { t e s t } } = \\left[ \\underset { \\beta = 1 } { \\overset { \\overline { { N } } _ { 2 } } { \\sum _ { \\beta } } } \\overline { { s } } _ { \\beta } ^ { 2 } \\right] ^ { - 1 } \\left[ \\underset { \\alpha = 1 } { \\overset { N _ { 2 } } { \\sum _ { \\alpha } } } s _ { \\alpha } ^ { 2 } + \\underset { \\beta = 1 } { \\overset { \\overline { { N } } _ { 2 } } { \\sum _ { \\beta } } } \\frac { 1 } { s _ { \\beta } ^ { 2 } } - 2 \\underset { \\alpha = 1 } { \\overset { N _ { 2 } } { \\sum _ { \\alpha } } } \\sum _ { \\beta = 1 } ^ { \\overline { { N _ { 2 } } } } s _ { \\alpha } \\overline { { s } } _ { \\beta } \\left( \\mathbf { u } ^ { \\alpha } \\cdot \\overline { { \\mathbf { u } } } ^ { \\beta } \\right) \\left( \\mathbf { v } ^ { \\alpha } \\cdot \\overline { { \\mathbf { v } } } ^ { \\beta } \\right) \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 585, + 766, + 693 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Thus as the student learns, its training and test error dynamics depends on the alignment of the time-evolving student singular modes $\\{ s ^ { \\alpha } , \\mathbf { u } ^ { \\alpha } , \\mathbf { v } ^ { \\alpha } \\}$ with the fixed training data $\\{ \\hat { s } ^ { \\alpha } , \\hat { \\mathbf { u } } ^ { \\alpha } , \\hat { \\mathbf { v } } ^ { \\alpha } \\}$ and teacher $\\{ \\overline { { s } } ^ { \\alpha } , \\bar { \\overline { { \\mathbf { u } } } } ^ { \\alpha } , \\overline { { \\mathbf { v } } } ^ { \\alpha } \\}$ singular modes respectively. ", + "bbox": [ + 176, + 698, + 823, + 741 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 SINGLE TASK GENERALIZATION DYNAMICS: THEORY AND EXPERIMENT", + "text_level": 1, + "bbox": [ + 171, + 762, + 800, + 779 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here we derive and numerically test analytic formulas for both the training and test errors of a student network as it learns from training data generated from a teacher network. We explore the dependence of these quantitites on the student network size, student initialization, teacher SNR, and training time. ", + "bbox": [ + 174, + 794, + 826, + 837 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 STUDENT TRAINING DYNAMICS AND TRAINING-ALIGNED (TA) NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 854, + 733, + 869 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We assume the student weights undergo batch gradient descent with learning rate $\\lambda$ on the training error $\\begin{array} { r } { \\sum _ { \\mu } | | \\hat { \\mathbf { y } } ^ { \\mu } - \\mathbf { W } ^ { 3 2 } \\mathbf { W } ^ { 2 1 } \\bar { \\hat { \\mathbf { x } } ^ { \\mu } } | | _ { 2 } ^ { 2 } } \\end{array}$ , which for small $\\lambda$ is well approximated by the differential equations: ", + "bbox": [ + 174, + 881, + 821, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/df635d5402308d3a9a9f10644ccaa9227749fea9232affb61828b212aa929642.jpg", + "image_caption": [ + "Figure 1: Learning dynamics as a function of singular dimension strength. (a) shows how modes of different singular value are learned, (b) shows that there is a wave of learning that picks up singular dimensions with smaller and smaller singular values as $t \\to \\infty$ . " + ], + "image_footnote": [], + "bbox": [ + 178, + 102, + 602, + 226 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8db3ff2fa5cb7fdcf05f43aacd234760cc422e880240ce8bc05e642b67ae268f.jpg", + "text": "$$\n\\tau \\frac { d } { d t } \\mathbf { W } ^ { 2 1 } = \\mathbf { W } ^ { 3 2 ^ { T } } \\left( \\boldsymbol { \\Sigma } ^ { 3 1 } - \\mathbf { W } ^ { 3 2 } \\mathbf { W } ^ { 2 1 } \\boldsymbol { \\Sigma } ^ { 1 1 } \\right) , \\qquad \\tau \\frac { d } { d t } \\mathbf { W } ^ { 3 2 } = \\left( \\boldsymbol { \\Sigma } ^ { 3 1 } - \\mathbf { W } ^ { 3 2 } \\mathbf { W } ^ { 2 1 } \\boldsymbol { \\Sigma } ^ { 1 1 } \\right) \\mathbf { W } ^ { 2 1 ^ { T } } ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 247, + 816, + 280 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(where $\\tau \\equiv 1 / \\lambda$ ), which must be solved from an initial set of student weights at time $t = 0$ (Saxe et al., 2013a). We consider two classes of student initializations. The first initialization corresponds to a random student where the weights $\\mathbf { W } ^ { 2 1 }$ and $\\mathbf { W ^ { 3 2 } }$ are chosen such that the composite map $\\mathbf { W } = \\mathbf { W ^ { 3 2 } W ^ { 2 1 } }$ has an SVD $\\mathbf { W } = \\epsilon \\mathbf { U } \\mathbf { V } ^ { \\mathbf { T } }$ , where $\\mathbf { U }$ and $\\mathbf { V }$ are random singular vector matrices and all student singular vala time dependent evolution $\\epsilon$ te map undergoes. For white inputs, $\\begin{array} { r } { \\mathbf { W } ( t ) = \\mathbf { U } ( t ) \\mathbf { S } ( t ) \\mathbf { V } ( t ) ^ { T } = \\sum _ { \\alpha = 1 } ^ { N _ { 2 } } \\mathbf { s } _ { \\alpha } ( t ) \\mathbf { u } ^ { \\alpha } ( t ) \\mathbf { v } ^ { \\alpha } ( t ) ^ { T } . } \\end{array}$ as , , and so the time-dependent student singular modes $\\{ s ^ { \\alpha } ( t ) , \\mathbf { u } ^ { \\alpha } ( \\mathbf { t } ) , \\mathbf { v } ^ { ( } \\mathbf { t } ) \\}$ converge to the training data singular modes $\\{ \\hat { s } ^ { \\alpha } , \\hat { \\mathbf { u } } ^ { \\alpha } , \\hat { \\mathbf { v } } ^ { \\alpha } \\}$ . However, the explicit dynamics of the student singular modes can be difficult to obtain analytically from random initial conditions. ", + "bbox": [ + 173, + 290, + 825, + 420 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Thus we also consider a special class of training aligned (TA) initial conditions in which $\\mathbf { W ^ { 2 1 } }$ and $\\mathbf { W ^ { 3 2 } }$ are chosen such that the composite map $\\mathbf { W } = \\mathbf { W ^ { 3 2 } W ^ { 2 1 } }$ has an SVD $\\mathbf { W } = \\epsilon \\hat { \\mathbf { U } } \\hat { \\mathbf { V } } ^ { \\mathbf { T } }$ . That is, the TA network (henceforth referred to simply as the TA) has the same singular vectors as the training data covariance $\\boldsymbol { \\Sigma } ^ { 3 1 }$ , but has all singular values equal to $\\epsilon$ . As shown in (Saxe et al., 2013a), as the TA learns according to (8), the singular vectors of its composite map W remain unchanged, while the singular values evolve as $s ^ { \\alpha } ( t ) \\stackrel { \\textstyle - } { = } s ( t , { \\hat { s } } ^ { \\alpha } )$ , where the learning curve function $s ( t , { \\hat { s } } )$ as well as its functional inverse $t ( s , { \\hat { s } } )$ is given by ", + "bbox": [ + 173, + 426, + 825, + 526 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7500cbc7044b60e71245e55285a419d9036853bbd5df18959fcf957b69514c1b.jpg", + "text": "$$\ns ( t , \\hat { s } ) = \\frac { \\hat { s } e ^ { 2 \\hat { s } t / \\tau } } { e ^ { 2 \\hat { s } t / \\tau } - 1 + \\hat { s } / \\epsilon } , \\qquad t ( s , \\hat { s } ) = \\frac { \\tau } { 2 \\hat { s } } \\ln \\frac { \\hat { s } / \\epsilon - 1 } { \\hat { s } / s - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 308, + 527, + 691, + 564 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here the function $s ( t , { \\hat { s } } )$ describes analytically how each training set singular value $\\hat { s }$ drives the dynamics of the corresponding TA singular value $s$ , and for notational simplicity, we have suppressed the dependence of $s ( t , { \\hat { s } } )$ on $\\tau$ and the initial condition $\\epsilon$ . As shown in Fig. 1A, for each $\\hat { s }$ , $s ( t , { \\hat { s } } )$ is a sigmoidal learning curve that undergoes aat which it rises from its small initial value of haat sition around time to its asymptotic v $\\begin{array} { r } { t / \\tau = \\frac { 1 } { 2 \\hat { s } } \\ln { \\left( \\hat { s } / \\epsilon - \\mathrm { 1 } \\right) } } \\end{array}$ $\\epsilon$ $t = 0$ $\\hat { s }$ $t / \\tau \\to \\infty$ Alternatively, we can plot $s ( t , { \\hat { s } } ) / { \\hat { s } }$ as a function of $\\hat { s }$ for different training times $t / \\tau$ , as in Fig. 1B. This shows that TA learning corresponds to a singular mode detection wave which progressively sweeps from large to small singular values. At any given training time $t$ , training data modes with singular values $\\hat { s } > t / \\tau$ have been learned, while those with singular values $\\hat { s } < t / \\tau$ have not. ", + "bbox": [ + 173, + 565, + 826, + 691 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "While the TA is more sophisticated than the random student, since it already knows the singular vectors of the training data before learning, we will see that the analytic solution for the TA learning dynamics provides a good approximation to the student learning dynamics, not only for the training error, as shown in (Saxe et al., 2013a), but also for the generalization error as shown below. ", + "bbox": [ + 173, + 698, + 825, + 755 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The results in this section assume a single hidden layer, but Saxe et al. (2013a) derived $t ( s , { \\hat { s } } )$ for networks of arbitrary depth and we apply our theory to some deeper networks. The general differential equation and derivations for deeper networks can be found in Appendix A. ", + "bbox": [ + 174, + 761, + 825, + 804 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 HOW THE TEACHER IS BURIED IN THE TRAINING DATA: A RANDOM MATRIX ANALYSIS ", + "text_level": 1, + "bbox": [ + 173, + 819, + 808, + 833 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the previous section, we reviewed an exact analytic solution for the composite map of a TA network, namely that its singular modes are related to those of the training data through the relation ", + "bbox": [ + 173, + 844, + 825, + 873 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2b6b38ab2445136253e3c2da9eb4b5601b0e0bf19d86f31d2d59f8f956174964.jpg", + "text": "$$\n\\begin{array} { r } { s _ { \\alpha } ( t ) = s ( t , \\hat { s } _ { \\alpha } ) , \\qquad { \\bf u } ^ { \\alpha } ( t ) = \\hat { \\bf u } ^ { \\alpha } , \\qquad { \\bf v } ^ { \\alpha } ( t ) = \\hat { \\bf v } ^ { \\alpha } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 321, + 876, + 676, + 892 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "However, computation of the generalization error through (5) then requires understanding how the teacher singular modes of $\\overline { { \\mathbf { W } } }$ are buried within the noisy training data singular modes of $\\pmb { \\Sigma } ^ { 3 1 }$ through the relation (3). Since the input matrix $\\hat { \\mathbf X }$ is orthonormal, $\\pmb { \\Sigma } ^ { 3 1 }$ is simply a perturbation of the low rank teacher $\\overline { { \\mathbf { W } } }$ by a high dimensional noise matrix $\\mathbf { Z }$ . The relation between the singular modes of a low rank matrix and its noise perturbed version has been studied extensively in Benaych-Georges $\\&$ Nadakuditi (2012), in the high dimensional limit we are working in, namely $\\overline { { N _ { 1 } } } , \\overline { { N _ { 3 } } } \\overline { { } } \\infty$ with the aspect ratio $\\mathcal { A } = \\overline { { N _ { 3 } } } / \\overline { { N _ { 1 } } } \\in \\overline { { ( 0 , 1 ] } }$ , and $\\overline { { N } } _ { 2 } \\sim O ( 1 )$ . ", + "bbox": [ + 173, + 895, + 828, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/74aa5b143963ba7d2cf4b03e7444309ed0f7fd59b33a147d86b619f88f63f9e7.jpg", + "image_caption": [ + "Figure 2: The teacher’s signal through the noise. Theoretical vs. empirical (a) histogram of singular values of noisy teacher $\\hat { s }$ . (b) $\\hat { s }$ as a function of $\\overline { { s } }$ . (c) alignment of noisy teacher and noiseless teacher singular vectors as a function of $\\overline { { s } }$ . ${ \\widetilde { N _ { 1 } } } = { \\overline { { N _ { 3 } } } } = 1 0 0 .$ ) " + ], + "image_footnote": [], + "bbox": [ + 181, + 102, + 802, + 209 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 266, + 825, + 340 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this limit, the top $\\overline { { N } } _ { 2 }$ singular values and vectors of $\\pmb { \\Sigma } ^ { 3 1 }$ converge to $\\hat { s } ( \\overline { { s } } _ { \\alpha } )$ , where the transfer function from a teacher singular value $\\overline { { s } }$ to a training data singular value $\\hat { s }$ is given by the function ", + "bbox": [ + 173, + 347, + 823, + 377 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2ba1154839d1485804a1e1f85264464ed778061a1065b612c52372f9c5c065a6.jpg", + "text": "$$\n{ \\hat { s } } ( { \\overline { { s } } } ) = { \\left\\{ \\begin{array} { l l } { ( { \\overline { { s } } } ) ^ { - 1 } { \\sqrt { ( 1 + { \\overline { { s } } } ^ { 2 } ) ( A + { \\overline { { s } } } ^ { 2 } ) } } } & { { \\mathrm { ~ i f ~ } } { \\overline { { s } } } > A ^ { 1 / 4 } } \\\\ { 1 + { \\sqrt { A } } } & { { \\mathrm { ~ o t h e r w i s e . } } } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 331, + 383, + 665, + 426 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The associated top $\\overline { { N } } _ { 2 }$ singular vectors of $\\pmb { \\Sigma } ^ { 3 1 }$ can also acquire a nontrivial overlap with the $\\overline { { N } } _ { 2 }$ modes of the teacher through the relation $\\left| \\hat { \\mathbf { u } } ^ { \\alpha } \\cdot \\overline { { \\mathbf { u } } } ^ { \\alpha } \\right| \\left| \\hat { \\mathbf { v } } ^ { \\alpha } \\cdot \\overline { { \\mathbf { v } } } ^ { \\alpha } \\right| = \\mathcal { O } ( \\overline { { s } } _ { \\alpha } )$ , where the singular vector overlap function is given by ", + "bbox": [ + 174, + 434, + 825, + 477 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d5c4b3b9ac8ae1dc6fcedacfb57f1842d2fd7cce16e70a28d6b8b5ae04de86ab.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { O } ( \\overline { { s } } ) = \\left\\{ \\begin{array} { l l } { \\left[ 1 - \\frac { \\mathcal { A } ( 1 + \\overline { { s } } ^ { 2 } ) } { \\overline { { s } } ^ { 2 } ( A + \\overline { { s } } ^ { 2 } ) } \\right] ^ { 1 / 2 } \\left[ 1 - \\frac { ( \\mathcal { A } + \\overline { { s } } ^ { 2 } ) } { \\overline { { s } } ^ { 2 } ( 1 + \\overline { { s } } ^ { 2 } ) } \\right] ^ { 1 / 2 } } & { \\mathrm { i f } \\overline { { s } } > \\mathcal { A } ^ { 1 / 4 } } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 290, + 483, + 704, + 534 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The rest of the $N _ { 3 } - { \\overline { { N _ { 2 } } } }$ singular vectors of $\\pmb { \\Sigma } ^ { 3 1 }$ are orthogonal to the top $\\overline { { N } } _ { 2 }$ ones, and their singular values are distributed according to the the Marchenko-Pastur (MP) distribution: ", + "bbox": [ + 171, + 541, + 825, + 570 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7cbae00fa09fe50ba7290dcc0e7ab93f7dd633fd65337439d8702ccd8624c824.jpg", + "text": "$$\nP ( \\hat { s } ) = \\left\\{ \\begin{array} { l l } { \\frac { \\sqrt { 4 A - ( \\hat { s } ^ { 2 } - ( 1 + A ) ) ^ { 2 } } } { \\pi A \\hat { s } } } & { \\hat { s } \\in [ 1 - \\sqrt { \\mathcal { A } } , 1 + \\sqrt { \\mathcal { A } } ] } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 316, + 577, + 679, + 619 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Overall, these equations describe a singular vector phase transition in the training data, as illustrated in Fig. 2BC. For example in the case of no teacher, the training data is simply noise and the singular values of $\\pmb { \\Sigma } ^ { 3 1 }$ are distributed as an MP sea spread between $1 \\pm { \\sqrt { A } }$ . When one adds a teacher, how each teacher singular mode is imprinted on the training data depends crucially on the teacher singular value $\\overline { { s } }$ , and the nature of this imprinting undergoes a phase transition at $\\overline { { s } } = \\mathcal { A } ^ { 1 / 4 }$ . For $\\overline { { s } } \\leq { \\mathcal { A } } ^ { 1 / 4 }$ , the teacher mode SNR is too low and this mode is not imprinted in the noisy training data; the associated√ training data singular value $\\hat { s }$ remains at the edge of the MP sea at $1 + { \\sqrt { A } }$ , and the overlap $\\mathcal { O } ( \\overline { { s } } )$ between training and teacher singular vectors remains zero. ", + "bbox": [ + 173, + 626, + 825, + 743 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "However, when $\\overline { { s } } > \\mathcal { A } ^ { 1 / 4 }$ , this teacher mode is imprinted in the training data; there is an associated training data singular value $\\hat { s }$ that pops out of the MP sea (Fig. 2AB). However, the training data singular value emerges at a position $\\hat { s } > \\overline { { s } }$ that is inflated by the noise, though the inflation effect decreases at larger $\\overline { { s } }$ , with the ratio $\\hat { s } / \\overline { { s } }$ approaching the unity line as $\\overline { { s } }$ becomes large (Fig. 2B). Similarly, the corresponding training data singular vectors acquire a non-trivial overlap with the teacher singular vectors when $\\overline { { s } } > \\mathcal { A } ^ { 1 / 4 }$ , and the alignment approaches unity as $\\overline { { s } }$ increases (Fig. 2C). ", + "bbox": [ + 174, + 750, + 825, + 837 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 PUTTING IT ALL TOGETHER: AN ANALYTIC THEORY OF GENERALIZATION DYNAMICS ", + "text_level": 1, + "bbox": [ + 171, + 853, + 805, + 868 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Based on an analytic understanding of how the singular mode structure $\\{ \\overline { { s } } ^ { \\alpha } , \\overline { { \\mathbf { u } } } ^ { \\alpha } , \\overline { { \\mathbf { v } } } ^ { \\alpha } \\}$ of the teacher $\\overline { { \\mathbf { W } } }$ is imprinted in the modes $\\{ \\hat { s } ^ { \\alpha } , \\hat { \\mathbf { u } } ^ { \\alpha } , \\hat { \\mathbf { v } } ^ { \\alpha } \\}$ of the training data covariance $\\pmb { \\Sigma } ^ { 3 1 }$ through (11), (12) and (13), and in turn how this training data singular structure drives the time evolving singular modes of a ", + "bbox": [ + 174, + 880, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/314c03e7d659a75d3bce2929de3fdf61f6a46155c699119ff46ac7743510ffe3.jpg", + "image_caption": [ + "Figure 3: Match between theory and experiment for rank 1 (row 1, a-d) and rank 3 (row 2, e-h) teachers with single-hidden-layer students: (a-b, e-f) log train and test error, respectively, showing very close match between theory and experiment for TA, and close match for the random student. (c,g) comparing TA and randomly initialized students minimum generalization errors, showing almost perfect match. (d,h) comparing TA and randomly initialized students optimal stopping times, showing small lag due to alignment. ( $N _ { 1 } = 1 0 0$ , $N _ { 2 } = 5 0$ , $N _ { 3 } = 5 0 .$ ) " + ], + "image_footnote": [], + "bbox": [ + 179, + 98, + 812, + 310 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "TA network $\\{ s ^ { \\alpha } ( t ) , \\hat { \\mathbf { u } } ^ { \\alpha } , \\hat { \\mathbf { v } } ^ { \\alpha } \\}$ of through (9), we can now derive analytic expressions for $\\varepsilon _ { \\mathrm { t r a i n } }$ and $\\varepsilon _ { \\mathrm { t e s t } }$ in (6) and (7), for a TA network. We will also show that these learning curves closely approximate those of a random student with time-evolving singular vectors $\\{ \\mathbf { u } ^ { \\alpha } ( t ) , \\mathbf { \\bar { v } } ^ { \\alpha } ( t ) \\}$ , and match on several key aspects. First, inserting the TA dynamics in (10) into $\\varepsilon _ { \\mathrm { t r a i n } }$ in (6), we obtain ", + "bbox": [ + 173, + 434, + 825, + 492 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/5aab040792c1a8dce6a9075a4cc0440ba793d936170b926d5ad03ab0fe901b4f.jpg", + "text": "$$\n\\mathfrak { c } _ { \\mathrm { r e a i n } } ( t ) = \\left[ \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 3 } } \\hat { s } _ { \\alpha } ^ { 2 } \\right] ^ { - 1 } \\left[ ( N _ { 3 } - N _ { 2 } ) \\langle \\hat { s } ^ { 2 } \\rangle _ { \\mathcal { R } _ { o u t } } + ( N _ { 2 } - \\overline { { N } } _ { 2 } ) \\langle ( s ( \\hat { s } , t ) - \\hat { s } ) ^ { 2 } \\rangle _ { \\mathcal { R } _ { i n } } + \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 2 } } \\left[ s _ { \\alpha } ( t ) - \\hat { s } _ { \\alpha } \\right] ^ { 2 } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 181, + 512, + 821, + 564 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here, $s _ { \\alpha } ( t ) = s ( \\hat { s } _ { \\alpha } , t )$ as defined in (9) are the TA singular values, and $\\hat { s } _ { \\alpha } = \\hat { s } ( \\overline { { s } } _ { \\alpha } )$ as defined in (11) are the training data singular values associated with the teacher singular values $\\overline { { s } } _ { \\alpha }$ . Also $\\langle \\cdot \\rangle _ { \\mathcal { R } }$ denotes an average with respect to the MP distribution in (13) over a region $\\mathcal { R }$ . Two distinct regions contribute to training error. First $\\mathcal { R } _ { i n }$ contains those top $N _ { 2 } - \\overline { { N } } _ { 2 }$ training data singular values that do not correspond to the $\\overline { { N } } _ { 2 }$ singular values of the teacher but will be learned by a rank $N _ { 2 }$ student. Second, $\\mathcal { R } _ { o u t }$ corresponds to the remaining $N _ { 3 } - N _ { 2 }$ lowest training data singular values√ that cannot be learned by a rank √ $N _ { 2 }$ student. In terms of the MP distribution, $\\mathcal { R } _ { o u t } = [ 1 - \\sqrt { \\mathcal { A } } , f ]$ and $\\mathcal { R } _ { i n } = [ f , 1 + \\sqrt { \\mathcal { A } } ]$ , where $f$ is the point at which the MP density has $1 - N _ { 2 } / N _ { 3 }$ of its mass to the left and $N _ { 2 } / N _ { 3 }$ of its mass to the right. In the simple case of a full rank student, $f = 1 - \\sqrt { \\mathcal { A } }$ , and one need only integrate over $\\mathcal { R } _ { i n }$ which is the entire range. Equation (14) for $\\varepsilon _ { \\mathrm { t r a i n } }$ makes it manifest that it will go to zero for a full rank student as its singular values approach those of the training data. ", + "bbox": [ + 173, + 574, + 825, + 734 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Of course the test error can behave very differently. Inserting the TA training dynamics in (10) into $\\varepsilon _ { \\mathrm { t e s t } }$ in (7), and using (11), (12) and (13) to relate training data to the teacher, we find ", + "bbox": [ + 173, + 741, + 823, + 770 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0ec73d593caa6ef7fd97cce903ec4844dce356c4c4f139eb7e32b3acc481a204.jpg", + "text": "$$\n\\varepsilon _ { \\mathrm { t e x t } } ( t ) = \\left[ \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 2 } } \\overline { { s } } _ { \\alpha } ^ { 2 } \\right] ^ { - 1 } \\left[ ( N _ { 2 } - \\overline { { N } } _ { 2 } ) \\langle s ( \\hat { s } , t ) ^ { 2 } \\rangle _ { \\mathcal { R } _ { i n } } + \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 2 } } \\left[ ( s _ { \\alpha } ( t ) - \\overline { { s } } _ { \\alpha } ) ^ { 2 } + 2 s _ { \\alpha } ( t ) \\overline { { s } } _ { \\alpha } ( 1 - \\mathcal { O } ( \\overline { { s } } _ { \\alpha } ) ) \\right] \\right]\n$$", + "text_format": "latex", + "bbox": [ + 181, + 789, + 821, + 842 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Together (14) and (15) constitute a complete theory of generalization dynamics in terms of the structure of the data distribution (i.e. the teacher rank $\\overline { { N } } _ { 2 }$ , teacher SNRs $\\left\\{ \\overline { { s } } _ { \\alpha } \\right\\}$ , and the teacher aspect ratio $\\mathcal { A } = \\overline { { N } } _ { 3 } / \\overline { { N } } _ { 1 } )$ , the architectural complexity of the student (i.e. its rank $N _ { 2 }$ , its number of layers $N _ { l }$ , and the norm $\\epsilon$ of its initialization), and the training time $t$ . They yield considerable insight into the dynamics of good generalization early in learning and overfitting later, as we show below. ", + "bbox": [ + 173, + 852, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/be9716a9b1de318c5906d81cbfaaab3bf19f9183b7b3ce2f88ef2d6bf3ba7e80.jpg", + "image_caption": [ + "Figure 4: Our theory applies to deeper networks: match between theory and simulation for rank 1 (row 1, a-d) and rank 3 (row 2, e-h) teachers with $n _ { l } = 5$ students: (a-b, e-f) log train and test error, respectively, showing very close match between theory and experiment for TA. (c,g) comparing TA and randomly initialized students minimum generalization errors, showing almost perfect match. (d,h) comparing TA and randomly initialized students optimal stopping times, showing large lag due to slower alignment in deeper networks. $N _ { 1 } = 1 0 0$ , $N _ { 2 } = 5 0$ , $N _ { 3 } = 5 0 .$ .) " + ], + "image_footnote": [], + "bbox": [ + 179, + 99, + 812, + 310 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.4 NUMERICAL TESTS OF THE THEORY OF NEURAL NETWORK GENERALIZATION DYNAMICS ", + "text_level": 1, + "bbox": [ + 173, + 428, + 823, + 440 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Fig. 3 demonstrates an excellent match between the theory and simulations for the TA, and a close match for random students, for single-hidden-layer students and various teacher ranks $\\overline { { N } } _ { 2 }$ . Intuitively, as time $t$ proceeds, learning corresponds to singular mode detection wave sweeping from large to small training data singular values (i.e. the wave in Fig. 1B sweeps across the training data spectrum in Fig 2A). Initially, strong singular values associated with large SNR teacher modes are learned and both $\\varepsilon _ { \\mathrm { t r a i n } }$ and $\\varepsilon _ { \\mathrm { t e s t } }$ drop. Fig. 3A-D are for a rank 1 teacher, and so in Fig 3AB we see a single sharp drop early on, if the teacher SNR is sufficiently high. By contrast, with a rank 3 teacher in Fig. 3E-H, there are several early drops as the three modes are picked up. However, as time progresses, the singular mode detection wave penetrates the MP sea, and the student picks up noise structure in the data, so $\\varepsilon _ { \\mathrm { t r a i n } }$ drops but $\\varepsilon _ { \\mathrm { t e s t } }$ rises, indicating the onset of overfitting. ", + "bbox": [ + 173, + 454, + 825, + 593 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The main difference between the random student and TA learning curves is that the random student learning is slightly delayed relative to the TA, especially late in training. This is understandable because the TA already knows the singular vectors of the training data, while the random student must learn them. Nevertheless, two of the most important aspects of learning, namely the optimal early stopping time $t _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } } \\equiv \\mathrm { a r g m i n } _ { t } \\varepsilon _ { \\mathrm { t e s t } } ( t )$ and the minimal test error achieved at this time $\\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } } \\equiv$ $\\mathrm { m i n } _ { t } \\varepsilon _ { \\mathrm { t e s t } } ( t )$ , match well between TA and random student, as shown in Fig. 3CD. At low teacher SNRs, the student takes a little longer to learn than the TA, but their optimal test errors match. ", + "bbox": [ + 174, + 599, + 825, + 702 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our theory can also be easily extended to describe the learning dynamics deeper networks. Saxe et al. (2013a) derived $t ( s , { \\hat { s } } )$ for networks of arbitrary depth, so we only need to adjust this factor in our formulas, see App. A for details. In Fig. 4 we show that again there is an excellent match between TA networks and theory for student networks with $N _ { l } = 5$ layers (i.e. 3 hidden layers). Randomly-initialized networks show a much longer alignment lag for deeper networks (see App. B for details), but the curves are qualitatively similar and optimal stopping errors match. We also demonstrate extensions of our theory to different numbers of training examples (App. G). ", + "bbox": [ + 174, + 708, + 825, + 805 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Importantly, many of the phenomena we observe in linear networks are qualitatively replicated in nonlinear networks (Fig. 5), suggesting that our theory may help guide understanding of the nonlinear case. In particular, features such as stage-like initial learning, followed by a plateau if SNR is high, and finally followed by overfitting, are replicated. However, there are some discrepancies, in particular nonlinear networks (especially deeper ones) begin overfitting earlier than linear networks. This is likely because a mode in a non-linear network can be co-opted by an orthogonal mode, while in a linear network it cannot. Thus noise modes are able to “stow away” on the strong signal modes once they are learned. However, overall learning patterns are similar, and we show below that many interesting phenomena in nonlinear networks are understandable in the linear case, such as the (non-)effects of overparameterization, the dynamics of memorization, and the benefits of transfer. ", + "bbox": [ + 174, + 813, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 132 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/4c96fe2cc1b69b79f16e01bb40a77f5c5b9805452e0fd502c4194097a289ca0c.jpg", + "image_caption": [ + "Figure 5: Train (first row, A-D) and test (second row, E-H) error for nonlinear networks (leaky relu at all hidden layers) with one hidden layer (first two columns) or three hidden layers (last two columns) trained on the tasks above, with a rank 1 teacher (first and third columns) or a rank 3 teacher (second and fourth columns). Note that many of the qualitative phenomena observed in linear networks, such as stage-like improvement in the errors, followed by a plateau, followed by overfitting, also appear in nonlinear networks. Compare the first column to Fig. 3AB, the second column to Fig. 3EF, the third to Fig. 4AB, and the fourth to Fig. 4EF. ( $N _ { 1 } = 1 0 0$ , $N _ { 2 } = 5 0$ , $N _ { 3 } = 5 0 .$ ) " + ], + "image_footnote": [], + "bbox": [ + 178, + 136, + 785, + 367 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": ".5 RANDOMIZED DATA VS. REAL DATA: A LEARNING TIME PUZZLE ", + "text_level": 1, + "bbox": [ + 189, + 503, + 651, + 517 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "An intriguing observation that resurrected the generalization puzzle in deep learning was the observation by Zhang et al. (2016) that deep networks can memorize data with the labels randomly permuted. However, as Arpit et al. (2017) pointed out, the learning dynamics of training error for randomized labels can be slower than than for structured data. This phenomenon also arises in deep linear networks, and our theory yields an analytic explanation for why. We randomize data by choosing orthonormal inputs $\\hat { \\mathbf { x } } ^ { \\mu }$ as in the structured case, but we choose the outputs ${ \\hat { \\mathbf { y } } } ^ { \\mu }$ to be i.i.d. Gaussian with zero mean and the same diagonal variance as the structured training data generated by the teacher. For structured data generated by a low rank teacher with singular values $\\overline { { s } } _ { \\alpha }$ , the diagonal output variance is given by $\\begin{array} { r } { \\sigma _ { r } ^ { 2 } = \\frac { 1 } { N _ { 3 } } \\left[ \\sum _ { i = \\alpha } ^ { \\overline { { N } } _ { 2 } } \\bar { s } _ { \\alpha } ^ { 2 } \\right] + \\frac { 1 } { \\overline { { N } } _ { 1 } } \\sigma _ { z } ^ { 2 } } \\end{array}$ , where $\\sigma _ { z }$ is the noise variance, as before. Since there is no relation between input and output, $\\pmb { \\Sigma } ^ { 3 1 }$ is now distributed as a MP distribution whose support is $[ ( \\sigma _ { r } ( 1 - \\sqrt { \\mathcal { A } } ) , \\sigma _ { r } ( 1 + \\sqrt { \\mathcal { A } } ) ]$ . Thus randomization essentially destroys the outlier signal singular values in $\\pmb { \\Sigma } ^ { 3 1 }$ reflecting the teacher, and distributes them across all randomized data modes, yielding this stretched MP distribution (compare 6A top and bottom). However, even on this stretched MP distribution, the right edge will be much smaller than the signal singular values, since the signal variance will be diluted by spreading it out over many more modes in the randomized data. Thus the randomized data will lead to slower initial training error drops relative to the structured data (Fig. 6B) since the singular mode detection wave encounters the first signal singular values in structured data earlier than it encounters the edge of the stretched MP sea in randomized data. ", + "bbox": [ + 173, + 529, + 826, + 709 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/214421209393a1f729207d74055cfc8b2447742fb08e8821164fae3a79d95a61.jpg", + "image_caption": [ + "Figure 6: Learning randomized data: Comparing (a) singular value distributions and (b) learning curves for data with a signal vs. random data that preserves basic statistics (mean, variance). Randomizing the data dilutes the signal singular values, spreading their variance out over many modes, hence randomly labelled data is learned more slowly. $N _ { 1 } = 1 0 0$ , $N _ { 2 } = 5 0$ , $N _ { 3 } = 5 0 .$ .) " + ], + "image_footnote": [], + "bbox": [ + 176, + 731, + 771, + 871 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/6f50ba8e93b7f0f0f5390f316c743f888d62c7d9566d1790b18925c63d2e3904.jpg", + "image_caption": [ + "Figure 7: Transfer setting– If two different tasks are combined, how well students of the combined teacher perform on each task depends on the alignment and SNRs of the teachers. " + ], + "image_footnote": [], + "bbox": [ + 194, + 101, + 807, + 244 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/0a5b9a9dbd5641d64b838b1df6081e9aedcd5fcf53c8a71c1f83e3bbc6d1722c.jpg", + "image_caption": [ + "Figure 8: Transfer benefit $\\mathcal { T } ^ { A B } ( \\overline { { s } } _ { A } , \\overline { { s } } _ { B } , q )$ plotted at different values of ${ \\overline { { s } } } _ { A }$ . (a) $\\overline { { s } } _ { A } = 0 . 8 4 = \\sqrt [ 4 ] { A }$ . Although this task is impossible to learn on its own, with support from another aligned task, especially one with high SNR, learning can occur. (b) ${ \\overline { { s } } } _ { A } = 3$ . Tasks with modest signals will face interference from poorly aligned tasks, but benefits from well aligned tasks. These effects are amplified by SNR. (c) $\\overline { { s } } _ { A } = 1 0 0$ . Tasks with very strong signals will show little effect from other tasks (note y-axis scales), but any impact will be negative unless the tasks are very well aligned. $N _ { 1 } = 1 0 0$ $\\dot { \\overline { { { N } } } } _ { 2 } ^ { A } = \\overline { { { N } } } _ { 2 } ^ { B } = 1 .$ , $N _ { 2 } = 5 0$ , $N _ { 3 } = 5 0 .$ ) " + ], + "image_footnote": [], + "bbox": [ + 173, + 289, + 803, + 424 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 561, + 826, + 645 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3.6 OUT-PERFORMING OPTIMAL EARLY STOPPING THROUGH A NON-GRADIENT ALGORITHM ", + "text_level": 1, + "bbox": [ + 174, + 666, + 823, + 680 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For the case of a rank 1 teacher, it is straightforward to derive a good analytic approximation to the important quantities $\\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }$ and topt $t _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }$ . We assume the teacher SNR is beyond the phase transition point so its unique singular value $\\overline { { s } } _ { 1 } > A ^ { 1 / 4 }$ , yielding a separation between the training data singular value $\\hat { s _ { 1 } }$ in (11) and the edge of the MP sea. In this scenario, optimal early stopping will occur at a time before the detection wave in Fig. 1B penetrates the MP sea, so to minimize test error, we can neglect the first term in (15). Then optimizing the second term yields the optimal student singular value $s _ { 1 } = \\overline { { s } } _ { 1 } \\mathcal { O } ( \\overline { { s } } _ { 1 } )$ . Inserting this value into (15) yields εoptgradient = 1 − O(s1)2, and inserting it into (9) yields $t _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }$ . Thus the optimal generalization error with a rank 1 teacher is very simply related to the alignment of the top training data singular vectors with the teacher singular vectors, and it decreases as this alignment increases. In App. E, we show this match in the rank 1 case. ", + "bbox": [ + 173, + 691, + 825, + 844 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "With higher rank teachers, teacher modes with differen $\\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }$ and For must negotiate a more complex trade-off betweene, as the singular mode detection wave passes the top training data singular value, $s _ { 1 } ( t ) \\to \\hat { s } _ { 1 }$ which is greater than the optimal $s _ { 1 } = \\overline { { s } } _ { 1 } \\mathcal { O } ( \\overline { { s } } _ { 1 } )$ for mode 1. Thus as learning progresses, the student overfits on the first mode but learns lower modes. However, this neural generalization dynamics suggests a superior non-gradient training algorithm that simply optimally sets each $s _ { \\alpha }$ to $\\overline { { s } } _ { \\alpha } \\mathcal { O } ( \\overline { { s } } _ { \\alpha } )$ in (15), yielding an optimal generalization error: ", + "bbox": [ + 173, + 851, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 102, + 722, + 119 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/570f4b4fad0704cfefbb961c062c1f4efb04e2f7fe004d72fb17f3060a73f853.jpg", + "text": "$$\n\\varepsilon _ { \\mathrm { n o n - g r a d i e n t } } ^ { \\mathrm { o p t } } = \\left[ \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 2 } } \\overline { { s } } _ { \\alpha } ^ { 2 } \\right] ^ { - 1 } \\left[ \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 2 } } \\overline { { s } } _ { \\alpha } ^ { 2 } ( 1 - \\mathcal { O } ( \\overline { { s } } _ { \\alpha } ) ^ { 2 } ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 123, + 669, + 176 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Standard gradient descent learning cannot achieve this low generalization error because it cannot independently adjust all student singular values. A simple algorithm that achieves $\\varepsilon _ { \\mathrm { n o n - g r a d i e n t } } ^ { \\mathrm { o p t } }$ is as follows. From the training data covariance $\\pmb { \\Sigma } ^ { 3 1 }$ , extract the top singular values $\\hat { s } _ { \\alpha }$ that pop-out of the MP sea, use the functional inverse of (11) to compute $\\overline { { s } } _ { a } ( \\widehat { s } _ { \\alpha } )$ , use (12) to compute the optimal $s _ { \\alpha }$ , and then construct a matrix $\\mathbf { W }$ with the same top singular vectors as $\\pmb { \\Sigma } ^ { 3 1 }$ , but with the outlier singular values shrunk from $\\hat { s } _ { \\alpha }$ to $s _ { \\alpha }$ and the rest set to zero. This non-gradient singular value shrinkage algorithm provably outperforms neural network training with $\\varepsilon _ { \\mathrm { n o n - g r a d i e n t } } ^ { \\mathrm { o p t } ^ { - } } \\le \\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } } .$ . ", + "bbox": [ + 173, + 180, + 826, + 286 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4 A THEORY FOR THE TRANSFER OF KNOWLEDGE ACROSS MULTIPLE TASKS ", + "text_level": 1, + "bbox": [ + 176, + 304, + 820, + 320 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Consider two tasks $A$ and $B$ , described by $\\overline { { N } } _ { 3 }$ by $\\overline { { N } } _ { 1 }$ teacher maps $\\overline { { \\mathbf { W } } } ^ { A }$ and $\\overline { { \\mathbf { W } } } ^ { B }$ , of ranks $\\overline { { N } } _ { 2 } ^ { A }$ and $\\overline { { N } } _ { 2 } ^ { B }$ , respectively. Now two student networks can learn from the two teacher networks separately, each achieving optimal early stopping test errors composite teacher (and student) that concatenate $\\varepsilon _ { A } ^ { \\mathrm { { o p t } } }$ and hid $\\varepsilon _ { B } ^ { \\mathrm { { o p t } } }$ . Alternatively, one could construct a and output units, but shares the same $\\overline { { N } } _ { 3 }$ ut units (Fig. 7). The composite student and teacher each have two heads, one for each tasneurons per head. Optimal early stopping on each head of the student yields test errors opt opt $\\varepsilon _ { A B } ^ { \\mathrm { o p t } }$ and . We define the transfer benefit that task B confers on task A to be $\\mathcal { T } ^ { A B } \\equiv \\varepsilon _ { A } ^ { \\mathrm { o p t } } - \\varepsilon _ { A B } ^ { \\mathrm { o p t } }$ A postive (negative) transfer benefit implies learning tasks A and B simultaneously yields a lower (higher) optimal test error on task A compared to just learning task A alone. ", + "bbox": [ + 173, + 330, + 826, + 469 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A foundational question is how the transfer benefit $\\scriptstyle { \\mathcal { T } } ^ { A B }$ depends on the two tasks defined by the teachers $\\overline { { \\mathbf { W } } } ^ { A }$ and $\\overline { { \\mathbf { W } } } ^ { B }$ . To answer this, consider the SVDs of each teacher alone: $\\begin{array} { r } { \\overline { { \\mathbf { W } } } ^ { A } = } \\end{array}$ $\\overline { { \\mathbf { U } } } ^ { A } \\overline { { \\mathbf { S } } } ^ { A } \\overline { { \\mathbf { V } } } ^ { A ^ { T } }$ and $\\overline { { \\mathbf { W } } } ^ { B } = \\overline { { \\mathbf { U } } } ^ { B } \\overline { { \\mathbf { S } } } ^ { B } \\overline { { \\mathbf { V } } } ^ { B ^ { T } }$ . From the above, we know that $\\varepsilon _ { A } ^ { \\mathrm { { o p t } } }$ depends on $\\overline { { \\mathbf { W } } } ^ { A }$ only through $\\overline { { \\mathbf { S } } } ^ { A }$ . In App. D we show that the transfer benefit depends on both $\\overline { { \\mathbf { W } } } ^ { A }$ and $\\overline { { \\mathbf { W } } } ^ { B }$ only through $\\overline { { \\mathbf { S } } } ^ { A } , \\overline { { \\mathbf { S } } } ^ { B }$ , and the $\\overline { { N } } _ { 2 } ^ { A }$ by $\\overline { { N } } _ { 2 } ^ { B }$ similarity matrix $\\mathbf { \\overline { { Q } } } = \\mathbf { \\overline { { V } } } ^ { A ^ { T } } \\mathbf { \\overline { { V } } } ^ { B }$ . If we think of the columns of each $\\overline { { \\mathbf { V } } }$ as spanning a low dimensional feature space in $\\overline { { N } } _ { 1 }$ dimensional input space that is important for each task, then $\\overline { { \\mathbf { Q } } }$ reflects the input feature subspace similarity matrix. Interestingly, the transfer benefit is independent of output singular vectors $\\bar { \\mathbf { U } } ^ { A }$ and $\\overline { { \\mathbf { U } } } ^ { B }$ . What matters for knowledge transfer in this setting are the relevant input features, not how you must respond to them. ", + "bbox": [ + 173, + 474, + 826, + 631 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We describe the transfer benefit for the simple case of two rank one teachers. Then $\\overline { { \\mathbf { S } } } ^ { A } , \\overline { { \\mathbf { S } } } ^ { B }$ , and $\\overline { { \\mathbf { Q } } }$ are simply scalars $s _ { A } , s _ { B }$ and $q$ , and we explore the function $\\mathcal { T } ^ { A B } ( \\overline { { s } } _ { A } , \\overline { { s } } _ { B } , q )$ in Fig. 5ABC, which reveals several interesting features. First, knowledge can be transferred from a high SNR task to a low SNR task (Fig. 5A) and the degree of transfer increases with task alignment $q$ . This can make it possible to capture signals from task $A$ which would otherwise sink into the MP sea by learning jointly with a related task, even if the tasks are only weakly aligned (Fig. 5A). However, if task $A$ already has a high SNR, task $B$ must be very well aligned to it for transfer to be beneficial – otherwise there will be interference. The degree of alignment required increases as the task $A$ SNR increases, but the quantity of benefit or interference decreases correspondingly (Fig. 5BC). In Appendix D we explain why our theory predicts these results. Furthermore, in Appendix F we demonstrate these phenomena are qualitatively recapitulated in nonlinear networks, which suggests that our theory may give insight into how to choose auxiliary tasks. ", + "bbox": [ + 173, + 640, + 825, + 809 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 829, + 310, + 844 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In summary, our analytic theory of generalization dynamics in deep linear networks reveals that many puzzling aspects of generalization in deep learning already arise in the simple linear setting, where the puzzles can be understood analytically. In particular, deep linear networks learn more important structure in data first, leading to generalization errors that depend on task structure much more than network size. Our theory explains why deep linear networks learn randomized data more slowly than structured data, and provides a non-gradient based learning method that out-performs gradient descent learning in the linear case. Finally, we provide an analytic theory of how knowledge is transferred from one task to another, demonstrating that the degree of alignment of input features important for each task, but not how one must respond to these features, is critical for facilitating knowledge transfer. We think these analytic results provide useful insight into the similar generalization and transfer phenomena observed in the nonlinear case. Among other things, we hope our work will motivate and enable: (1) the search for tighter upper bounds on generalization error that take into account task structure; (2) the design of non gradient based training algorithms that outperform gradient-based learning; and (3) the theory-driven selection of auxiliary tasks that maximize knowledge transfer. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 250, + 285, + 265 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Madhu S. Advani and Andrew M. Saxe. High-dimensional dynamics of generalization error in neural networks. arXiv, pp. 1–32, 2017. ", + "bbox": [ + 174, + 272, + 823, + 301 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Sanjeev Arora, Rong Ge, Behnam Neyshabur, and Yi Zhang. Stronger generalization bounds for deep nets via a compression approach. arXiv preprint, pp. 1–39, 2018. URL http://arxiv. org/abs/1802.05296. 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", + "bbox": [ + 174, + 611, + 826, + 654 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A LEARNING DYNAMICS FOR DEEPER NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 680, + 602, + 695 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In the main text, we described the dynamics of how a single-hidden-layer network converges toward the training data singular modes $\\{ \\hat { s } ^ { \\alpha } , \\hat { \\mathbf { u } } ^ { \\alpha } , \\hat { \\mathbf { v } } ^ { \\alpha } \\}$ , which were originally derived in Saxe et al. (2013a). There it was also proven that for a network with $N _ { l }$ layers (i.e. $N _ { l } - 2$ hidden layers), the strength of the mode obeys the differential equation: ", + "bbox": [ + 173, + 710, + 826, + 767 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/e8382140836b5cfe34671dd944c538f5a90a3735e89a8a2bf073af8301a4e25f.jpg", + "text": "$$\n\\tau { \\frac { d } { d t } } u = ( N _ { l } - 1 ) u ^ { 2 - 2 / ( N _ { l } - 1 ) } ( s - u )\n$$", + "text_format": "latex", + "bbox": [ + 370, + 773, + 625, + 804 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "This equation is separable and can be integrated for any integer number of layers. In particular, we consider the case of 5 layers (3 hidden), in which case: ", + "bbox": [ + 173, + 809, + 823, + 839 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/73166119140c9bab6af2d2ea3b4f65abbb44c69b0e083ef0870b03452afa4a5e.jpg", + "text": "$$\nt ( s , \\hat { s } ) = \\frac { \\tau } { 2 } \\left[ \\frac { \\operatorname { t a n h } ^ { - 1 } \\left( \\sqrt { \\frac { u } { \\hat { s } } } \\right) } { \\hat { s } ^ { 3 / 2 } } - \\frac { 1 } { \\hat { s } \\sqrt { u } } \\right] _ { \\epsilon } ^ { s }\n$$", + "text_format": "latex", + "bbox": [ + 367, + 844, + 629, + 888 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "This expression cannot be analytically inverted to find $s ( t , { \\hat { s } } )$ , so we numerically invert it where necessary. ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B ALIGNMENT LAG IN RANDOMLY INITIALIZED NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 103, + 686, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "As noted in the main text, the randomly-initialized networks behave quite similarly to the TA networks, except that the randomly-initialized networks show a lag due to the time it takes for the network’s modes to align with the data modes. In fig. 9 we explore this lag by plotting the alignment of the modes and the increase in the singular value for several randomly initialized networks. ", + "bbox": [ + 174, + 133, + 825, + 189 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Notice that stronger modes align more quickly. Furthermore, the mode alignment is relatively independent – whether the teacher is rank 1 or rank 3, the alignment of the modes is similar for the mode of singular value 2. Most importantly, note how the deeper networks show substantially slower mode alignment, with alignment not completed until around when the singular value increases. This explains why deeper networks show a larger lag between randomly-initialized and TA networks – the alignment process is much slower for deeper networks. ", + "bbox": [ + 174, + 195, + 825, + 280 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C TRAIN AND TEST ERRORS AFTER A PROJECTION ", + "text_level": 1, + "bbox": [ + 173, + 301, + 607, + 318 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In the case of transfer learning, or more generally when we want to evaluate a network’s loss on a subset of its outputs, we need to use a slight generalization of the train and test error formulas given in the main text. Suppose we are interested in the train and test errors after applying a projection operator $\\mathbf { P }$ : ", + "bbox": [ + 176, + 332, + 823, + 388 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/824d5e77da2fc9bf0002ff555d48fd7e0608505786d086f2d942b570d2f3b38b.jpg", + "text": "$$\n\\varepsilon _ { \\mathrm { t r a i n } } \\equiv \\frac { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\mathbf { P } \\mathbf { W } \\hat { \\mathbf { x } } ^ { \\mu } - \\mathbf { P } \\hat { \\mathbf { y } } ^ { \\mu } | | _ { 2 } ^ { 2 } } { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\mathbf { P } \\hat { \\mathbf { y } } ^ { \\mu } | | _ { 2 } ^ { 2 } } , \\varepsilon _ { \\mathrm { t e s t } } \\equiv \\frac { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\mathbf { P } \\mathbf { W } \\overline { { \\mathbf { x } } } ^ { \\mu } - \\mathbf { P } \\overline { { \\mathbf { y } } } ^ { \\mu } | | _ { 2 } ^ { 2 } } { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\mathbf { P } \\overline { { \\mathbf { y } } } ^ { \\mu } | | _ { 2 } ^ { 2 } } ,\n$$", + "text_format": "latex", + "bbox": [ + 232, + 395, + 764, + 444 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "respectively. As in the main text, we can rexpress these as: ", + "bbox": [ + 173, + 449, + 558, + 464 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/b3dfe306ce0e4b0416590bb0eb5e48c67ca0e50453c4ad1d27c97a97d4efeb9e.jpg", + "text": "$$\n\\varepsilon _ { \\mathrm { t r a i n } } = \\frac { \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } { \\mathbf { W } } - 2 \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\Sigma ^ { 3 1 } + \\mathrm { T r } { \\Sigma ^ { 3 1 } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\Sigma ^ { 3 1 } } { \\mathrm { T r } \\Sigma ^ { 3 1 } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\Sigma ^ { 3 1 } } ,\n$$", + "text_format": "latex", + "bbox": [ + 276, + 469, + 720, + 510 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/a7f3c63e6dd22c620bfb08e215774d513e2fb6bf53b51956396165b08996231d.jpg", + "text": "$$\n\\varepsilon _ { \\mathrm { t e s t } } = \\frac { \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } { \\mathbf { W } } - 2 \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\overline { { { \\mathbf { W } } } } + \\mathrm { T r } \\overline { { { \\mathbf { W } } } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\overline { { { \\mathbf { W } } } } } { \\mathrm { T r } \\overline { { { \\mathbf { W } } } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\overline { { { \\mathbf { W } } } } } .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 515, + 705, + 558 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Using the cyclic property of the trace, we can modify these to get: ", + "bbox": [ + 173, + 559, + 606, + 574 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/c7c12ca256b899740c8be0c793209c9a8228e253de168ef8f6fd3701fafdbc93.jpg", + "text": "$$\n\\varepsilon _ { \\mathrm { t r a i n } } = \\frac { \\mathrm { T r } { \\bf P } { \\bf W } { \\bf W } ^ { T } { \\bf P } ^ { T } - 2 \\mathrm { T r } { \\bf P } \\Sigma ^ { 3 1 } { \\bf W } ^ { T } { \\bf P } ^ { T } + \\mathrm { T r } { \\bf P } \\Sigma ^ { 3 1 } \\Sigma ^ { 3 1 ^ { T } } { \\bf P } ^ { T } } { \\mathrm { T r } { \\bf P } \\Sigma ^ { 3 1 } \\Sigma ^ { 3 1 ^ { T } } { \\bf P } ^ { T } } ,\n$$", + "text_format": "latex", + "bbox": [ + 277, + 579, + 720, + 618 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/b809fb7c45678781dfedbcdedaa34d7e51911d701c9fda075999b1f71d247a8e.jpg", + "text": "$$\n\\varepsilon _ { \\mathrm { t e s t } } = \\frac { \\mathrm { T r } \\mathbf { P } \\mathbf { W } \\mathbf { W } ^ { T } \\mathbf { P } ^ { T } - 2 \\mathrm { T r } \\mathbf { P } \\overline { { \\mathbf { W } } } \\mathbf { W } ^ { T } \\mathbf { P } ^ { T } + \\mathrm { T r } \\mathbf { P } \\overline { { \\mathbf { W } } } \\mathbf { W } ^ { T } \\mathbf { P } ^ { T } } { \\mathrm { T r } \\mathbf { P } \\overline { { \\mathbf { W } } } \\mathbf { W } ^ { T } \\mathbf { P } ^ { T } } .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 625, + 705, + 667 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "As before, we express these in terms of the student, training data and teacher SVDs, $\\mathbf { W } = \\mathbf { U } \\mathbf { S } \\mathbf { V } ^ { T }$ $\\begin{array} { r } { \\pmb { \\Sigma } ^ { 3 1 } = \\hat { \\mathbf { U } } \\hat { \\mathbf { S } } \\hat { \\mathbf { V } } ^ { T } } \\end{array}$ , and $\\overline { { \\mathbf { W } } } = \\overline { { \\mathbf { U } } } \\overline { { \\mathbf { S } } } \\overline { { \\mathbf { V } } } ^ { T }$ respectively. Specifically, ", + "bbox": [ + 173, + 670, + 823, + 704 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/e2a2b7673c35307ab430f1855d1ff64e839930bd9e9c11e872e6526a7ca5afdb.jpg", + "text": "$$\n\\begin{array} { r } { \\operatorname { t r a i n } = \\Big [ \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 3 } } \\hat { s } _ { \\beta } ^ { 2 } \\vert \\vert { \\mathbf { P } } \\hat { \\mathbf { a } } ^ { \\alpha } \\vert \\vert _ { 2 } ^ { 2 } \\Big ] ^ { - 1 } \\Big [ \\sum _ { \\alpha = 1 } ^ { N _ { 2 } } s _ { \\alpha } ^ { 2 } \\vert \\vert { \\mathbf { P } } \\mathbf { u } ^ { \\alpha } \\vert \\vert _ { 2 } ^ { 2 } + \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 3 } } \\hat { s } _ { \\beta } ^ { 2 } \\vert \\vert { \\mathbf { P } } \\hat { \\mathbf { a } } ^ { \\alpha } \\vert \\vert _ { 2 } ^ { 2 } - 2 \\sum _ { \\alpha = 1 } ^ { N _ { 2 } } \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 3 } } s _ { \\alpha } \\hat { s } _ { \\beta } \\big ( { \\mathbf { P } } \\mathbf { u } ^ { \\alpha } \\cdot { \\mathbf { P } } \\hat { \\mathbf { u } } ^ { \\beta } \\big ) \\big ( { \\mathbf { v } } ^ { \\alpha } \\cdot { \\mathbf { \\hat { v } } } ^ { \\beta } \\big ) \\Big ] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 710, + 823, + 732 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/4b62348655d38b06bf41391656fd181444d4be54d4d2a458526a43428d448f9c.jpg", + "text": "$$\n\\begin{array} { r } { \\varepsilon _ { \\mathrm { t e s t } } = \\Big [ \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 2 } } \\bar { s } _ { \\beta } ^ { 2 } \\big | \\big | { \\bf P } \\bar { \\bf u } ^ { \\alpha } \\big | \\big | _ { 2 } ^ { 2 } \\Big ] ^ { - 1 } \\Big [ \\sum _ { \\alpha = 1 } ^ { N _ { 2 } } s _ { \\alpha } ^ { 2 } \\big | \\big | { \\bf P } { \\bf u } ^ { \\alpha } \\big | \\big | _ { 2 } ^ { 2 } + \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 2 } } \\bar { s } _ { \\beta } ^ { 2 } \\big | \\big | { \\bf P } \\bar { \\bf u } ^ { \\alpha } \\big | \\big | _ { 2 } ^ { 2 } - 2 \\sum _ { \\alpha = 1 } ^ { N _ { 2 } } \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 2 } } s _ { \\alpha } \\bar { s } _ { \\beta } \\big ( { \\bf P } { \\bf u } ^ { \\alpha } \\cdot { \\bf P } { \\bf { \\bar { u } } } ^ { \\beta } \\big ) \\big ( { \\bf v } ^ { \\alpha } \\cdot { \\bf \\bar { v } } ^ { \\beta } \\big ) \\Big ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 747, + 823, + 768 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "D TRANSFER LEARNING DERIVATIONS & DETAILS ", + "text_level": 1, + "bbox": [ + 173, + 800, + 607, + 818 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Thm 1 (Transfer theorem) The transfer benefit $\\scriptstyle { \\mathcal { T } } ^ { A B }$ : ", + "bbox": [ + 173, + 830, + 550, + 848 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• Is unaffected by the $\\overline { { \\mathbf { U } } } ^ { A }$ and $\\overline { { \\mathbf { U } } } ^ { B }$ . \n• Is completely determined by only $\\sigma _ { z } ^ { 2 }$ $, \\overline { { \\mathbf { S } } } ^ { A } , \\overline { { \\mathbf { S } } } ^ { B }$ , and the $\\overline { { N } } _ { 2 } ^ { A }$ by $\\overline { { N } } _ { 2 } ^ { B }$ similarity matrix $\\overline { { \\mathbf { Q } } } =$ $\\overline { { \\mathbf { V } } } ^ { A T } \\overline { { \\mathbf { V } } } ^ { B }$ . ", + "bbox": [ + 214, + 857, + 825, + 922 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/f7d69e78559f4561bf6dde026948060bf03ae8cb1ead40ce9131543b8df1331b.jpg", + "image_caption": [ + "Figure 9: Alignment of randomly-initialized network modes to data modes and growth of singular values, plotted for 1 hidden layer (first two rows, a-d) and 3 hidden layers (last two rows, e-h), and for a rank 1 teacher (first and third rows, a & e), or a rank 3 teacher (second and fourth rows, b-d & f-h). The columns are the different modes, with respective singular values of 6, 4, and 2. $\\sigma _ { z }$ was set to 1. The deeper networks show substantially slower mode alignment, with alignment not completed until around when the singular value increases. " + ], + "image_footnote": [], + "bbox": [ + 174, + 165, + 808, + 758 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof: We define ", + "bbox": [ + 173, + 103, + 290, + 117 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f14751854781d9ed6b6cd2d58f59abe99d16da60103bf5deb51a18fe492fe695.jpg", + "text": "$$\n\\begin{array} { l l } { { { \\overline { { { \\cal { \\bf { U } } } } } } ^ { A B } = \\begin{array} { c } { { { \\overline { { { \\cal { N } } } } } _ { 3 } } } \\\\ { { { \\overline { { { \\cal { N } } } } } _ { 3 } } } \\\\ { { { \\overline { { { \\cal { N } } } } } _ { 3 } } } \\end{array} } } & { { { \\overline { { { \\cal { N } } } } } _ { 3 } ^ { A } \\begin{array} { c } { { { \\overline { { { \\cal { N } } } } } _ { 2 } ^ { B } } } \\\\ { { \\left| \\begin{array} { c } { { { \\bf { \\overline { { { \\bf { U } } } } } } ^ { A } } } \\end{array} \\right| } } } \\\\ { { { \\overline { { { \\bf { U } } } } } ^ { B } } } \\end{array} } & { { { \\overline { { { \\bf { S } } } } } ^ { A B } = \\begin{array} { c } { { { \\overline { { { \\cal { N } } } } } _ { 2 } ^ { A } } } \\\\ { { { \\overline { { { \\cal { N } } } } } _ { 2 } ^ { B } } } \\end{array} } } \\\\ { { { \\overline { { { \\cal { N } } } } } _ { 2 } ^ { A B } = \\begin{array} { c } { { { \\overline { { { \\cal { N } } } } } _ { 3 } ^ { A } } } \\end{array} } } & { { { \\overline { { { \\bf { U } } } } } _ { 2 } ^ { B } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 271, + 122, + 725, + 243 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/a18b9f94544a5a673db1bcfe1b5857da9b17361d350e76db494cf0796df70657.jpg", + "text": "$$\n\\mathbf { \\overline { { W } } } ^ { A + B } = [ \\frac { \\mathbf { \\overline { { U } } } ^ { A } } { \\mathbf { 0 } } | \\begin{array} { c } { \\mathbf { 0 } } \\\\ { \\mathbf { \\overline { { U } } } ^ { B } } \\end{array} ] [ \\frac { \\mathbf { \\overline { { S } } } ^ { A } } { \\mathbf { 0 } } | \\begin{array} { c } { \\mathbf { 0 } } \\\\ { \\mathbf { \\overline { { S } } } ^ { B } } \\end{array} ] [ \\frac { \\mathbf { \\overline { { V } } } ^ { A ^ { T } } } { \\mathbf { \\overline { { V } } } ^ { B ^ { T } } } ] \n$$", + "text_format": "latex", + "bbox": [ + 320, + 260, + 674, + 310 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Because of the 0 blocks in UAB, the vectors in blocks corresponding to task $A$ and task $B$ are completely orthogonal, so UAB remains orthonormal. Thus the relationship between the $\\overline { { \\mathbf { U } } } ^ { A }$ and $\\overline { { \\mathbf { U } } } ^ { A }$ is irrelevant to the transfer. (In our simulations we use arbitrary orthonormal matrices for $\\overline { { \\mathbf { U } } } ^ { A }$ and $\\overline { { \\mathbf { U } } } ^ { B }$ .) Therefore the transfer effects will be entirely driven by the relationship between the matrices $\\overline { { \\mathbf { V } } } ^ { A }$ and $\\overline { { \\mathbf { V } } } ^ { B }$ and the singular values. ", + "bbox": [ + 171, + 313, + 826, + 405 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We define $\\overline { { N } } _ { 2 } ^ { A }$ by $\\overline { { N } } _ { 2 } ^ { B }$ similarity matrix $\\mathbf { \\overline { { Q } } } = \\mathbf { \\overline { { V } } } ^ { A ^ { T } } \\mathbf { \\overline { { V } } } ^ { B }$ . If we think of the columns of each $\\overline { { \\mathbf { V } } }$ as spanning a low dimensional feature space in $\\overline { { N } } _ { 1 }$ dimensional input space that is important for each task, then $\\overline { { \\mathbf { Q } } }$ reflects the input feature subspace similarity matrix. We can now calculate the singular values of WA+ B. First, note that the input singular modes of WA+B a re eigenvectors of $\\overline { { \\mathbf { W } } } ^ { A + B ^ { T } } \\overline { { \\mathbf { W } } } ^ { A + B }$ B, and the associated singular values are square roots of the eigenvalues of WA+B. Now ", + "bbox": [ + 173, + 410, + 826, + 511 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/43f3e333d102893ea85bcb339755c45e779e6ca5d1bd65d76ea9b096dcd55ec1.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathbf { W } } } ^ { A + B ^ { T } } \\overline { { \\mathbf { W } } } ^ { A + B } = \\overline { { \\mathbf { V } } } ^ { A B } \\overline { { \\mathbf { S } } } ^ { A B } \\overline { { \\mathbf { U } } } ^ { A B ^ { T } } \\overline { { \\mathbf { U } } } ^ { A B } \\overline { { \\mathbf { S } } } ^ { A B } \\overline { { \\mathbf { V } } } ^ { A B ^ { T } } = \\overline { { \\mathbf { V } } } ^ { A B } \\overline { { \\mathbf { S } } } ^ { A B ^ { 2 } } \\overline { { \\mathbf { V } } } ^ { A B ^ { T } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 245, + 510, + 751, + 534 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Now if $\\vec { c }$ is an eigenvector of this matrix: ", + "bbox": [ + 176, + 537, + 442, + 553 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/d7fbbfee188d62cbef18a339da60f5fecd653384c9ddb9f8966295ffd74cec4d.jpg", + "text": "$$\n\\overline { { { \\bf V } } } ^ { A B } \\overline { { { \\bf S } } } ^ { A B ^ { 2 } } \\overline { { { \\bf V } } } ^ { A B ^ { T } } \\vec { c } = \\lambda \\vec { c }\n$$", + "text_format": "latex", + "bbox": [ + 413, + 559, + 586, + 582 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "This implies that ", + "bbox": [ + 173, + 590, + 285, + 604 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/50b46fc1cb03b08ff6212cd1f7f0a846462b9c8bc1bced40cebff3590a8bc23f.jpg", + "text": "$$\n\\mathbf { \\overline { { { V } } } } ^ { A B ^ { T } } \\mathbf { \\overline { { { V } } } } ^ { A B } \\mathbf { \\overline { { { S } } } } ^ { A B ^ { 2 } } \\mathbf { \\overline { { { V } } } } ^ { A B ^ { T } } { \\vec { c } } = \\lambda \\mathbf { \\overline { { { V } } } } ^ { A B ^ { T } } { \\vec { c } }\n$$", + "text_format": "latex", + "bbox": [ + 367, + 602, + 632, + 625 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Hence eigenvalues of VABSAB2V T are also eigenvalues of VABT V $\\overline { { { \\bf V } } } ^ { A B ^ { T } } \\overline { { { \\bf V } } } ^ { A B } \\overline { { { \\bf S } } } ^ { A B ^ { 2 } }$ , with the mapping between the eigenvectors given by $\\overline { { \\mathbf { V } } } ^ { A B }$ . Furthermore, this mapping must be a bijection for eigenvectors with non-zero eigenvalues, since the matrices have the same rank (the rank of $\\overline { { \\mathbf { V } } } ^ { A B }$ ). To see this, note that SAB 2 i s full rank. From this, it is clear that ", + "bbox": [ + 173, + 627, + 826, + 705 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/0bc80a4ea89838e575d1e39ac60c1781ae8dad24763c055da29210f52e7d5ca3.jpg", + "text": "$$\n\\mathrm { r a n k } \\overline { { { \\mathbf { V } } } } ^ { A B ^ { T } } \\overline { { { \\mathbf { V } } } } ^ { A B } \\overline { { { \\mathbf { S } } } } ^ { A B ^ { 2 } } = \\mathrm { r a n k } \\overline { { { \\mathbf { V } } } } ^ { A B ^ { T } } \\overline { { { \\mathbf { V } } } } ^ { A B } = \\mathrm { r a n k } \\overline { { { \\mathbf { V } } } } ^ { A B } .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 710, + 686, + 733 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Furthermore, SAB2 is positive definite, so ", + "bbox": [ + 176, + 739, + 450, + 761 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/6024782c7c6e7e4621afe41cd86c3d33045b9dfe4a3339b1e81d109d9f9f61ae.jpg", + "text": "$$\n\\operatorname { r a n k } { \\overline { { \\mathbf { V } } } } ^ { A B } { \\overline { { \\mathbf { S } } } } ^ { A B ^ { 2 } } { \\overline { { \\mathbf { V } } } } ^ { A B ^ { T } } = \\operatorname { r a n k } { \\overline { { \\mathbf { V } } } } ^ { A B } .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 766, + 620, + 790 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Now that we know the eigenvectors of these matrices are in bijection, note that: ", + "bbox": [ + 173, + 804, + 694, + 820 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/be28d71d674191e649188adf52e87ce586d0dcb0b12b5f5659e8081cf5581141.jpg", + "text": "$$\n\\begin{array}{c} \\overline { { \\mathbf { V } } } ^ { A B ^ { T } } \\overline { { \\mathbf { V } } } ^ { A B } \\overline { { \\mathbf { S } } } ^ { A B ^ { 2 } } = [ \\frac { \\overline { { \\mathbf { V } } } ^ { A ^ { T } } } { \\overline { { \\mathbf { V } } } ^ { B ^ { T } } } ] [ \\begin{array} { c } { \\overline { { \\mathbf { V } } } ^ { A } \\begin{array} { c } { \\overline { { \\mathbf { V } } } ^ { B } } \\end{array} } \\end{array} ] [ \\frac { \\overline { { \\mathbf { S } } } ^ { A ^ { 2 } } } { \\mathbf { 0 } } \\end{array} ] \\frac { \\mathbf { 0 } } { \\overline { { \\mathbf { S } } } ^ { B ^ { 2 } } } ] = [ \\begin{array} { c c } { \\mathbf { I } } & { \\mathbf { Q } } \\\\ { \\mathbf { Q } ^ { \\mathbf { T } } } & { \\mathbf { I } } \\end{array} ] [ \\frac { \\overline { { \\mathbf { S } } } ^ { A ^ { 2 } } } { \\mathbf { 0 } } \\begin{array} { c } { \\mathbf { [ \\begin{array} { c } { \\mathbf { 0 } } \\\\ { \\overline { { \\mathbf { S } } } ^ { B ^ { 2 } } } \\end{array} ] } \\overline { { \\mathbf { S } } } ^ { B ^ { 2 } } } \\end{array} ]\n$$", + "text_format": "latex", + "bbox": [ + 181, + 824, + 823, + 876 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Because the output modes don’t matter (as noted above), the alignment between the eigenvectors of VABSAB2V ABT and $\\overline { { \\mathbf { V } } } ^ { A }$ , weighted by their respective eigenvalues, gives the transfer benefit. ", + "bbox": [ + 173, + 887, + 825, + 925 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For any given tasks, the transfer benefit can be calculated using our theory. However, in certain special cases, we can give exact answers. For example, in the rank one case with equal singular values between the tasks $\\mathit { \\Pi } _ { \\overline { { { s } } } _ { A } } ^ { \\prime } = \\mathit { \\Pi } _ { \\overline { { { s } } } _ { B } } ^ { \\prime } = \\mathit { \\Pi } _ { \\overline { { { s } } } } ^ { \\prime }$ ), the matrix ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/6aae4f1a123bce7b66db026b59e4796fc3c20b5647fa05e85504bf5559ce2853.jpg", + "text": "$$\n\\left[ \\begin{array} { l l } { \\mathbf { I } } & { \\mathbf { Q } } \\\\ { \\mathbf { Q } ^ { \\mathbf { T } } } & { \\mathbf { I } } \\end{array} \\right] \\left[ \\frac { \\mathbf { \\overline { { S } } } ^ { A ^ { 2 } } \\mathbf { \\Phi } } { \\mathbf { 0 } } \\right] \\mathbf { \\overline { { S } } } ^ { B ^ { 2 } } \\mathbf { \\Phi } ]\n$$", + "text_format": "latex", + "bbox": [ + 393, + 156, + 602, + 208 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "reduces to ", + "bbox": [ + 173, + 218, + 243, + 233 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/6a2c53846f1ca48cec4d80ce2c1220c9e646aad6bf64eda364d75f6fa394d57f.jpg", + "text": "$$\n\\left[ \\begin{array} { l l } { 1 } & { q } \\\\ { q } & { 1 } \\end{array} \\right] \\overline { { s } } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 454, + 236, + 544, + 271 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "with eigenvalues $s \\sqrt { 1 \\pm q }$ and eigenvectors ", + "bbox": [ + 176, + 282, + 460, + 297 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/2f0d2041763343936b837f370cfb8ba4aca59bf616c69f3d2049dadc2ca9c356.jpg", + "text": "$$\n\\left[ \\begin{array} { c c } { 1 } & { 1 } \\\\ { 1 } & { - 1 } \\end{array} \\right]\n$$", + "text_format": "latex", + "bbox": [ + 455, + 308, + 540, + 344 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Corresponding to the shared structure between the tasks and the differences between them. We note that the sign of the alignment $q$ is irrelevant as a special case of the fact (noted above) that any orthogonal transformation on the output modes does not affect transfer. ", + "bbox": [ + 174, + 354, + 825, + 397 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "D.1 MISALIGNMENT AND INTERFERENCE", + "text_level": 1, + "bbox": [ + 176, + 420, + 477, + 434 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Why is there interference between tasks which are not well aligned? In the rank one case, we are effectively changing the (input) singular dimensions of $\\overline { { \\mathbf { Y } } } _ { A }$ from $\\overline { { \\mathbf { V } } } ^ { A }$ to $\\overline { { \\mathbf { V } } } ^ { A B }$ . The two singular modes of VAB correspond to the shared structure between the tasks (weighted by the relative signal strengths), and the differences between them, respectively. Although we may be improving our estimates of the shared mode if $q > 0$ (by increasing its singular value relative to ${ \\overline { { s } } } _ { A }$ ), we are actually decreasing its alignment with $\\overline { { \\mathbf { V } } } ^ { A }$ unless $q = 1$ . This misalignment is captured by the second mode of $\\overline { { \\mathbf { V } } } ^ { A B }$ , but the increase in the singular value of the first mode must come at the cost of a decrease in the singular value of the second mode. See Fig. 10 for a conceptual illustration of this. This means that the multi-task setting allows the distinctions between the tasks to sink towards the sea of noise, while pulling out the common structure. In other words, transferring knowledge from one task always comes at the cost of ignoring differences between the tasks. Furthermore, incorporating a task $B$ allows its noise to seep into the task $A$ signal. Together, these two effects help to explain why transfer can be sometimes beneficial but sometimes detrimental. ", + "bbox": [ + 173, + 446, + 825, + 647 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/2ff9eeb17e34a1c5c018c10339262e9b2643717551b0503ce8da75485a108ec8.jpg", + "image_caption": [ + "Figure 10: Conceptual cartoon of how $\\mathcal { T } ^ { A B }$ , the transfer benefit (or cost) arises from alignment between the task’s input modes. " + ], + "image_footnote": [], + "bbox": [ + 379, + 671, + 619, + 765 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "E NON-GRADIENT TRAINING ALGORITHM ", + "text_level": 1, + "bbox": [ + 174, + 847, + 539, + 863 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In Fig. 11 we show the match between the error achieved by training the student by gradient descent and the optimal stopping error predicted by the non-gradient shrinkage algorithm in the case of a rank-1 teacher. ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/8fe699450a90e0a1da920afb212efc64c0b70cce8fa419ef6569d47e281026cc.jpg", + "image_caption": [ + "Figure 11: Match between optimal stopping error prediction from non-gradient training algorithm and empirical optimal stopping error for a rank-1 teacher. " + ], + "image_footnote": [], + "bbox": [ + 369, + 103, + 624, + 250 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "F TRANSFER RESULTS GENERALIZE TO NON-LINEAR NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 323, + 725, + 338 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Since most deep learning practitioners do not train linear networks, it is important that our theoretical insights generalize beyond this simple case. In this section we show that the transfer patterns qualitatively generalize to non-linear networks. ", + "bbox": [ + 176, + 354, + 823, + 397 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Here, we show results from teacher networks with $\\overline { { N } } _ { 1 } = 1 0 0 \\overline { { N } } _ { 3 } = 5 0$ , $\\overline { { N } } _ { 2 } = 4$ (thus the task is higher rank) and leaky relu non-linearities at the hidden and output layers. We train a student with leaky relu units and $N _ { 2 } = N _ { 3 }$ to solve this task. Results qualitatively look quite similar to those in Fig 5. of the main text for rank one linear teachers, see below. Thus our insights into transfer may help to understand multi-task benefits in more complicated architectures. ", + "bbox": [ + 173, + 402, + 825, + 474 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/3aa21b341a46731773f7366f14f4c4df08b4e4ea25985a881fa5ca23d11fc270.jpg", + "image_caption": [ + "Figure 12: Transfer benefit $\\mathcal { T } ^ { A B } ( \\overline { { s } } _ { A } , \\overline { { s } } _ { B } , q )$ for non-linear teachers and students, plotted at different values of ${ \\overline { { s } } } _ { A }$ . (a) $\\overline { { s } } _ { A } = 0 . 8 4 = \\sqrt [ 4 ] { A }$ . With support from another aligned task, especially one with moderately higher SNR, performance on a low SNR task will improve. (b) ${ \\overline { { s } } } _ { A } = 3$ . Tasks with modest signals will face interference from poorly aligned tasks, but benefits from well aligned tasks. These effects are amplified by SNR. (c) $\\overline { { s } } _ { A } = 1 0 0$ . Tasks with very strong signals will show little effect from other tasks (note y-axis scale), but any impact will be negative unless the tasks are very well aligned. " + ], + "image_footnote": [], + "bbox": [ + 176, + 489, + 810, + 613 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "G VARYING THE NUMBER OF TRAINING EXAMPLES", + "text_level": 1, + "bbox": [ + 173, + 761, + 614, + 776 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In the main text, we focused on the test error dynamics in the case in which the number of examples equalled the number of inputs. Here we show how the formula for test error curves is modified as the number of training examples $P$ is varied. For simplicity, when $P \\neq N _ { 1 }$ , we focus on the case of a full rank student with aspect ratio $A = 1$ (so that $N _ { 1 } = N _ { 2 } = N _ { 3 }$ ). The more general case of lower rank students with non-unity aspect ratios can be easily found from this case, but with some additional bookkeeping. ", + "bbox": [ + 173, + 792, + 826, + 877 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "As before, we assume the teacher generates noisy outputs from a set of $P$ inputs: ", + "bbox": [ + 173, + 883, + 702, + 898 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/017aac481874cee2e8ced8eecd7adc08c4e109af6d0b198562927d7c19ea5c3d.jpg", + "text": "$$\n\\hat { \\mathbf { y } } ^ { \\mu } = \\overline { { \\mathbf { W } } } \\hat { \\mathbf { x } } ^ { \\mu } + \\mathbf { z } ^ { \\mu } \\qquad \\mathrm { f o r } \\quad \\mu = 1 , \\ldots , \\mathbf { P } .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 906, + 640, + 924 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/cdfbc5d1756ec978413726647b732f78c395fed4934518082d5a82318c0a50f6.jpg", + "image_caption": [ + "Figure 13: The effects of varying the number of training examples $P$ . (a) Test error for a student learning from a rank-1 teacher with an SNR of 3, with different numbers of inputs. (b,c) Minimum generalization error plotted against $\\sqrt { P / N _ { 1 } }$ and SNR · $\\sqrt { P / N _ { 1 } }$ , respectively, at different SNRs. When $P \\geq N _ { 1 }$ , the minimum generalization error is simply determined by $\\mathrm { S N R } \\sqrt { P / N _ { 1 } }$ , so all curves converge to a single asymptotic line in (c) as $P$ increases. When $P < N _ { 1 }$ , however, the curves for different SNRs separate because the projection and noise effects depend on initial SNR. (d) Optimal stopping error for gaussian vs. orthogonal inputs, showing a strong correlation. Thus our use of orthogonal inputs in the theory also yields insight into the more general case of approximately unit norm Gaussian inputs. (For all panels $N _ { 1 } = N _ { 2 } = N _ { 3 } = 1 0 0$ , ${ \\overline { { N } } } _ { 2 } ^ { - } = 1 .$ ) " + ], + "image_footnote": [], + "bbox": [ + 178, + 99, + 803, + 354 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "This training set yields important second-order training statistics that will guide student learning: ", + "bbox": [ + 171, + 522, + 805, + 537 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/209c20bdf91e361008c626888198448737b108bc9fa9e2b412c9e72f9d64346e.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { \\Sigma } ^ { 1 1 } \\equiv \\hat { \\mathbf { X } } \\hat { \\mathbf { X } } ^ { T } \\qquad \\pmb { \\Sigma } ^ { 3 1 } \\equiv \\hat { \\mathbf { Y } } \\hat { \\mathbf { X } } ^ { T } = \\overline { { \\mathbf { W } } } \\hat { \\mathbf { X } } \\hat { \\mathbf { X } } ^ { T } + \\mathbf { Z } \\hat { \\mathbf { X } } ^ { T } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 540, + 674, + 558 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Here $\\hat { \\mathbf { X } } , \\hat { \\mathbf { Y } }$ , and $\\mathbf { Z }$ are each $\\overline { { N } } _ { 1 }$ by $P$ , $\\overline { { N } } _ { 3 }$ by $P$ , and $\\overline { { N } } _ { 3 }$ by $P$ matrices respectively, whose $\\mu ^ { \\mathrm { i } }$ ’th columns are $\\hat { \\mathbf { x } } ^ { \\mu } , \\hat { \\mathbf { y } } ^ { \\mu }$ , and $\\hat { \\mathbf { z } } ^ { \\mu }$ , respectively. $\\dot { \\mathbf { \\Sigma } } ^ { 1 \\mathrm { 1 } }$ is an $\\overline { { N } } _ { 1 }$ by $\\overline { { N } } _ { 1 }$ input correlation matrix, and $\\pmb { \\Sigma } ^ { 3 1 }$ is an $\\overline { { N } } _ { 3 }$ by $\\overline { { N } } _ { 1 }$ the input-output correlation matrix. We choose the matrix elements $z _ { i } ^ { \\mu }$ of the noise matrix $\\mathbf { Z }$ to be drawn iid from a Gaussian with zero mean and variance $\\sigma _ { z } ^ { 2 } / \\overline { { N } } _ { 1 }$ . The noise scaling is chosen so the singular values of the teacher $\\overline { { \\mathbf { W } } }$ and the noise $\\mathbf { Z }$ are both $O ( 1 )$ , leading to non-trivial generalization effects. Furthermore, we chose training inputs to be close to unit-norm, and make the input covariance matrix $\\pmb { \\Sigma } ^ { 1 1 }$ as white as possible (whitening is a common pre-processing step for inputs). When $P > \\overline { { N } } _ { 1 }$ , this can be done by choosing the rows of $\\hat { \\bf X }$ to be orthonormal and then scaling up by $\\sqrt { P / \\overline { { N } } _ { 1 } }$ , so the columns are approximately unit norm. Then $\\Sigma ^ { 1 1 } = P / \\overline { { N } } _ { 1 } \\mathbf { I }$ is proportional to the identity. On the otherhand, if $P < \\overline { { N } } _ { 1 }$ , we choose the columns of $\\hat { \\bf X }$ to be orthonormal, so that $\\pmb { \\Sigma } ^ { 1 1 } = \\pmb { \\mathcal { P } } ^ { | | }$ , where $\\mathcal { P } ^ { | | }$ is a projection operator onto the $P$ dimensional column space of $\\hat { \\mathbf X }$ spanned by the input examples. Both these choices are intended to approximate the situation in which the columns of $\\hat { \\bf X }$ are chosen to be iid unit-norm vectors. Finally, as generalization performance will depend on the ratio of teacher singular values to the noise variance parameter $\\sigma _ { z } ^ { 2 }$ we simply set $\\sigma _ { z } = 1$ as in the main text. Thus, given the unit-norm inputs, we can think of the teacher singular values as signal to noise ratios (SNRs). We now examine how the dynamics of the test error evolves as we vary the number of training examples $P$ . We split our analyses into two distinct regimes: (1) the oversampled regime in which the data density $\\bar { \\mathcal { D } } \\equiv P / N _ { 1 } > \\mathrm { ~ \\bar { ~ } { ~ 1 ~ } ~ }$ , and (2) the undersampled regime in which $\\mathcal { D } < 1$ . ", + "bbox": [ + 173, + 561, + 826, + 852 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "G.1 THE OVERSAMPLED REGIME ", + "text_level": 1, + "bbox": [ + 176, + 867, + 415, + 882 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The oversampled regime $\\mathcal { D } > 1 \\vert$ ) is relatively simple. First $\\pmb { \\Sigma } ^ { 1 1 }$ is scaled up by a factor of $\\mathcal { D }$ . And in the input-output covariance matrix, ${ \\boldsymbol { \\Sigma } } ^ { 3 1 } = \\overline { { \\mathbf { W } } } \\hat { \\mathbf { X } } \\hat { \\mathbf { X } } ^ { T } + \\mathbf { Z } \\hat { \\mathbf { X } } ^ { T }$ , the signal component, $\\overline { { \\mathbf { W } } } \\hat { \\mathbf { X } } \\hat { \\mathbf { X } } ^ { T }$ is scaled up by a factor of $\\mathcal { D }$ while the noise component √ $\\mathbf { Z } \\hat { \\mathbf { X } } ^ { T }$ has the same singular value spectrum as the $\\mathcal { D } = 1$ case, up to an overall scaling by √ $\\sqrt { \\mathcal { D } }$ (since the rows of $\\hat { \\mathbf X }$ are orthogonal and all its√ singular values are equal to $\\sqrt { \\mathcal { D } }$ ). This leads to an increase in the effective SNR by a factor of $\\sqrt { \\mathcal { D } }$ . Thus overall, the test error curves for the case of $\\mathcal { D } > 1$ can be simply obtained from the theory of the test error curves for $\\mathcal { D } = 1$ through two modifications: (1) a boost in the SNR for the $\\mathcal { D } = 1$ case by a multiplicative factor of $\\sqrt { \\mathcal { D } }$ , and (2) and an overall speed up in the learning time by a multiplicative factor of $\\mathcal { D }$ . ", + "bbox": [ + 176, + 892, + 823, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 207 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "G.2 THE UNDERSAMPLED REGIME ", + "text_level": 1, + "bbox": [ + 176, + 223, + 426, + 238 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "For the undersampled regime $\\mathcal { D } < 1 \\dot { }$ ), we must account for the fact that the $P$ training inputs do not span the full $N _ { 1 }$ dimensional space of all inputs. Thus the projection operator $\\mathcal { P } ^ { | | }$ onto the $P$ dimensional column space of $\\hat { \\mathbf X }$ plays a crucial role. Indeed the input-correlation $\\pmb { \\Sigma } ^ { 1 1 } = \\pmb { \\mathcal { P } } ^ { | | }$ . And ${ \\boldsymbol { \\Sigma } } ^ { 3 1 } = \\overline { { \\mathbf { W } } } \\mathbf { \\mathcal { P } } ^ { | | } + \\mathbf { Z } \\hat { \\mathbf { X } } ^ { T }$ . This implies that the learning dynamics only transforms the composite student map W from the $P$ dimensional subspace spanned by the inputs to the $N _ { 3 }$ dimensional output space. In contrast, the student map from the $N _ { 1 } - P$ dimensional subspace orthogonal to the image of $\\mathcal { P } ^ { | | }$ remains frozen. Tracing through the equations of the main paper and accounting for the projection operator $\\mathcal { P } ^ { | | }$ , we find the effective aspect ratio for this undersampled learning problem (when $N _ { 3 } = N _ { 2 } = N _ { 1 } ;$ is no longer $\\mathcal { A } = N _ { 3 } / \\bar { N _ { 1 } }$ but rather $\\mathcal { D } = P / N _ { 1 }$ . Furthermore, in the limit $\\overline { { N } } _ { 3 } , \\overline { { N } } _ { 1 } \\infty$ while $\\overline { { N } } _ { 2 }$ remains √ $O ( 1 )$ , the singular values of the signal component $\\overline { { \\mathbf { W } } } \\mathcal { P } ^ { | | }$ of $\\pmb { \\Sigma } ^ { 3 1 }$ are attenuated by a factor of $\\sqrt { \\mathcal { D } }$ , making the associated singular vectors more susceptible to noise. Again tracing through the equations of the main paper, with all of these modifications, we find the final formula for test error curves in the undersampled measurement regime: ", + "bbox": [ + 173, + 248, + 825, + 445 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/8f6ceadd27457f5652cdaa6ec4f561c6d4443cbcd30ad5996998202abd0ca431.jpg", + "text": "$$\n\\varepsilon _ { \\mathsf { t e s t } } ( t ) = { \\frac { \\left[ ( N _ { 3 } - P ) \\epsilon ^ { 2 } + ( P - { \\overline { { N } } } _ { 2 } ) \\langle s ( { \\hat { s } } , t ) ^ { 2 } \\rangle + \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 2 } } \\left[ ( s _ { \\alpha } ( t ) - { \\overline { { s } } } _ { \\alpha } ) ^ { 2 } + 2 s _ { \\alpha } ( t ) { \\overline { { s } } } _ { \\alpha } ( 1 - { \\mathcal { O } } ( { \\sqrt { \\overline { { D } } } } { \\overline { { s } } } _ { \\alpha } ) ) \\right] \\right] } { \\left[ \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 2 } } { \\overline { { s } } } _ { \\alpha } ^ { 2 } \\right] } }\n$$", + "text_format": "latex", + "bbox": [ + 222, + 449, + 776, + 489 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "This equation has several modifications compared to the case $P = N _ { 1 }$ in (15). First the term in the numerator involving $N _ { 3 } - P$ reflects generalization error due to the $N _ { 3 } - P$ dimensional frozen subspace, and the initial weight variance $\\epsilon ^ { 2 }$ contributes to this generalization error. The second term in the numerator involves all the $P - { \\overline { { N } } } _ { 2 }$ training modes which cannot be correlated with the teacher, and the average $\\langle \\cdot \\rangle$ is over a Marcenko-Pasteur distribution of singular values (see (13)) except with the aspect ratio $\\mathcal { A }$ replaced by $\\mathcal { D }$ . The third term accounts for learned correlations between the student and teacher. It involves the transformation from teacher singular values $\\overline { { s } }$ to training data singular values $\\hat { s }$ through the formula (11) except with the aspect ratio replacement √ $A \\mathcal { D }$ , and the effective teacher singular value attenuation $\\overline { { s } } \\sqrt { D } \\overline { { s } }$ . Similarly, the computation of the singular vector overlap is done through (12) also with the replacements $A \\mathcal { D }$ and $\\overline { { s } } \\sqrt { D } \\overline { { s } }$ . ", + "bbox": [ + 173, + 491, + 825, + 635 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "G.3 COMPARISON OF THEORY AND EXPERIMENT FOR UNDER AND OVER SAMPLED MEASUREMENT REGIMES ", + "text_level": 1, + "bbox": [ + 178, + 651, + 756, + 678 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In Fig. 13, we show an excellent match between our theory and empirical simulations for varying values of $P$ , both in the oversampled and undersampled measurement regimes. There are a number of interesting features to note. First, although the minimum generalization error improves monotonically with $P$ , the asymptotic $t \\to \\infty$ ) generalization error does not, because of a frozen subspace (Advani & Saxe, 2017) of the modes that are not overfit when $P < N _ { 1 }$ , because the training data rank is $\\le P$ . Second, when $P \\geq N _ { 1 }$ , the minimum generalization error is simply determined by $\\mathbf { S N R } \\sqrt { P / N _ { 1 } }$ , so all curves converge to a single asymptotic line as $P$ increases. When $P < N _ { 1 }$ , however, the curves for different SNRs separate because the projection and noise effects depend on initial SNR. Finally, in Fig. 13D we show that approximately unit norm i.i.d. gaussian inputs yield similar results to the orthogonalized data matrices we employed in the theory, although the gaussian inputs do result in slightly higher optimal stopping error. ", + "bbox": [ + 173, + 689, + 826, + 845 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "H LESS THAN FULL RANK STUDENTS ", + "text_level": 1, + "bbox": [ + 176, + 866, + 498, + 881 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Although we generally assumed students were full rank in the main text to simplify the calculations, our theory remains exact for TA networks of any rank. Furthermore, as shown in Fig. 14, the TA and random networks again show very similar optimal stopping generalization error, but with the optimal stopping time of the random networks lagging behind that of the TA networks. Furthermore, this lag increases as the rank of the random network decreases (because a low rank network will have less initial projection onto the random modes, there is is more alignment to be done). However, reducing the student rank does not change the optimal stopping error (as long as it is still greater than the teacher rank). ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 826, + 188 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/6f1091544ee918b4ed7b6456f495e4dd5c10e618ea9cc7217b0d7c4d68a60fb8.jpg", + "image_caption": [ + "(c) Optimal generalization error vs. optimal(d) Optimal generalization error vs. optimal stopping time for stopping time for randomly initialized networksinitially aligned networks ", + "Figure 14: Empirical verification that the simplifying assumptions of our theory are approximately valid in the regime we are considering at different student ranks. Initializations with random initial weights (random init.) and initializations with initial weight aligned to the noisy data SVD (aligned init.) are compared across varying student ranks. (a) The minimum generalization errors are almost identical between the different initializations and different student ranks. (b) The optimal stopping time in the randomly initialized networks consistently lags behind the aligned networks, because it takes time for the alignment to occur. This lag increases as the students rank decreases. (c) Randomly initialized networks of varying ranks obey qualitatively similar trends of increase in optimal stopping error and optimal stopping time as SNR decreases. (d) The theory predicts the aligned networks trends of increase in optimal stopping error and optimal stopping time with decreasing SNR almost perfectly. (All plots are made with a rank 1 teacher and $N _ { 1 } = N _ { 3 } = 1 0 0 \\rangle$ ) " + ], + "image_footnote": [], + "bbox": [ + 173, + 204, + 825, + 633 + ], + "page_idx": 19 + } +] \ No newline at end of file diff --git a/parse/train/ryfMLoCqtQ/ryfMLoCqtQ_middle.json b/parse/train/ryfMLoCqtQ/ryfMLoCqtQ_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..2655837f0002f246076887bc00a8a0850b86e3b5 --- /dev/null +++ b/parse/train/ryfMLoCqtQ/ryfMLoCqtQ_middle.json @@ -0,0 +1,61646 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 80, + 507, + 135 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 509, + 100 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 509, + 100 + ], + "score": 1.0, + "content": "AN ANALYTIC THEORY OF GENERALIZATION DYNAM-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 101, + 508, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 508, + 118 + ], + "score": 1.0, + "content": "ICS AND TRANSFER LEARNING IN DEEP LINEAR NET-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 120, + 162, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 162, + 139 + ], + "score": 1.0, + "content": "WORKS", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 113, + 155, + 240, + 199 + ], + "lines": [ + { + "bbox": [ + 112, + 154, + 208, + 167 + ], + "spans": [ + { + "bbox": [ + 112, + 154, + 208, + 167 + ], + "score": 1.0, + "content": "Andrew K. 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Finally,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 472, + 470, + 484 + ], + "spans": [ + { + "bbox": [ + 141, + 472, + 470, + 484 + ], + "score": 1.0, + "content": "for transfer learning, our theory reveals that knowledge transfer depends sensitively,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 482, + 447, + 495 + ], + "spans": [ + { + "bbox": [ + 141, + 482, + 447, + 495 + ], + "score": 1.0, + "content": "but computably, on the SNRs and input feature alignments of pairs of tasks.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 508, + 206, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 208, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 208, + 524 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "Many deep learning practitioners closely monitor both training and test errors, hoping to achieve", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 538, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 553 + ], + "score": 1.0, + "content": "both a small training error and a small generalization error, or gap between testing and training", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 551, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 507, + 563 + ], + "score": 1.0, + "content": "errors. Training is usually stopped early, before overfitting sets in and increases the test error.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "This procedure often results in large networks that generalize well on structured tasks, raising an", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "important generalization puzzle (Zhang et al., 2016): many existing theories that upper bound", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "generalization error (Bartlett & Mendelson, 2002; Neyshabur et al., 2015; Dziugaite & Roy, 2017;", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Golowich et al., 2017; Neyshabur et al., 2017; Bartlett et al., 2017; Arora et al., 2018, e.g) in terms of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 605, + 504, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 504, + 618 + ], + "score": 1.0, + "content": "various measures of network complexity yield very loose bounds. Therefore they cannot explain the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 616, + 312, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 312, + 629 + ], + "score": 1.0, + "content": "impressive generalization capabilities of deep nets.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 633, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "In the absence of any such tight and computable theory of deep network generalization error, we", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "develop an analytic theory of generalization error for deep linear networks. Such networks exhibit", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "highly nonlinear learning dynamics (Saxe et al., 2013a;b) including many prominent phenomena", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "like learning plateaus, saddle points, and sudden drops in training error. Moreover, theory developed", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "for the learning dynamics of deep linear networks directly inspired better initialization schemes", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "for nonlinear networks (Schoenholz et al., 2016; Pennington et al., 2017; 2018). Here we show", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "that deep linear networks also provide a good theoretical model for generalization dynamics. 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Furthermore, a major hope is that knowledge may transfer across", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 318, + 470, + 330 + ], + "spans": [ + { + "bbox": [ + 141, + 318, + 470, + 330 + ], + "score": 1.0, + "content": "tasks, so that multi-task learning can improve generalization on individual tasks.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 329, + 470, + 342 + ], + "spans": [ + { + "bbox": [ + 141, + 329, + 470, + 342 + ], + "score": 1.0, + "content": "However we lack analytic theories that can quantitatively predict how the degree of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 340, + 469, + 353 + ], + "spans": [ + { + "bbox": [ + 141, + 340, + 469, + 353 + ], + "score": 1.0, + "content": "knowledge transfer depends on the relationship between the tasks. We develop an", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 351, + 470, + 363 + ], + "spans": [ + { + "bbox": [ + 142, + 351, + 470, + 363 + ], + "score": 1.0, + "content": "analytic theory of the nonlinear dynamics of generalization in deep linear networks,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 362, + 469, + 375 + ], + "spans": [ + { + "bbox": [ + 141, + 362, + 469, + 375 + ], + "score": 1.0, + "content": "both within and across tasks. In particular, our theory provides analytic solutions", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 372, + 470, + 386 + ], + "spans": [ + { + "bbox": [ + 141, + 372, + 470, + 386 + ], + "score": 1.0, + "content": "to the training and testing error of deep networks as a function of training time,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 384, + 470, + 396 + ], + "spans": [ + { + "bbox": [ + 141, + 384, + 470, + 396 + ], + "score": 1.0, + "content": "number of examples, network size and initialization, and the task structure and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 396, + 469, + 407 + ], + "spans": [ + { + "bbox": [ + 141, + 396, + 469, + 407 + ], + "score": 1.0, + "content": "SNR. Our theory reveals that deep networks progressively learn the most important", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 405, + 469, + 419 + ], + "spans": [ + { + "bbox": [ + 141, + 405, + 469, + 419 + ], + "score": 1.0, + "content": "task structure first, so that generalization error at the early stopping time primarily", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 417, + 469, + 429 + ], + "spans": [ + { + "bbox": [ + 141, + 417, + 469, + 429 + ], + "score": 1.0, + "content": "depends on task structure and is independent of network size. This suggests", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 142, + 429, + 470, + 439 + ], + "spans": [ + { + "bbox": [ + 142, + 429, + 470, + 439 + ], + "score": 1.0, + "content": "any tight bound on generalization error must take into account task structure,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 142, + 439, + 471, + 451 + ], + "spans": [ + { + "bbox": [ + 142, + 439, + 471, + 451 + ], + "score": 1.0, + "content": "and explains observations about real data being learned faster than random data.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 450, + 470, + 462 + ], + "spans": [ + { + "bbox": [ + 142, + 450, + 470, + 462 + ], + "score": 1.0, + "content": "Intriguingly our theory also reveals the existence of a learning algorithm that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 461, + 470, + 473 + ], + "spans": [ + { + "bbox": [ + 141, + 461, + 470, + 473 + ], + "score": 1.0, + "content": "proveably out-performs neural network training through gradient descent. Finally,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 472, + 470, + 484 + ], + "spans": [ + { + "bbox": [ + 141, + 472, + 470, + 484 + ], + "score": 1.0, + "content": "for transfer learning, our theory reveals that knowledge transfer depends sensitively,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 482, + 447, + 495 + ], + "spans": [ + { + "bbox": [ + 141, + 482, + 447, + 495 + ], + "score": 1.0, + "content": "but computably, on the SNRs and input feature alignments of pairs of tasks.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 23.5, + "bbox_fs": [ + 140, + 272, + 471, + 495 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 508, + 206, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 208, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 208, + 524 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "Many deep learning practitioners closely monitor both training and test errors, hoping to achieve", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 538, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 553 + ], + "score": 1.0, + "content": "both a small training error and a small generalization error, or gap between testing and training", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 551, + 507, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 507, + 563 + ], + "score": 1.0, + "content": "errors. Training is usually stopped early, before overfitting sets in and increases the test error.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "This procedure often results in large networks that generalize well on structured tasks, raising an", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "important generalization puzzle (Zhang et al., 2016): many existing theories that upper bound", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "generalization error (Bartlett & Mendelson, 2002; Neyshabur et al., 2015; Dziugaite & Roy, 2017;", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Golowich et al., 2017; Neyshabur et al., 2017; Bartlett et al., 2017; Arora et al., 2018, e.g) in terms of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 605, + 504, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 504, + 618 + ], + "score": 1.0, + "content": "various measures of network complexity yield very loose bounds. Therefore they cannot explain the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 616, + 312, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 312, + 629 + ], + "score": 1.0, + "content": "impressive generalization capabilities of deep nets.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 529, + 507, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 633, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "In the absence of any such tight and computable theory of deep network generalization error, we", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "develop an analytic theory of generalization error for deep linear networks. Such networks exhibit", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "highly nonlinear learning dynamics (Saxe et al., 2013a;b) including many prominent phenomena", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "like learning plateaus, saddle points, and sudden drops in training error. Moreover, theory developed", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 506, + 691 + ], + "score": 1.0, + "content": "for the learning dynamics of deep linear networks directly inspired better initialization schemes", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "for nonlinear networks (Schoenholz et al., 2016; Pennington et al., 2017; 2018). Here we show", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "that deep linear networks also provide a good theoretical model for generalization dynamics. In", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "particular we develop an analytic theory for both the training and test error of a deep linear network", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "as a function of training time, number of training examples, network architecture, initialization, and", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "task structure and SNR. Our theory matches simulations and reveals that deep networks with small", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "weight initialization learn the most important aspects of a task first. Thus the optimal test error at the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "early stopping time depends largely on task structure and SNR, and not on network architecture, as", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "long as the architecture is expressive enough to attain small training error. Thus our exact analysis", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "of generalization dynamics reveals the important lesson that any theory that seeks to upper bound", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "generalization error based only on network architecture, and not on task structure, is likely to yield", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "exceedingly loose upper bounds. Intriguingly our theory also reveals a non-gradient-descent learning", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 461, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 461, + 172 + ], + "score": 1.0, + "content": "algorithm that proveably out-performs neural network training through gradient descent.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 48, + "bbox_fs": [ + 104, + 633, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "task structure and SNR. Our theory matches simulations and reveals that deep networks with small", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "weight initialization learn the most important aspects of a task first. Thus the optimal test error at the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "early stopping time depends largely on task structure and SNR, and not on network architecture, as", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "long as the architecture is expressive enough to attain small training error. Thus our exact analysis", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "of generalization dynamics reveals the important lesson that any theory that seeks to upper bound", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "generalization error based only on network architecture, and not on task structure, is likely to yield", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "exceedingly loose upper bounds. Intriguingly our theory also reveals a non-gradient-descent learning", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 461, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 461, + 172 + ], + "score": 1.0, + "content": "algorithm that proveably out-performs neural network training through gradient descent.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 176, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 189 + ], + "score": 1.0, + "content": "We also apply our theory to multi-task learning, which enables knowledge transfer from one task to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 201 + ], + "score": 1.0, + "content": "another, thereby further lowering generalization error (Dong et al., 2015; Rusu et al., 2015; Luong", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "et al., 2016, e.g.). Moreover, knowledge transfer across tasks may be key to human generalization", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "capabilities (Hansen et al., 2017; Lampinen et al., 2017). We provide an analytic theory for how", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "much knowledge is transferred between pairs of tasks, and we find that it displays a sensitive but", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "computable dependence on the relationship between pairs of tasks, in particular, their SNRs and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 210, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 210, + 255 + ], + "score": 1.0, + "content": "feature space alignments.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 272 + ], + "score": 1.0, + "content": "We note that a related prior work (Advani & Saxe, 2017) studied generalization in shallow and deep", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "linear networks, but that work was limited to networks with a single output, thereby precluding the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "possibility of addressing the issue of transfer learning. Moreover, analyzing networks with a single", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "score": 1.0, + "content": "output also precludes the possibility of addressing interesting tasks that require higher dimensional", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "score": 1.0, + "content": "outputs, for example in language (Dong et al., 2015, e.g.), generative models (Goodfellow et al.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 434, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 434, + 326 + ], + "score": 1.0, + "content": "2014, e.g), and reinforcement learning (Mnih et al., 2015; Silver et al., 2016, e.g).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 341, + 270, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 339, + 273, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 273, + 356 + ], + "score": 1.0, + "content": "2 THEORETICAL FRAMEWORK", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "We work in a student-teacher scenario in which we consider an ensemble of low rank, noisy teacher", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "networks that generate training data for a potentially more complex student network, and define the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 388, + 361, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 361, + 399 + ], + "score": 1.0, + "content": "training and test errors whose dynamics we wish to understand.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 107, + 413, + 334, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 335, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 335, + 425 + ], + "score": 1.0, + "content": "2.1 AN ENSEMBLE OF LOW-RANK NOISY TEACHERS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 381, + 446 + ], + "score": 1.0, + "content": "We first consider an ensemble of 3-layer linear teacher networks with", + "type": "text" + }, + { + "bbox": [ + 382, + 433, + 395, + 444 + ], + "score": 0.89, + "content": "\\overline { { N } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 432, + 450, + 446 + ], + "score": 1.0, + "content": "units in layer", + "type": "text" + }, + { + "bbox": [ + 451, + 435, + 455, + 444 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 432, + 505, + 446 + ], + "score": 1.0, + "content": ", and weight", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 443, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 144, + 462 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 144, + 444, + 212, + 458 + ], + "score": 0.92, + "content": "\\overline { { \\mathbf { W } } } ^ { 2 1 } \\in \\mathbb { R } ^ { \\overline { { N _ { 2 } } } \\times \\overline { { N } } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 443, + 231, + 462 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 231, + 444, + 299, + 458 + ], + "score": 0.93, + "content": "\\overline { { \\mathbf { W } } } ^ { 3 2 } \\in \\mathbb { R } ^ { \\overline { { N _ { 3 } } } \\times \\overline { { N _ { 2 } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 443, + 506, + 462 + ], + "score": 1.0, + "content": "between the input to hidden, and hidden to output", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 429, + 472 + ], + "score": 1.0, + "content": "layers, respectively. The teacher network thus computes the composite map", + "type": "text" + }, + { + "bbox": [ + 429, + 459, + 473, + 471 + ], + "score": 0.93, + "content": "\\overline { { \\mathbf { y } } } = \\overline { { \\mathbf { W } } } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 459, + 505, + 472 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 469, + 470, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 173, + 484 + ], + "score": 0.91, + "content": "\\dot { \\overline { { \\mathbf { W } } } } \\equiv \\overline { { \\mathbf { W } } } ^ { 3 \\bar { 2 } } \\overline { { \\mathbf { W } } } ^ { 2 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 469, + 452, + 488 + ], + "score": 1.0, + "content": ". 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We provide an analytic theory for how", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "much knowledge is transferred between pairs of tasks, and we find that it displays a sensitive but", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "computable dependence on the relationship between pairs of tasks, in particular, their SNRs and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 210, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 210, + 255 + ], + "score": 1.0, + "content": "feature space alignments.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 175, + 506, + 255 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 272 + ], + "score": 1.0, + "content": "We note that a related prior work (Advani & Saxe, 2017) studied generalization in shallow and deep", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "linear networks, but that work was limited to networks with a single output, thereby precluding the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 294 + ], + "score": 1.0, + "content": "possibility of addressing the issue of transfer learning. Moreover, analyzing networks with a single", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "score": 1.0, + "content": "output also precludes the possibility of addressing interesting tasks that require higher dimensional", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "score": 1.0, + "content": "outputs, for example in language (Dong et al., 2015, e.g.), generative models (Goodfellow et al.,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 434, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 434, + 326 + ], + "score": 1.0, + "content": "2014, e.g), and reinforcement learning (Mnih et al., 2015; Silver et al., 2016, e.g).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 257, + 506, + 326 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 341, + 270, + 354 + ], + "lines": [ + { + "bbox": [ + 105, + 339, + 273, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 273, + 356 + ], + "score": 1.0, + "content": "2 THEORETICAL FRAMEWORK", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 366, + 505, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "We work in a student-teacher scenario in which we consider an ensemble of low rank, noisy teacher", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 389 + ], + "score": 1.0, + "content": "networks that generate training data for a potentially more complex student network, and define the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 388, + 361, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 361, + 399 + ], + "score": 1.0, + "content": "training and test errors whose dynamics we wish to understand.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 366, + 505, + 399 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 413, + 334, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 335, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 335, + 425 + ], + "score": 1.0, + "content": "2.1 AN ENSEMBLE OF LOW-RANK NOISY TEACHERS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 381, + 446 + ], + "score": 1.0, + "content": "We first consider an ensemble of 3-layer linear teacher networks with", + "type": "text" + }, + { + "bbox": [ + 382, + 433, + 395, + 444 + ], + "score": 0.89, + "content": "\\overline { { N } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 432, + 450, + 446 + ], + "score": 1.0, + "content": "units in layer", + "type": "text" + }, + { + "bbox": [ + 451, + 435, + 455, + 444 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 432, + 505, + 446 + ], + "score": 1.0, + "content": ", and weight", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 443, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 144, + 462 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 144, + 444, + 212, + 458 + ], + "score": 0.92, + "content": "\\overline { { \\mathbf { W } } } ^ { 2 1 } \\in \\mathbb { R } ^ { \\overline { { N _ { 2 } } } \\times \\overline { { N } } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 443, + 231, + 462 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 231, + 444, + 299, + 458 + ], + "score": 0.93, + "content": "\\overline { { \\mathbf { W } } } ^ { 3 2 } \\in \\mathbb { R } ^ { \\overline { { N _ { 3 } } } \\times \\overline { { N _ { 2 } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 443, + 506, + 462 + ], + "score": 1.0, + "content": "between the input to hidden, and hidden to output", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 429, + 472 + ], + "score": 1.0, + "content": "layers, respectively. The teacher network thus computes the composite map", + "type": "text" + }, + { + "bbox": [ + 429, + 459, + 473, + 471 + ], + "score": 0.93, + "content": "\\overline { { \\mathbf { y } } } = \\overline { { \\mathbf { W } } } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 459, + 505, + 472 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 469, + 470, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 173, + 484 + ], + "score": 0.91, + "content": "\\dot { \\overline { { \\mathbf { W } } } } \\equiv \\overline { { \\mathbf { W } } } ^ { 3 \\bar { 2 } } \\overline { { \\mathbf { W } } } ^ { 2 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 469, + 452, + 488 + ], + "score": 1.0, + "content": ". 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As generalization performance will depend on", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 507, + 133 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 375, + 133 + ], + "score": 1.0, + "content": "the ratio of teacher singular values to the noise variance parameter", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 376, + 117, + 387, + 129 + ], + "score": 0.9, + "content": "\\sigma _ { z } ^ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 388, + 115, + 448, + 133 + ], + "score": 1.0, + "content": ", we simply set", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 449, + 118, + 479, + 128 + ], + "score": 0.91, + "content": "\\sigma _ { z } = 1", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 479, + 115, + 507, + 133 + ], + "score": 1.0, + "content": "in the", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 461, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 461, + 140 + ], + "score": 1.0, + "content": "following. Thus we can think of teacher singular values as signal to noise ratios (SNRs).", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 44.5, + "bbox_fs": [ + 104, + 705, + 507, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 140 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 148, + 94 + ], + "score": 1.0, + "content": "column is", + "type": "text" + }, + { + "bbox": [ + 149, + 83, + 160, + 92 + ], + "score": 0.87, + "content": "\\mathbf { z } ^ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 81, + 245, + 94 + ], + "score": 1.0, + "content": ". Its matrix elements", + "type": "text" + }, + { + "bbox": [ + 245, + 82, + 257, + 95 + ], + "score": 0.9, + "content": "z _ { i } ^ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 81, + 505, + 94 + ], + "score": 1.0, + "content": "are drawn iid. from a Gaussian with zero mean and variance", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 506, + 109 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 137, + 107 + ], + "score": 0.91, + "content": "\\sigma _ { z } ^ { 2 } / \\overline { { N } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 93, + 392, + 109 + ], + "score": 1.0, + "content": ". The noise scaling is chosen so the singular values of the teacher", + "type": "text" + }, + { + "bbox": [ + 393, + 94, + 406, + 105 + ], + "score": 0.78, + "content": "\\overline { { \\mathbf { W } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 93, + 461, + 109 + ], + "score": 1.0, + "content": "and the noise", + "type": "text" + }, + { + "bbox": [ + 461, + 95, + 470, + 105 + ], + "score": 0.51, + "content": "\\mathbf { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 93, + 506, + 109 + ], + "score": 1.0, + "content": "are both", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 107, + 107, + 128, + 118 + ], + "score": 0.89, + "content": "O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 105, + 505, + 119 + ], + "score": 1.0, + "content": ", leading to non-trivial generalization effects. As generalization performance will depend on", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 507, + 133 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 375, + 133 + ], + "score": 1.0, + "content": "the ratio of teacher singular values to the noise variance parameter", + "type": "text" + }, + { + "bbox": [ + 376, + 117, + 387, + 129 + ], + "score": 0.9, + "content": "\\sigma _ { z } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 115, + 448, + 133 + ], + "score": 1.0, + "content": ", we simply set", + "type": "text" + }, + { + "bbox": [ + 449, + 118, + 479, + 128 + ], + "score": 0.91, + "content": "\\sigma _ { z } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 115, + 507, + 133 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 461, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 461, + 140 + ], + "score": 1.0, + "content": "following. Thus we can think of teacher singular values as signal to noise ratios (SNRs).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "score": 1.0, + "content": "Finally, we note that while we focus for ease of exposition in the main paper on the case of one", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 330, + 169 + ], + "score": 1.0, + "content": "hidden layer networks and a full orthonormal basis of", + "type": "text" + }, + { + "bbox": [ + 330, + 156, + 366, + 167 + ], + "score": 0.92, + "content": "P = { \\overline { { N _ { 1 } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 154, + 506, + 169 + ], + "score": 1.0, + "content": "training inputs in the main paper,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 506, + 180 + ], + "score": 1.0, + "content": "neither of these assumptions are essential to our theory. Indeed in Section 3.4 and App. A we extend", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 506, + 191 + ], + "score": 1.0, + "content": "our theory to networks of arbitrary depth, and in App. G we extend our theory to the case of white", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 474, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 154, + 203 + ], + "score": 1.0, + "content": "inputs with", + "type": "text" + }, + { + "bbox": [ + 154, + 189, + 189, + 200 + ], + "score": 0.91, + "content": "P \\neq \\overline { { N } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 187, + 474, + 203 + ], + "score": 1.0, + "content": ", obtaining a good match between theory and experiment in both cases.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 215, + 293, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 215, + 295, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 295, + 228 + ], + "score": 1.0, + "content": "2.2 STUDENT TRAINING AND TEST ERROR", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 506, + 291 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 254, + 248 + ], + "score": 1.0, + "content": "Now consider a student network with", + "type": "text" + }, + { + "bbox": [ + 254, + 237, + 266, + 246 + ], + "score": 0.87, + "content": "N _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "units in each layer. We assume the first and last layers match", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 245, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 172, + 259 + ], + "score": 1.0, + "content": "the teacher (i.e.", + "type": "text" + }, + { + "bbox": [ + 173, + 246, + 212, + 258 + ], + "score": 0.91, + "content": "N _ { 1 } = \\overline { { N _ { 1 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 245, + 231, + 259 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 231, + 247, + 271, + 258 + ], + "score": 0.9, + "content": "N _ { 3 } = \\overline { { N _ { 3 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 245, + 291, + 259 + ], + "score": 1.0, + "content": ") but", + "type": "text" + }, + { + "bbox": [ + 291, + 246, + 330, + 258 + ], + "score": 0.92, + "content": "N _ { 2 } \\geq \\overline { { N _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 245, + 505, + 259 + ], + "score": 1.0, + "content": ", allowing the student to have more hidden", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 272 + ], + "score": 1.0, + "content": "units than the teacher. We also consider deeper students (see below and App. A). Now consider any", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 266, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 282, + 282 + ], + "score": 1.0, + "content": "student whose input-output map is given by", + "type": "text" + }, + { + "bbox": [ + 282, + 268, + 376, + 280 + ], + "score": 0.9, + "content": "\\mathbf { \\dot { y } } = \\mathbf { W } ^ { 3 2 } \\mathbf { W } ^ { 2 1 } \\equiv \\mathbf { W } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 266, + 506, + 282 + ], + "score": 1.0, + "content": ". Its training error on the teacher", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 279, + 411, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 411, + 293 + ], + "score": 1.0, + "content": "dataset in (2) and its test error over a distribution of new inputs are given by", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 297, + 438, + 335 + ], + "lines": [ + { + "bbox": [ + 172, + 297, + 438, + 335 + ], + "spans": [ + { + "bbox": [ + 172, + 297, + 438, + 335 + ], + "score": 0.93, + "content": "\\varepsilon _ { \\mathrm { t r a i n } } \\equiv \\frac { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\mathbf { W } \\hat { \\mathbf { x } } ^ { \\mu } - \\hat { \\mathbf { y } } ^ { \\mu } | | _ { 2 } ^ { 2 } } { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\hat { \\mathbf { y } } ^ { \\mu } | | _ { 2 } ^ { 2 } } , \\varepsilon _ { \\mathrm { t e s t } } \\equiv \\frac { \\langle | | \\mathbf { W } \\overline { { \\mathbf { x } } } - \\overline { { \\mathbf { y } } } | | _ { 2 } ^ { 2 } \\rangle } { \\langle | | \\overline { { \\mathbf { y } } } | | _ { 2 } ^ { 2 } \\rangle } ,", + "type": "interline_equation", + "image_path": "ec10a694331c27b34d51ae231b55eb44404ae9aa4b000936042e75c64c0f3aab.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 172, + 297, + 438, + 309.6666666666667 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 172, + 309.6666666666667, + 438, + 322.33333333333337 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 172, + 322.33333333333337, + 438, + 335.00000000000006 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 340, + 505, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 179, + 353 + ], + "score": 1.0, + "content": "respectively. 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Due to the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 503, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 368, + 388 + ], + "score": 1.0, + "content": "orthonormality of the training and isotropy of the test inputs, both", + "type": "text" + }, + { + "bbox": [ + 368, + 376, + 388, + 386 + ], + "score": 0.88, + "content": "\\varepsilon _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 375, + 405, + 388 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 405, + 376, + 421, + 386 + ], + "score": 0.88, + "content": "{ \\varepsilon } _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 375, + 503, + 388 + ], + "score": 1.0, + "content": "can be expressed as", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 392, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 111, + 392, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 111, + 392, + 505, + 424 + ], + "score": 0.92, + "content": "\\mathrm { ~ \\xi ~ } _ { \\mathrm { t r a i n } } = \\frac { \\mathrm { T } \\mathbf { F } \\mathbf { W } ^ { T } \\mathbf { W } - 2 \\mathrm { T r } \\mathbf { W } ^ { T } { \\boldsymbol { \\Sigma } } ^ { 3 1 } + \\mathrm { T r } { \\boldsymbol { \\Sigma } } ^ { 3 1 T } { \\boldsymbol { \\Sigma } } ^ { 3 1 } } { \\mathrm { T r } \\boldsymbol { \\Sigma } ^ { 3 1 T } \\boldsymbol { \\Sigma } ^ { 3 1 } } , \\mathrm { ~ \\xi ~ } _ { \\mathrm { f e s t } } = \\frac { \\mathrm { T r } \\mathbf { W } ^ { T } \\mathbf { W } - 2 \\mathrm { T r } \\mathbf { W } ^ { T } \\overline { { \\mathbf { W } } } + \\mathrm { T r } \\overline { { \\mathbf { W } } } ^ { T } \\overline { { \\mathbf { W } } } } { \\mathrm { T r } \\overline { { \\mathbf { W } } } ^ { T } \\overline { { \\mathbf { W } } } } .", + "type": "interline_equation", + "image_path": "2164005cd357741499916e1ea4597b4f17af0fa591d2c70c9b6722b832ef8c1d.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 111, + 392, + 505, + 402.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 111, + 402.6666666666667, + 505, + 413.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 111, + 413.33333333333337, + 505, + 424.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 432, + 509, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 128, + 445 + ], + "score": 1.0, + "content": "Both", + "type": "text" + }, + { + "bbox": [ + 128, + 434, + 148, + 444 + ], + "score": 0.87, + "content": "\\varepsilon _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 432, + 165, + 445 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 166, + 434, + 182, + 444 + ], + "score": 0.88, + "content": "\\varepsilon _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "can be further expressed in terms of the student, training data and teacher SVDs,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 441, + 498, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 441, + 189, + 462 + ], + "score": 1.0, + "content": "which we denote by", + "type": "text" + }, + { + "bbox": [ + 189, + 446, + 245, + 457 + ], + "score": 0.84, + "content": "\\mathbf { W } = \\mathbf { U } \\mathbf { S } \\mathbf { V } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 441, + 249, + 462 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 249, + 444, + 311, + 457 + ], + "score": 0.84, + "content": "\\begin{array} { r } { \\pmb { \\Sigma } ^ { 3 1 } = \\hat { \\mathbf { U } } \\hat { \\mathbf { S } } \\hat { \\mathbf { V } } ^ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 441, + 331, + 462 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 332, + 443, + 391, + 457 + ], + "score": 0.91, + "content": "\\overline { { \\mathbf { W } } } = \\overline { { \\mathbf { U } } } \\overline { { \\mathbf { S } } } \\overline { { \\mathbf { V } } } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 441, + 498, + 462 + ], + "score": 1.0, + "content": "respectively. 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1 } \\left[ \\underset { \\alpha = 1 } { \\overset { N _ { 2 } } { \\sum _ { \\alpha } } } s _ { \\alpha } ^ { 2 } + \\underset { \\beta = 1 } { \\overset { \\overline { { N } } _ { 2 } } { \\sum _ { \\beta } } } \\frac { 1 } { s _ { \\beta } ^ { 2 } } - 2 \\underset { \\alpha = 1 } { \\overset { N _ { 2 } } { \\sum _ { \\alpha } } } \\sum _ { \\beta = 1 } ^ { \\overline { { N _ { 2 } } } } s _ { \\alpha } \\overline { { s } } _ { \\beta } \\left( \\mathbf { u } ^ { \\alpha } \\cdot \\overline { { \\mathbf { u } } } ^ { \\beta } \\right) \\left( \\mathbf { v } ^ { \\alpha } \\cdot \\overline { { \\mathbf { v } } } ^ { \\beta } \\right) \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "9924429ca3ea20bc11ef1ca8fe58cf05d95015678f3a03aceca1e14b494d6ac4.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 142, + 464, + 469, + 492.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 142, + 492.3333333333333, + 469, + 520.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 142, + 520.6666666666666, + 469, + 549.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 553, + 504, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "Thus as the student learns, its training and test error dynamics depends on the alignment of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 263, + 577 + ], + "score": 1.0, + "content": "time-evolving student singular modes", + "type": "text" + }, + { + "bbox": [ + 263, + 564, + 317, + 577 + ], + "score": 0.93, + "content": "\\{ s ^ { \\alpha } , \\mathbf { u } ^ { \\alpha } , \\mathbf { v } ^ { \\alpha } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 564, + 433, + 577 + ], + "score": 1.0, + "content": "with the fixed training data", + "type": "text" + }, + { + "bbox": [ + 433, + 564, + 486, + 577 + ], + "score": 0.92, + "content": "\\{ \\hat { s } ^ { \\alpha } , \\hat { \\mathbf { u } } ^ { \\alpha } , \\hat { \\mathbf { v } } ^ { \\alpha } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 575, + 309, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 138, + 588 + ], + "score": 1.0, + "content": "teacher", + "type": "text" + }, + { + "bbox": [ + 138, + 575, + 192, + 587 + ], + "score": 0.93, + "content": "\\{ \\overline { { s } } ^ { \\alpha } , \\bar { \\overline { { \\mathbf { u } } } } ^ { \\alpha } , \\overline { { \\mathbf { v } } } ^ { \\alpha } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 575, + 309, + 588 + ], + "score": 1.0, + "content": "singular modes respectively.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 105, + 604, + 490, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 492, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 492, + 618 + ], + "score": 1.0, + "content": "3 SINGLE TASK GENERALIZATION DYNAMICS: THEORY AND EXPERIMENT", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 506, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "score": 1.0, + "content": "Here we derive and numerically test analytic formulas for both the training and test errors of a student", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "network as it learns from training data generated from a teacher network. We explore the dependence", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 506, + 664 + ], + "score": 1.0, + "content": "of these quantitites on the student network size, student initialization, teacher SNR, and training time.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 108, + 677, + 449, + 689 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 450, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 450, + 691 + ], + "score": 1.0, + "content": "3.1 STUDENT TRAINING DYNAMICS AND TRAINING-ALIGNED (TA) NETWORKS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 435, + 713 + ], + "score": 1.0, + "content": "We assume the student weights undergo batch gradient descent with learning rate", + "type": "text" + }, + { + "bbox": [ + 435, + 700, + 443, + 709 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 697, + 505, + 713 + ], + "score": 1.0, + "content": "on the training", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 707, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 707, + 127, + 724 + ], + "score": 1.0, + "content": "error", + "type": "text" + }, + { + "bbox": [ + 128, + 709, + 236, + 723 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { \\mu } | | \\hat { \\mathbf { y } } ^ { \\mu } - \\mathbf { W } ^ { 3 2 } \\mathbf { W } ^ { 2 1 } \\bar { \\hat { \\mathbf { x } } ^ { \\mu } } | | _ { 2 } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 707, + 302, + 724 + ], + "score": 1.0, + "content": ", which for small", + "type": "text" + }, + { + "bbox": [ + 303, + 711, + 309, + 720 + ], + "score": 0.83, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 707, + 506, + 724 + ], + "score": 1.0, + "content": "is well approximated by the differential equations:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 140 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 104, + 81, + 507, + 140 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 505, + 201 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 158 + ], + "score": 1.0, + "content": "Finally, we note that while we focus for ease of exposition in the main paper on the case of one", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 330, + 169 + ], + "score": 1.0, + "content": "hidden layer networks and a full orthonormal basis of", + "type": "text" + }, + { + "bbox": [ + 330, + 156, + 366, + 167 + ], + "score": 0.92, + "content": "P = { \\overline { { N _ { 1 } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 154, + 506, + 169 + ], + "score": 1.0, + "content": "training inputs in the main paper,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 506, + 180 + ], + "score": 1.0, + "content": "neither of these assumptions are essential to our theory. Indeed in Section 3.4 and App. A we extend", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 506, + 191 + ], + "score": 1.0, + "content": "our theory to networks of arbitrary depth, and in App. G we extend our theory to the case of white", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 474, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 154, + 203 + ], + "score": 1.0, + "content": "inputs with", + "type": "text" + }, + { + "bbox": [ + 154, + 189, + 189, + 200 + ], + "score": 0.91, + "content": "P \\neq \\overline { { N } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 187, + 474, + 203 + ], + "score": 1.0, + "content": ", obtaining a good match between theory and experiment in both cases.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 144, + 506, + 203 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 215, + 293, + 226 + ], + "lines": [ + { + "bbox": [ + 105, + 215, + 295, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 295, + 228 + ], + "score": 1.0, + "content": "2.2 STUDENT TRAINING AND TEST ERROR", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 506, + 291 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 254, + 248 + ], + "score": 1.0, + "content": "Now consider a student network with", + "type": "text" + }, + { + "bbox": [ + 254, + 237, + 266, + 246 + ], + "score": 0.87, + "content": "N _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "units in each layer. We assume the first and last layers match", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 245, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 172, + 259 + ], + "score": 1.0, + "content": "the teacher (i.e.", + "type": "text" + }, + { + "bbox": [ + 173, + 246, + 212, + 258 + ], + "score": 0.91, + "content": "N _ { 1 } = \\overline { { N _ { 1 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 245, + 231, + 259 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 231, + 247, + 271, + 258 + ], + "score": 0.9, + "content": "N _ { 3 } = \\overline { { N _ { 3 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 245, + 291, + 259 + ], + "score": 1.0, + "content": ") but", + "type": "text" + }, + { + "bbox": [ + 291, + 246, + 330, + 258 + ], + "score": 0.92, + "content": "N _ { 2 } \\geq \\overline { { N _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 245, + 505, + 259 + ], + "score": 1.0, + "content": ", allowing the student to have more hidden", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 506, + 272 + ], + "score": 1.0, + "content": "units than the teacher. We also consider deeper students (see below and App. A). Now consider any", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 266, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 282, + 282 + ], + "score": 1.0, + "content": "student whose input-output map is given by", + "type": "text" + }, + { + "bbox": [ + 282, + 268, + 376, + 280 + ], + "score": 0.9, + "content": "\\mathbf { \\dot { y } } = \\mathbf { W } ^ { 3 2 } \\mathbf { W } ^ { 2 1 } \\equiv \\mathbf { W } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 266, + 506, + 282 + ], + "score": 1.0, + "content": ". 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Here", + "type": "text" + }, + { + "bbox": [ + 180, + 342, + 192, + 351 + ], + "score": 0.86, + "content": "\\hat { \\mathbf { x } } ^ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 341, + 209, + 353 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 210, + 342, + 223, + 352 + ], + "score": 0.89, + "content": "{ \\hat { \\mathbf { y } } } ^ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 341, + 457, + 353 + ], + "score": 1.0, + "content": "are the noisy training set inputs and outputs in (2), whereas", + "type": "text" + }, + { + "bbox": [ + 457, + 342, + 465, + 351 + ], + "score": 0.51, + "content": "\\overline { { \\mathbf { x } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 341, + 505, + 353 + ], + "score": 1.0, + "content": "denotes a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 419, + 365 + ], + "score": 1.0, + "content": "random test input drawn from zero mean Gaussian with identity covariance,", + "type": "text" + }, + { + "bbox": [ + 420, + 352, + 470, + 364 + ], + "score": 0.92, + "content": "\\overline { { \\mathbf { y } } } ^ { \\mu } = \\overline { { \\mathbf { W } } } \\overline { { \\mathbf { x } } } ^ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "is noise", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 364, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 205, + 378 + ], + "score": 1.0, + "content": "free teacher output, and", + "type": "text" + }, + { + "bbox": [ + 205, + 364, + 217, + 376 + ], + "score": 0.89, + "content": "\\langle \\cdot \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 364, + 448, + 378 + ], + "score": 1.0, + "content": "denotes an average w.r.t the distribution of the test input", + "type": "text" + }, + { + "bbox": [ + 448, + 366, + 456, + 374 + ], + "score": 0.64, + "content": "\\overline { { \\mathbf { x } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 364, + 505, + 378 + ], + "score": 1.0, + "content": ". 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1 } \\left[ \\underset { \\alpha = 1 } { \\overset { N _ { 2 } } { \\sum _ { \\alpha } } } s _ { \\alpha } ^ { 2 } + \\underset { \\beta = 1 } { \\overset { \\overline { { N } } _ { 2 } } { \\sum _ { \\beta } } } \\frac { 1 } { s _ { \\beta } ^ { 2 } } - 2 \\underset { \\alpha = 1 } { \\overset { N _ { 2 } } { \\sum _ { \\alpha } } } \\sum _ { \\beta = 1 } ^ { \\overline { { N _ { 2 } } } } s _ { \\alpha } \\overline { { s } } _ { \\beta } \\left( \\mathbf { u } ^ { \\alpha } \\cdot \\overline { { \\mathbf { u } } } ^ { \\beta } \\right) \\left( \\mathbf { v } ^ { \\alpha } \\cdot \\overline { { \\mathbf { v } } } ^ { \\beta } \\right) \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "9924429ca3ea20bc11ef1ca8fe58cf05d95015678f3a03aceca1e14b494d6ac4.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 142, + 464, + 469, + 492.3333333333333 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 142, + 492.3333333333333, + 469, + 520.6666666666666 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 142, + 520.6666666666666, + 469, + 549.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 553, + 504, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "Thus as the student learns, its training and test error dynamics depends on the alignment of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 263, + 577 + ], + "score": 1.0, + "content": "time-evolving student singular modes", + "type": "text" + }, + { + "bbox": [ + 263, + 564, + 317, + 577 + ], + "score": 0.93, + "content": "\\{ s ^ { \\alpha } , \\mathbf { u } ^ { \\alpha } , \\mathbf { v } ^ { \\alpha } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 564, + 433, + 577 + ], + "score": 1.0, + "content": "with the fixed training data", + "type": "text" + }, + { + "bbox": [ + 433, + 564, + 486, + 577 + ], + "score": 0.92, + "content": "\\{ \\hat { s } ^ { \\alpha } , \\hat { \\mathbf { u } } ^ { \\alpha } , \\hat { \\mathbf { v } } ^ { \\alpha } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 575, + 309, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 138, + 588 + ], + "score": 1.0, + "content": "teacher", + "type": "text" + }, + { + "bbox": [ + 138, + 575, + 192, + 587 + ], + "score": 0.93, + "content": "\\{ \\overline { { s } } ^ { \\alpha } , \\bar { \\overline { { \\mathbf { u } } } } ^ { \\alpha } , \\overline { { \\mathbf { v } } } ^ { \\alpha } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 575, + 309, + 588 + ], + "score": 1.0, + "content": "singular modes respectively.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 553, + 505, + 588 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 604, + 490, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 603, + 492, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 492, + 618 + ], + "score": 1.0, + "content": "3 SINGLE TASK GENERALIZATION DYNAMICS: THEORY AND EXPERIMENT", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 506, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 506, + 643 + ], + "score": 1.0, + "content": "Here we derive and numerically test analytic formulas for both the training and test errors of a student", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "network as it learns from training data generated from a teacher network. 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We consider two classes of student initializations. The first initialization corresponds", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 250, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 104, + 250, + 270, + 265 + ], + "score": 1.0, + "content": "to a random student where the weights", + "type": "text" + }, + { + "bbox": [ + 271, + 251, + 293, + 262 + ], + "score": 0.87, + "content": "\\mathbf { W } ^ { 2 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 250, + 313, + 265 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 313, + 251, + 335, + 262 + ], + "score": 0.86, + "content": "\\mathbf { W ^ { 3 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 250, + 505, + 265 + ], + "score": 1.0, + "content": "are chosen such that the composite map", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 261, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 176, + 273 + ], + "score": 0.9, + "content": "\\mathbf { W } = \\mathbf { W ^ { 3 2 } W ^ { 2 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 261, + 229, + 276 + ], + "score": 1.0, + "content": "has an SVD", + "type": "text" + }, + { + "bbox": [ + 229, + 262, + 284, + 273 + ], + "score": 0.86, + "content": "\\mathbf { W } = \\epsilon \\mathbf { U } \\mathbf { V } ^ { \\mathbf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 261, + 316, + 276 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 316, + 263, + 326, + 273 + ], + "score": 0.59, + "content": "\\mathbf { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 261, + 344, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 345, + 263, + 355, + 273 + ], + "score": 0.6, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 261, + 506, + 276 + ], + "score": 1.0, + "content": "are random singular vector matrices", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 100, + 273, + 511, + 307 + ], + "spans": [ + { + "bbox": [ + 100, + 273, + 214, + 307 + ], + "score": 1.0, + "content": "and all student singular vala time dependent evolution", + "type": "text" + }, + { + "bbox": [ + 243, + 276, + 248, + 284 + ], + "score": 0.5, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 273, + 511, + 307 + ], + "score": 1.0, + "content": "te map undergoes. For white inputs,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 215, + 284, + 434, + 300 + ], + "spans": [ + { + "bbox": [ + 215, + 284, + 434, + 300 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbf { W } ( t ) = \\mathbf { U } ( t ) \\mathbf { S } ( t ) \\mathbf { V } ( t ) ^ { T } = \\sum _ { \\alpha = 1 } ^ { N _ { 2 } } \\mathbf { s } _ { \\alpha } ( t ) \\mathbf { u } ^ { \\alpha } ( t ) \\mathbf { v } ^ { \\alpha } ( t ) ^ { T } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 298, + 504, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 118, + 312 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 151, + 298, + 156, + 312 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 204, + 298, + 420, + 312 + ], + "score": 1.0, + "content": ", and so the time-dependent student singular modes", + "type": "text" + }, + { + "bbox": [ + 420, + 299, + 504, + 312 + ], + "score": 0.92, + "content": "\\{ s ^ { \\alpha } ( t ) , \\mathbf { u } ^ { \\alpha } ( \\mathbf { t } ) , \\mathbf { v } ^ { ( } \\mathbf { t } ) \\}", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 310, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 290, + 323 + ], + "score": 1.0, + "content": "converge to the training data singular modes", + "type": "text" + }, + { + "bbox": [ + 290, + 310, + 343, + 322 + ], + "score": 0.93, + "content": "\\{ \\hat { s } ^ { \\alpha } , \\hat { \\mathbf { u } } ^ { \\alpha } , \\hat { \\mathbf { v } } ^ { \\alpha } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 310, + 506, + 323 + ], + "score": 1.0, + "content": ". However, the explicit dynamics of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 322, + 476, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 476, + 334 + ], + "score": 1.0, + "content": "student singular modes can be difficult to obtain analytically from random initial conditions.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 338, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 104, + 337, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 104, + 337, + 464, + 351 + ], + "score": 1.0, + "content": "Thus we also consider a special class of training aligned (TA) initial conditions in which", + "type": "text" + }, + { + "bbox": [ + 464, + 338, + 486, + 349 + ], + "score": 0.87, + "content": "\\mathbf { W ^ { 2 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 337, + 506, + 351 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 348, + 507, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 128, + 361 + ], + "score": 0.83, + "content": "\\mathbf { W ^ { 3 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 348, + 293, + 364 + ], + "score": 1.0, + "content": "are chosen such that the composite map", + "type": "text" + }, + { + "bbox": [ + 293, + 350, + 362, + 361 + ], + "score": 0.92, + "content": "\\mathbf { W } = \\mathbf { W ^ { 3 2 } W ^ { 2 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 348, + 414, + 364 + ], + "score": 1.0, + "content": "has an SVD", + "type": "text" + }, + { + "bbox": [ + 415, + 349, + 469, + 362 + ], + "score": 0.76, + "content": "\\mathbf { W } = \\epsilon \\hat { \\mathbf { U } } \\hat { \\mathbf { V } } ^ { \\mathbf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 348, + 507, + 364 + ], + "score": 1.0, + "content": ". 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The general differential", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 625, + 406, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 406, + 637 + ], + "score": 1.0, + "content": "equation and derivations for deeper networks can be found in Appendix A.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + }, + { + "type": "title", + "bbox": [ + 106, + 649, + 495, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 497, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 497, + 661 + ], + "score": 1.0, + "content": "3.2 HOW THE TEACHER IS BURIED IN THE TRAINING DATA: A RANDOM MATRIX ANALYSIS", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 106, + 669, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 106, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "In the previous section, we reviewed an exact analytic 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We consider two classes of student initializations. The first initialization corresponds", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 250, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 104, + 250, + 270, + 265 + ], + "score": 1.0, + "content": "to a random student where the weights", + "type": "text" + }, + { + "bbox": [ + 271, + 251, + 293, + 262 + ], + "score": 0.87, + "content": "\\mathbf { W } ^ { 2 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 250, + 313, + 265 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 313, + 251, + 335, + 262 + ], + "score": 0.86, + "content": "\\mathbf { W ^ { 3 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 250, + 505, + 265 + ], + "score": 1.0, + "content": "are chosen such that the composite map", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 261, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 176, + 273 + ], + "score": 0.9, + "content": "\\mathbf { W } = \\mathbf { W ^ { 3 2 } W ^ { 2 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 261, + 229, + 276 + ], + "score": 1.0, + "content": "has an SVD", + "type": "text" + }, + { + "bbox": [ + 229, + 262, + 284, + 273 + ], + "score": 0.86, + "content": "\\mathbf { W } = \\epsilon \\mathbf { U } \\mathbf { V } ^ { \\mathbf { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 261, + 316, + 276 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 316, + 263, + 326, + 273 + ], + "score": 0.59, + "content": "\\mathbf { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 261, + 344, + 276 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 345, + 263, + 355, + 273 + ], + "score": 0.6, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 261, + 506, + 276 + ], + "score": 1.0, + "content": "are random singular vector matrices", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 100, + 273, + 511, + 307 + ], + "spans": [ + { + "bbox": [ + 100, + 273, + 214, + 307 + ], + "score": 1.0, + "content": "and all student singular vala time dependent evolution", + "type": "text" + }, + { + "bbox": [ + 243, + 276, + 248, + 284 + ], + "score": 0.5, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 273, + 511, + 307 + ], + "score": 1.0, + "content": "te map undergoes. For white inputs,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 215, + 284, + 434, + 300 + ], + "spans": [ + { + "bbox": [ + 215, + 284, + 434, + 300 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbf { W } ( t ) = \\mathbf { U } ( t ) \\mathbf { S } ( t ) \\mathbf { V } ( t ) ^ { T } = \\sum _ { \\alpha = 1 } ^ { N _ { 2 } } \\mathbf { s } _ { \\alpha } ( t ) \\mathbf { u } ^ { \\alpha } ( t ) \\mathbf { v } ^ { \\alpha } ( t ) ^ { T } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 298, + 504, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 118, + 312 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 151, + 298, + 156, + 312 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 204, + 298, + 420, + 312 + ], + "score": 1.0, + "content": ", and so the time-dependent student singular modes", + "type": "text" + }, + { + "bbox": [ + 420, + 299, + 504, + 312 + ], + "score": 0.92, + "content": "\\{ s ^ { \\alpha } ( t ) , \\mathbf { u } ^ { \\alpha } ( \\mathbf { t } ) , \\mathbf { v } ^ { ( } \\mathbf { t } ) \\}", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 310, + 506, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 290, + 323 + ], + "score": 1.0, + "content": "converge to the training data singular modes", + "type": "text" + }, + { + "bbox": [ + 290, + 310, + 343, + 322 + ], + "score": 0.93, + "content": "\\{ \\hat { s } ^ { \\alpha } , \\hat { \\mathbf { u } } ^ { \\alpha } , \\hat { \\mathbf { v } } ^ { \\alpha } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 310, + 506, + 323 + ], + "score": 1.0, + "content": ". 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That is,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "the TA network (henceforth referred to simply as the TA) has the same singular vectors as the training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 371, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 171, + 385 + ], + "score": 1.0, + "content": "data covariance", + "type": "text" + }, + { + "bbox": [ + 172, + 372, + 189, + 383 + ], + "score": 0.88, + "content": "\\boldsymbol { \\Sigma } ^ { 3 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 371, + 334, + 385 + ], + "score": 1.0, + "content": ", but has all singular values equal to", + "type": "text" + }, + { + "bbox": [ + 335, + 375, + 340, + 383 + ], + "score": 0.6, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 371, + 505, + 385 + ], + "score": 1.0, + "content": ". 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(", + "type": "text" + }, + { + "bbox": [ + 222, + 312, + 263, + 323 + ], + "score": 0.86, + "content": "N _ { 1 } = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 312, + 266, + 324 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 267, + 312, + 304, + 323 + ], + "score": 0.89, + "content": "N _ { 2 } = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 312, + 307, + 324 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 307, + 312, + 346, + 323 + ], + "score": 0.88, + "content": "N _ { 3 } = 5 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 312, + 352, + 324 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 104, + 343, + 504, + 359 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 155, + 359 + ], + "score": 1.0, + "content": "TA network", + "type": "text" + }, + { + "bbox": [ + 155, + 345, + 221, + 357 + ], + "score": 0.94, + "content": "\\{ s ^ { \\alpha } ( t ) , \\hat { \\mathbf { u } } ^ { \\alpha } , \\hat { \\mathbf { v } } ^ { \\alpha } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 343, + 451, + 359 + ], + "score": 1.0, + "content": "of through (9), we can now derive analytic expressions for", + "type": "text" + }, + { + "bbox": [ + 451, + 347, + 470, + 357 + ], + "score": 0.88, + "content": "\\varepsilon _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 343, + 487, + 359 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 488, + 347, + 504, + 357 + ], + "score": 0.84, + "content": "\\varepsilon _ { \\mathrm { t e s t } }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "in (6) and (7), for a TA network. We will also show that these learning curves closely approximate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 354, + 380 + ], + "score": 1.0, + "content": "those of a random student with time-evolving singular vectors", + "type": "text" + }, + { + "bbox": [ + 354, + 367, + 415, + 379 + ], + "score": 0.93, + "content": "\\{ \\mathbf { u } ^ { \\alpha } ( t ) , \\mathbf { \\bar { v } } ^ { \\alpha } ( t ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 367, + 506, + 380 + ], + "score": 1.0, + "content": ", and match on several", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 378, + 425, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 335, + 391 + ], + "score": 1.0, + "content": "key aspects. 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Also", + "type": "text" + }, + { + "bbox": [ + 485, + 466, + 503, + 479 + ], + "score": 0.89, + "content": "\\langle \\cdot \\rangle _ { \\mathcal { R } }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 408, + 490 + ], + "score": 1.0, + "content": "denotes an average with respect to the MP distribution in (13) over a region", + "type": "text" + }, + { + "bbox": [ + 408, + 478, + 417, + 487 + ], + "score": 0.8, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 477, + 505, + 490 + ], + "score": 1.0, + "content": ". Two distinct regions", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 245, + 502 + ], + "score": 1.0, + "content": "contribute to training error. First", + "type": "text" + }, + { + "bbox": [ + 245, + 489, + 263, + 500 + ], + "score": 0.9, + "content": "\\mathcal { R } _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 488, + 343, + 502 + ], + "score": 1.0, + "content": "contains those top", + "type": "text" + }, + { + "bbox": [ + 343, + 488, + 384, + 500 + ], + "score": 0.94, + "content": "N _ { 2 } - \\overline { { N } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "training data singular values", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 499, + 504, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 228, + 512 + ], + "score": 1.0, + "content": "that do not correspond to the", + "type": "text" + }, + { + "bbox": [ + 228, + 499, + 243, + 510 + ], + "score": 0.9, + "content": "\\overline { { N } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 499, + 490, + 512 + ], + "score": 1.0, + "content": "singular values of the teacher but will be learned by a rank", + "type": "text" + }, + { + "bbox": [ + 490, + 500, + 504, + 510 + ], + "score": 0.87, + "content": "N _ { 2 }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 509, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 176, + 524 + ], + "score": 1.0, + "content": "student. Second,", + "type": "text" + }, + { + "bbox": [ + 177, + 511, + 198, + 522 + ], + "score": 0.9, + "content": "\\mathcal { R } _ { o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 509, + 320, + 524 + ], + "score": 1.0, + "content": "corresponds to the remaining", + "type": "text" + }, + { + "bbox": [ + 320, + 511, + 358, + 522 + ], + "score": 0.92, + "content": "N _ { 3 } - N _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 509, + 506, + 524 + ], + "score": 1.0, + "content": "lowest training data singular values√", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 522, + 504, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 238, + 536 + ], + "score": 1.0, + "content": "that cannot be learned by a rank √", + "type": "text" + }, + { + "bbox": [ + 239, + 523, + 252, + 534 + ], + "score": 0.89, + "content": "N _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 522, + 419, + 536 + ], + "score": 1.0, + "content": "student. In terms of the MP distribution,", + "type": "text" + }, + { + "bbox": [ + 420, + 522, + 504, + 535 + ], + "score": 0.93, + "content": "\\mathcal { R } _ { o u t } = [ 1 - \\sqrt { \\mathcal { A } } , f ]", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 123, + 548 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 534, + 203, + 547 + ], + "score": 0.93, + "content": "\\mathcal { R } _ { i n } = [ f , 1 + \\sqrt { \\mathcal { A } } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 534, + 233, + 548 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 233, + 536, + 241, + 547 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 534, + 402, + 548 + ], + "score": 1.0, + "content": "is the point at which the MP density has", + "type": "text" + }, + { + "bbox": [ + 402, + 535, + 449, + 547 + ], + "score": 0.93, + "content": "1 - N _ { 2 } / N _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "of its mass to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 152, + 561 + ], + "score": 1.0, + "content": "the left and", + "type": "text" + }, + { + "bbox": [ + 153, + 548, + 183, + 560 + ], + "score": 0.92, + "content": "N _ { 2 } / N _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 547, + 433, + 561 + ], + "score": 1.0, + "content": "of its mass to the right. In the simple case of a full rank student,", + "type": "text" + }, + { + "bbox": [ + 433, + 547, + 485, + 560 + ], + "score": 0.92, + "content": "f = 1 - \\sqrt { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 547, + 506, + 561 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 220, + 573 + ], + "score": 1.0, + "content": "one need only integrate over", + "type": "text" + }, + { + "bbox": [ + 221, + 560, + 238, + 570 + ], + "score": 0.9, + "content": "\\mathcal { R } _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 559, + 412, + 573 + ], + "score": 1.0, + "content": "which is the entire range. Equation (14) for", + "type": "text" + }, + { + "bbox": [ + 413, + 561, + 432, + 570 + ], + "score": 0.87, + "content": "\\varepsilon _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "makes it manifest", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "that it will go to zero for a full rank student as its singular values approach those of the training data.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 587, + 504, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "Of course the test error can behave very differently. 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They yield considerable insight into", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 721, + 481, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 481, + 733 + ], + "score": 1.0, + "content": "the dynamics of good generalization early in learning and overfitting later, as we show below.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + } + ], + "page_idx": 5, + "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 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 78, + 497, + 246 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 78, + 497, + 246 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 78, + 497, + 246 + ], + "spans": [ + { + "bbox": [ + 110, + 78, + 497, + 246 + ], + "score": 0.974, + "type": "image", + "image_path": "314c03e7d659a75d3bce2929de3fdf61f6a46155c699119ff46ac7743510ffe3.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 78, + 497, + 134.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 134.0, + 497, + 190.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 190.0, + 497, + 246.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 256, + 506, + 324 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 256, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 506, + 269 + ], + "score": 1.0, + "content": "Figure 3: Match between theory and experiment for rank 1 (row 1, a-d) and rank 3 (row 2, e-h)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 266, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 506, + 282 + ], + "score": 1.0, + "content": "teachers with single-hidden-layer students: (a-b, e-f) log train and test error, respectively, showing", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "very close match between theory and experiment for TA, and close match for the random student.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "(c,g) comparing TA and randomly initialized students minimum generalization errors, showing almost", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 299, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 315 + ], + "score": 1.0, + "content": "perfect match. (d,h) comparing TA and randomly initialized students optimal stopping times, showing", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 312, + 352, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 221, + 324 + ], + "score": 1.0, + "content": "small lag due to alignment. (", + "type": "text" + }, + { + "bbox": [ + 222, + 312, + 263, + 323 + ], + "score": 0.86, + "content": "N _ { 1 } = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 312, + 266, + 324 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 267, + 312, + 304, + 323 + ], + "score": 0.89, + "content": "N _ { 2 } = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 312, + 307, + 324 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 307, + 312, + 346, + 323 + ], + "score": 0.88, + "content": "N _ { 3 } = 5 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 312, + 352, + 324 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 104, + 343, + 504, + 359 + ], + "spans": [ + { + "bbox": [ + 104, + 343, + 155, + 359 + ], + "score": 1.0, + "content": "TA network", + "type": "text" + }, + { + "bbox": [ + 155, + 345, + 221, + 357 + ], + "score": 0.94, + "content": "\\{ s ^ { \\alpha } ( t ) , \\hat { \\mathbf { u } } ^ { \\alpha } , \\hat { \\mathbf { v } } ^ { \\alpha } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 343, + 451, + 359 + ], + "score": 1.0, + "content": "of through (9), we can now derive analytic expressions for", + "type": "text" + }, + { + "bbox": [ + 451, + 347, + 470, + 357 + ], + "score": 0.88, + "content": "\\varepsilon _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 343, + 487, + 359 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 488, + 347, + 504, + 357 + ], + "score": 0.84, + "content": "\\varepsilon _ { \\mathrm { t e s t } }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "in (6) and (7), for a TA network. We will also show that these learning curves closely approximate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 367, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 354, + 380 + ], + "score": 1.0, + "content": "those of a random student with time-evolving singular vectors", + "type": "text" + }, + { + "bbox": [ + 354, + 367, + 415, + 379 + ], + "score": 0.93, + "content": "\\{ \\mathbf { u } ^ { \\alpha } ( t ) , \\mathbf { \\bar { v } } ^ { \\alpha } ( t ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 367, + 506, + 380 + ], + "score": 1.0, + "content": ", and match on several", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 378, + 425, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 335, + 391 + ], + "score": 1.0, + "content": "key aspects. 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Also", + "type": "text" + }, + { + "bbox": [ + 485, + 466, + 503, + 479 + ], + "score": 0.89, + "content": "\\langle \\cdot \\rangle _ { \\mathcal { R } }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 408, + 490 + ], + "score": 1.0, + "content": "denotes an average with respect to the MP distribution in (13) over a region", + "type": "text" + }, + { + "bbox": [ + 408, + 478, + 417, + 487 + ], + "score": 0.8, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 477, + 505, + 490 + ], + "score": 1.0, + "content": ". Two distinct regions", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 245, + 502 + ], + "score": 1.0, + "content": "contribute to training error. 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In the simple case of a full rank student,", + "type": "text" + }, + { + "bbox": [ + 433, + 547, + 485, + 560 + ], + "score": 0.92, + "content": "f = 1 - \\sqrt { \\mathcal { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 547, + 506, + 561 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 220, + 573 + ], + "score": 1.0, + "content": "one need only integrate over", + "type": "text" + }, + { + "bbox": [ + 221, + 560, + 238, + 570 + ], + "score": 0.9, + "content": "\\mathcal { R } _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 559, + 412, + 573 + ], + "score": 1.0, + "content": "which is the entire range. Equation (14) for", + "type": "text" + }, + { + "bbox": [ + 413, + 561, + 432, + 570 + ], + "score": 0.87, + "content": "\\varepsilon _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "makes it manifest", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "that it will go to zero for a full rank student as its singular values approach those of the training data.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 454, + 506, + 583 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 587, + 504, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "Of course the test error can behave very differently. 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(c,g) comparing TA", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "and randomly initialized students minimum generalization errors, showing almost perfect match.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "(d,h) comparing TA and randomly initialized students optimal stopping times, showing large lag due", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 312, + 403, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 272, + 324 + ], + "score": 1.0, + "content": "to slower alignment in deeper networks.", + "type": "text" + }, + { + "bbox": [ + 272, + 312, + 314, + 323 + ], + "score": 0.87, + "content": "N _ { 1 } = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 312, + 318, + 324 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 318, + 312, + 355, + 323 + ], + "score": 0.9, + "content": "N _ { 2 } = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 312, + 359, + 324 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 359, + 312, + 397, + 323 + ], + "score": 0.89, + "content": "N _ { 3 } = 5 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 312, + 403, + 324 + ], + "score": 1.0, + "content": ".)", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "title", + "bbox": [ + 106, + 339, + 504, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 504, + 349 + ], + "score": 1.0, + "content": "3.4 NUMERICAL TESTS OF THE THEORY OF NEURAL NETWORK GENERALIZATION DYNAMICS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 360, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "Fig. 3 demonstrates an excellent match between the theory and simulations for the TA, and a close", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 369, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 442, + 384 + ], + "score": 1.0, + "content": "match for random students, for single-hidden-layer students and various teacher ranks", + "type": "text" + }, + { + "bbox": [ + 443, + 370, + 457, + 382 + ], + "score": 0.89, + "content": "\\overline { { N } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 369, + 506, + 384 + ], + "score": 1.0, + "content": ". Intuitively,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 138, + 395 + ], + "score": 1.0, + "content": "as time", + "type": "text" + }, + { + "bbox": [ + 138, + 383, + 144, + 392 + ], + "score": 0.62, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "proceeds, learning corresponds to singular mode detection wave sweeping from large to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "small training data singular values (i.e. the wave in Fig. 1B sweeps across the training data spectrum", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "in Fig 2A). Initially, strong singular values associated with large SNR teacher modes are learned", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 144, + 428 + ], + "score": 1.0, + "content": "and both", + "type": "text" + }, + { + "bbox": [ + 144, + 416, + 163, + 426 + ], + "score": 0.88, + "content": "\\varepsilon _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 414, + 181, + 428 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 181, + 416, + 198, + 426 + ], + "score": 0.88, + "content": "\\varepsilon _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "drop. Fig. 3A-D are for a rank 1 teacher, and so in Fig 3AB we see a single", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 424, + 507, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 507, + 439 + ], + "score": 1.0, + "content": "sharp drop early on, if the teacher SNR is sufficiently high. By contrast, with a rank 3 teacher in Fig.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "score": 1.0, + "content": "3E-H, there are several early drops as the three modes are picked up. However, as time progresses,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 448, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 460 + ], + "score": 1.0, + "content": "the singular mode detection wave penetrates the MP sea, and the student picks up noise structure in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 458, + 392, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 154, + 472 + ], + "score": 1.0, + "content": "the data, so", + "type": "text" + }, + { + "bbox": [ + 154, + 460, + 173, + 470 + ], + "score": 0.82, + "content": "\\varepsilon _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 458, + 214, + 472 + ], + "score": 1.0, + "content": "drops but", + "type": "text" + }, + { + "bbox": [ + 214, + 460, + 230, + 470 + ], + "score": 0.88, + "content": "\\varepsilon _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 458, + 392, + 472 + ], + "score": 1.0, + "content": "rises, indicating the onset of overfitting.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "The main difference between the random student and TA learning curves is that the random student", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "score": 1.0, + "content": "learning is slightly delayed relative to the TA, especially late in training. This is understandable", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "because the TA already knows the singular vectors of the training data, while the random student must", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "learn them. Nevertheless, two of the most important aspects of learning, namely the optimal early", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 518, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 165, + 538 + ], + "score": 1.0, + "content": "stopping time", + "type": "text" + }, + { + "bbox": [ + 165, + 519, + 266, + 534 + ], + "score": 0.93, + "content": "t _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } } \\equiv \\mathrm { a r g m i n } _ { t } \\varepsilon _ { \\mathrm { t e s t } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 518, + 463, + 538 + ], + "score": 1.0, + "content": "and the minimal test error achieved at this time", + "type": "text" + }, + { + "bbox": [ + 463, + 519, + 505, + 534 + ], + "score": 0.91, + "content": "\\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } } \\equiv", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 153, + 545 + ], + "score": 0.94, + "content": "\\mathrm { m i n } _ { t } \\varepsilon _ { \\mathrm { t e s t } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 533, + 506, + 546 + ], + "score": 1.0, + "content": ", match well between TA and random student, as shown in Fig. 3CD. At low teacher", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 543, + 483, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 483, + 557 + ], + "score": 1.0, + "content": "SNRs, the student takes a little longer to learn than the TA, but their optimal test errors match.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "Our theory can also be easily extended to describe the learning dynamics deeper networks. Saxe", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 196, + 585 + ], + "score": 1.0, + "content": "et al. (2013a) derived", + "type": "text" + }, + { + "bbox": [ + 196, + 572, + 222, + 584 + ], + "score": 0.93, + "content": "t ( s , { \\hat { s } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "for networks of arbitrary depth, so we only need to adjust this factor", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "in our formulas, see App. A for details. In Fig. 4 we show that again there is an excellent match", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 593, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 354, + 606 + ], + "score": 1.0, + "content": "between TA networks and theory for student networks with", + "type": "text" + }, + { + "bbox": [ + 354, + 594, + 387, + 605 + ], + "score": 0.92, + "content": "N _ { l } = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 593, + 507, + 606 + ], + "score": 1.0, + "content": "layers (i.e. 3 hidden layers).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 603, + 507, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 507, + 618 + ], + "score": 1.0, + "content": "Randomly-initialized networks show a much longer alignment lag for deeper networks (see App.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "B for details), but the curves are qualitatively similar and optimal stopping errors match. We also", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 627, + 466, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 466, + 640 + ], + "score": 1.0, + "content": "demonstrate extensions of our theory to different numbers of training examples (App. G).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Importantly, many of the phenomena we observe in linear networks are qualitatively replicated", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "in nonlinear networks (Fig. 5), suggesting that our theory may help guide understanding of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "nonlinear case. In particular, features such as stage-like initial learning, followed by a plateau if SNR", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "is high, and finally followed by overfitting, are replicated. However, there are some discrepancies, in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "particular nonlinear networks (especially deeper ones) begin overfitting earlier than linear networks.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "This is likely because a mode in a non-linear network can be co-opted by an orthogonal mode,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "while in a linear network it cannot. Thus noise modes are able to “stow away” on the strong signal", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "modes once they are learned. However, overall learning patterns are similar, and we show below that", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5 + } + ], + "page_idx": 6, + "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 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 79, + 497, + 246 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 79, + 497, + 246 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 79, + 497, + 246 + ], + "spans": [ + { + "bbox": [ + 110, + 79, + 497, + 246 + ], + "score": 0.974, + "type": "image", + "image_path": "be9716a9b1de318c5906d81cbfaaab3bf19f9183b7b3ce2f88ef2d6bf3ba7e80.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 79, + 497, + 134.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 134.66666666666666, + 497, + 190.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 190.33333333333331, + 497, + 245.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 256, + 506, + 323 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 268 + ], + "score": 1.0, + "content": "Figure 4: Our theory applies to deeper networks: match between theory and simulation for rank 1", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 267, + 507, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 306, + 281 + ], + "score": 1.0, + "content": "(row 1, a-d) and rank 3 (row 2, e-h) teachers with", + "type": "text" + }, + { + "bbox": [ + 306, + 268, + 335, + 279 + ], + "score": 0.91, + "content": "n _ { l } = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 267, + 507, + 281 + ], + "score": 1.0, + "content": "students: (a-b, e-f) log train and test error,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 506, + 291 + ], + "score": 1.0, + "content": "respectively, showing very close match between theory and experiment for TA. (c,g) comparing TA", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "and randomly initialized students minimum generalization errors, showing almost perfect match.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 505, + 313 + ], + "score": 1.0, + "content": "(d,h) comparing TA and randomly initialized students optimal stopping times, showing large lag due", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 312, + 403, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 272, + 324 + ], + "score": 1.0, + "content": "to slower alignment in deeper networks.", + "type": "text" + }, + { + "bbox": [ + 272, + 312, + 314, + 323 + ], + "score": 0.87, + "content": "N _ { 1 } = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 312, + 318, + 324 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 318, + 312, + 355, + 323 + ], + "score": 0.9, + "content": "N _ { 2 } = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 312, + 359, + 324 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 359, + 312, + 397, + 323 + ], + "score": 0.89, + "content": "N _ { 3 } = 5 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 312, + 403, + 324 + ], + "score": 1.0, + "content": ".)", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "title", + "bbox": [ + 106, + 339, + 504, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 504, + 349 + ], + "score": 1.0, + "content": "3.4 NUMERICAL TESTS OF THE THEORY OF NEURAL NETWORK GENERALIZATION DYNAMICS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 360, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 372 + ], + "score": 1.0, + "content": "Fig. 3 demonstrates an excellent match between the theory and simulations for the TA, and a close", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 369, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 442, + 384 + ], + "score": 1.0, + "content": "match for random students, for single-hidden-layer students and various teacher ranks", + "type": "text" + }, + { + "bbox": [ + 443, + 370, + 457, + 382 + ], + "score": 0.89, + "content": "\\overline { { N } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 369, + 506, + 384 + ], + "score": 1.0, + "content": ". Intuitively,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 381, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 138, + 395 + ], + "score": 1.0, + "content": "as time", + "type": "text" + }, + { + "bbox": [ + 138, + 383, + 144, + 392 + ], + "score": 0.62, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 381, + 506, + 395 + ], + "score": 1.0, + "content": "proceeds, learning corresponds to singular mode detection wave sweeping from large to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "small training data singular values (i.e. the wave in Fig. 1B sweeps across the training data spectrum", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "in Fig 2A). Initially, strong singular values associated with large SNR teacher modes are learned", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 414, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 144, + 428 + ], + "score": 1.0, + "content": "and both", + "type": "text" + }, + { + "bbox": [ + 144, + 416, + 163, + 426 + ], + "score": 0.88, + "content": "\\varepsilon _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 414, + 181, + 428 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 181, + 416, + 198, + 426 + ], + "score": 0.88, + "content": "\\varepsilon _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 414, + 506, + 428 + ], + "score": 1.0, + "content": "drop. Fig. 3A-D are for a rank 1 teacher, and so in Fig 3AB we see a single", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 424, + 507, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 507, + 439 + ], + "score": 1.0, + "content": "sharp drop early on, if the teacher SNR is sufficiently high. By contrast, with a rank 3 teacher in Fig.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "score": 1.0, + "content": "3E-H, there are several early drops as the three modes are picked up. However, as time progresses,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 448, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 460 + ], + "score": 1.0, + "content": "the singular mode detection wave penetrates the MP sea, and the student picks up noise structure in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 458, + 392, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 154, + 472 + ], + "score": 1.0, + "content": "the data, so", + "type": "text" + }, + { + "bbox": [ + 154, + 460, + 173, + 470 + ], + "score": 0.82, + "content": "\\varepsilon _ { \\mathrm { t r a i n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 458, + 214, + 472 + ], + "score": 1.0, + "content": "drops but", + "type": "text" + }, + { + "bbox": [ + 214, + 460, + 230, + 470 + ], + "score": 0.88, + "content": "\\varepsilon _ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 458, + 392, + 472 + ], + "score": 1.0, + "content": "rises, indicating the onset of overfitting.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 360, + 507, + 472 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 475, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "The main difference between the random student and TA learning curves is that the random student", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "score": 1.0, + "content": "learning is slightly delayed relative to the TA, especially late in training. This is understandable", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "because the TA already knows the singular vectors of the training data, while the random student must", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "learn them. Nevertheless, two of the most important aspects of learning, namely the optimal early", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 518, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 165, + 538 + ], + "score": 1.0, + "content": "stopping time", + "type": "text" + }, + { + "bbox": [ + 165, + 519, + 266, + 534 + ], + "score": 0.93, + "content": "t _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } } \\equiv \\mathrm { a r g m i n } _ { t } \\varepsilon _ { \\mathrm { t e s t } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 518, + 463, + 538 + ], + "score": 1.0, + "content": "and the minimal test error achieved at this time", + "type": "text" + }, + { + "bbox": [ + 463, + 519, + 505, + 534 + ], + "score": 0.91, + "content": "\\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } } \\equiv", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 153, + 545 + ], + "score": 0.94, + "content": "\\mathrm { m i n } _ { t } \\varepsilon _ { \\mathrm { t e s t } } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 533, + 506, + 546 + ], + "score": 1.0, + "content": ", match well between TA and random student, as shown in Fig. 3CD. At low teacher", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 543, + 483, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 483, + 557 + ], + "score": 1.0, + "content": "SNRs, the student takes a little longer to learn than the TA, but their optimal test errors match.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 475, + 506, + 557 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "Our theory can also be easily extended to describe the learning dynamics deeper networks. Saxe", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 196, + 585 + ], + "score": 1.0, + "content": "et al. (2013a) derived", + "type": "text" + }, + { + "bbox": [ + 196, + 572, + 222, + 584 + ], + "score": 0.93, + "content": "t ( s , { \\hat { s } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "for networks of arbitrary depth, so we only need to adjust this factor", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "in our formulas, see App. A for details. In Fig. 4 we show that again there is an excellent match", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 593, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 354, + 606 + ], + "score": 1.0, + "content": "between TA networks and theory for student networks with", + "type": "text" + }, + { + "bbox": [ + 354, + 594, + 387, + 605 + ], + "score": 0.92, + "content": "N _ { l } = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 593, + 507, + 606 + ], + "score": 1.0, + "content": "layers (i.e. 3 hidden layers).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 603, + 507, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 507, + 618 + ], + "score": 1.0, + "content": "Randomly-initialized networks show a much longer alignment lag for deeper networks (see App.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "B for details), but the curves are qualitatively similar and optimal stopping errors match. We also", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 627, + 466, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 466, + 640 + ], + "score": 1.0, + "content": "demonstrate extensions of our theory to different numbers of training examples (App. G).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 561, + 507, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Importantly, many of the phenomena we observe in linear networks are qualitatively replicated", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "in nonlinear networks (Fig. 5), suggesting that our theory may help guide understanding of the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "nonlinear case. In particular, features such as stage-like initial learning, followed by a plateau if SNR", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "is high, and finally followed by overfitting, are replicated. However, there are some discrepancies, in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "particular nonlinear networks (especially deeper ones) begin overfitting earlier than linear networks.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "This is likely because a mode in a non-linear network can be co-opted by an orthogonal mode,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "while in a linear network it cannot. Thus noise modes are able to “stow away” on the strong signal", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "modes once they are learned. However, overall learning patterns are similar, and we show below that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "many interesting phenomena in nonlinear networks are understandable in the linear case, such as the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 497, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 497, + 106 + ], + "score": 1.0, + "content": "(non-)effects of overparameterization, the dynamics of memorization, and the benefits of transfer.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 644, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "many interesting phenomena in nonlinear networks are understandable in the linear case, such as the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 497, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 497, + 106 + ], + "score": 1.0, + "content": "(non-)effects of overparameterization, the dynamics of memorization, and the benefits of transfer.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "image", + "bbox": [ + 109, + 108, + 481, + 291 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 108, + 481, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 108, + 481, + 291 + ], + "spans": [ + { + "bbox": [ + 109, + 108, + 481, + 291 + ], + "score": 0.974, + "type": "image", + "image_path": "4c96fe2cc1b69b79f16e01bb40a77f5c5b9805452e0fd502c4194097a289ca0c.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 109, + 108, + 481, + 169.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 109, + 169.0, + 481, + 230.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 109, + 230.0, + 481, + 291.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 299, + 506, + 378 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 312 + ], + "score": 1.0, + "content": "Figure 5: Train (first row, A-D) and test (second row, E-H) error for nonlinear networks (leaky relu at", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 311, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 506, + 322 + ], + "score": 1.0, + "content": "all hidden layers) with one hidden layer (first two columns) or three hidden layers (last two columns)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "trained on the tasks above, with a rank 1 teacher (first and third columns) or a rank 3 teacher (second", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "and fourth columns). Note that many of the qualitative phenomena observed in linear networks, such", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "as stage-like improvement in the errors, followed by a plateau, followed by overfitting, also appear in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 355, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 366 + ], + "score": 1.0, + "content": "nonlinear networks. Compare the first column to Fig. 3AB, the second column to Fig. 3EF, the third", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 365, + 402, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 271, + 378 + ], + "score": 1.0, + "content": "to Fig. 4AB, and the fourth to Fig. 4EF. (", + "type": "text" + }, + { + "bbox": [ + 271, + 366, + 313, + 377 + ], + "score": 0.87, + "content": "N _ { 1 } = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 366, + 316, + 378 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 317, + 366, + 353, + 376 + ], + "score": 0.89, + "content": "N _ { 2 } = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 366, + 357, + 378 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 357, + 365, + 396, + 377 + ], + "score": 0.87, + "content": "N _ { 3 } = 5 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 366, + 402, + 378 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 116, + 399, + 399, + 410 + ], + "lines": [ + { + "bbox": [ + 114, + 399, + 401, + 411 + ], + "spans": [ + { + "bbox": [ + 114, + 399, + 401, + 411 + ], + "score": 1.0, + "content": ".5 RANDOMIZED DATA VS. REAL DATA: A LEARNING TIME PUZZLE", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 506, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 433 + ], + "score": 1.0, + "content": "An intriguing observation that resurrected the generalization puzzle in deep learning was the ob-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "servation by Zhang et al. (2016) that deep networks can memorize data with the labels randomly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 104, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "permuted. However, as Arpit et al. (2017) pointed out, the learning dynamics of training error for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "randomized labels can be slower than than for structured data. This phenomenon also arises in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "score": 1.0, + "content": "deep linear networks, and our theory yields an analytic explanation for why. We randomize data by", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 225, + 487 + ], + "score": 1.0, + "content": "choosing orthonormal inputs", + "type": "text" + }, + { + "bbox": [ + 226, + 475, + 238, + 485 + ], + "score": 0.86, + "content": "\\hat { \\mathbf { x } } ^ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 474, + 449, + 487 + ], + "score": 1.0, + "content": "as in the structured case, but we choose the outputs", + "type": "text" + }, + { + "bbox": [ + 449, + 475, + 462, + 486 + ], + "score": 0.89, + "content": "{ \\hat { \\mathbf { y } } } ^ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "to be i.i.d.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "score": 1.0, + "content": "Gaussian with zero mean and the same diagonal variance as the structured training data generated by", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 438, + 509 + ], + "score": 1.0, + "content": "the teacher. For structured data generated by a low rank teacher with singular values", + "type": "text" + }, + { + "bbox": [ + 439, + 497, + 450, + 507 + ], + "score": 0.88, + "content": "\\overline { { s } } _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 496, + 505, + 509 + ], + "score": 1.0, + "content": ", the diagonal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 103, + 504, + 510, + 530 + ], + "spans": [ + { + "bbox": [ + 103, + 504, + 215, + 530 + ], + "score": 1.0, + "content": "output variance is given by", + "type": "text" + }, + { + "bbox": [ + 216, + 507, + 337, + 527 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\sigma _ { r } ^ { 2 } = \\frac { 1 } { N _ { 3 } } \\left[ \\sum _ { i = \\alpha } ^ { \\overline { { N } } _ { 2 } } \\bar { s } _ { \\alpha } ^ { 2 } \\right] + \\frac { 1 } { \\overline { { N } } _ { 1 } } \\sigma _ { z } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 504, + 367, + 530 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 368, + 513, + 379, + 522 + ], + "score": 0.84, + "content": "\\sigma _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 504, + 510, + 530 + ], + "score": 1.0, + "content": "is the noise variance, as before.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 322, + 539 + ], + "score": 1.0, + "content": "Since there is no relation between input and output,", + "type": "text" + }, + { + "bbox": [ + 323, + 526, + 340, + 537 + ], + "score": 0.87, + "content": "\\pmb { \\Sigma } ^ { 3 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 525, + 505, + 539 + ], + "score": 1.0, + "content": "is now distributed as a MP distribution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 177, + 552 + ], + "score": 1.0, + "content": "whose support is", + "type": "text" + }, + { + "bbox": [ + 177, + 538, + 293, + 551 + ], + "score": 0.89, + "content": "[ ( \\sigma _ { r } ( 1 - \\sqrt { \\mathcal { A } } ) , \\sigma _ { r } ( 1 + \\sqrt { \\mathcal { A } } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 539, + 506, + 552 + ], + "score": 1.0, + "content": ". 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Note that many of the qualitative phenomena observed in linear networks, such", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "as stage-like improvement in the errors, followed by a plateau, followed by overfitting, also appear in", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 355, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 366 + ], + "score": 1.0, + "content": "nonlinear networks. Compare the first column to Fig. 3AB, the second column to Fig. 3EF, the third", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 365, + 402, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 271, + 378 + ], + "score": 1.0, + "content": "to Fig. 4AB, and the fourth to Fig. 4EF. (", + "type": "text" + }, + { + "bbox": [ + 271, + 366, + 313, + 377 + ], + "score": 0.87, + "content": "N _ { 1 } = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 366, + 316, + 378 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 317, + 366, + 353, + 376 + ], + "score": 0.89, + "content": "N _ { 2 } = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 366, + 357, + 378 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 357, + 365, + 396, + 377 + ], + "score": 0.87, + "content": "N _ { 3 } = 5 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 366, + 402, + 378 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 116, + 399, + 399, + 410 + ], + "lines": [ + { + "bbox": [ + 114, + 399, + 401, + 411 + ], + "spans": [ + { + "bbox": [ + 114, + 399, + 401, + 411 + ], + "score": 1.0, + "content": ".5 RANDOMIZED DATA VS. REAL DATA: A LEARNING TIME PUZZLE", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 506, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 419, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 433 + ], + "score": 1.0, + "content": "An intriguing observation that resurrected the generalization puzzle in deep learning was the ob-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "servation by Zhang et al. (2016) that deep networks can memorize data with the labels randomly", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 104, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "permuted. However, as Arpit et al. (2017) pointed out, the learning dynamics of training error for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "randomized labels can be slower than than for structured data. This phenomenon also arises in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "score": 1.0, + "content": "deep linear networks, and our theory yields an analytic explanation for why. We randomize data by", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 225, + 487 + ], + "score": 1.0, + "content": "choosing orthonormal inputs", + "type": "text" + }, + { + "bbox": [ + 226, + 475, + 238, + 485 + ], + "score": 0.86, + "content": "\\hat { \\mathbf { x } } ^ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 474, + 449, + 487 + ], + "score": 1.0, + "content": "as in the structured case, but we choose the outputs", + "type": "text" + }, + { + "bbox": [ + 449, + 475, + 462, + 486 + ], + "score": 0.89, + "content": "{ \\hat { \\mathbf { y } } } ^ { \\mu }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "to be i.i.d.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "score": 1.0, + "content": "Gaussian with zero mean and the same diagonal variance as the structured training data generated by", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 496, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 438, + 509 + ], + "score": 1.0, + "content": "the teacher. For structured data generated by a low rank teacher with singular values", + "type": "text" + }, + { + "bbox": [ + 439, + 497, + 450, + 507 + ], + "score": 0.88, + "content": "\\overline { { s } } _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 496, + 505, + 509 + ], + "score": 1.0, + "content": ", the diagonal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 103, + 504, + 510, + 530 + ], + "spans": [ + { + "bbox": [ + 103, + 504, + 215, + 530 + ], + "score": 1.0, + "content": "output variance is given by", + "type": "text" + }, + { + "bbox": [ + 216, + 507, + 337, + 527 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\sigma _ { r } ^ { 2 } = \\frac { 1 } { N _ { 3 } } \\left[ \\sum _ { i = \\alpha } ^ { \\overline { { N } } _ { 2 } } \\bar { s } _ { \\alpha } ^ { 2 } \\right] + \\frac { 1 } { \\overline { { N } } _ { 1 } } \\sigma _ { z } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 504, + 367, + 530 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 368, + 513, + 379, + 522 + ], + "score": 0.84, + "content": "\\sigma _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 504, + 510, + 530 + ], + "score": 1.0, + "content": "is the noise variance, as before.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 525, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 322, + 539 + ], + "score": 1.0, + "content": "Since there is no relation between input and output,", + "type": "text" + }, + { + "bbox": [ + 323, + 526, + 340, + 537 + ], + "score": 0.87, + "content": "\\pmb { \\Sigma } ^ { 3 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 525, + 505, + 539 + ], + "score": 1.0, + "content": "is now distributed as a MP distribution", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 177, + 552 + ], + "score": 1.0, + "content": "whose support is", + "type": "text" + }, + { + "bbox": [ + 177, + 538, + 293, + 551 + ], + "score": 0.89, + "content": "[ ( \\sigma _ { r } ( 1 - \\sqrt { \\mathcal { A } } ) , \\sigma _ { r } ( 1 + \\sqrt { \\mathcal { A } } ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 539, + 506, + 552 + ], + "score": 1.0, + "content": ". Thus randomization essentially destroys the outlier", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 205, + 563 + ], + "score": 1.0, + "content": "signal singular values in", + "type": "text" + }, + { + "bbox": [ + 206, + 550, + 223, + 560 + ], + "score": 0.85, + "content": "\\pmb { \\Sigma } ^ { 3 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "reflecting the teacher, and distributes them across all randomized data", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "modes, yielding this stretched MP distribution (compare 6A top and bottom). 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Tasks with very strong signals will show little effect from other tasks (note", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 406, + 460, + 419 + ], + "score": 1.0, + "content": "y-axis scales), but any impact will be negative unless the tasks are very well aligned.", + "type": "text" + }, + { + "bbox": [ + 461, + 406, + 505, + 417 + ], + "score": 0.9, + "content": "N _ { 1 } = 1 0 0", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 416, + 259, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 171, + 432 + ], + "score": 0.89, + "content": "\\dot { \\overline { { { N } } } } _ { 2 } ^ { A } = \\overline { { { N } } } _ { 2 } ^ { B } = 1 .", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 417, + 174, + 433 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 174, + 419, + 212, + 432 + ], + "score": 0.79, + "content": "N _ { 2 } = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 417, + 215, + 433 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 215, + 420, + 253, + 432 + ], + "score": 0.79, + "content": "N _ { 3 } = 5 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 417, + 259, + 433 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 506, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "modes, yielding this stretched MP distribution (compare 6A top and bottom). However, even on this", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "stretched MP distribution, the right edge will be much smaller than the signal singular values, since", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "the signal variance will be diluted by spreading it out over many more modes in the randomized data.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "Thus the randomized data will lead to slower initial training error drops relative to the structured", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "data (Fig. 6B) since the singular mode detection wave encounters the first signal singular values in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 501, + 482, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 482, + 512 + ], + "score": 1.0, + "content": "structured data earlier than it encounters the edge of the stretched MP sea in randomized data.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 107, + 528, + 504, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 504, + 538 + ], + "score": 1.0, + "content": "3.6 OUT-PERFORMING OPTIMAL EARLY STOPPING THROUGH A NON-GRADIENT ALGORITHM", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 548, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "For the case of a rank 1 teacher, it is straightforward to derive a good analytic approximation to the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 559, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 189, + 575 + ], + "score": 1.0, + "content": "important quantities", + "type": "text" + }, + { + "bbox": [ + 189, + 560, + 218, + 575 + ], + "score": 0.89, + "content": "\\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 559, + 254, + 572 + ], + "score": 1.0, + "content": "and topt", + "type": "text" + }, + { + "bbox": [ + 236, + 560, + 264, + 574 + ], + "score": 0.89, + "content": "t _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 561, + 505, + 576 + ], + "score": 1.0, + "content": ". We assume the teacher SNR is beyond the phase transition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 573, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 238, + 590 + ], + "score": 1.0, + "content": "point so its unique singular value", + "type": "text" + }, + { + "bbox": [ + 238, + 574, + 282, + 586 + ], + "score": 0.92, + "content": "\\overline { { s } } _ { 1 } > A ^ { 1 / 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 573, + 506, + 590 + ], + "score": 1.0, + "content": ", yielding a separation between the training data singular", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 130, + 599 + ], + "score": 1.0, + "content": "value", + "type": "text" + }, + { + "bbox": [ + 130, + 587, + 141, + 597 + ], + "score": 0.87, + "content": "\\hat { s _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "in (11) and the edge of the MP sea. In this scenario, optimal early stopping will occur at a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "time before the detection wave in Fig. 1B penetrates the MP sea, so to minimize test error, we can", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "neglect the first term in (15). Then optimizing the second term yields the optimal student singular", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 618, + 507, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 130, + 633 + ], + "score": 1.0, + "content": "value", + "type": "text" + }, + { + "bbox": [ + 130, + 620, + 187, + 632 + ], + "score": 0.93, + "content": "s _ { 1 } = \\overline { { s } } _ { 1 } \\mathcal { O } ( \\overline { { s } } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 619, + 333, + 633 + ], + "score": 1.0, + "content": ". Inserting this value into (15) yields", + "type": "text" + }, + { + "bbox": [ + 333, + 618, + 507, + 635 + ], + "score": 1.0, + "content": "εoptgradient = 1 − O(s1)2, and inserting it into", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 100, + 628, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 100, + 628, + 146, + 653 + ], + "score": 1.0, + "content": "(9) yields", + "type": "text" + }, + { + "bbox": [ + 147, + 633, + 174, + 648 + ], + "score": 0.89, + "content": "t _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 628, + 505, + 653 + ], + "score": 1.0, + "content": ". Thus the optimal generalization error with a rank 1 teacher is very simply related", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 648, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 659 + ], + "score": 1.0, + "content": "to the alignment of the top training data singular vectors with the teacher singular vectors, and it", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 658, + 459, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 459, + 669 + ], + "score": 1.0, + "content": "decreases as this alignment increases. In App. E, we show this match in the rank 1 case.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 674, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 101, + 667, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 101, + 667, + 218, + 701 + ], + "score": 1.0, + "content": "With higher rank teachers, teacher modes with differen", + "type": "text" + }, + { + "bbox": [ + 218, + 675, + 248, + 688 + ], + "score": 0.87, + "content": "\\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 667, + 267, + 701 + ], + "score": 1.0, + "content": "and For", + "type": "text" + }, + { + "bbox": [ + 296, + 667, + 506, + 701 + ], + "score": 1.0, + "content": "must negotiate a more complex trade-off betweene, as the singular mode detection wave passes the top", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 219, + 711 + ], + "score": 1.0, + "content": "training data singular value,", + "type": "text" + }, + { + "bbox": [ + 219, + 699, + 265, + 711 + ], + "score": 0.92, + "content": "s _ { 1 } ( t ) \\to \\hat { s } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 699, + 399, + 711 + ], + "score": 1.0, + "content": "which is greater than the optimal", + "type": "text" + }, + { + "bbox": [ + 399, + 699, + 457, + 711 + ], + "score": 0.93, + "content": "s _ { 1 } = \\overline { { s } } _ { 1 } \\mathcal { O } ( \\overline { { s } } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "for mode 1.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "Thus as learning progresses, the student overfits on the first mode but learns lower modes. 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(a)", + "type": "text" + }, + { + "bbox": [ + 429, + 349, + 503, + 362 + ], + "score": 0.92, + "content": "\\overline { { s } } _ { A } = 0 . 8 4 = \\sqrt [ 4 ] { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 348, + 507, + 365 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "score": 1.0, + "content": "Although this task is impossible to learn on its own, with support from another aligned task, especially", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 278, + 385 + ], + "score": 1.0, + "content": "one with high SNR, learning can occur. (b)", + "type": "text" + }, + { + "bbox": [ + 279, + 374, + 309, + 384 + ], + "score": 0.9, + "content": "{ \\overline { { s } } } _ { A } = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 373, + 505, + 385 + ], + "score": 1.0, + "content": ". Tasks with modest signals will face interference", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 382, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 397 + ], + "score": 1.0, + "content": "from poorly aligned tasks, but benefits from well aligned tasks. These effects are amplified by", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 145, + 407 + ], + "score": 1.0, + "content": "SNR. 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Tasks with very strong signals will show little effect from other tasks (note", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 104, + 406, + 460, + 419 + ], + "score": 1.0, + "content": "y-axis scales), but any impact will be negative unless the tasks are very well aligned.", + "type": "text" + }, + { + "bbox": [ + 461, + 406, + 505, + 417 + ], + "score": 0.9, + "content": "N _ { 1 } = 1 0 0", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 416, + 259, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 171, + 432 + ], + "score": 0.89, + "content": "\\dot { \\overline { { { N } } } } _ { 2 } ^ { A } = \\overline { { { N } } } _ { 2 } ^ { B } = 1 .", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 417, + 174, + 433 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 174, + 419, + 212, + 432 + ], + "score": 0.79, + "content": "N _ { 2 } = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 417, + 215, + 433 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 215, + 420, + 253, + 432 + ], + "score": 0.79, + "content": "N _ { 3 } = 5 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 417, + 259, + 433 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 506, + 511 + ], + "lines": [], + "index": 17.5, + "bbox_fs": [ + 105, + 445, + 506, + 512 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 528, + 504, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 527, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 504, + 538 + ], + "score": 1.0, + "content": "3.6 OUT-PERFORMING OPTIMAL EARLY STOPPING THROUGH A NON-GRADIENT ALGORITHM", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 548, + 505, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "For the case of a rank 1 teacher, it is straightforward to derive a good analytic approximation to the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 559, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 189, + 575 + ], + "score": 1.0, + "content": "important quantities", + "type": "text" + }, + { + "bbox": [ + 189, + 560, + 218, + 575 + ], + "score": 0.89, + "content": "\\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 559, + 254, + 572 + ], + "score": 1.0, + "content": "and topt", + "type": "text" + }, + { + "bbox": [ + 236, + 560, + 264, + 574 + ], + "score": 0.89, + "content": "t _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 561, + 505, + 576 + ], + "score": 1.0, + "content": ". We assume the teacher SNR is beyond the phase transition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 573, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 238, + 590 + ], + "score": 1.0, + "content": "point so its unique singular value", + "type": "text" + }, + { + "bbox": [ + 238, + 574, + 282, + 586 + ], + "score": 0.92, + "content": "\\overline { { s } } _ { 1 } > A ^ { 1 / 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 573, + 506, + 590 + ], + "score": 1.0, + "content": ", yielding a separation between the training data singular", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 130, + 599 + ], + "score": 1.0, + "content": "value", + "type": "text" + }, + { + "bbox": [ + 130, + 587, + 141, + 597 + ], + "score": 0.87, + "content": "\\hat { s _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 586, + 506, + 599 + ], + "score": 1.0, + "content": "in (11) and the edge of the MP sea. In this scenario, optimal early stopping will occur at a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "time before the detection wave in Fig. 1B penetrates the MP sea, so to minimize test error, we can", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "neglect the first term in (15). Then optimizing the second term yields the optimal student singular", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 618, + 507, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 130, + 633 + ], + "score": 1.0, + "content": "value", + "type": "text" + }, + { + "bbox": [ + 130, + 620, + 187, + 632 + ], + "score": 0.93, + "content": "s _ { 1 } = \\overline { { s } } _ { 1 } \\mathcal { O } ( \\overline { { s } } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 619, + 333, + 633 + ], + "score": 1.0, + "content": ". Inserting this value into (15) yields", + "type": "text" + }, + { + "bbox": [ + 333, + 618, + 507, + 635 + ], + "score": 1.0, + "content": "εoptgradient = 1 − O(s1)2, and inserting it into", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 100, + 628, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 100, + 628, + 146, + 653 + ], + "score": 1.0, + "content": "(9) yields", + "type": "text" + }, + { + "bbox": [ + 147, + 633, + 174, + 648 + ], + "score": 0.89, + "content": "t _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 628, + 505, + 653 + ], + "score": 1.0, + "content": ". Thus the optimal generalization error with a rank 1 teacher is very simply related", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 648, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 506, + 659 + ], + "score": 1.0, + "content": "to the alignment of the top training data singular vectors with the teacher singular vectors, and it", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 658, + 459, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 459, + 669 + ], + "score": 1.0, + "content": "decreases as this alignment increases. In App. E, we show this match in the rank 1 case.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5, + "bbox_fs": [ + 100, + 549, + 507, + 669 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 674, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 101, + 667, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 101, + 667, + 218, + 701 + ], + "score": 1.0, + "content": "With higher rank teachers, teacher modes with differen", + "type": "text" + }, + { + "bbox": [ + 218, + 675, + 248, + 688 + ], + "score": 0.87, + "content": "\\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 667, + 267, + 701 + ], + "score": 1.0, + "content": "and For", + "type": "text" + }, + { + "bbox": [ + 296, + 667, + 506, + 701 + ], + "score": 1.0, + "content": "must negotiate a more complex trade-off betweene, as the singular mode detection wave passes the top", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 219, + 711 + ], + "score": 1.0, + "content": "training data singular value,", + "type": "text" + }, + { + "bbox": [ + 219, + 699, + 265, + 711 + ], + "score": 0.92, + "content": "s _ { 1 } ( t ) \\to \\hat { s } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 699, + 399, + 711 + ], + "score": 1.0, + "content": "which is greater than the optimal", + "type": "text" + }, + { + "bbox": [ + 399, + 699, + 457, + 711 + ], + "score": 0.93, + "content": "s _ { 1 } = \\overline { { s } } _ { 1 } \\mathcal { O } ( \\overline { { s } } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "for mode 1.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "Thus as learning progresses, the student overfits on the first mode but learns lower modes. However,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "this neural generalization dynamics suggests a superior non-gradient training algorithm that simply", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 81, + 443, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 185, + 96 + ], + "score": 1.0, + "content": "optimally sets each", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 185, + 84, + 197, + 93 + ], + "score": 0.86, + "content": "s _ { \\alpha }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 197, + 81, + 208, + 96 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 208, + 83, + 246, + 95 + ], + "score": 0.93, + "content": "\\overline { { s } } _ { \\alpha } \\mathcal { O } ( \\overline { { s } } _ { \\alpha } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 246, + 81, + 443, + 96 + ], + "score": 1.0, + "content": "in (15), yielding an optimal generalization error:", + "type": "text", + "cross_page": true + } + ], + "index": 0 + } + ], + "index": 33.5, + "bbox_fs": [ + 101, + 667, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 442, + 95 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 443, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 185, + 96 + ], + "score": 1.0, + "content": "optimally sets each", + "type": "text" + }, + { + "bbox": [ + 185, + 84, + 197, + 93 + ], + "score": 0.86, + "content": "s _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 81, + 208, + 96 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 208, + 83, + 246, + 95 + ], + "score": 0.93, + "content": "\\overline { { s } } _ { \\alpha } \\mathcal { O } ( \\overline { { s } } _ { \\alpha } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 81, + 443, + 96 + ], + "score": 1.0, + "content": "in (15), yielding an optimal generalization error:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 98, + 410, + 140 + ], + "lines": [ + { + "bbox": [ + 199, + 98, + 410, + 140 + ], + "spans": [ + { + "bbox": [ + 199, + 98, + 410, + 140 + ], + "score": 0.95, + "content": "\\varepsilon _ { \\mathrm { n o n - g r a d i e n t } } ^ { \\mathrm { o p t } } = \\left[ \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 2 } } \\overline { { s } } _ { \\alpha } ^ { 2 } \\right] ^ { - 1 } \\left[ \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 2 } } \\overline { { s } } _ { \\alpha } ^ { 2 } ( 1 - \\mathcal { O } ( \\overline { { s } } _ { \\alpha } ) ^ { 2 } ) \\right] .", + "type": "interline_equation", + "image_path": "570f4b4fad0704cfefbb961c062c1f4efb04e2f7fe004d72fb17f3060a73f853.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 98, + 410, + 119.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 199, + 119.0, + 410, + 140.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 506, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "Standard gradient descent learning cannot achieve this low generalization error because it cannot", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 103, + 150, + 508, + 174 + ], + "spans": [ + { + "bbox": [ + 103, + 150, + 441, + 174 + ], + "score": 1.0, + "content": "independently adjust all student singular values. A simple algorithm that achieves", + "type": "text" + }, + { + "bbox": [ + 441, + 154, + 483, + 169 + ], + "score": 0.87, + "content": "\\varepsilon _ { \\mathrm { n o n - g r a d i e n t } } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 150, + 508, + 174 + ], + "score": 1.0, + "content": "is as", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 166, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 275, + 182 + ], + "score": 1.0, + "content": "follows. From the training data covariance", + "type": "text" + }, + { + "bbox": [ + 276, + 168, + 293, + 179 + ], + "score": 0.88, + "content": "\\pmb { \\Sigma } ^ { 3 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 166, + 416, + 182 + ], + "score": 1.0, + "content": ", extract the top singular values", + "type": "text" + }, + { + "bbox": [ + 417, + 169, + 428, + 180 + ], + "score": 0.89, + "content": "\\hat { s } _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 166, + 506, + 182 + ], + "score": 1.0, + "content": "that pop-out of the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 178, + 507, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 326, + 194 + ], + "score": 1.0, + "content": "MP sea, use the functional inverse of (11) to compute", + "type": "text" + }, + { + "bbox": [ + 326, + 180, + 355, + 192 + ], + "score": 0.93, + "content": "\\overline { { s } } _ { a } ( \\widehat { s } _ { \\alpha } )", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 178, + 491, + 194 + ], + "score": 1.0, + "content": ", use (12) to compute the optimal", + "type": "text" + }, + { + "bbox": [ + 491, + 182, + 502, + 191 + ], + "score": 0.84, + "content": "s _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 178, + 507, + 194 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 188, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 214, + 204 + ], + "score": 1.0, + "content": "and then construct a matrix", + "type": "text" + }, + { + "bbox": [ + 214, + 191, + 228, + 201 + ], + "score": 0.26, + "content": "\\mathbf { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 188, + 374, + 204 + ], + "score": 1.0, + "content": "with the same top singular vectors as", + "type": "text" + }, + { + "bbox": [ + 374, + 191, + 391, + 201 + ], + "score": 0.89, + "content": "\\pmb { \\Sigma } ^ { 3 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 188, + 506, + 204 + ], + "score": 1.0, + "content": ", but with the outlier singular", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 202, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 188, + 215 + ], + "score": 1.0, + "content": "values shrunk from", + "type": "text" + }, + { + "bbox": [ + 189, + 203, + 200, + 213 + ], + "score": 0.88, + "content": "\\hat { s } _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 202, + 212, + 215 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 213, + 204, + 224, + 213 + ], + "score": 0.86, + "content": "s _ { \\alpha }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 202, + 505, + 215 + ], + "score": 1.0, + "content": "and the rest set to zero. This non-gradient singular value shrinkage", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 208, + 439, + 232 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 352, + 232 + ], + "score": 1.0, + "content": "algorithm provably outperforms neural network training with", + "type": "text" + }, + { + "bbox": [ + 353, + 212, + 437, + 227 + ], + "score": 0.7, + "content": "\\varepsilon _ { \\mathrm { n o n - g r a d i e n t } } ^ { \\mathrm { o p t } ^ { - } } \\le \\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } } .", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 218, + 439, + 228 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 241, + 502, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 502, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 502, + 255 + ], + "score": 1.0, + "content": "4 A THEORY FOR THE TRANSFER OF KNOWLEDGE ACROSS MULTIPLE TASKS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 506, + 372 + ], + "lines": [ + { + "bbox": [ + 104, + 260, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 183, + 279 + ], + "score": 1.0, + "content": "Consider two tasks", + "type": "text" + }, + { + "bbox": [ + 184, + 266, + 192, + 275 + ], + "score": 0.8, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 260, + 210, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 210, + 266, + 219, + 275 + ], + "score": 0.8, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 260, + 276, + 279 + ], + "score": 1.0, + "content": ", described by", + "type": "text" + }, + { + "bbox": [ + 276, + 264, + 290, + 276 + ], + "score": 0.9, + "content": "\\overline { { N } } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 260, + 304, + 279 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 304, + 264, + 319, + 276 + ], + "score": 0.89, + "content": "\\overline { { N } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 260, + 374, + 279 + ], + "score": 1.0, + "content": "teacher maps", + "type": "text" + }, + { + "bbox": [ + 374, + 261, + 394, + 276 + ], + "score": 0.87, + "content": "\\overline { { \\mathbf { W } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 260, + 412, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 412, + 261, + 432, + 276 + ], + "score": 0.86, + "content": "\\overline { { \\mathbf { W } } } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 260, + 470, + 279 + ], + "score": 1.0, + "content": ", of ranks", + "type": "text" + }, + { + "bbox": [ + 470, + 261, + 487, + 277 + ], + "score": 0.91, + "content": "\\overline { { N } } _ { 2 } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 260, + 506, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 276, + 507, + 293 + ], + "spans": [ + { + "bbox": [ + 107, + 276, + 123, + 291 + ], + "score": 0.9, + "content": "\\overline { { N } } _ { 2 } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 276, + 507, + 293 + ], + "score": 1.0, + "content": ", respectively. Now two student networks can learn from the two teacher networks separately,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 101, + 286, + 510, + 315 + ], + "spans": [ + { + "bbox": [ + 101, + 286, + 305, + 315 + ], + "score": 1.0, + "content": "each achieving optimal early stopping test errors composite teacher (and student) that concatenate", + "type": "text" + }, + { + "bbox": [ + 306, + 291, + 321, + 304 + ], + "score": 0.9, + "content": "\\varepsilon _ { A } ^ { \\mathrm { { o p t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 286, + 339, + 315 + ], + "score": 1.0, + "content": "and hid", + "type": "text" + }, + { + "bbox": [ + 339, + 290, + 354, + 304 + ], + "score": 0.92, + "content": "\\varepsilon _ { B } ^ { \\mathrm { { o p t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 286, + 510, + 315 + ], + "score": 1.0, + "content": ". Alternatively, one could construct a and output units, but shares the same", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 313, + 504, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 121, + 337 + ], + "score": 0.88, + "content": "\\overline { { N } } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 313, + 477, + 347 + ], + "score": 1.0, + "content": "ut units (Fig. 7). The composite student and teacher each have two heads, one for each tasneurons per head. Optimal early stopping on each head of the student yields test errors opt opt", + "type": "text" + }, + { + "bbox": [ + 477, + 325, + 504, + 338 + ], + "score": 0.87, + "content": "\\varepsilon _ { A B } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 101, + 331, + 503, + 358 + ], + "spans": [ + { + "bbox": [ + 101, + 331, + 123, + 358 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 150, + 331, + 408, + 358 + ], + "score": 1.0, + "content": ". We define the transfer benefit that task B confers on task A to be", + "type": "text" + }, + { + "bbox": [ + 409, + 337, + 503, + 351 + ], + "score": 0.9, + "content": "\\mathcal { T } ^ { A B } \\equiv \\varepsilon _ { A } ^ { \\mathrm { o p t } } - \\varepsilon _ { A B } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "A postive (negative) transfer benefit implies learning tasks A and B simultaneously yields a lower", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 361, + 412, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 412, + 373 + ], + "score": 1.0, + "content": "(higher) optimal test error on task A compared to just learning task A alone.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 506, + 500 + ], + "lines": [ + { + "bbox": [ + 104, + 373, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 373, + 320, + 391 + ], + "score": 1.0, + "content": "A foundational question is how the transfer benefit", + "type": "text" + }, + { + "bbox": [ + 320, + 376, + 350, + 388 + ], + "score": 0.9, + "content": "\\scriptstyle { \\mathcal { T } } ^ { A B }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 373, + 506, + 391 + ], + "score": 1.0, + "content": "depends on the two tasks defined by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 387, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 159, + 403 + ], + "score": 1.0, + "content": "the teachers", + "type": "text" + }, + { + "bbox": [ + 159, + 388, + 179, + 402 + ], + "score": 0.88, + "content": "\\overline { { \\mathbf { W } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 387, + 199, + 405 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 199, + 388, + 219, + 402 + ], + "score": 0.86, + "content": "\\overline { { \\mathbf { W } } } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 387, + 471, + 405 + ], + "score": 1.0, + "content": ". To answer this, consider the SVDs of each teacher alone:", + "type": "text" + }, + { + "bbox": [ + 471, + 388, + 505, + 403 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\overline { { \\mathbf { W } } } ^ { A } = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 401, + 507, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 157, + 418 + ], + "score": 0.88, + "content": "\\overline { { \\mathbf { U } } } ^ { A } \\overline { { \\mathbf { S } } } ^ { A } \\overline { { \\mathbf { V } } } ^ { A ^ { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 401, + 177, + 418 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 177, + 403, + 262, + 418 + ], + "score": 0.91, + "content": "\\overline { { \\mathbf { W } } } ^ { B } = \\overline { { \\mathbf { U } } } ^ { B } \\overline { { \\mathbf { S } } } ^ { B } \\overline { { \\mathbf { V } } } ^ { B ^ { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 402, + 397, + 424 + ], + "score": 1.0, + "content": ". From the above, we know that", + "type": "text" + }, + { + "bbox": [ + 397, + 406, + 413, + 420 + ], + "score": 0.88, + "content": "\\varepsilon _ { A } ^ { \\mathrm { { o p t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 402, + 463, + 424 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 463, + 404, + 483, + 418 + ], + "score": 0.89, + "content": "\\overline { { \\mathbf { W } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 402, + 507, + 424 + ], + "score": 1.0, + "content": "only", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 419, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 139, + 437 + ], + "score": 1.0, + "content": "through", + "type": "text" + }, + { + "bbox": [ + 139, + 419, + 153, + 433 + ], + "score": 0.84, + "content": "\\overline { { \\mathbf { S } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 419, + 394, + 437 + ], + "score": 1.0, + "content": ". In App. D we show that the transfer benefit depends on both", + "type": "text" + }, + { + "bbox": [ + 395, + 420, + 414, + 433 + ], + "score": 0.87, + "content": "\\overline { { \\mathbf { W } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 419, + 432, + 437 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 432, + 419, + 452, + 433 + ], + "score": 0.85, + "content": "\\overline { { \\mathbf { W } } } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 419, + 506, + 437 + ], + "score": 1.0, + "content": "only through", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 430, + 504, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 138, + 451 + ], + "score": 0.81, + "content": "\\overline { { \\mathbf { S } } } ^ { A } , \\overline { { \\mathbf { S } } } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 430, + 173, + 456 + ], + "score": 1.0, + "content": ", and the", + "type": "text" + }, + { + "bbox": [ + 174, + 435, + 190, + 451 + ], + "score": 0.91, + "content": "\\overline { { N } } _ { 2 } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 430, + 204, + 456 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 204, + 435, + 221, + 451 + ], + "score": 0.91, + "content": "\\overline { { N } } _ { 2 } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 430, + 291, + 456 + ], + "score": 1.0, + "content": "similarity matrix", + "type": "text" + }, + { + "bbox": [ + 291, + 434, + 351, + 451 + ], + "score": 0.92, + "content": "\\mathbf { \\overline { { Q } } } = \\mathbf { \\overline { { V } } } ^ { A ^ { T } } \\mathbf { \\overline { { V } } } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 430, + 493, + 456 + ], + "score": 1.0, + "content": ". If we think of the columns of each", + "type": "text" + }, + { + "bbox": [ + 494, + 438, + 504, + 450 + ], + "score": 0.73, + "content": "\\overline { { \\mathbf { V } } }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 291, + 464 + ], + "score": 1.0, + "content": "as spanning a low dimensional feature space in", + "type": "text" + }, + { + "bbox": [ + 292, + 451, + 307, + 462 + ], + "score": 0.89, + "content": "\\overline { { N } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "dimensional input space that is important for each", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 147, + 477 + ], + "score": 1.0, + "content": "task, then", + "type": "text" + }, + { + "bbox": [ + 148, + 462, + 158, + 475 + ], + "score": 0.84, + "content": "\\overline { { \\mathbf { Q } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "reflects the input feature subspace similarity matrix. Interestingly, the transfer benefit", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 474, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 474, + 272, + 491 + ], + "score": 1.0, + "content": "is independent of output singular vectors", + "type": "text" + }, + { + "bbox": [ + 272, + 474, + 289, + 488 + ], + "score": 0.86, + "content": "\\bar { \\mathbf { U } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 474, + 307, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 307, + 474, + 323, + 488 + ], + "score": 0.85, + "content": "\\overline { { \\mathbf { U } } } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 474, + 506, + 491 + ], + "score": 1.0, + "content": ". What matters for knowledge transfer in this", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 488, + 402, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 402, + 501 + ], + "score": 1.0, + "content": "setting are the relevant input features, not how you must respond to them.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 103, + 504, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 103, + 504, + 440, + 523 + ], + "score": 1.0, + "content": "We describe the transfer benefit for the simple case of two rank one teachers. Then", + "type": "text" + }, + { + "bbox": [ + 441, + 505, + 473, + 520 + ], + "score": 0.52, + "content": "\\overline { { \\mathbf { S } } } ^ { A } , \\overline { { \\mathbf { S } } } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 504, + 494, + 523 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 494, + 507, + 505, + 520 + ], + "score": 0.68, + "content": "\\overline { { \\mathbf { Q } } }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 517, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 517, + 178, + 534 + ], + "score": 1.0, + "content": "are simply scalars", + "type": "text" + }, + { + "bbox": [ + 178, + 522, + 207, + 531 + ], + "score": 0.29, + "content": "s _ { A } , s _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 517, + 225, + 534 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 522, + 231, + 532 + ], + "score": 0.79, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 517, + 345, + 534 + ], + "score": 1.0, + "content": ", and we explore the function", + "type": "text" + }, + { + "bbox": [ + 346, + 519, + 419, + 532 + ], + "score": 0.93, + "content": "\\mathcal { T } ^ { A B } ( \\overline { { s } } _ { A } , \\overline { { s } } _ { B } , q )", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 517, + 506, + 534 + ], + "score": 1.0, + "content": "in Fig. 5ABC, which", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "reveals several interesting features. First, knowledge can be transferred from a high SNR task to a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 433, + 554 + ], + "score": 1.0, + "content": "low SNR task (Fig. 5A) and the degree of transfer increases with task alignment", + "type": "text" + }, + { + "bbox": [ + 433, + 543, + 439, + 553 + ], + "score": 0.75, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 542, + 505, + 554 + ], + "score": 1.0, + "content": ". This can make", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 551, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 264, + 567 + ], + "score": 1.0, + "content": "it possible to capture signals from task", + "type": "text" + }, + { + "bbox": [ + 265, + 553, + 273, + 563 + ], + "score": 0.72, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 551, + 506, + 567 + ], + "score": 1.0, + "content": "which would otherwise sink into the MP sea by learning", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 563, + 504, + 576 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 495, + 576 + ], + "score": 1.0, + "content": "jointly with a related task, even if the tasks are only weakly aligned (Fig. 5A). However, if task", + "type": "text" + }, + { + "bbox": [ + 495, + 564, + 504, + 574 + ], + "score": 0.58, + "content": "A", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 220, + 587 + ], + "score": 1.0, + "content": "already has a high SNR, task", + "type": "text" + }, + { + "bbox": [ + 221, + 575, + 230, + 585 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "must be very well aligned to it for transfer to be beneficial – otherwise", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 584, + 507, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 432, + 599 + ], + "score": 1.0, + "content": "there will be interference. The degree of alignment required increases as the task", + "type": "text" + }, + { + "bbox": [ + 433, + 586, + 442, + 596 + ], + "score": 0.65, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 584, + 507, + 599 + ], + "score": 1.0, + "content": "SNR increases,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 595, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 610 + ], + "score": 1.0, + "content": "but the quantity of benefit or interference decreases correspondingly (Fig. 5BC). In Appendix D we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 608, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 620 + ], + "score": 1.0, + "content": "explain why our theory predicts these results. Furthermore, in Appendix F we demonstrate these", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 618, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 633 + ], + "score": 1.0, + "content": "phenomena are qualitatively recapitulated in nonlinear networks, which suggests that our theory may", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 630, + 296, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 296, + 642 + ], + "score": 1.0, + "content": "give insight into how to choose auxiliary tasks.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 107, + 657, + 190, + 669 + ], + "lines": [ + { + "bbox": [ + 104, + 656, + 192, + 672 + ], + "spans": [ + { + "bbox": [ + 104, + 656, + 192, + 672 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "In summary, our analytic theory of generalization dynamics in deep linear networks reveals that many", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "puzzling aspects of generalization in deep learning already arise in the simple linear setting, where", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "the puzzles can be understood analytically. In particular, deep linear networks learn more important", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "structure in data first, leading to generalization errors that depend on task structure much more than", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "network size. Our theory explains why deep linear networks learn randomized data more slowly than", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "10", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 442, + 95 + ], + "lines": [], + "index": 0, + "bbox_fs": [ + 106, + 81, + 443, + 96 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 98, + 410, + 140 + ], + "lines": [ + { + "bbox": [ + 199, + 98, + 410, + 140 + ], + "spans": [ + { + "bbox": [ + 199, + 98, + 410, + 140 + ], + "score": 0.95, + "content": "\\varepsilon _ { \\mathrm { n o n - g r a d i e n t } } ^ { \\mathrm { o p t } } = \\left[ \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 2 } } \\overline { { s } } _ { \\alpha } ^ { 2 } \\right] ^ { - 1 } \\left[ \\sum _ { \\alpha = 1 } ^ { \\overline { { N } } _ { 2 } } \\overline { { s } } _ { \\alpha } ^ { 2 } ( 1 - \\mathcal { O } ( \\overline { { s } } _ { \\alpha } ) ^ { 2 } ) \\right] .", + "type": "interline_equation", + "image_path": "570f4b4fad0704cfefbb961c062c1f4efb04e2f7fe004d72fb17f3060a73f853.jpg" + } + ] + } + ], + "index": 1.5, + "virtual_lines": [ + { + "bbox": [ + 199, + 98, + 410, + 119.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 199, + 119.0, + 410, + 140.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 506, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 506, + 156 + ], + "score": 1.0, + "content": "Standard gradient descent learning cannot achieve this low generalization error because it cannot", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 103, + 150, + 508, + 174 + ], + "spans": [ + { + "bbox": [ + 103, + 150, + 441, + 174 + ], + "score": 1.0, + "content": "independently adjust all student singular values. A simple algorithm that achieves", + "type": "text" + }, + { + "bbox": [ + 441, + 154, + 483, + 169 + ], + "score": 0.87, + "content": "\\varepsilon _ { \\mathrm { n o n - g r a d i e n t } } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 150, + 508, + 174 + ], + "score": 1.0, + "content": "is as", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 166, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 166, + 275, + 182 + ], + "score": 1.0, + "content": "follows. 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This non-gradient singular value shrinkage", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 208, + 439, + 232 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 352, + 232 + ], + "score": 1.0, + "content": "algorithm provably outperforms neural network training with", + "type": "text" + }, + { + "bbox": [ + 353, + 212, + 437, + 227 + ], + "score": 0.7, + "content": "\\varepsilon _ { \\mathrm { n o n - g r a d i e n t } } ^ { \\mathrm { o p t } ^ { - } } \\le \\varepsilon _ { \\mathrm { g r a d i e n t } } ^ { \\mathrm { o p t } } .", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 218, + 439, + 228 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6, + "bbox_fs": [ + 103, + 142, + 508, + 232 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 241, + 502, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 502, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 502, + 255 + ], + "score": 1.0, + "content": "4 A THEORY FOR THE TRANSFER OF KNOWLEDGE ACROSS MULTIPLE TASKS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 262, + 506, + 372 + ], + "lines": [ + { + "bbox": [ + 104, + 260, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 183, + 279 + ], + "score": 1.0, + "content": "Consider two tasks", + "type": "text" + }, + { + "bbox": [ + 184, + 266, + 192, + 275 + ], + "score": 0.8, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 260, + 210, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 210, + 266, + 219, + 275 + ], + "score": 0.8, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 260, + 276, + 279 + ], + "score": 1.0, + "content": ", described by", + "type": "text" + }, + { + "bbox": [ + 276, + 264, + 290, + 276 + ], + "score": 0.9, + "content": "\\overline { { N } } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 260, + 304, + 279 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 304, + 264, + 319, + 276 + ], + "score": 0.89, + "content": "\\overline { { N } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 260, + 374, + 279 + ], + "score": 1.0, + "content": "teacher maps", + "type": "text" + }, + { + "bbox": [ + 374, + 261, + 394, + 276 + ], + "score": 0.87, + "content": "\\overline { { \\mathbf { W } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 260, + 412, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 412, + 261, + 432, + 276 + ], + "score": 0.86, + "content": "\\overline { { \\mathbf { W } } } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 260, + 470, + 279 + ], + "score": 1.0, + "content": ", of ranks", + "type": "text" + }, + { + "bbox": [ + 470, + 261, + 487, + 277 + ], + "score": 0.91, + "content": "\\overline { { N } } _ { 2 } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 260, + 506, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 276, + 507, + 293 + ], + "spans": [ + { + "bbox": [ + 107, + 276, + 123, + 291 + ], + "score": 0.9, + "content": "\\overline { { N } } _ { 2 } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 276, + 507, + 293 + ], + "score": 1.0, + "content": ", respectively. Now two student networks can learn from the two teacher networks separately,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 101, + 286, + 510, + 315 + ], + "spans": [ + { + "bbox": [ + 101, + 286, + 305, + 315 + ], + "score": 1.0, + "content": "each achieving optimal early stopping test errors composite teacher (and student) that concatenate", + "type": "text" + }, + { + "bbox": [ + 306, + 291, + 321, + 304 + ], + "score": 0.9, + "content": "\\varepsilon _ { A } ^ { \\mathrm { { o p t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 286, + 339, + 315 + ], + "score": 1.0, + "content": "and hid", + "type": "text" + }, + { + "bbox": [ + 339, + 290, + 354, + 304 + ], + "score": 0.92, + "content": "\\varepsilon _ { B } ^ { \\mathrm { { o p t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 286, + 510, + 315 + ], + "score": 1.0, + "content": ". Alternatively, one could construct a and output units, but shares the same", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 313, + 504, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 121, + 337 + ], + "score": 0.88, + "content": "\\overline { { N } } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 313, + 477, + 347 + ], + "score": 1.0, + "content": "ut units (Fig. 7). The composite student and teacher each have two heads, one for each tasneurons per head. Optimal early stopping on each head of the student yields test errors opt opt", + "type": "text" + }, + { + "bbox": [ + 477, + 325, + 504, + 338 + ], + "score": 0.87, + "content": "\\varepsilon _ { A B } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 101, + 331, + 503, + 358 + ], + "spans": [ + { + "bbox": [ + 101, + 331, + 123, + 358 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 150, + 331, + 408, + 358 + ], + "score": 1.0, + "content": ". We define the transfer benefit that task B confers on task A to be", + "type": "text" + }, + { + "bbox": [ + 409, + 337, + 503, + 351 + ], + "score": 0.9, + "content": "\\mathcal { T } ^ { A B } \\equiv \\varepsilon _ { A } ^ { \\mathrm { o p t } } - \\varepsilon _ { A B } ^ { \\mathrm { o p t } }", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "A postive (negative) transfer benefit implies learning tasks A and B simultaneously yields a lower", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 361, + 412, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 412, + 373 + ], + "score": 1.0, + "content": "(higher) optimal test error on task A compared to just learning task A alone.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 101, + 260, + 510, + 373 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 506, + 500 + ], + "lines": [ + { + "bbox": [ + 104, + 373, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 373, + 320, + 391 + ], + "score": 1.0, + "content": "A foundational question is how the transfer benefit", + "type": "text" + }, + { + "bbox": [ + 320, + 376, + 350, + 388 + ], + "score": 0.9, + "content": "\\scriptstyle { \\mathcal { T } } ^ { A B }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 373, + 506, + 391 + ], + "score": 1.0, + "content": "depends on the two tasks defined by", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 387, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 159, + 403 + ], + "score": 1.0, + "content": "the teachers", + "type": "text" + }, + { + "bbox": [ + 159, + 388, + 179, + 402 + ], + "score": 0.88, + "content": "\\overline { { \\mathbf { W } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 387, + 199, + 405 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 199, + 388, + 219, + 402 + ], + "score": 0.86, + "content": "\\overline { { \\mathbf { W } } } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 387, + 471, + 405 + ], + "score": 1.0, + "content": ". 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From the above, we know that", + "type": "text" + }, + { + "bbox": [ + 397, + 406, + 413, + 420 + ], + "score": 0.88, + "content": "\\varepsilon _ { A } ^ { \\mathrm { { o p t } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 402, + 463, + 424 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 463, + 404, + 483, + 418 + ], + "score": 0.89, + "content": "\\overline { { \\mathbf { W } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 402, + 507, + 424 + ], + "score": 1.0, + "content": "only", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 419, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 139, + 437 + ], + "score": 1.0, + "content": "through", + "type": "text" + }, + { + "bbox": [ + 139, + 419, + 153, + 433 + ], + "score": 0.84, + "content": "\\overline { { \\mathbf { S } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 419, + 394, + 437 + ], + "score": 1.0, + "content": ". 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If we think of the columns of each", + "type": "text" + }, + { + "bbox": [ + 494, + 438, + 504, + 450 + ], + "score": 0.73, + "content": "\\overline { { \\mathbf { V } } }", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 291, + 464 + ], + "score": 1.0, + "content": "as spanning a low dimensional feature space in", + "type": "text" + }, + { + "bbox": [ + 292, + 451, + 307, + 462 + ], + "score": 0.89, + "content": "\\overline { { N } } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "dimensional input space that is important for each", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 147, + 477 + ], + "score": 1.0, + "content": "task, then", + "type": "text" + }, + { + "bbox": [ + 148, + 462, + 158, + 475 + ], + "score": 0.84, + "content": "\\overline { { \\mathbf { Q } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "reflects the input feature subspace similarity matrix. Interestingly, the transfer benefit", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 474, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 474, + 272, + 491 + ], + "score": 1.0, + "content": "is independent of output singular vectors", + "type": "text" + }, + { + "bbox": [ + 272, + 474, + 289, + 488 + ], + "score": 0.86, + "content": "\\bar { \\mathbf { U } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 474, + 307, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 307, + 474, + 323, + 488 + ], + "score": 0.85, + "content": "\\overline { { \\mathbf { U } } } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 474, + 506, + 491 + ], + "score": 1.0, + "content": ". What matters for knowledge transfer in this", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 488, + 402, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 402, + 501 + ], + "score": 1.0, + "content": "setting are the relevant input features, not how you must respond to them.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 373, + 507, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 103, + 504, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 103, + 504, + 440, + 523 + ], + "score": 1.0, + "content": "We describe the transfer benefit for the simple case of two rank one teachers. Then", + "type": "text" + }, + { + "bbox": [ + 441, + 505, + 473, + 520 + ], + "score": 0.52, + "content": "\\overline { { \\mathbf { S } } } ^ { A } , \\overline { { \\mathbf { S } } } ^ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 504, + 494, + 523 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 494, + 507, + 505, + 520 + ], + "score": 0.68, + "content": "\\overline { { \\mathbf { Q } } }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 517, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 517, + 178, + 534 + ], + "score": 1.0, + "content": "are simply scalars", + "type": "text" + }, + { + "bbox": [ + 178, + 522, + 207, + 531 + ], + "score": 0.29, + "content": "s _ { A } , s _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 517, + 225, + 534 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 522, + 231, + 532 + ], + "score": 0.79, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 517, + 345, + 534 + ], + "score": 1.0, + "content": ", and we explore the function", + "type": "text" + }, + { + "bbox": [ + 346, + 519, + 419, + 532 + ], + "score": 0.93, + "content": "\\mathcal { T } ^ { A B } ( \\overline { { s } } _ { A } , \\overline { { s } } _ { B } , q )", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 517, + 506, + 534 + ], + "score": 1.0, + "content": "in Fig. 5ABC, which", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "reveals several interesting features. First, knowledge can be transferred from a high SNR task to a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 433, + 554 + ], + "score": 1.0, + "content": "low SNR task (Fig. 5A) and the degree of transfer increases with task alignment", + "type": "text" + }, + { + "bbox": [ + 433, + 543, + 439, + 553 + ], + "score": 0.75, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 542, + 505, + 554 + ], + "score": 1.0, + "content": ". This can make", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 551, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 104, + 551, + 264, + 567 + ], + "score": 1.0, + "content": "it possible to capture signals from task", + "type": "text" + }, + { + "bbox": [ + 265, + 553, + 273, + 563 + ], + "score": 0.72, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 551, + 506, + 567 + ], + "score": 1.0, + "content": "which would otherwise sink into the MP sea by learning", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 563, + 504, + 576 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 495, + 576 + ], + "score": 1.0, + "content": "jointly with a related task, even if the tasks are only weakly aligned (Fig. 5A). However, if task", + "type": "text" + }, + { + "bbox": [ + 495, + 564, + 504, + 574 + ], + "score": 0.58, + "content": "A", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 220, + 587 + ], + "score": 1.0, + "content": "already has a high SNR, task", + "type": "text" + }, + { + "bbox": [ + 221, + 575, + 230, + 585 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "must be very well aligned to it for transfer to be beneficial – otherwise", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 584, + 507, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 432, + 599 + ], + "score": 1.0, + "content": "there will be interference. The degree of alignment required increases as the task", + "type": "text" + }, + { + "bbox": [ + 433, + 586, + 442, + 596 + ], + "score": 0.65, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 584, + 507, + 599 + ], + "score": 1.0, + "content": "SNR increases,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 595, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 610 + ], + "score": 1.0, + "content": "but the quantity of benefit or interference decreases correspondingly (Fig. 5BC). In Appendix D we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 608, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 506, + 620 + ], + "score": 1.0, + "content": "explain why our theory predicts these results. Furthermore, in Appendix F we demonstrate these", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 618, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 633 + ], + "score": 1.0, + "content": "phenomena are qualitatively recapitulated in nonlinear networks, which suggests that our theory may", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 630, + 296, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 296, + 642 + ], + "score": 1.0, + "content": "give insight into how to choose auxiliary tasks.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32.5, + "bbox_fs": [ + 103, + 504, + 507, + 642 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 657, + 190, + 669 + ], + "lines": [ + { + "bbox": [ + 104, + 656, + 192, + 672 + ], + "spans": [ + { + "bbox": [ + 104, + 656, + 192, + 672 + ], + "score": 1.0, + "content": "5 DISCUSSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "In summary, our analytic theory of generalization dynamics in deep linear networks reveals that many", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "puzzling aspects of generalization in deep learning already arise in the simple linear setting, where", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "the puzzles can be understood analytically. In particular, deep linear networks learn more important", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "structure in data first, leading to generalization errors that depend on task structure much more than", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "network size. Our theory explains why deep linear networks learn randomized data more slowly than", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "structured data, and provides a non-gradient based learning method that out-performs gradient descent", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "learning in the linear case. 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Among other things, we hope our work will motivate", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "and enable: (1) the search for tighter upper bounds on generalization error that take into account task", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "structure; (2) the design of non gradient based training algorithms that outperform gradient-based", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 495, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 495, + 183 + ], + "score": 1.0, + "content": "learning; and (3) the theory-driven selection of auxiliary tasks that maximize knowledge transfer.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 108, + 198, + 175, + 210 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 176, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 176, + 211 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 504, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "Madhu S. 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(2013a).", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 585, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 297, + 599 + ], + "score": 1.0, + "content": "There it was also proven that for a network with", + "type": "text" + }, + { + "bbox": [ + 297, + 586, + 309, + 597 + ], + "score": 0.86, + "content": "N _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 585, + 354, + 599 + ], + "score": 1.0, + "content": "layers (i.e.", + "type": "text" + }, + { + "bbox": [ + 355, + 586, + 384, + 597 + ], + "score": 0.88, + "content": "N _ { l } - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 585, + 506, + 599 + ], + "score": 1.0, + "content": "hidden layers), the strength of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 597, + 272, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 272, + 609 + ], + "score": 1.0, + "content": "the mode obeys the differential equation:", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 564, + 507, + 609 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 227, + 613, + 383, + 637 + ], + "lines": [ + { + "bbox": [ + 227, + 613, + 383, + 637 + ], + "spans": [ + { + "bbox": [ + 227, + 613, + 383, + 637 + ], + "score": 0.93, + "content": "\\tau { \\frac { d } { d t } } u = ( N _ { l } - 1 ) u ^ { 2 - 2 / ( N _ { l } - 1 ) } ( s - u )", + "type": "interline_equation", + "image_path": "e8382140836b5cfe34671dd944c538f5a90a3735e89a8a2bf073af8301a4e25f.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 227, + 613, + 383, + 637 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 641, + 504, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 506, + 656 + ], + "score": 1.0, + "content": "This equation is separable and can be integrated for any integer number of layers. In particular, we", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 653, + 328, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 328, + 666 + ], + "score": 1.0, + "content": "consider the case of 5 layers (3 hidden), in which case:", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 640, + 506, + 666 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 669, + 385, + 704 + ], + "lines": [ + { + "bbox": [ + 225, + 669, + 385, + 704 + ], + "spans": [ + { + "bbox": [ + 225, + 669, + 385, + 704 + ], + "score": 0.94, + "content": "t ( s , \\hat { s } ) = \\frac { \\tau } { 2 } \\left[ \\frac { \\operatorname { t a n h } ^ { - 1 } \\left( \\sqrt { \\frac { u } { \\hat { s } } } \\right) } { \\hat { s } ^ { 3 / 2 } } - \\frac { 1 } { \\hat { s } \\sqrt { u } } \\right] _ { \\epsilon } ^ { s }", + "type": "interline_equation", + "image_path": "73166119140c9bab6af2d2ea3b4f65abbb44c69b0e083ef0870b03452afa4a5e.jpg" + } + ] + } + ], + "index": 41.5, + "virtual_lines": [ + { + "bbox": [ + 225, + 669, + 385, + 686.5 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 225, + 686.5, + 385, + 704.0 + ], + "spans": [], + "index": 42 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 333, + 723 + ], + "score": 1.0, + "content": "This expression cannot be analytically inverted to find", + "type": "text" + }, + { + "bbox": [ + 333, + 709, + 359, + 722 + ], + "score": 0.93, + "content": "s ( t , { \\hat { s } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 708, + 505, + 723 + ], + "score": 1.0, + "content": ", so we numerically invert it where", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 719, + 150, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 150, + 734 + ], + "score": 1.0, + "content": "necessary.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 708, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 420, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 421, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 421, + 96 + ], + "score": 1.0, + "content": "B ALIGNMENT LAG IN RANDOMLY INITIALIZED NETWORKS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "As noted in the main text, the randomly-initialized networks behave quite similarly to the TA networks,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "except that the randomly-initialized networks show a lag due to the time it takes for the network’s", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "score": 1.0, + "content": "modes to align with the data modes. In fig. 9 we explore this lag by plotting the alignment of the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 453, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 453, + 151 + ], + "score": 1.0, + "content": "modes and the increase in the singular value for several randomly initialized networks.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 155, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 169 + ], + "score": 1.0, + "content": "Notice that stronger modes align more quickly. Furthermore, the mode alignment is relatively", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "independent – whether the teacher is rank 1 or rank 3, the alignment of the modes is similar for the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "mode of singular value 2. Most importantly, note how the deeper networks show substantially slower", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 504, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 504, + 201 + ], + "score": 1.0, + "content": "mode alignment, with alignment not completed until around when the singular value increases. This", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "explains why deeper networks show a larger lag between randomly-initialized and TA networks – the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 329, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 329, + 224 + ], + "score": 1.0, + "content": "alignment process is much slower for deeper networks.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 106, + 239, + 372, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 237, + 374, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 374, + 253 + ], + "score": 1.0, + "content": "C TRAIN AND TEST ERRORS AFTER A PROJECTION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 263, + 504, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "In the case of transfer learning, or more generally when we want to evaluate a network’s loss on a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "subset of its outputs, we need to use a slight generalization of the train and test error formulas given", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 104, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "in the main text. Suppose we are interested in the train and test errors after applying a projection", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 297, + 155, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 142, + 308 + ], + "score": 1.0, + "content": "operator", + "type": "text" + }, + { + "bbox": [ + 142, + 297, + 151, + 307 + ], + "score": 0.65, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 297, + 155, + 308 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 313, + 468, + 352 + ], + "lines": [ + { + "bbox": [ + 142, + 313, + 468, + 352 + ], + "spans": [ + { + "bbox": [ + 142, + 313, + 468, + 352 + ], + "score": 0.93, + "content": "\\varepsilon _ { \\mathrm { t r a i n } } \\equiv \\frac { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\mathbf { P } \\mathbf { W } \\hat { \\mathbf { x } } ^ { \\mu } - \\mathbf { P } \\hat { \\mathbf { y } } ^ { \\mu } | | _ { 2 } ^ { 2 } } { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\mathbf { P } \\hat { \\mathbf { y } } ^ { \\mu } | | _ { 2 } ^ { 2 } } , \\varepsilon _ { \\mathrm { t e s t } } \\equiv \\frac { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\mathbf { P } \\mathbf { W } \\overline { { \\mathbf { x } } } ^ { \\mu } - \\mathbf { P } \\overline { { \\mathbf { y } } } ^ { \\mu } | | _ { 2 } ^ { 2 } } { \\sum _ { \\mu = 1 } ^ { \\overline { { N _ { 1 } } } } | | \\mathbf { P } \\overline { { \\mathbf { y } } } ^ { \\mu } | | _ { 2 } ^ { 2 } } ,", + "type": "interline_equation", + "image_path": "824d5e77da2fc9bf0002ff555d48fd7e0608505786d086f2d942b570d2f3b38b.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 142, + 313, + 468, + 326.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 142, + 326.0, + 468, + 339.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 142, + 339.0, + 468, + 352.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 342, + 368 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 343, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 343, + 370 + ], + "score": 1.0, + "content": "respectively. As in the main text, we can rexpress these as:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 372, + 441, + 404 + ], + "lines": [ + { + "bbox": [ + 169, + 372, + 441, + 404 + ], + "spans": [ + { + "bbox": [ + 169, + 372, + 441, + 404 + ], + "score": 0.9, + "content": "\\varepsilon _ { \\mathrm { t r a i n } } = \\frac { \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } { \\mathbf { W } } - 2 \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\Sigma ^ { 3 1 } + \\mathrm { T r } { \\Sigma ^ { 3 1 } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\Sigma ^ { 3 1 } } { \\mathrm { T r } \\Sigma ^ { 3 1 } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\Sigma ^ { 3 1 } } ,", + "type": "interline_equation", + "image_path": "b3dfe306ce0e4b0416590bb0eb5e48c67ca0e50453c4ad1d27c97a97d4efeb9e.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 169, + 372, + 441, + 382.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 169, + 382.6666666666667, + 441, + 393.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 169, + 393.33333333333337, + 441, + 404.00000000000006 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 408, + 432, + 442 + ], + "lines": [ + { + "bbox": [ + 178, + 408, + 432, + 442 + ], + "spans": [ + { + "bbox": [ + 178, + 408, + 432, + 442 + ], + "score": 0.9, + "content": "\\varepsilon _ { \\mathrm { t e s t } } = \\frac { \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } { \\mathbf { W } } - 2 \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\overline { { { \\mathbf { W } } } } + \\mathrm { T r } \\overline { { { \\mathbf { W } } } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\overline { { { \\mathbf { W } } } } } { \\mathrm { T r } \\overline { { { \\mathbf { W } } } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\overline { { { \\mathbf { W } } } } } .", + "type": "interline_equation", + "image_path": "a7f3c63e6dd22c620bfb08e215774d513e2fb6bf53b51956396165b08996231d.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 178, + 408, + 432, + 419.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 178, + 419.3333333333333, + 432, + 430.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 178, + 430.66666666666663, + 432, + 441.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 371, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 442, + 372, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 372, + 457 + ], + "score": 1.0, + "content": "Using the cyclic property of the trace, we can modify these to get:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 459, + 441, + 490 + ], + "lines": [ + { + "bbox": [ + 170, + 459, + 441, + 490 + ], + "spans": [ + { + "bbox": [ + 170, + 459, + 441, + 490 + ], + "score": 0.91, + "content": "\\varepsilon _ { \\mathrm { t r a i n } } = \\frac { \\mathrm { T r } { \\bf P } { \\bf W } { \\bf W } ^ { T } { \\bf P } ^ { T } - 2 \\mathrm { T r } { \\bf P } \\Sigma ^ { 3 1 } { \\bf W } ^ { T } { \\bf P } ^ { T } + \\mathrm { T r } { \\bf P } \\Sigma ^ { 3 1 } \\Sigma ^ { 3 1 ^ { T } } { \\bf P } ^ { T } } { \\mathrm { T r } { \\bf P } \\Sigma ^ { 3 1 } \\Sigma ^ { 3 1 ^ { T } } { \\bf P } ^ { T } } ,", + "type": "interline_equation", + "image_path": "c7c12ca256b899740c8be0c793209c9a8228e253de168ef8f6fd3701fafdbc93.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 170, + 459, + 441, + 469.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 170, + 469.3333333333333, + 441, + 479.66666666666663 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 170, + 479.66666666666663, + 441, + 489.99999999999994 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 495, + 432, + 529 + ], + "lines": [ + { + "bbox": [ + 178, + 495, + 432, + 529 + ], + "spans": [ + { + "bbox": [ + 178, + 495, + 432, + 529 + ], + "score": 0.89, + "content": "\\varepsilon _ { \\mathrm { t e s t } } = \\frac { \\mathrm { T r } \\mathbf { P } \\mathbf { W } \\mathbf { W } ^ { T } \\mathbf { P } ^ { T } - 2 \\mathrm { T r } \\mathbf { P } \\overline { { \\mathbf { W } } } \\mathbf { W } ^ { T } \\mathbf { P } ^ { T } + \\mathrm { T r } \\mathbf { P } \\overline { { \\mathbf { W } } } \\mathbf { W } ^ { T } \\mathbf { P } ^ { T } } { \\mathrm { T r } \\mathbf { P } \\overline { { \\mathbf { W } } } \\mathbf { W } ^ { T } \\mathbf { P } ^ { T } } .", + "type": "interline_equation", + "image_path": "b809fb7c45678781dfedbcdedaa34d7e51911d701c9fda075999b1f71d247a8e.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 178, + 495, + 432, + 506.3333333333333 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 178, + 506.3333333333333, + 432, + 517.6666666666666 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 178, + 517.6666666666666, + 432, + 529.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 504, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 502, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 446, + 544 + ], + "score": 1.0, + "content": "As before, we express these in terms of the student, training data and teacher SVDs,", + "type": "text" + }, + { + "bbox": [ + 446, + 531, + 502, + 542 + ], + "score": 0.87, + "content": "\\mathbf { W } = \\mathbf { U } \\mathbf { S } \\mathbf { V } ^ { T }", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 541, + 354, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 167, + 556 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\pmb { \\Sigma } ^ { 3 1 } = \\hat { \\mathbf { U } } \\hat { \\mathbf { S } } \\hat { \\mathbf { V } } ^ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 541, + 188, + 559 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 189, + 542, + 248, + 556 + ], + "score": 0.92, + "content": "\\overline { { \\mathbf { W } } } = \\overline { { \\mathbf { U } } } \\overline { { \\mathbf { S } } } \\overline { { \\mathbf { V } } } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 541, + 354, + 559 + ], + "score": 1.0, + "content": "respectively. Specifically,", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 563, + 504, + 580 + ], + "lines": [ + { + "bbox": [ + 111, + 563, + 504, + 580 + ], + "spans": [ + { + "bbox": [ + 111, + 563, + 504, + 580 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\operatorname { t r a i n } = \\Big [ \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 3 } } \\hat { s } _ { \\beta } ^ { 2 } \\vert \\vert { \\mathbf { P } } \\hat { \\mathbf { a } } ^ { \\alpha } \\vert \\vert _ { 2 } ^ { 2 } \\Big ] ^ { - 1 } \\Big [ \\sum _ { \\alpha = 1 } ^ { N _ { 2 } } s _ { \\alpha } ^ { 2 } \\vert \\vert { \\mathbf { P } } \\mathbf { u } ^ { \\alpha } \\vert \\vert _ { 2 } ^ { 2 } + \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 3 } } \\hat { s } _ { \\beta } ^ { 2 } \\vert \\vert { \\mathbf { P } } \\hat { \\mathbf { a } } ^ { \\alpha } \\vert \\vert _ { 2 } ^ { 2 } - 2 \\sum _ { \\alpha = 1 } ^ { N _ { 2 } } \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 3 } } s _ { \\alpha } \\hat { s } _ { \\beta } \\big ( { \\mathbf { P } } \\mathbf { u } ^ { \\alpha } \\cdot { \\mathbf { P } } \\hat { \\mathbf { u } } ^ { \\beta } \\big ) \\big ( { \\mathbf { v } } ^ { \\alpha } \\cdot { \\mathbf { \\hat { v } } } ^ { \\beta } \\big ) \\Big ] , } \\end{array}", + "type": "interline_equation", + "image_path": "e2a2b7673c35307ab430f1855d1ff64e839930bd9e9c11e872e6526a7ca5afdb.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 111, + 563, + 504, + 580 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 592, + 504, + 609 + ], + "lines": [ + { + "bbox": [ + 111, + 592, + 504, + 609 + ], + "spans": [ + { + "bbox": [ + 111, + 592, + 504, + 609 + ], + "score": 0.86, + "content": "\\begin{array} { r } { \\varepsilon _ { \\mathrm { t e s t } } = \\Big [ \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 2 } } \\bar { s } _ { \\beta } ^ { 2 } \\big | \\big | { \\bf P } \\bar { \\bf u } ^ { \\alpha } \\big | \\big | _ { 2 } ^ { 2 } \\Big ] ^ { - 1 } \\Big [ \\sum _ { \\alpha = 1 } ^ { N _ { 2 } } s _ { \\alpha } ^ { 2 } \\big | \\big | { \\bf P } { \\bf u } ^ { \\alpha } \\big | \\big | _ { 2 } ^ { 2 } + \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 2 } } \\bar { s } _ { \\beta } ^ { 2 } \\big | \\big | { \\bf P } \\bar { \\bf u } ^ { \\alpha } \\big | \\big | _ { 2 } ^ { 2 } - 2 \\sum _ { \\alpha = 1 } ^ { N _ { 2 } } \\sum _ { \\beta = 1 } ^ { \\mathbb { N } _ { 2 } } s _ { \\alpha } \\bar { s } _ { \\beta } \\big ( { \\bf P } { \\bf u } ^ { \\alpha } \\cdot { \\bf P } { \\bf { \\bar { u } } } ^ { \\beta } \\big ) \\big ( { \\bf v } ^ { \\alpha } \\cdot { \\bf \\bar { v } } ^ { \\beta } \\big ) \\Big ] . } \\end{array}", + "type": "interline_equation", + "image_path": "4b62348655d38b06bf41391656fd181444d4be54d4d2a458526a43428d448f9c.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 111, + 592, + 504, + 609 + ], + 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506, + 130 + ], + "score": 1.0, + "content": "except that the randomly-initialized networks show a lag due to the time it takes for the network’s", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "score": 1.0, + "content": "modes to align with the data modes. In fig. 9 we explore this lag by plotting the alignment of the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 140, + 453, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 453, + 151 + ], + "score": 1.0, + "content": "modes and the increase in the singular value for several randomly initialized networks.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 106, + 506, + 151 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 155, + 505, + 222 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 169 + ], + "score": 1.0, + "content": "Notice that stronger modes align more quickly. 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This", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "explains why deeper networks show a larger lag between randomly-initialized and TA networks – the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 329, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 329, + 224 + ], + "score": 1.0, + "content": "alignment process is much slower for deeper networks.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 155, + 505, + 224 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 239, + 372, + 252 + ], + "lines": [ + { + "bbox": [ + 106, + 237, + 374, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 374, + 253 + ], + "score": 1.0, + "content": "C TRAIN AND TEST ERRORS AFTER A PROJECTION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 263, + 504, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 276 + ], + "score": 1.0, + "content": "In the case of transfer learning, or more generally when we want to evaluate a network’s loss on a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "subset of its outputs, we need to use a slight generalization of the train and test error formulas given", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 104, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "in the main text. 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As in the main text, we can rexpress these as:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 355, + 343, + 370 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 372, + 441, + 404 + ], + "lines": [ + { + "bbox": [ + 169, + 372, + 441, + 404 + ], + "spans": [ + { + "bbox": [ + 169, + 372, + 441, + 404 + ], + "score": 0.9, + "content": "\\varepsilon _ { \\mathrm { t r a i n } } = \\frac { \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } { \\mathbf { W } } - 2 \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\Sigma ^ { 3 1 } + \\mathrm { T r } { \\Sigma ^ { 3 1 } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\Sigma ^ { 3 1 } } { \\mathrm { T r } \\Sigma ^ { 3 1 } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\Sigma ^ { 3 1 } } ,", + "type": "interline_equation", + "image_path": "b3dfe306ce0e4b0416590bb0eb5e48c67ca0e50453c4ad1d27c97a97d4efeb9e.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 169, + 372, + 441, + 382.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 169, + 382.6666666666667, + 441, + 393.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 169, + 393.33333333333337, + 441, + 404.00000000000006 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 408, + 432, + 442 + ], + "lines": [ + { + "bbox": [ + 178, + 408, + 432, + 442 + ], + "spans": [ + { + "bbox": [ + 178, + 408, + 432, + 442 + ], + "score": 0.9, + "content": "\\varepsilon _ { \\mathrm { t e s t } } = \\frac { \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } { \\mathbf { W } } - 2 \\mathrm { T r } { \\mathbf { W } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\overline { { { \\mathbf { W } } } } + \\mathrm { T r } \\overline { { { \\mathbf { W } } } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\overline { { { \\mathbf { W } } } } } { \\mathrm { T r } \\overline { { { \\mathbf { W } } } } ^ { T } { \\mathbf { P } } ^ { T } { \\mathbf { P } } \\overline { { { \\mathbf { W } } } } } .", + "type": "interline_equation", + "image_path": "a7f3c63e6dd22c620bfb08e215774d513e2fb6bf53b51956396165b08996231d.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 178, + 408, + 432, + 419.3333333333333 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 178, + 419.3333333333333, + 432, + 430.66666666666663 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 178, + 430.66666666666663, + 432, + 441.99999999999994 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 371, + 455 + ], + "lines": [ + { + "bbox": [ + 106, + 442, + 372, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 372, + 457 + ], + "score": 1.0, + "content": "Using the cyclic property of the trace, we can modify these to get:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 442, + 372, + 457 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 459, + 441, + 490 + ], + "lines": [ + { + "bbox": [ + 170, + 459, + 441, + 490 + ], + "spans": [ + { + "bbox": [ + 170, + 459, + 441, + 490 + ], + "score": 0.91, + "content": "\\varepsilon _ { \\mathrm { t r a i n } } = \\frac { \\mathrm { T r } { \\bf P } { \\bf W } { \\bf W } ^ { T } { \\bf P } ^ { T } - 2 \\mathrm { T r } { \\bf P } \\Sigma ^ { 3 1 } { \\bf W } ^ { T } { \\bf P } ^ { T } + \\mathrm { T r } { \\bf P } \\Sigma ^ { 3 1 } \\Sigma ^ { 3 1 ^ { T } } { \\bf P } ^ { T } } { \\mathrm { T r } { \\bf P } \\Sigma ^ { 3 1 } \\Sigma ^ { 3 1 ^ { T } } { \\bf P } ^ { T } } ,", + "type": "interline_equation", + "image_path": "c7c12ca256b899740c8be0c793209c9a8228e253de168ef8f6fd3701fafdbc93.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 170, + 459, + 441, + 469.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 170, + 469.3333333333333, + 441, + 479.66666666666663 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 170, + 479.66666666666663, + 441, + 489.99999999999994 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 495, + 432, + 529 + ], + "lines": [ + { + "bbox": [ + 178, + 495, + 432, + 529 + ], + "spans": [ + { + "bbox": [ + 178, + 495, + 432, + 529 + ], + "score": 0.89, + "content": "\\varepsilon _ { \\mathrm { t e s t } } = \\frac { \\mathrm { T r } \\mathbf { P } \\mathbf { W } \\mathbf { W } ^ { T } \\mathbf { P } ^ { T } - 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Thus the relationship between the", + "type": "text" + }, + { + "bbox": [ + 470, + 263, + 487, + 276 + ], + "score": 0.89, + "content": "\\overline { { \\mathbf { U } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 264, + 506, + 278 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 275, + 507, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 123, + 290 + ], + "score": 0.88, + "content": "\\overline { { \\mathbf { U } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 275, + 507, + 296 + ], + "score": 1.0, + "content": "is irrelevant to the transfer. 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We can now calculate the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 102, + 362, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 102, + 364, + 218, + 384 + ], + "score": 1.0, + "content": "singular values of WA+", + "type": "text" + }, + { + "bbox": [ + 206, + 362, + 430, + 385 + ], + "score": 1.0, + "content": "B. 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\\mathbf { \\overline { { { V } } } } ^ { A B ^ { T } } { \\vec { c } }", + "type": "interline_equation", + "image_path": "50b46fc1cb03b08ff6212cd1f7f0a846462b9c8bc1bced40cebff3590a8bc23f.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 225, + 477, + 387, + 495 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 506, + 559 + ], + "lines": [ + { + "bbox": [ + 101, + 493, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 101, + 495, + 250, + 518 + ], + "score": 1.0, + "content": "Hence eigenvalues of VABSAB2V", + "type": "text" + }, + { + "bbox": [ + 267, + 496, + 403, + 517 + ], + "score": 1.0, + "content": "T are also eigenvalues of VABT V", + "type": "text" + }, + { + "bbox": [ + 369, + 497, + 443, + 513 + ], + "score": 0.9, + "content": "\\overline { { { \\bf V } } } ^ { A B ^ { T } } \\overline { { { \\bf V } } } ^ { A B } \\overline { { { \\bf S } } } ^ { A B ^ { 2 } }", + "type": "inline_equation" + }, + { + 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Furthermore, this mapping must be a bijection for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 527, + 507, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 462, + 543 + ], + "score": 1.0, + "content": "eigenvectors with non-zero eigenvalues, since the matrices have the same rank (the rank of", + "type": "text" + }, + { + "bbox": [ + 462, + 527, + 485, + 541 + ], + "score": 0.87, + "content": "\\overline { { \\mathbf { V } } } ^ { A B }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 528, + 507, + 543 + ], + "score": 1.0, + "content": "). To", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 101, + 540, + 354, + 563 + ], + "spans": [ + { + "bbox": [ + 101, + 540, + 210, + 563 + ], + "score": 1.0, + "content": "see this, note that SAB 2 i", + "type": "text" + }, + { + "bbox": [ + 203, + 546, + 354, + 559 + ], + "score": 1.0, + "content": "s full rank. From this, it is clear that", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 563, + 420, + 581 + ], + "lines": [ + { + "bbox": [ + 190, + 563, + 420, + 581 + ], + "spans": [ + { + "bbox": [ + 190, + 563, + 420, + 581 + ], + "score": 0.89, + "content": "\\mathrm { r a n k } \\overline { { { \\mathbf { V } } } } ^ { A B ^ { T } } \\overline { { { \\mathbf { V } } } } ^ { A B } \\overline { { { \\mathbf { S } } } } ^ { A B ^ { 2 } } = \\mathrm { r a n k } \\overline { { { \\mathbf { V } } } } ^ { A B ^ { T } } \\overline { { { \\mathbf { V } } } } ^ { A B } = \\mathrm { r a n k } \\overline { { { \\mathbf { V } } } } ^ { A B } .", + "type": "interline_equation", + "image_path": "0bc80a4ea89838e575d1e39ac60c1781ae8dad24763c055da29210f52e7d5ca3.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 190, + 563, + 420, + 581 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", 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Furthermore, this mapping must be a bijection for", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 527, + 507, + 543 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 462, + 543 + ], + "score": 1.0, + "content": "eigenvectors with non-zero eigenvalues, since the matrices have the same rank (the rank of", + "type": "text" + }, + { + "bbox": [ + 462, + 527, + 485, + 541 + ], + "score": 0.87, + "content": "\\overline { { \\mathbf { V } } } ^ { A B }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 528, + 507, + 543 + ], + "score": 1.0, + "content": "). To", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 101, + 540, + 354, + 563 + ], + "spans": [ + { + "bbox": [ + 101, + 540, + 210, + 563 + ], + "score": 1.0, + "content": "see this, note that SAB 2 i", + "type": "text" + }, + { + "bbox": [ + 203, + 546, + 354, + 559 + ], + "score": 1.0, + "content": "s full rank. 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We", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 245, + 306 + ], + "score": 1.0, + "content": "note that the sign of the alignment", + "type": "text" + }, + { + "bbox": [ + 246, + 294, + 252, + 304 + ], + "score": 0.8, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 292, + 505, + 306 + ], + "score": 1.0, + "content": "is irrelevant as a special case of the fact (noted above) that any", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 304, + 393, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 393, + 315 + ], + "score": 1.0, + "content": "orthogonal transformation on the output modes does not affect transfer.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 333, + 292, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 333, + 293, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 293, + 346 + ], + "score": 1.0, + "content": "D.1 MISALIGNMENT AND INTERFERENCE", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 354, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "score": 1.0, + "content": "Why is there interference between tasks which are not well aligned? 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The two singular", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 380, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 380, + 172, + 397 + ], + "score": 1.0, + "content": "modes of VAB", + "type": "text" + }, + { + "bbox": [ + 169, + 381, + 506, + 398 + ], + "score": 1.0, + "content": "correspond to the shared structure between the tasks (weighted by the relative signal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 408 + ], + "score": 1.0, + "content": "strengths), and the differences between them, respectively. Although we may be improving our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 230, + 419 + ], + "score": 1.0, + "content": "estimates of the shared mode if", + "type": "text" + }, + { + "bbox": [ + 231, + 407, + 255, + 417 + ], + "score": 0.89, + "content": "q > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 406, + 424, + 419 + ], + "score": 1.0, + "content": "(by increasing its singular value relative to", + "type": "text" + }, + { + "bbox": [ + 424, + 407, + 437, + 418 + ], + "score": 0.84, + "content": "{ \\overline { { s } } } _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "), we are actually", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 417, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 226, + 434 + ], + "score": 1.0, + "content": "decreasing its alignment with", + "type": "text" + }, + { + "bbox": [ + 226, + 418, + 243, + 431 + ], + "score": 0.88, + "content": "\\overline { { \\mathbf { V } } } ^ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 417, + 271, + 434 + ], + "score": 1.0, + "content": "unless", + "type": "text" + }, + { + "bbox": [ + 271, + 421, + 295, + 432 + ], + "score": 0.91, + "content": "q = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 417, + 506, + 434 + ], + "score": 1.0, + "content": ". This misalignment is captured by the second mode", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 431, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 117, + 446 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 431, + 140, + 445 + ], + "score": 0.9, + "content": "\\overline { { \\mathbf { V } } } ^ { A B }", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 433, + 506, + 448 + ], + "score": 1.0, + "content": ", but the increase in the singular value of the first mode must come at the cost of a decrease", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "in the singular value of the second mode. See Fig. 10 for a conceptual illustration of this. 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Thus our insights into transfer may", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 363, + 399, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 399, + 377 + ], + "score": 1.0, + "content": "help to understand multi-task benefits in more complicated architectures.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "image", + "bbox": [ + 108, + 388, + 496, + 486 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 388, + 496, + 486 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 388, + 496, + 486 + ], + "spans": [ + { + "bbox": [ + 108, + 388, + 496, + 486 + ], + "score": 0.969, + "type": "image", + "image_path": "3aa21b341a46731773f7366f14f4c4df08b4e4ea25985a881fa5ca23d11fc270.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 108, + 388, + 496, + 420.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 108, + 420.6666666666667, + 496, + 453.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 108, + 453.33333333333337, + 496, + 486.00000000000006 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 498, + 506, + 579 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 497, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 213, + 513 + ], + "score": 1.0, + "content": "Figure 12: Transfer benefit", + "type": "text" + }, + { + "bbox": [ + 213, + 499, + 287, + 512 + ], + "score": 0.92, + "content": "\\mathcal { T } ^ { A B } ( \\overline { { s } } _ { A } , \\overline { { s } } _ { B } , q )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 497, + 507, + 513 + ], + "score": 1.0, + "content": "for non-linear teachers and students, plotted at different", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 145, + 525 + ], + "score": 1.0, + "content": "values of", + "type": "text" + }, + { + "bbox": [ + 146, + 513, + 158, + 523 + ], + "score": 0.85, + "content": "{ \\overline { { s } } } _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 511, + 177, + 525 + ], + "score": 1.0, + "content": ". 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In this section we show that the transfer patterns", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 303, + 297, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 297, + 315 + ], + "score": 1.0, + "content": "qualitatively generalize to non-linear networks.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 281, + 506, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 315, + 332 + ], + "score": 1.0, + "content": "Here, we show results from teacher networks with", + "type": "text" + }, + { + "bbox": [ + 315, + 319, + 399, + 331 + ], + "score": 0.7, + "content": "\\overline { { N } } _ { 1 } = 1 0 0 \\overline { { N } } _ { 3 } = 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 319, + 403, + 332 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 403, + 319, + 437, + 331 + ], + "score": 0.89, + "content": "\\overline { { N } } _ { 2 } = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "(thus the task is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 330, + 504, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 504, + 343 + ], + "score": 1.0, + "content": "higher rank) and leaky relu non-linearities at the hidden and output layers. We train a student with", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 341, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 188, + 355 + ], + "score": 1.0, + "content": "leaky relu units and", + "type": "text" + }, + { + "bbox": [ + 188, + 342, + 228, + 353 + ], + "score": 0.93, + "content": "N _ { 2 } = N _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 341, + 505, + 355 + ], + "score": 1.0, + "content": "to solve this task. Results qualitatively look quite similar to those in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 351, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 367 + ], + "score": 1.0, + "content": "Fig 5. of the main text for rank one linear teachers, see below. Thus our insights into transfer may", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 363, + 399, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 399, + 377 + ], + "score": 1.0, + "content": "help to understand multi-task benefits in more complicated architectures.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 319, + 506, + 377 + ] + }, + { + "type": "image", + "bbox": [ + 108, + 388, + 496, + 486 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 388, + 496, + 486 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 388, + 496, + 486 + ], + "spans": [ + { + "bbox": [ + 108, + 388, + 496, + 486 + ], + "score": 0.969, + "type": "image", + "image_path": "3aa21b341a46731773f7366f14f4c4df08b4e4ea25985a881fa5ca23d11fc270.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 108, + 388, + 496, + 420.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 108, + 420.6666666666667, + 496, + 453.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 108, + 453.33333333333337, + 496, + 486.00000000000006 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 498, + 506, + 579 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 497, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 213, + 513 + ], + "score": 1.0, + "content": "Figure 12: Transfer benefit", + "type": "text" + }, + { + "bbox": [ + 213, + 499, + 287, + 512 + ], + "score": 0.92, + "content": "\\mathcal { T } ^ { A B } ( \\overline { { s } } _ { A } , \\overline { { s } } _ { B } , q )", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 497, + 507, + 513 + ], + "score": 1.0, + "content": "for non-linear teachers and students, plotted at different", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 145, + 525 + ], + "score": 1.0, + "content": "values of", + "type": "text" + }, + { + "bbox": [ + 146, + 513, + 158, + 523 + ], + "score": 0.85, + "content": "{ \\overline { { s } } } _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 511, + 177, + 525 + ], + "score": 1.0, + "content": ". (a)", + "type": "text" + }, + { + "bbox": [ + 177, + 511, + 253, + 523 + ], + "score": 0.92, + "content": "\\overline { { s } } _ { A } = 0 . 8 4 = \\sqrt [ 4 ] { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 511, + 506, + 525 + ], + "score": 1.0, + "content": ". With support from another aligned task, especially one with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 419, + 536 + ], + "score": 1.0, + "content": "moderately higher SNR, performance on a low SNR task will improve. (b)", + "type": "text" + }, + { + "bbox": [ + 420, + 524, + 452, + 534 + ], + "score": 0.91, + "content": "{ \\overline { { s } } } _ { A } = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 523, + 505, + 536 + ], + "score": 1.0, + "content": ". Tasks with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 535, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 506, + 546 + ], + "score": 1.0, + "content": "modest signals will face interference from poorly aligned tasks, but benefits from well aligned tasks.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 270, + 558 + ], + "score": 1.0, + "content": "These effects are amplified by SNR. (c)", + "type": "text" + }, + { + "bbox": [ + 270, + 545, + 311, + 556 + ], + "score": 0.91, + "content": "\\overline { { s } } _ { A } = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 544, + 506, + 558 + ], + "score": 1.0, + "content": ". Tasks with very strong signals will show little", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 555, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 570 + ], + "score": 1.0, + "content": "effect from other tasks (note y-axis scale), but any impact will be negative unless the tasks are very", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 567, + 160, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 160, + 579 + ], + "score": 1.0, + "content": "well aligned.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 106, + 603, + 376, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 378, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 378, + 618 + ], + "score": 1.0, + "content": "G VARYING THE NUMBER OF TRAINING EXAMPLES", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 506, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 504, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 504, + 640 + ], + "score": 1.0, + "content": "In the main text, we focused on the test error dynamics in the case in which the number of examples", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "score": 1.0, + "content": "equalled the number of inputs. Here we show how the formula for test error curves is modified as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 650, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 241, + 662 + ], + "score": 1.0, + "content": "the number of training examples", + "type": "text" + }, + { + "bbox": [ + 241, + 651, + 250, + 660 + ], + "score": 0.84, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 651, + 379, + 662 + ], + "score": 1.0, + "content": "is varied. For simplicity, when", + "type": "text" + }, + { + "bbox": [ + 379, + 650, + 414, + 662 + ], + "score": 0.91, + "content": "P \\neq N _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 651, + 505, + 662 + ], + "score": 1.0, + "content": ", we focus on the case", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 661, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 264, + 674 + ], + "score": 1.0, + "content": "of a full rank student with aspect ratio", + "type": "text" + }, + { + "bbox": [ + 264, + 662, + 291, + 672 + ], + "score": 0.89, + "content": "A = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 661, + 326, + 674 + ], + "score": 1.0, + "content": "(so that", + "type": "text" + }, + { + "bbox": [ + 326, + 662, + 392, + 672 + ], + "score": 0.91, + "content": "N _ { 1 } = N _ { 2 } = N _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 661, + 506, + 674 + ], + "score": 1.0, + "content": "). 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(b,c) Minimum", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 315, + 507, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 253, + 330 + ], + "score": 1.0, + "content": "generalization error plotted against", + "type": "text" + }, + { + "bbox": [ + 254, + 315, + 289, + 329 + ], + "score": 0.94, + "content": "\\sqrt { P / N _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 315, + 336, + 330 + ], + "score": 1.0, + "content": "and SNR ·", + "type": "text" + }, + { + "bbox": [ + 336, + 315, + 371, + 329 + ], + "score": 0.72, + "content": "\\sqrt { P / N _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 315, + 507, + 330 + ], + "score": 1.0, + "content": ", respectively, at different SNRs.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 327, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 133, + 343 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 133, + 329, + 171, + 341 + ], + "score": 0.92, + "content": "P \\geq N _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 329, + 421, + 343 + ], + "score": 1.0, + "content": ", the minimum generalization error is simply determined by", + "type": "text" + }, + { + "bbox": [ + 421, + 327, + 475, + 342 + ], + "score": 0.92, + "content": "\\mathrm { S N R } \\sqrt { P / N _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 329, + 506, + 343 + ], + "score": 1.0, + "content": ", so all", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 341, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 325, + 353 + ], + "score": 1.0, + "content": "curves converge to a single asymptotic line in (c) as", + "type": "text" + }, + { + "bbox": [ + 325, + 341, + 334, + 351 + ], + "score": 0.81, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 341, + 408, + 353 + ], + "score": 1.0, + "content": "increases. 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Both these choices are intended to approximate the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 240, + 609 + ], + "score": 1.0, + "content": "situation in which the columns of", + "type": "text" + }, + { + "bbox": [ + 240, + 596, + 250, + 607 + ], + "score": 0.81, + "content": "\\hat { \\bf X }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 597, + 505, + 609 + ], + "score": 1.0, + "content": "are chosen to be iid unit-norm vectors. 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Thus, given the unit-norm inputs, we can think of the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 630, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 506, + 643 + ], + "score": 1.0, + "content": "teacher singular values as signal to noise ratios (SNRs). We now examine how the dynamics of the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 642, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 363, + 653 + ], + "score": 1.0, + "content": "test error evolves as we vary the number of training examples", + "type": "text" + }, + { + "bbox": [ + 363, + 642, + 372, + 651 + ], + "score": 0.79, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 642, + 505, + 653 + ], + "score": 1.0, + "content": ". 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There are a number of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 570, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 581 + ], + "score": 1.0, + "content": "interesting features to note. 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When", + "type": "text" + }, + { + "bbox": [ + 385, + 614, + 419, + 625 + ], + "score": 0.91, + "content": "P < N _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 614, + 506, + 627 + ], + "score": 1.0, + "content": ", however, the curves", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 624, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 639 + ], + "score": 1.0, + "content": "for different SNRs separate because the projection and noise effects depend on initial SNR. Finally,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "in Fig. 13D we show that approximately unit norm i.i.d. gaussian inputs yield similar results to the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "score": 1.0, + "content": "orthogonalized data matrices we employed in the theory, although the gaussian inputs do result in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 658, + 260, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 260, + 671 + ], + "score": 1.0, + "content": "slightly higher optimal stopping error.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41 + }, + { + "type": "title", + "bbox": [ + 108, + 686, + 305, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 307, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 307, + 699 + ], + "score": 1.0, + "content": "H LESS THAN FULL RANK STUDENTS", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Although we generally assumed students were full rank in the main text to simplify the calculations,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "our theory remains exact for TA networks of any rank. Furthermore, as shown in Fig. 14, the TA", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48.5 + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 164 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 104, + 81, + 506, + 164 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 177, + 261, + 189 + ], + "lines": [ + { + "bbox": [ + 106, + 177, + 263, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 263, + 189 + ], + "score": 1.0, + "content": "G.2 THE UNDERSAMPLED REGIME", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 197, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 231, + 210 + ], + "score": 1.0, + "content": "For the undersampled regime", + "type": "text" + }, + { + "bbox": [ + 232, + 198, + 261, + 209 + ], + "score": 0.85, + "content": "\\mathcal { D } < 1 \\dot { }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 197, + 420, + 210 + ], + "score": 1.0, + "content": "), we must account for the fact that the", + "type": "text" + }, + { + "bbox": [ + 420, + 198, + 429, + 208 + ], + "score": 0.83, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "training inputs do", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 177, + 223 + ], + "score": 1.0, + "content": "not span the full", + "type": "text" + }, + { + "bbox": [ + 178, + 210, + 191, + 221 + ], + "score": 0.87, + "content": "N _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 209, + 451, + 223 + ], + "score": 1.0, + "content": "dimensional space of all inputs. 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The third term accounts for learned correlations between the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 433, + 468 + ], + "score": 1.0, + "content": "student and teacher. It involves the transformation from teacher singular values", + "type": "text" + }, + { + "bbox": [ + 433, + 456, + 439, + 465 + ], + "score": 0.28, + "content": "\\overline { { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "to training data", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 168, + 479 + ], + "score": 1.0, + "content": "singular values", + "type": "text" + }, + { + "bbox": [ + 168, + 467, + 174, + 476 + ], + "score": 0.78, + "content": "\\hat { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 466, + 437, + 479 + ], + "score": 1.0, + "content": "through the formula (11) except with the aspect ratio replacement √", + "type": "text" + }, + { + "bbox": [ + 438, + 466, + 470, + 477 + ], + "score": 0.91, + "content": "A \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 466, + 505, + 479 + ], + "score": 1.0, + "content": ", and the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 284, + 491 + ], + "score": 1.0, + "content": "effective teacher singular value attenuation", + "type": "text" + }, + { + "bbox": [ + 284, + 477, + 327, + 489 + ], + "score": 0.93, + "content": "\\overline { { s } } \\sqrt { D } \\overline { { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 478, + 505, + 491 + ], + "score": 1.0, + "content": ". Similarly, the computation of the singular", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 455, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 358, + 503 + ], + "score": 1.0, + "content": "vector overlap is done through (12) also with the replacements", + "type": "text" + }, + { + "bbox": [ + 358, + 491, + 390, + 501 + ], + "score": 0.91, + "content": "A \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 491, + 408, + 503 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 408, + 489, + 451, + 502 + ], + "score": 0.93, + "content": "\\overline { { s } } \\sqrt { D } \\overline { { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 491, + 455, + 503 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 389, + 506, + 503 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 516, + 463, + 537 + ], + "lines": [ + { + "bbox": [ + 107, + 515, + 465, + 527 + ], + "spans": [ + { + "bbox": [ + 107, + 515, + 128, + 527 + ], + "score": 1.0, + "content": "G.3", + "type": "text" + }, + { + "bbox": [ + 131, + 515, + 465, + 527 + ], + "score": 1.0, + "content": "COMPARISON OF THEORY AND EXPERIMENT FOR UNDER AND OVER SAMPLED", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 132, + 528, + 242, + 537 + ], + "spans": [ + { + "bbox": [ + 132, + 528, + 242, + 537 + ], + "score": 1.0, + "content": "MEASUREMENT REGIMES", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 546, + 506, + 670 + ], + "lines": [ + { + "bbox": [ + 104, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "In Fig. 13, we show an excellent match between our theory and empirical simulations for varying", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 144, + 570 + ], + "score": 1.0, + "content": "values of", + "type": "text" + }, + { + "bbox": [ + 144, + 559, + 153, + 568 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 558, + 506, + 570 + ], + "score": 1.0, + "content": ", both in the oversampled and undersampled measurement regimes. There are a number of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 570, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 581 + ], + "score": 1.0, + "content": "interesting features to note. First, although the minimum generalization error improves monotonically", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 126, + 593 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 581, + 135, + 590 + ], + "score": 0.8, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 579, + 201, + 593 + ], + "score": 1.0, + "content": ", the asymptotic", + "type": "text" + }, + { + "bbox": [ + 202, + 581, + 232, + 591 + ], + "score": 0.86, + "content": "t \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 579, + 505, + 593 + ], + "score": 1.0, + "content": ") generalization error does not, because of a frozen subspace (Advani", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 590, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 316, + 604 + ], + "score": 1.0, + "content": "& Saxe, 2017) of the modes that are not overfit when", + "type": "text" + }, + { + "bbox": [ + 317, + 591, + 351, + 602 + ], + "score": 0.92, + "content": "P < N _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 590, + 483, + 604 + ], + "score": 1.0, + "content": ", because the training data rank is", + "type": "text" + }, + { + "bbox": [ + 484, + 591, + 503, + 602 + ], + "score": 0.88, + "content": "\\le P", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 590, + 506, + 604 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 602, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 164, + 616 + ], + "score": 1.0, + "content": "Second, when", + "type": "text" + }, + { + "bbox": [ + 164, + 603, + 199, + 614 + ], + "score": 0.91, + "content": "P \\geq N _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 603, + 435, + 616 + ], + "score": 1.0, + "content": ", the minimum generalization error is simply determined by", + "type": "text" + }, + { + "bbox": [ + 436, + 602, + 490, + 616 + ], + "score": 0.92, + "content": "\\mathbf { S N R } \\sqrt { P / N _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 603, + 506, + 616 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 305, + 627 + ], + "score": 1.0, + "content": "all curves converge to a single asymptotic line as", + "type": "text" + }, + { + "bbox": [ + 305, + 615, + 314, + 624 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 614, + 384, + 627 + ], + "score": 1.0, + "content": "increases. When", + "type": "text" + }, + { + "bbox": [ + 385, + 614, + 419, + 625 + ], + "score": 0.91, + "content": "P < N _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 614, + 506, + 627 + ], + "score": 1.0, + "content": ", however, the curves", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 624, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 639 + ], + "score": 1.0, + "content": "for different SNRs separate because the projection and noise effects depend on initial SNR. Finally,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "in Fig. 13D we show that approximately unit norm i.i.d. gaussian inputs yield similar results to the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "score": 1.0, + "content": "orthogonalized data matrices we employed in the theory, although the gaussian inputs do result in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 658, + 260, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 260, + 671 + ], + "score": 1.0, + "content": "slightly higher optimal stopping error.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 546, + 506, + 671 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 686, + 305, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 685, + 307, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 685, + 307, + 699 + ], + "score": 1.0, + "content": "H LESS THAN FULL RANK STUDENTS", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Although we generally assumed students were full rank in the main text to simplify the calculations,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "our theory remains exact for TA networks of any rank. Furthermore, as shown in Fig. 14, the TA", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "and random networks again show very similar optimal stopping generalization error, but with the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "optimal stopping time of the random networks lagging behind that of the TA networks. 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