diff --git a/parse/train/BJg866NFvB/BJg866NFvB.md b/parse/train/BJg866NFvB/BJg866NFvB.md new file mode 100644 index 0000000000000000000000000000000000000000..a09d0f4656efcf87489bbb478b3b2a96b6189ce7 --- /dev/null +++ b/parse/train/BJg866NFvB/BJg866NFvB.md @@ -0,0 +1,665 @@ +# ESTIMATING COUNTERFACTUAL TREATMENT OUTCOMES OVER TIME THROUGH ADVERSARIALLY BALANCED REPRESENTATIONS + +# Ioana Bica + +Department of Engineering Science University of Oxford, Oxford, UK The Alan Turing Institute, London, UK ioana.bica@eng.ox.ac.uk + +Ahmed M. Alaa Department of Electrical Engineering University of California, Los Angeles, USA ahmedmalaa@ucla.edu + +# James Jordon + +# Mihaela van der Schaar + +Department of Engineering Science University of Oxford, Oxford, UK james.jordon@wolfson.ox.ac.uk + +University of Cambridge, Cambridge, UK University of California, Los Angeles, USA The Alan Turing Institute, London, UK mv472@cam.ac.uk + +# ABSTRACT + +Identifying when to give treatments to patients and how to select among multiple treatments over time are important medical problems with a few existing solutions. In this paper, we introduce the Counterfactual Recurrent Network (CRN), a novel sequence-to-sequence model that leverages the increasingly available patient observational data to estimate treatment effects over time and answer such medical questions. To handle the bias from time-varying confounders, covariates affecting the treatment assignment policy in the observational data, CRN uses domain adversarial training to build balancing representations of the patient history. At each timestep, CRN constructs a treatment invariant representation which removes the association between patient history and treatment assignments and thus can be reliably used for making counterfactual predictions. On a simulated model of tumour growth, with varying degree of time-dependent confounding, we show how our model achieves lower error in estimating counterfactuals and in choosing the correct treatment and timing of treatment than current state-of-the-art methods. + +# 1 INTRODUCTION + +As clinical decision-makers are often faced with the problem of choosing between treatment alternatives for patients, reliably estimating their effects is paramount. While clinical trials represent the gold standard for causal inference, they are expensive, have a few patients and narrow inclusion criteria (Booth & Tannock, 2014). Leveraging the increasingly available observational data about patients, such as electronic health records, represents a more viable alternative for estimating treatment effects. + +A large number of methods have been proposed for performing causal inference using observational data in the static setting (Johansson et al., 2016; Shalit et al., 2017; Alaa & van der Schaar, 2017; Li & Fu, 2017; Yoon et al., 2018; Alaa & van der Schaar, 2018; Yao et al., 2018) and only a few methods address the longitudinal setting (Xu et al., 2016; Roy et al., 2016; Soleimani et al., 2017; Schulam & Saria, 2017; Lim et al., 2018). However, estimating the effects of treatments over time poses unique opportunities such as understanding how diseases evolve under different treatment plans, how individual patients respond to medication over time, but also which are optimal timings for assigning treatments, thus providing new tools to improve clinical decision support systems. + +The biggest challenge when estimating the effects of time-dependent treatments from observational data involves correctly handling the time-dependent confounders: patient covariates that are affected by past treatments which then influence future treatments and outcomes (Platt et al., 2009). For instance, consider that treatment A is given when a certain patient covariate (e.g. white blood cell count) has been outside of normal range values for several consecutive timesteps. Suppose also that this patient covariate was itself affected by the past administration of treatment B. If these patients are more likely to die, without adjusting for the time-dependent confounding (e.g. the changes in the white blood cell count over time), we will incorrectly conclude that treatment A is harmful to patients. Moreover, estimating the effect of a different sequence of treatments on the patient outcome would require not only adjusting for the bias at the current step (in treatment A), but also for the bias introduced by the previous application of treatment B. + +![](images/48102b092b83fa34ae96204e15fa6bcdf8c5bf203ea2fe1f45443282de05bbd2.jpg) +Figure 1: Applicability of CRN in cancer treatment planning. We illustrate 3 patients with different covariate and treatment histories $\bar { \mathbf { H } } _ { t }$ . For a current time $t$ , CRN can predict counterfactual trajectories (the coloured dashed branches) for planned treatments in the future. Through the counterfactual predictions, we can decide which treatment plan results in the best patient outcome (in this case, the lowest tumour volume). This way, CRN can be used to perform all of the following: choose optimal treatments (a), find timing when treatment is most effective (b) decide when to stop treatment (c). + +Existing methods for causal inference in the static setting cannot be applied in this longitudinal setting since they are designed to handle the cross-sectional set-up, where the treatment and outcome depend only on a static value of the patient covariates. If we consider again the above example, these methods would not be able to model how the changes in patient covariates over time affect the assignment of treatments and they would also not be able to estimate the effect of a sequence of treatments on the patient outcome (e.g. sequential application of treatment A followed by treatment B). Different models that can handle these temporal dependencies in the observational data and varying-length patient histories are needed for estimating treatment effects over time. + +Time-dependent confounders are present in observational data because doctors follow policies: the history of the patients’ covariates and the patients’ response to past treatments are used to decide future treatments (Mansournia et al., 2012). The direct use of supervised learning methods will be biased by the treatment policies present in the observational data and will not be able to correctly estimate counterfactuals for different treatment assignment policies. + +Standard methods for adjusting for time-varying confounding and estimating the effects of timevarying exposures are based on ideas from epidemiology. The most widely used among these are Marginal Structural Models (MSMs) (Robins et al., 2000; Mansournia et al., 2012) which use the inverse probability of treatment weighting (IPTW) to adjust for the time-dependent confounding bias. Through IPTW, MSMs create a pseudo-population where the probability of treatment does not depend on the time-varying confounders. However, MSMs are not robust to model misspecification in computing the IPTWs. MSMs can also give high-variance estimates due to extreme weights; computing the IPTW involves dividing by probability of assigning a treatment conditional on patient history which can be numerically unstable if the probability is small. + +We introduce the Counterfactual Recurrent Network (CRN), a novel sequence-to-sequence architecture for estimating treatment effects over time. CRN leverages recent advances in representation learning (Bengio et al., 2012) and domain adversarial training (Ganin et al., 2016) to overcome the problems of existing methods for causal inference over time. Our main contributions are as follows. + +Treatment invariant representations over time. CRN constructs treatment invariant representations at each timestep in order to break the association between patient history and treatment assignment and thus removes the bias from time-dependent confounders. For this, CRN uses domain adversarial training (Ganin et al., 2016; Li et al., 2018; Sebag et al., 2019) to trade-off between building this balancing representation and predicting patient outcomes. We show that these representations remove the bias from time-varying confounders and can be reliably used for estimating counterfactual outcomes. This represents the first work that introduces ideas from domain adaptation to the area of estimating treatment effects over time. In addition, by building balancing representations, we propose a novel way of removing the bias introduced by time-varying confounders. + +Counterfactual estimation of future outcomes. To estimate counterfactual outcomes for treatment plans (and not just single treatments), we integrate the domain adversarial training procedure as part of a sequence-to-sequence architecture. CRN consists of an encoder network which builds treatment invariant representations of the patient history that are used to initialize the decoder. The decoder network estimates outcomes under an intended sequence of future treatments, while also updating the balanced representation. By performing counterfactual estimation of future treatment outcomes, CRN can be used to answer critical medical questions such as deciding when to give treatments to patients, when to start and stop treatment regimes, and also how to select from multiple treatments over time. We illustrate in Figure 1 the applicability of our method in choosing optimal cancer treatments. + +In our experiments, we evaluate CRN in a realistic set-up using a model of tumour growth (Geng et al., 2017). We show that CRN achieves better performance in predicting counterfactual outcomes, but also in choosing the right treatment and timing of treatment than current state-of-the-art methods. + +# 2 RELATED WORK + +We focus on methods for estimating treatment effects over time and for building balancing representations for causal inference. A more in-depth review of related work is in Appendix A. + +Treatment effects over time. Standard methods for estimating the effects of time-varying exposures were first developed in the epidemiology literature and include the $\mathrm { g }$ -computation formula, Structural Nested Models and Marginal Structural Models (MSMs) (Robins, 1986; 1994; Robins et al., 2000; Robins & Hernán, 2008). Originally, these methods have used predictors performing logistic/linear regression which makes them unsuitable for handling complex time-dependencies (Hernán et al., 2001; Mansournia et al., 2012; Mortimer et al., 2005). To address these limitations, methods that use Bayesian non-parametrics or recurrent neural networks as part of these frameworks have been proposed. (Xu et al., 2016; Roy et al., 2016; Lim et al., 2018). + +To begin with, Xu et al. (2016) use Gaussian processes to model discrete patient outcomes as a generalized mixed-effects model and uses the $g$ -computation method to handle time-varying confounders. Soleimani et al. (2017) extend the approach in $\mathrm { X u }$ et al. (2016) to the continuous time-setting and model treatment responses using linear time-invariant dynamical systems. Roy et al. (2016) use Dirichlet and Gaussian processes to model the observational data and estimate the IPTW in Marginal Structural Models. Schulam & Saria (2017) build upon work from Lok et al. (2008); Arjas & Parner (2004) and use marked point processes and Gaussian processes to learn causal effects in continuous-time data. These Bayesian non-parametric methods make strong assumptions about model structure and consequently cannot handle well heterogeneous treatment effects arising from baseline variables (Soleimani et al., 2017; Schulam & Saria, 2017) and multiple treatment outcomes (Xu et al., 2016; Schulam & Saria, 2017). + +The work most related to ours is the one of Lim et al. (2018) which improves on the standard MSMs by using recurrent neural networks to estimate the inverse probability of treatment weights (IPTWs). Lim et al. (2018) introduces Recurrent Marginal Structural Networks (RMSNs) which also use a sequence-to-sequence deep learning architecture to forecast treatment responses in a similar fashion to our model. However, RMSNs require training additional RNNs to estimate the propensity weights and does not overcome the fundamental problems with IPTWs, such as the high-variance of the weights. Conversely, CRN takes advantage of the recent advances in machine learning, in particular, representation learning to propose a novel way of handling time-varying confounders. + +Balancing representations for treatment effect estimation. Balancing the distribution of control and treated groups has been used for counterfactual estimation in the static setting. The methods proposed in the static setting for balancing representations are based on using discrepancy measures in the representation space between treated and untreated patients, which do not generalize to multiple treatments (Johansson et al., 2016; Shalit et al., 2017; Li & Fu, 2017; Yao et al., 2018). Moreover, due to the sequential assignment of treatments in the longitudinal setting, and due to the change of patient covariates over time according to previous treatments, the methods for the static setting are not directly applicable to the time-varying setting (Hernán et al., 2000; Mansournia et al., 2012). + +# 3 PROBLEM FORMULATION + +Consider an observational dataset $\mathcal { D } = \left\{ \{ \mathbf { x } _ { t } ^ { ( i ) } , \mathbf { a } _ { t } ^ { ( i ) } , \mathbf { y } _ { t + 1 } ^ { ( i ) } \} _ { t = 1 } ^ { T ^ { ( i ) } } \cup \{ \mathbf { v } ^ { ( i ) } \} \right\} _ { i = 1 } ^ { N }$ consisting of information about $N$ independent patients. For each patient $( i )$ , we observe time-dependent covariates $\mathbf { X } _ { t } ^ { ( i ) } \in \mathcal { X } _ { t }$ , treatment received $\mathbf { A } _ { t } ^ { ( i ) } \in \{ A _ { 1 } , . . . A _ { K } \} = \mathcal { A }$ and outcomes $\mathbf { Y } _ { t + 1 } ^ { ( i ) } \in \mathcal { V } _ { t + 1 }$ for $T ^ { ( i ) }$ discrete timesteps. The patient can also have baseline covariates $\mathbf { V } ^ { ( i ) } \in \mathcal { V }$ such as gender and genetic information. Note that the ou me $\mathbf { Y } _ { t + 1 } ^ { ( i ) }$ will be part of the observed covariates $\mathbf { X } _ { t + 1 } ^ { ( i ) }$ . For simplicity, the patient $( i )$ + +We adopt the potential outcomes framework proposed by (Neyman, 1923; Rubin, 1978) and extended by (Robins & Hernán, 2008) to account for time-varying treatments. Let $\mathbf { Y } [ \bar { \mathbf { a } } ]$ be the potential outcomes, either factual or counterfactual, for each possible course of treatment a¯. Let $\mathbf { \bar { H } } _ { t } = ( \bar { \mathbf { X } } _ { t } , \bar { \mathbf { A } } _ { \underline { { t } } - 1 } , \mathbf { V } )$ represent the history of the patient covariates $\bar { \mathbf X } _ { t } = ( \mathbf X _ { 1 } , \dots , \mathbf X _ { t } )$ , treatment assignments $\bar { \mathbf { A } } _ { t } = ( \mathbf { A } _ { 1 } , \ldots , \mathbf { A } _ { t } )$ and static features $\mathbf { V }$ . We want to estimate: + +$$ +\mathbb { E } ( \mathbf { Y } _ { t + \tau } [ \bar { \mathbf { a } } ( t , t + \tau - 1 ) ] | \bar { \mathbf { H } } _ { t } ) , +$$ + +where $\bar { \mathbf { a } } ( t , t + \tau - 1 ) = [ \mathbf { a } _ { t } , \dots \mathbf { a } _ { t + \tau - 1 } ]$ represents a possible sequence of treatments from timestep $t$ just until before the potential outcome $\mathbf { Y } _ { t + \tau }$ is observed. We make the standard assumptions (Robins et al., 2000; Lim et al., 2018) needed to identify the treatment effects: consistency, positivity and no hidden confounders (sequential strong ignorability). See Appendix B for more more details. + +# 4 COUNTERFACTUAL RECURRENT NETWORK + +The observational data can be used to train a supervised learning model to forecast: $\mathbb { E } ( \mathbf { Y } _ { t + \tau } \ )$ $\bar { \mathbf { A } } ( t , t + \tau - 1 ) = \bar { \mathbf { a } } ( t , t + \tau - 1 ) , \bar { \mathbf { H } } _ { t } )$ . However, without adjusting for the bias introduced by timevarying confounders, this model cannot be reliably used for making causal predictions (Robins et al., 2000; Robins & Hernán, 2008; Schulam $\&$ Saria, 2017). The Counterfactual Recurrent Network (CRN) removes this bias through domain adversarial training and estimates the counterfactual outcomes $\mathbb { E } ( \mathbf { Y } _ { t + \tau } [ \bar { \mathbf { a } } ( t , t + \tau - 1 \bar { ) } ] | \bar { \mathbf { H } } _ { t } )$ , for any intended future treatment assignment $\bar { \mathbf { a } } ( t , t + \tau - 1 )$ . + +Balancing representations. The history $\bar { \mathbf { H } } _ { t } = ( \bar { \mathbf { X } } _ { t } , \bar { \mathbf { A } } _ { t - 1 } , \mathbf { V } )$ of the patient contains the timevarying confounders $\bar { \mathbf { X } } _ { t }$ which bias the treatment assignment $\mathbf { A } _ { t } \in \{ A _ { 1 } , \dotsc . . . A _ { K } \}$ in the observational dataset. Inverse probability of treatment weighting, as performed by MSMs, creates a pseudopopulation where the probability of treatment ${ \bf A } _ { t }$ does not depend on the time-varying confounders (Robins et al., 2000). In this paper, we propose instead building a representation of the history $\bar { \mathbf { H } } _ { t }$ that is not predictive of the treatment ${ \bf A } _ { t }$ . This way, we remove the association between history, containing the time-varying confounders $\bar { \mathbf { X } } _ { t }$ , and current treatment ${ \bf A } _ { t }$ . Robins (1999) shows that in this case, the estimation of counterfactual treatment outcomes is unbiased. See Appendix $\textrm { C }$ for details and for an example of a causal graph with time-dependent confounders. + +Let $\Phi$ be the representation function that maps the patient history $\bar { \mathbf { H } } _ { t }$ to a representation space $\mathcal { R }$ . To obtain unbiased treatment effects, $\Phi$ needs to construct treatment invariant representations such that $P ( \Phi ( { \bar { \mathbf { H } } } _ { t } ) \mid \mathbf { A } _ { t } = A _ { 1 } ) = \cdots = P ( \Phi ( { \bar { \mathbf { H } } } _ { t } ) \mid \mathbf { A } _ { t } = A _ { K } )$ . To achieve this and to estimate counterfactual outcomes under a planned sequence of treatments, we integrate the domain adversarial training framework proposed by Ganin et al. (2016) and extended by Sebag et al. (2019) to the multi-domain learning setting, into a sequence-to-sequence architecture. In our case, the different treatments at each timestep are considered the different domains. Note that the novelty here comes from the use of domain adversarial training to handle the bias from the time-dependent confounders, rather than the use of sequence-to-sequence models, which have already been applied to forecast treatment responses (Lim et al., 2018). Figure 2 illustrates our model architecture. + +Encoder. The encoder network uses an RNN, with LSTM unit (Hochreiter & Schmidhuber, 1997), to process the history of treatments $\bar { \mathbf { A } } _ { t - 1 }$ , covariates $\bar { \mathbf { X } } _ { t }$ and baseline features $\mathbf { V }$ to build a treatment invariant representation $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ , but also to predict one-step-ahead outcomes $\mathbf { Y } _ { t + 1 }$ . To achieve this, the encoder network aims to maximize the loss of the treatment classifier $G _ { a }$ and minimize the loss of the outcome predictor network $G _ { y }$ . This way, the balanced representation $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ is not predictive of the assigned treatment ${ \bf A } _ { t }$ , but is discriminative enough to estimate the outcome $\mathbf { Y } _ { t + 1 }$ . To train this model using gradient descent, we use the Gradient Reversal Layer (Ganin et al., 2016). + +![](images/39de32791ca33f62a2ed333d3baf6dd543573abf1582505564a41dc63718a59d.jpg) +Figure 2: CRN architecture. Encoder builds representation $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ that maximizes loss of treatment classifier $G _ { a }$ and minimizes loss of outcome predictor $G _ { y }$ . $\bar { \Phi ( \bar { \mathbf { H } } _ { t } ) }$ is used to initialize the decoder, which continues to update it to predict counterfactual outcomes of a sequence of future treatments. + +Decoder. The decoder network uses the balanced representation computed by the encoder to initialize the state of an RNN that predicts the counterfactual outcomes for a sequence of future treatments. During training, the decoder uses as input the outcomes from the observational data $( \mathbf { Y } _ { t + 1 } , . . . \mathbf { Y } _ { t + \tau - 1 } )$ , the static patient features $\mathbf { V }$ and the intended sequence of treatments $\bar { \mathbf { a } } ( t , t + \tau - 1 )$ . The decoder is trained in a similar way to the encoder to update the balanced representation and to estimate the outcomes. During testing, we do not have access to ground-truth outcomes; thus, the outcomes predicted by the decoder $( \mathsf { \bar { Y } } _ { t + 1 } , \dots , \hat { \mathbf { Y } } _ { t + \tau - 1 } )$ are auto-regressively used instead as inputs. By running the decoder with different treatment settings, and by auto-regressively feeding back the outcomes, we can determine when to start and end different treatments, which is the optimal time to give the treatment and which treatments to give over time to obtain the best patient outcomes. + +The representation $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ is built by applying a fully connected layer, with Exponential Linear Unit (ELU) activation to the output of the LSTM. The treatment classifier $G _ { a }$ and the predictor network $G _ { y }$ consist of a hidden layer each, also with ELU activation. The output layer of $G _ { a }$ uses softmax activation, while the output layer of $G _ { y }$ uses linear activation for continuous predictions. For categorical outcomes, softmax activation can be used. We follow an approach similar to Lim et al. (2018) and we split the encoder and decoder training into separate steps. See Appendix E for details. + +The encoder and decoder networks use variational dropout (Gal & Ghahramani, 2016) such that the CRN can also give uncertainty intervals for the treatment outcomes. This is particularity important in the estimation of treatment effects, since the model predictions should only be used when they have high confidence. Our model can also be modified to allow for irregular samplings of observations by using a PhasedLSTM (Neil et al., 2016). + +# 5 ADVERSARIALLY BALANCED REPRESENTATION OVER TIME + +At each timestep $t$ , let the $K$ different possible treatments $\mathbf { A } _ { t } \in \{ A _ { 1 } , \dotsc . . . A _ { K } \}$ represent our domains. As described in Section 4, to remove the bias from time-dependent confounders, we build a representation of history $\bar { \mathbf { H } } _ { t }$ that is invariant across treatments: $P ( \mathbf { \bar { \Phi } } ( \mathbf { \bar { H } } _ { t } ) \mid A _ { 1 } ) = \cdots = P ( \Phi ( \mathbf { \bar { H } } _ { t } ) \mid \mathbf { \bar { \Phi } } _ { X } )$ . + +This requirement can be enforced by minimizing the distance in the distribution of $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ between any two pairs of treatments. Kifer et al. (2004); Ben-David et al. (2007), propose measuring the disparity between distributions based on their separability by a discriminatively-trained classifier. Let the symmetric hypothesis class $\mathcal { H }$ consist of the set of symmetric multiclass classifiers, such as neural network architectures. The $\mathcal { H }$ -divergence between all pairs of two distributions is defined in terms of the capacity of the hypothesis class $\mathcal { H }$ to discriminate between examples from the multiple distributions. Empirically, minimizing the $\varkappa -$ divergence involves building a representation where examples from the multiple domains are as indistinguishable as possible (Ben-David et al., 2007; Li et al., 2018; Sebag et al., 2019). Ganin et al. (2016) use this idea to propose an adversarial framework for domain adaptation involving building a representation which achieves maximum error on a domain classifier and minimum error on an outcome predictor. Similarly, in our case, we use domain adversarial training to build a representation of the patient history $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ that is both invariant to the treatment given at timestep $t$ , ${ \bf A } _ { t }$ and that achieves low error in estimating the outcome $\mathbf { Y } _ { t + 1 }$ . + +Let $G _ { a } \big ( \Phi ( \bar { \mathbf { H } } _ { t } ) ; \theta _ { a } \big )$ be the treatment classifier with parameters $\theta _ { a }$ and let $G _ { a } ^ { j } ( \Phi ( { \bar { \mathbf { H } } } _ { t } ) ; \theta _ { a } )$ be the output corresponding to treatment $A _ { j }$ . Let $\hat { G } _ { y } ( \dot { \Phi } ( \bar { \mathbf { H } } _ { t } ) ; \theta _ { y } )$ be the predictor network with parameters $\theta _ { y }$ . The representation function $\Phi$ is parameterized by the parameters $\theta _ { r }$ in the RNN: $\Phi ( { \bar { \mathbf { H } } } _ { t } ; \theta _ { r } )$ . Figure 3 shows the adversarial training procedure used. + +For timestep $t$ and patient $( i )$ , let $\mathcal { L } _ { t , a } ^ { ( i ) } ( \theta _ { r } , \theta _ { a } )$ be the treatment (domain) loss and let $\mathcal { L } _ { t , y } ^ { ( i ) } ( \theta _ { r } , \theta _ { y } )$ the outcome loss, defined as follows: + +$$ +\mathcal { L } _ { t , a } ^ { ( i ) } ( \theta _ { r } , \theta _ { a } ) = - \sum _ { j = 1 } ^ { K } \mathbb { I } _ { \{ \mathbf { a } _ { t } ^ { ( i ) } = a _ { j } \} } \log ( G _ { a } ^ { j } ( \Phi ( \bar { \mathbf { H } } _ { t } ; \theta _ { r } ) ; \theta _ { a } ) ) +$$ + +$$ +\mathcal { L } _ { t , y } ^ { ( i ) } ( \theta _ { r } , \theta _ { y } ) = \| \mathbf { Y } _ { t + 1 } ^ { ( i ) } - \left( G _ { y } ( \Phi ( \bar { \mathbf { H } } _ { t } ; \theta _ { r } ) , \theta _ { y } ) \right) \| ^ { 2 } . +$$ + +If the outcome is binary, the cross-entropy loss can be used instead for $\mathcal { L } _ { t , y }$ . To build treatment invariant representations and to also estimate patient outcomes, we aim to maximize treatment loss and minimize outcome loss. + +![](images/d393ec25b88d42f7d5cc610cfeb256d83fddebdab392f202255cb7d1638d9cd5.jpg) +Figure 3: Training procedure for building balancing representation. + +$\mathcal { L } _ { t , y } ^ { ( i ) }$ at timestep $t$ is given by: + +$$ +\mathcal { L } _ { t } ^ { ( i ) } ( \theta _ { r } , \theta _ { y } , \theta _ { a } ) = \sum _ { i = 1 } ^ { N } \mathcal { L } _ { t , y } ^ { ( i ) } ( \theta _ { r } , \theta _ { y } ) - \lambda \mathcal { L } _ { t , a } ^ { ( i ) } ( \theta _ { r } , \theta _ { a } ) , +$$ + +where the hyperparameter $\lambda$ controls this trade-off between domain discrimination and outcome prediction. We use the standard procedure for training domain adversarial networks from Ganin et al. (2016) and we start off with an initial value for $\lambda$ and use an exponentially increasing schedule during training. To train the model using backpropagation, we use the Gradient Reversal Layer (GRL) (Ganin et al., 2016). For more details about the training procedure, see Appendix $\mathrm { E }$ . + +By using the objective $\mathcal { L } _ { t } ^ { ( i ) } ( \theta _ { r } , \theta _ { y } , \theta _ { a } )$ , we reach the saddle point $( \hat { \theta } _ { r } , \hat { \theta } _ { y } , \hat { \theta } _ { a } )$ that achieves the equilibrium between domain discrimination and outcome estimation. + +$$ +( \hat { \theta } _ { r } , \hat { \theta } _ { y } ) = \arg \operatorname* { m i n } _ { \theta _ { r } , \theta _ { y } } \mathcal { L } _ { t } ^ { ( i ) } ( \theta _ { r } , \theta _ { y } , \hat { \theta } _ { a } ) \qquad \hat { \theta } _ { a } = \arg \operatorname* { m a x } _ { \theta _ { a } } \mathcal { L } _ { t } ^ { ( i ) } ( \hat { \theta } _ { r } , \hat { \theta } _ { y } , \theta _ { a } ) . +$$ + +The result stated in Theorem 1 proves that the treatment (domain) loss part of our objective (from equation 2) aims to remove the time-dependent confounding bias. + +Theorem 1. Let $t \in \{ 1 , 2 , \dots \}$ . For each $j = 1 , . . . , K$ , let $P _ { j }$ denote the distribution of $\bar { \mathbf { H } } _ { t }$ conditional on $\mathbf { A } _ { t } = A _ { j }$ and let $P _ { j } ^ { \Phi }$ denote the distribution of $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ conditional on $\mathbf { A } _ { t } = A _ { j }$ . Let $G _ { a } ^ { j }$ denote the output of $G _ { a }$ corresponding to treatment $A _ { j }$ . Then the minimax game defined by + +$$ +\underset { \Phi } { \mathop { \operatorname* { m i n } } } \underset { G _ { a } } { \mathop { \operatorname* { m a x } } } \sum _ { j = 1 } ^ { K } \mathbb { E } _ { \bar { \mathbf { H } } _ { t } \sim P _ { j } } \left[ \log ( G _ { a } ^ { j } ( \Phi ( \bar { \mathbf { H } } _ { t } ) ; \theta _ { a } ) ) \right] \qquad \quad s u b j e c t t o \sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( \Phi ( \bar { \mathbf { H } } _ { t } ) ) = 1 +$$ + +has a global minimum which is attained if and only if $P _ { 1 } ^ { \Phi } = P _ { 2 } ^ { \Phi } = \ldots = P _ { K } ^ { \Phi }$ , i.e. when the learned representations are invariant across all treatments. + +Proof. This result is a restatement of the one in Li et al. (2018). For details, see the Appendix D. + +A good representation allows us to obtain a low error in estimating counterfactuals for all treatments, while at the same time to minimize the $\mathcal { H }$ -divergence between induced marginal distributions of all the domains. We use an algorithm that directly minimizes a combination of the $\varkappa -$ divergence and the empirical training margin. + +# 6 EXPERIMENTS + +In real datasets, counterfactual outcomes and the degree of time-dependent confounding are not known (Schulam & Saria, 2017; Lim et al., 2018). To validate the $\mathrm { C R N ^ { 1 } }$ , we evaluate it on a Pharmacokinetic-Pharmacodynamic model of tumour growth (Geng et al., 2017), which uses a stateof-the-art bio-mathematical model to simulate the combined effects of chemotherapy and radiotherapy in lung cancer patients. The same model was used by Lim et al. (2018) to evaluate RMSNs. + +Model of tumour growth The volume of tumour $t$ days after diagnosis is modelled as follows: + +$$ +V ( t + 1 ) = \Big ( 1 + \underbrace { \rho \mathrm { l o g } \big ( \frac { K } { V ( t ) } \big ) } _ { \mathrm { T u m o r ~ g r o w t h } } - \underbrace { \beta _ { c } C ( t ) } _ { \mathrm { C h e m o t h e r a p y } } - \underbrace { \big ( \alpha _ { r } d ( t ) + \beta _ { r } d ( t ) ^ { 2 } \big ) } _ { \mathrm { R a d i o t h e r a p y } } + \underbrace { e _ { t } } _ { \mathrm { N o i s e } } \Big ) V ( t ) +$$ + +where $K , \rho , \beta _ { c } , \alpha _ { r } , \beta _ { r } , e _ { t }$ are sampled as described in Geng et al. (2017). To incorporate heterogeneity in patient responses, the prior means for $\beta _ { c }$ and $\alpha _ { r }$ are adjusted to create patient subgroups, which are used as baseline features. The chemotherapy concentration $C ( t )$ and radiotherapy dose $d ( t )$ are modelled as described in Appendix F. Time-varying confounding is introduced by modelling emotherapy and radiotherapy assigndepending on the tumour diameter: ables, and $p _ { c }$ $p _ { r }$ $\begin{array} { r } { p _ { c } ( t ) = \sigma \big ( \frac { \gamma _ { c } } { D _ { \mathrm { m a x } } } ( \bar { D } ( t ) - \delta _ { c } ) \big ) } \end{array}$ $\begin{array} { r } { p _ { r } ( t ) \dot { = } \sigma \big ( \frac { \gamma _ { r } } { D _ { \mathrm { m a x } } } ( \hat { \bar { D } } ( t ) - } \end{array}$ $\delta _ { r } )$  where is the average diameter over the last 15 days, $D _ { \mathrm { m a x } } = 1 3 \mathrm { c m } , \sigma ( \cdot )$ is the sigmoid and $\delta _ { c } = \delta _ { r } = D _ { \operatorname* { m a x } } / 2$ . The amount of time-dependent confounding is controlled through $\gamma _ { c } , \gamma _ { r }$ ; the higher $\gamma _ { \star }$ is, the more important the history is in assigning treatments. At each timestep, there are four treatment options: no treatment, chemotherapy, radiotherapy, combined chemotherapy and radiotherapy. For details about data simulation, see Appendix F. + +Benchmarks We used the following benchmarks for performance comparison: Marginal Structural Models (MSMs) (Robins et al., 2000), which use logistic regression for estimating the IPTWs and linear regression for prediction (see Appendix G for details). We also compare against the Recurrent Marginal Structural Networks (RMSNs) Lim et al. (2018), which is the current state-of-the-art model in estimating treatment responses. RMSNs use RNNs to estimate the IPTWs and the patient outcomes (details in Appendix H). To show that standard supervised learning models do not handle the timevarying confounders we compare against an RNN and a linear regression model, which receive as input treatments and covariates to predict the outcome (see Appendix I for details). Our model architecture follows the description in Sections 4 and 5, with full training details and hyperparameter optimization in Appendix J. To show the importance of adversarial training, we also benchmark against CRN $\lambda = 0$ ) a model with the same architecture, but with $\lambda = 0$ , i.e our model architecture without adversarial training. + +# 6.1 EVALUATE MODELS ON COUNTERFACTUAL PREDICTIONS + +Previous methods focused on evaluating the error only for factual outcomes (observed patient outcomes) (Lim et al., 2018). However, to build decision support systems, we need to evaluate how well the models estimate the counterfactual outcomes, i.e patient outcomes under alternative treatment options. The parameters $\gamma _ { c }$ and $\gamma _ { r }$ control the treatment assignment policy, i.e. the degree of time-dependent confounding present in the data. We evaluate the benchmarks under different degrees of time-dependent confounding by setting $\gamma = \gamma _ { c } = \gamma _ { r }$ . For each $\gamma$ we simulate a 10000 patients for training, 1000 for validation (hyperparameter tuning) and 1000 for out-of-sample testing. For the patients in the test set, for each time $t$ , we also simulate counterfactuals $\mathbf { Y } _ { t + 1 }$ , represented by tumour volume $V ( t + 1 )$ , under all possible treatment options. + +![](images/cfd2e5bc6015b41a94296fbf8b32467dc117f38d981d224e897f47516e68addb.jpg) +Figure 4: Results for prediction of patient counterfactuals. + +Figure 4 (a) shows the normalized root mean squared error (RMSE) for one-step ahead estimation of counterfactuals with varying degree of time-dependent confounding $\gamma$ . The RMSE is normalized by the maximum tumour volume: $V _ { m a x } = \mathrm { 1 \bar { 1 } 5 0 c m ^ { 3 } }$ . The linear and MSM models provide a baseline for performance as they achieve the highest RMSE. While the use of IPTW in MSMs helps when $\gamma$ increases, using linear modelling has severe limitations. When there is no time-dependent confounding, the machine learning methods achieve similar performance, close to $0 . 6 \%$ RMSE. As the bias in the dataset increases, the harder it becomes for the RNN and the CRN $\lambda = 0 )$ ) to generalize to estimate outcomes of treatments not matching the training policy. When $\gamma = 1 0$ , CRN improves by $4 8 . 1 \%$ on the same model architecture without domain adversarial training CRN $\lambda = 0$ ). + +Our proposed model achieves the lowest RMSE across all values of $\gamma$ . Compared to RMSNs, CRN improves by $\sim 1 7 \%$ when $\gamma > 6$ . To highlight the gains of our method even for smaller $\gamma$ , Figure 4 (b) shows the RMSE for five-step ahead prediction (with counterfactuals generated as described in Section 6.2 and Appendix L). RMSNs also use a decoder for sequence prediction. However, RMSNs require training additional RNNs to estimate the IPTW, which are used to weight each sample during the decoder training. For $\tau$ -step ahead prediction, IPTW involves multiplying $\tau$ weights which can result in high variance. The results in Figure 4 (b) show the problems with using IPTW to handle the time-dependent confounding bias. See Appendix K for more results on multi-step ahead prediction. + +Balancing representation: To evaluate whether the CRN has indeed learnt treatment invariant represenations, for $\gamma = 5$ , we illustrate in Figure 5 the T-SNE embeddings of the balancing representations $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ built by the CRN encoder for test patients. We color each point by the treatment $\mathbf { A } _ { t } \in \{ \mathrm { n o } \}$ treatment, chemotherapy, radiotherapy, combined chemotherapy and radiotherapy} received at timestep $t$ to highlight the invariance of $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ across the different treatments. In Figure 5(b), we show $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ only for chemotherapy and radiotherapy for better understanding. + +![](images/81db339d71547cdef7a4a6bfc60803c8c22dc7a97cbc2b111eb3ea319e11801f.jpg) +Figure 5: TSNE embedding of the balancing representation $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ learnt by the CRN encoder at different timesteps $t$ . Notice that $\Phi ( { \bar { \mathbf { H } } } _ { t } )$ is not predictive of the treatment ${ \bf A } _ { t }$ given at timestep $t$ . + +# 6.2 EVALUATE RECOMMENDING THE RIGHT TREATMENT AND TIMING OF TREATMENT + +Evaluating the models just in terms of the RMSE on counterfactual estimation is also not enough for assessing their reliability when used as part of decision support systems. In this section we assess how well the models can select the correct treatment and timing of treatment for several forecasting horizons $\tau$ . We generate test sets consisting of 1000 patients where for each horizon $\tau$ and for each time $t$ in a patient’s trajectory, there are $\tau$ options for giving chemotherapy at one of $t , \dots t + \tau - 1$ and $\tau$ options for giving radiotherapy at one of $t , \dots t + \tau - 1$ . At the rest of the future timesteps, no treatment is applied. These $2 \tau$ treatment plans are assessed in terms of the tumour volume outcome $\mathbf { Y } _ { t + \tau }$ . We select the treatment (chemotherapy or radiotherapy) that achieves lowest $\mathbf { Y } _ { t + \tau }$ , and within the correct treatment the timing with lowest $\mathbf { Y } _ { t + \tau }$ . We also compute the normalized RMSE for predicting $\mathbf { Y } _ { t + \tau }$ . See Appendix L for more details about the test set. The models are evaluated for 3 settings of $\gamma _ { c }$ and $\gamma _ { r }$ . + +Table 1: Results for recommending the correct treatment and timing of treatment. + +
%c=5,γr=5% =5,γr=01 Yc =0,γr =5
TCRNRMSNMSMCRN RMSNMSMCRNRMSNMSM
Normalized RMSE2.43%3.16%6.75%1.08% 1.35%3.68%1.54%1.59%3.23%
2.83%3.95%7.65% 1.21%1.81%3.84%1.81%2.25%3.52%
3４56７3.18%4.37% 7.95%1.33%2.13%3.91%2.03%2.71%3.63%
3.51%5.61%8.19%1.42% 2.41%3.97%2.23%2.73%3.71%
3.93%6.21% 8.52%1.53%2.43%4.04%2.43%2.88%3.79%
Treatment Accuracy83.1%75.3%73.9%83.2% 78.6%77.1%92.9%87.3%74.9%
82.5%74.1%68.5%81.3%77.7% 73.9%85.7%83.8%74.1%
3４56７73.5%72.7%63.2%78.3% 77.2%72.3%83.8%82.1%72.8%
69.4%66.7%62.7%79.5% 76.3%71.8%78.6%69.7%64.5%
71.2%68.8%62.4%72.7% 71.8%71.6%71.9%69.3%61.2%
Treatment Timing Accuracy79.6%78.1%67.6%80.5%76.8% 77.5%79.8%75.7%60.6%
73.9%70.3%63.1%79.0%77.2%73.4% 75.4%71.4%58.2%
69.8%68.6%62.4%78.3%73.3%63.6%66.9%31.3%29.5%
3456７66.9%66.2%62.6%73.5%72.1%63.9%65.8%24.2%15.5%
64.5%63.6%62.2%70.6%57.4%44.2%63.9%25.6%12.5%
+ +Table 1 shows the results for this evaluation set-up. The treatment accuracy denotes the percentage of patients for which the correct treatment was selected, while the treatment timing accuracy is the percentage for which the correct timing was selected. Note that when $\gamma _ { c } = 0$ and $\gamma _ { r } = 5$ , RMSN and MSM select the wrong treatment timing for projection horizons $\tau > 4$ . CRN performs similarly among the different policies present in the observational data and achieve the lowest RMSE and highest accuracy in selecting the correct treatment and timing of treatment. + +In Appendix M we also show the applicability of the CRN in more complex medical scenarios involving real data. We provide experimental results based on the Medical Information Mart for Intensive Care (MIMIC III) database (Johnson et al., 2016) consisting of electronic health records from patients in the ICU. + +# 7 CONCLUSION + +Despite its wide applicability, the problem of causal inference for time-dependent treatments has been relatively less studied compared to problem of causal inference in the static setting. Both new methods and theory are necessary to be able to harness the full potential of observational data for learning individualized effects of complex treatment scenarios. Further work in this direction is needed for proposing alternative methods for handling time-dependent confounders, for modelling combinations of treatments assigned over time or for estimating the individualized effects of time-dependent treatments with associated dosage. + +In this paper, we introduced the Counterfactual Recurrent Network (CRN), a model that estimates individualized effects of treatments over time using a novel way of handling the bias from timedependent confounders through adversarial training. Using a model of tumour growth, we validated CRN in realistic medical scenarios and we showed improvements over existing state-of-the-art methods. We also showed the applicability of the CRN a real dataset consiting of patient electronic health records. The counterfactual predictions of CRN have the potential to be used as part of clinical decision support systems to address relevant medical challenges involving selecting the best treatments for patients over time, identify optimal treatment timings but also when the treatment is no longer needed. In future work, we will aim to build better balancing representations and to provide theoretical guarantees for the expected error on the counterfactuals. + +# ACKNOWLEDGMENTS + +We would like to thank the reviewers for their valuable feedback. The research presented in this paper was supported by The Alan Turing Institute, under the EPSRC grant EP/N510129/1 and by the US Office of Naval Research (ONR). + +# REFERENCES + +Alberto Abadie and Guido W Imbens. Matching on the estimated propensity score. Econometrica, 84(2):781–807, 2016. + +Ahmed Alaa and Mihaela van der Schaar. Limits of estimating heterogeneous treatment effects: Guidelines for practical algorithm design. 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Several ways of handling the selection bias involve using propensity matching (Austin, 2011; Imai & Ratkovic, 2014; Abadie & Imbens, 2016), building representations where treated and un-treated populations had similar distributions (Johansson et al., 2016; Shalit et al., 2017; Li & Fu, 2017; Yao et al., 2018) or performing propensity-aware hyperparameter tuning (Alaa & van der Schaar, 2017; 2018). However, these methods for the static setting cannot be extended directly to time-varying treatments (Hernán et al., 2000; Schisterman et al., 2009). + +Learning optimal policies: A related problem to ours involves learning the optimal treatment policies from logged data (Swaminathan & Joachims, 2015a;b; Atan et al., 2018). That is, learning the treatment option that would give the best reward. Note the difference to the causal inference setting considered in this paper, where the aim is to learn the counterfactual patient outcomes under all possible treatment options. Learning all of the counterfactual outcomes is a harder problem and can also be used for finding the optimal treatment. + +A method for learning optimal policies, proposed by Atan et al. (2018) uses domain adversarial training to build a representation that is invariant to the following two domains: observational data and simulated randomized clinical trial data, where the treatments have equal probabilities. Atan et al. (2018) only considers the static setting and aims to choose the optimal treatment instead of estimating all of the counterfactual outcomes. In our paper the aim is to eliminate the bias from the time-dependent confounders and reliably estimate all of the potential outcomes; thus, at each timestep $t$ we build a representation that is invariant to the treatment. + +Off-policy evaluation in reinforcement learning: In reinforcement learning, a similar problem to ours is off-policy evaluation, which uses retrospective observational data, also known as logged bandit feedback (Hoiles & Van Der Schaar, 2016; Paduraru et al., 2013; Doroudi et al., 2017). In this case, ˘ the retrospective observational data consists of sequences of states, actions and rewards which were generated by an agent operating under an unknown policy. The off-policy evaluation methods aim to use this data to estimate the expected reward of a target policy. These methods use algorithms based on importance sampling (Precup, 2000; Thomas et al., 2015; Guo et al., 2017), action-value function approximation (model based) (Hallak et al., 2015) or doubly robust combination of both approaches (Jiang & Li, 2015). Nevertheless, these methods focus on obtaining average rewards of policies, while in our case the aim is to estimate individualized patient outcomes for future treatments. + +# B ASSUMPTIONS + +The standard assumptions needed for identifying the treatment effects are (Robins & Hernán, 2008; Lim et al., 2018; Schulam & Saria, 2017): + +Assumption 1: Consistency. If ${ \bf A } _ { t } = { \bf a } _ { t }$ for a given patient, then the potential outcome for treatment $\mathbf { a } _ { t }$ is the same as the observed (factual) outcome: $\mathbf Y _ { t + 1 } [ \mathbf a _ { t } ] = \mathbf Y _ { t + 1 }$ . + +Assumption 2: Positivity (Overlap) (Imai & Van Dyk, 2004): If $P ( \bar { \mathbf { A } } _ { t - 1 } = \bar { \mathbf { a } } _ { t - 1 } , \bar { \mathbf { X } } _ { t } = \bar { \mathbf { x } } _ { t } ) \neq 0$ then $P ( \mathbf { A } _ { t } = \mathbf { a } _ { t } \mid \bar { \mathbf { A } } _ { t - 1 } = \bar { \mathbf { a } } _ { t - 1 } , \bar { \mathbf { X } } _ { t } = \bar { \mathbf { x } } _ { t } ) > 0$ for all $\bar { \mathbf { a } } _ { t }$ . + +Assumption 3: Sequential strong ignorability. $\mathbf { Y } _ { t + 1 } [ \mathbf { a } _ { t } ] \perp \perp \mathbf { A } _ { t } \mid \bar { \mathbf { A } } _ { t - 1 } , \bar { \mathbf { X } } _ { t } , \forall \mathbf { a } _ { t } \in \mathcal { A } , \forall t .$ + +Assumption 2 means that, for each timestep, each treatment has non-zero probability of being assigned. Assumption 3 means that there are no hidden confounders, that is, all of covariates affecting both the treatment assignment and the outcomes are present in the the observational dataset. Note that while assumption 3 is standard across all methods for estimating treatment effects, it is not testable in practice (Robins et al., 2000; Pearl et al., 2009). + +# C TIME-DEPENDENT CONFOUNDING + +Figure 6 illustrates the causal graphs for a time-varying exposures with 2-steps (Robins et al., 2000). In Figure 6 (a), the covariate $X$ is a time-dependent confounder because it affects the treatment assignments and at the same time, its value is changed by past treatments (Mansournia et al., 2017), as illustrated by the red arrows. Thus, the treatment probabilities at each time $t$ depend on the history of covariate $X$ and past treatments. Note that $U _ { 0 }$ and $U _ { 1 }$ are hidden variables which only affect the covariates, i.e. they do not have arrows into the treatments. Thus, the no hidden confounders assumption (Assumption 3) is satisfied. + +Figure 6 (a) and (b) illustrate the two cases when there is no bias from time-dependent confounding. In Figure 6 (a) the treatment probabilities are independent, while in Figure 6 (b) they depend on past treatments. + +![](images/36d7077a848a6b126878a2fc9fc2c440777cd99b9c6aaab7a41d4644439ac523.jpg) +Figure 6: Causal graphs for 2-step time-varying exposures (Robins et al., 2000). $X _ { 0 } , X _ { 1 }$ are patient covariates, $A _ { 0 } , A _ { 1 }$ are treatments, $U _ { 0 }$ , $U _ { 1 }$ are unobserved variable and $Y _ { 2 }$ is the outcome. + +Marginal Structural Models Robins et al. (2000). To remove the association between timedependent confounders and time-varying treatments, Marginal Structural Models propose using inverse probability of treatment weighting (IPTW). Without loss of generality, consider the use of MSMs with univariate treatments, baseline variables and outcomes. The outcome after $t$ timesteps is parametrized as follows: $\mathbf { E } [ Y _ { t + 1 } \mid \mathbf { a } _ { 1 } , \dots \mathbf { a } _ { t } , V ] = g ( \mathbf { a } _ { 1 } , \dots \mathbf { a } _ { n } , V ; \theta )$ , where $g ( \cdot )$ is usually a linear function with parameters $\theta$ . To remove the bias from the time-dependent confounders present in the observational dataset, in the regression model $g ( \cdot )$ MSMs weights each patients using either stabilized weights: + +$$ +S W ( t ) = \prod _ { l = 1 } ^ { t } \frac { f ( \mathbf { A } _ { l } \mid \bar { \mathbf { A } } _ { l - 1 } ) } { f ( \mathbf { A } _ { l } \mid \bar { \mathbf { X } } _ { l } , \bar { \mathbf { A } } _ { l - 1 } , \mathbf { V } ) } +$$ + +or unstabilized weights: + +$$ +W ( t ) = \prod _ { l = 1 } ^ { t } { \frac { 1 } { f ( \mathbf { A } _ { l } \mid \bar { \mathbf { X } } _ { l } , \bar { \mathbf { A } } _ { l - 1 } , \mathbf { V } ) } } , +$$ + +where $f ( \cdot )$ represents the conditional probability mass function for discrete treatments. + +Inverse probability of treatment weighting (IPTW) creates a pseudo-population where each member consists of themselves and $W - 1$ (or $S W - 1 )$ copies added though weighting. In this pseudopopulation, Robins Robins (1999) shows that $\bar { \mathbf X } _ { t }$ does not predict treatment ${ \bf A } _ { t }$ , thus removing the bias from time-dependent confounders. + +When using unstabilized weights $W$ , the causal graph in the pseudo-population is the one in Figure 6 (a) where $\mathbf { \bar { \nabla } } P ( \mathbf { A } _ { t } \mid \bar { \mathbf { X } } _ { t } , \bar { \mathbf { A } } _ { t - 1 } , \mathbf { \bar { \xi } } V ) = P ( \mathbf { A } _ { t } )$ . On the other hand, when using stabilized weights $S W$ , causal graph in the pseudo-population is the one in Figure 6 (b) where $\mathbf { \bar { \nabla } } P ( \mathbf { A } _ { t } \mid \bar { \mathbf { X } } _ { t } , \bar { \mathbf { A } } _ { t - 1 } , V ) =$ $P ( \mathbf { A } _ { t } \mid \bar { \mathbf { A } } _ { t - 1 } )$ . + +Counterfactual Recurrent Networks. Instead of using IPTW, we proposed building a representation of $\bar { \mathbf { X } } _ { t } , \bar { \mathbf { A } } _ { t - 1 } , V$ that is not predictive of treatment ${ \bf A } _ { t }$ . At timestep $t$ , we have $k$ different possible treatments $\mathbf { A } _ { t } \in \{ A _ { 1 } , \dotsc . . . A _ { K } \}$ . We build a representation of the history and covariates and treatments that has the same distribution across the different possible treatments: $P ( \Phi ( { \bar { \mathbf { X } } } _ { t } , { \bar { \mathbf { A } } } _ { t - 1 } , \mathbf { V } ) \mid \mathbf { A } _ { t } =$ $A _ { 1 } ) = \cdots = P ( \Phi ( { \bar { \mathbf { X } } } _ { t } , { \bar { \mathbf { A } } } _ { t - 1 } , \mathbf { V } ) \mid \mathbf { A } _ { t } = A _ { K } )$ . By breaking the association between past exposure and current treatments ${ \bf A } _ { t }$ , we satisfy the causal graph in Figure 6 (a) and thus we remove the bias from time-dependent confounders. + +# D PROOF OF THEOREM 1 + +We first prove the following proposition. + +Proposition 1. For fixed $\Phi$ , let $x ^ { \prime } = \Phi ( \bar { \mathbf { h } } _ { t } )$ . Then the optimal prediction probabilities of $G _ { a }$ are given by + +$$ +{ G _ { a } ^ { j } } ^ { * } ( x ^ { \prime } ) = \frac { P _ { j } ^ { \Phi } ( x ^ { \prime } ) } { \sum _ { i = 1 } ^ { K } P _ { i } ^ { \Phi } ( x ^ { \prime } ) } . +$$ + +Proof. For fixed $\Phi$ , the optimal prediction probabilities are given by + +$$ +G _ { a } ^ { * } = \arg \operatorname* { m a x } _ { G _ { a } } \sum _ { j = 1 } ^ { K } \int _ { x ^ { \prime } } \log ( G _ { a } ^ { j } ( x ^ { \prime } ) ) P _ { j } ^ { \Phi } ( x ^ { \prime } ) d x ^ { \prime } \qquad \mathrm { s u b j e c t ~ t o } \sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( x ^ { \prime } ) = 1 . +$$ + +Maximising the value function pointwise and applying Lagrange multiplies, we get + +$$ +G _ { a } ^ { * } = \arg \operatorname* { m a x } _ { G _ { a } } \sum _ { j = 1 } ^ { K } \log ( G _ { a } ^ { j } ( x ^ { \prime } ) ) P _ { j } ^ { \Phi } ( x ^ { \prime } ) + \lambda \Bigg ( \sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( x ^ { \prime } ) - 1 \Bigg ) . +$$ + +Setting the derivative (w.r.t. $G _ { a } ^ { j } { } ^ { * } ( x ^ { \prime } ) )$ to 0 and solving for $G _ { a } ^ { j } { } ^ { * } ( x ^ { \prime } )$ we get + +$$ +{ G _ { a } ^ { j } } ^ { * } ( x ^ { \prime } ) = - \frac { P _ { j } ^ { \Phi } ( x ^ { \prime } ) } { \lambda } +$$ + +where $\lambda$ can now be solved for using the constraint to be $\begin{array} { r } { \lambda = - \sum _ { i = 1 } ^ { K } P _ { i } ^ { \Phi } ( x ^ { \prime } ) } \end{array}$ . This gives the result. + +Proof. (of Theorem 1) By substituting the expression from Proposition 1 into the minimax game defined in Eq. 6, the objective for $\Phi$ becomes + +$$ +\operatorname* { m i n } _ { \Phi } \sum _ { j = 1 } ^ { K } \mathbb { E } _ { x ^ { \prime } \sim P _ { j } ^ { \Phi } } \left[ \log \left( \frac { P _ { j } ^ { \Phi } ( x ^ { \prime } ) } { \sum _ { i = 1 } ^ { K } P _ { i } ^ { \Phi } ( x ^ { \prime } ) } \right) \right] . +$$ + +We then note that + +$$ +\sum _ { j = 1 } ^ { K } \mathbb { E } _ { x ^ { \prime } \sim P _ { j } ^ { \Phi } } \bigg [ \log \bigg ( \frac { P _ { j } ^ { \Phi } ( x ^ { \prime } ) } { \sum _ { i = 1 } ^ { K } P _ { i } ^ { \Phi } ( x ^ { \prime } ) } \bigg ) \bigg ] + K \log K = \sum _ { j = 1 } ^ { K } \Bigg ( \mathbb { E } _ { x ^ { \prime } \sim P _ { j } ^ { \Phi } } \bigg [ \log \bigg ( \frac { P _ { j } ^ { \Phi } ( x ^ { \prime } ) } { \sum _ { i = 1 } ^ { K } P _ { i } ^ { \Phi } ( x ^ { \prime } ) } \bigg ) \bigg ] + \log K \Bigg ) +$$ + +$$ +\begin{array} { l } { { \displaystyle = \sum _ { j = 1 } ^ { K } \mathbb { E } _ { x ^ { \prime } \sim P _ { j } ^ { \Phi } } \left[ \log \left( \frac { P _ { j } ^ { \Phi } ( x ^ { \prime } ) } { \frac { 1 } { K } \sum _ { i = 1 } ^ { K } P _ { i } ^ { \Phi } ( x ^ { \prime } ) } \right) \right] } } \\ { { \displaystyle = \sum _ { j = 1 } ^ { K } K L \biggl ( P _ { j } ^ { \Phi } ( x ^ { \prime } ) \biggl \| \frac { 1 } { K } \sum _ { i = 1 } ^ { K } P _ { i } ^ { \Phi } ( x ^ { \prime } ) \biggr ) } } \\ { { \displaystyle = K \cdot J S D ( P _ { 1 } ^ { \Phi } , . . . , P _ { K } ^ { \Phi } ) } } \end{array} +$$ + +where $K L ( \cdot | | \cdot )$ is the Kullback-Leibler divergence and $J S D ( \cdot , . . . , \cdot )$ is the multi-distribution Jensennon-negative and 0 if and only if all distributions are equal, we have that Shannon Divergence (Li et al., 2018). Since $K \log K$ is a constant and the multi-distribution JSD is $P _ { 1 } ^ { \Phi } = \ldots = P _ { K } ^ { \Phi }$ . □ + +# E TRAINING PROCEDURE FOR CRN + +Let $\mathcal { D } = \left\{ \{ \mathbf { x } _ { t } ^ { ( i ) } , \mathbf { a } _ { t } ^ { ( i ) } , \mathbf { y } _ { t + 1 } ^ { ( i ) } \} _ { t = 1 } ^ { T ^ { ( i ) } } \cup \{ \mathbf { v } ^ { ( i ) } \} \right\} _ { i = 1 } ^ { N }$ be an observational dataset consisting of information about $N$ independent patients that we use to train CRN. The encoder and decoder networks part of CRN are trained into two separate steps. + +To begin with, the encoder is trained to built treatment invariant representations of the patient history and to perform one-step ahead prediction. After the encoder is optimized, we use it to compute the balancing representation $\mathbf { b r } _ { t } ^ { ( i ) }$ for each timestep in the trajectory of patient $( i )$ . To train the decoder, we modify the training dataset as follows. For each patient $( i )$ , we split their trajectory into shorter sequences of the $\tau _ { \mathrm { m a x } }$ timesteps of the form: + +$$ +\left\{ \mathbf { b r } _ { l } ^ { ( i ) } \cup \{ \mathbf { y } _ { l + t } ^ { ( i ) } , \mathbf { a } _ { l + t } ^ { ( i ) } , \mathbf { y } _ { l + t + 1 } ^ { ( i ) } \} _ { t = 1 } ^ { \tau _ { m a x } } \cup \mathbf { v } ^ { ( i ) } \right\} , +$$ + +for $l = 1 , . . . T ^ { ( i ) } - \tau _ { \mathrm { m a x } }$ . Thus, each patients contributes with $T ^ { ( i ) } - \tau _ { \operatorname* { m a x } }$ examples in the dataset for training the decoder. The different sequences obtained for all patents are randomly grouped into minibatches and used for training. + +The pseudocode in Algorithm 1 shows the training procedure used for the encoder and decoder networks part of CRN. The model was implemented in TensorFlow and trained on an NVIDIA Tesla K80 GPU. The Adam optimizer (Kingma & Ba, 2014) was used for training and both the encoder and the decoder are trained for 100 epochs. + +# Algorithm 1 Pseudo-code for training CRN + +Input: Training data: $\mathcal { D } = \left\{ \{ \mathbf { x } _ { t } ^ { ( i ) } , \mathbf { a } _ { t } ^ { ( i ) } , \mathbf { y } _ { t + 1 } ^ { ( i ) } \} _ { t = 1 } ^ { T ^ { ( i ) } } \cup \mathbf { v } ^ { ( i ) } \right\} _ { i = 1 } ^ { N }$ + +(1) Encoder optimization: parameters $\theta _ { E , r } , \theta _ { E , a } , \theta _ { E , y }$ + +Learning rate: $\mu$ + +for $p = 1 , \ldots , \operatorname* { m a x }$ epochs do $\lambda _ { p } = \frac { 2 } { 1 + \exp ( - 1 0 \cdot p ) } - 1$ + +for Batch $\boldsymbol { B } = \left\{ \{ \mathbf { x } _ { t } ^ { ( i ) } , \mathbf { a } _ { t } ^ { ( i ) } , \mathbf { y } _ { t + 1 } ^ { ( i ) } \} _ { t = 0 } ^ { T ^ { ( i ) } } \cup \mathbf { v } ^ { ( i ) } \right\} _ { i = 1 } ^ { | \mathcal { B } | }$ in epoch do + +end for end for + +# (2) Compute the encoder balanced representation and use it to initialize the decoder hidden state. + +for $i = 1 , . . . , N$ do for $t = 1 , \ldots , T ^ { ( i ) }$ do 0 $\mathbf { b r } _ { t } ^ { ( i ) } = \operatorname { e n c o d e r } ( \bar { \mathbf { x } } _ { t } ^ { ( i ) } , \bar { \mathbf { a } } _ { t - 1 } ^ { ( i ) } , \mathbf { v } ^ { ( i ) } ; \boldsymbol { \theta } _ { E , r } )$ end for +end for + +# (3) Split dataset in sequences of $\tau _ { \mathrm { m a x } }$ timesteps: + +$$ +\left\{ \left\{ \mathbf { b r } _ { l } ^ { ( i ) } \cup \{ \mathbf { y } _ { l + t } ^ { ( i ) } , \mathbf { a } _ { l + t } ^ { ( i ) } , \mathbf { y } _ { l + t + 1 } ^ { ( i ) } \} _ { t = 1 } ^ { \tau _ { m a x } } \cup \mathbf { v } ^ { ( i ) } \right\} _ { l = 1 } ^ { T ^ { ( i ) } - \tau _ { \operatorname* { m a x } } } \right\} _ { i = 1 } ^ { N } +$$ + +(4) Optimize decoder: parameters $\theta _ { D , r } , \theta _ { D , a } , \theta _ { D , y }$ Learning rate: $\mu$ + +for $\mathrm { p } = 1 , \ldots$ , max epochs do + +$$ +\begin{array} { r l } & { \mathrm { ~ \gamma = \frac { 2 } { 1 + \exp ( - 1 0 \cdot p ) } - 1 } } \\ & { \mathrm { ~ } } \\ & { \mathrm { ~ \gamma ~ R a t c h ~ } \mathcal { B } = \left\{ \mathbf { b r } _ { l } ^ { ( i ) } \ \bigcup \left\{ \mathbf { y } _ { l + t } ^ { ( i ) } , \ \mathbf { a } _ { l + t } ^ { ( i ) } , \mathbf { y } _ { l + t + 1 } ^ { ( i ) } \right\} _ { t = 0 } ^ { \tau _ { m a x } } \ | \ \mathbf { y } | ^ { ( i ) } \right\} _ { i = 1 } ^ { | B | } \mathrm { ~ i n ~ e ~ } } \\ & { \mathrm { ~ C o m p u t ~ } \mathcal { L } _ { D , a } ^ { D } ( \theta _ { D , r } , \theta _ { D , a } ) = \frac { 1 } { | B | } \sum _ { i \in B } \sum _ { r = 1 } ^ { r _ { m a x } } \mathcal { L } _ { t , a } ^ { ( i ) } ( \theta _ { D , r } , \theta _ { D , a } ) } \\ & { \mathrm { ~ C o m p u t ~ } \mathcal { L } _ { D , y } ^ { D } ( \theta _ { D , r } , \theta _ { D , y } ) = \frac { 1 } { | B | } \sum _ { i \in B } \sum _ { r = 1 } ^ { r _ { m a x } } \mathcal { L } _ { t , y } ^ { ( i ) } ( \theta _ { D , r } , \theta _ { D , y } ) } \\ & { \mathrm { ~ \ } } \\ & { \theta _ { D , r } \cdot \theta _ { D , r } - \mu \left( \frac { \partial \mathcal { L } _ { D , y } ^ { D } ( \theta _ { D , r } , \theta _ { D , y } ) } { \partial \theta _ { D , r } } - \lambda _ { p } \frac { \partial \mathcal { L } _ { D , 0 } ^ { D } ( \theta _ { D , r } , \theta _ { D , a } ) } { \partial \theta _ { D , r } } \right) } \\ & { \theta _ { D , y } + \theta _ { D , y } - \mu \frac { \partial \mathcal { L } _ { D , y } ^ { D } ( \theta _ { D , r } , \theta _ { D , y } ) } { \partial \theta _ { D , r } } } \\ & { \theta _ { D , a } \gets \theta _ { D , \theta } - \mu \frac { \partial \mathcal { L } _ { D , x } ^ { D } ( \theta _ { D , r } , \theta _ { D , x } ) } { \partial \theta _ { D , a } } } \\ & \theta _ { D , a } \gets \theta _ { D , \theta } - \mu \frac \end{array} +$$ + +end for + +end for + +Output: Trained CRN encoder (parameters $\theta _ { E , r } , \theta _ { E , a } , \theta _ { E , y } )$ and trained CRN decoder (parameters $\theta _ { D , r } , \theta _ { D , a } , \theta _ { D , y } . ~ )$ ) + +# F PHARMACOKINETIC-PHARMACODYNAMIC MODEL OF TUMOUR GROWTH + +To evaluate the CRN on counterfactual estimation, we need access to the data generation mechanism to build a test set that consists of patient outcomes under all possible treatment options. For this purpose, we use the state-of-the-art pharmacokinetic-pharmacodynamic (PK-PD) model of tumour growth proposed by Geng et al. (2017) and also used by Lim et al. (2018) for evaluating RMSMs. The PK-PD model characterizes patients suffering from non-small cell lung cancer and models the evolution of their tumour under the combined effects of chemotherapy and radiotherapy. In addition, the model includes different distributions of tumour sizes based on the cancer stage at diagnosis. + +Model of tumour growth The volume of tumour $t$ days after diagnosis is modelled as follows: + +$$ +V ( t + 1 ) = \Big ( 1 + \underbrace { \rho \mathrm { l o g } ( \frac { K } { V ( t ) } ) } _ { \mathrm { T u m o r ~ g r o w t h } } - \underbrace { \beta _ { c } C ( t ) } _ { \mathrm { C h e m o t h e r a p y } } - \underbrace { \big ( \alpha _ { r } d ( t ) + \beta _ { r } d ( t ) ^ { 2 } \big ) } _ { \mathrm { R a d i o t h e r a p y } } + \underbrace { e _ { t } } _ { \mathrm { N o i s e } } \Big ) V ( t ) +$$ + +where the parameters $K , \rho , \beta _ { c } , \alpha _ { r } , \beta _ { r }$ are sampled from the prior distributions described in (Geng et al., 2017) and $e _ { t } \sim \mathcal { N } ( 0 , 0 . 0 1 ^ { 2 } )$ is a noise term that accounts for randomness in the tumour growth. + +To incorporate heterogeneity among patient responses, due to, for instance, gender or genetic factors Bartsch et al. (2007), the prior means for $\beta _ { c }$ and $\alpha _ { r }$ are adjusted to create three patient subgroups $S ^ { ( i ) } \in \{ 1 , 2 , 3 \}$ as described in Lim et al. (2018). This way, we incorporate in the model of tumour growth specific characteristics that affect the patient’s individualized response to treatments. Thus, the prior mean $\mu _ { \beta _ { c } }$ of $\beta _ { c }$ and the prior mean $\mu _ { \alpha _ { r } }$ of $\alpha _ { r }$ are augmented as follows. + +$$ +\begin{array} { r } { \mu _ { \beta _ { c } } ^ { \prime } ( i ) = \left\{ \begin{array} { l l } { 1 . 1 \mu _ { \beta _ { c } } , \mathrm { i f } \mathrm { S } ^ { ( i ) } = 3 } \\ { \mu _ { \beta _ { c } } , \mathrm { o t h e r w i s e } } \end{array} \right. \qquad \mu _ { \alpha _ { r } } ^ { \prime } ( i ) = \left\{ \begin{array} { l l } { 1 . 1 \mu _ { \alpha _ { r } } , \mathrm { i f } \mathrm { S } ^ { ( i ) } = 1 } \\ { \mu _ { \alpha _ { r } } , \mathrm { o t h e r w i s e } } \end{array} \right. } \end{array} +$$ + +where $\mu _ { \beta _ { c } }$ and $\mu _ { \alpha _ { r } }$ are the mean parameters from Geng et al. (2017) and $\mu _ { \beta _ { c } } ^ { \prime } ( i )$ and $\mu _ { \alpha _ { r } } ^ { \prime } ( i )$ are the parameters used in the data simulation. The patient subgroup $S ^ { ( i ) } \in \{ 1 , 2 , 3 \}$ is used as baseline features. + +The chemotherapy drug concentration follows an exponential decay with half life of 1 day: + +$$ +C ( t ) = \tilde { C } ( t ) + C ( t - 1 ) / 2 , +$$ + +where $\tilde { C } ( t ) = 5 . 0 m g / m ^ { 3 }$ of Vinblastine if chemotherapy is given at time $t$ . $d ( t ) = 2 . 0 G y$ fractions of radiotherapy if the radiotherapy treatment is applied at timestep $t$ . + +Time-varying confounding is introduced by modelling chemotherapy and radiotherapy assignment as Bernoulli random variables, with probabilities $p _ { c }$ and $p _ { r }$ depending on the tumour diameter: + +$$ +p _ { c } ( t ) = \sigma \left( \frac { \gamma _ { c } } { D _ { \mathrm { m a x } } } ( \bar { D } ( t ) - \delta _ { c } ) \right) \qquad p _ { r } ( t ) = \sigma \left( \frac { \gamma _ { r } } { D _ { \mathrm { m a x } } } ( \bar { D } ( t ) - \delta _ { r } ) \right) , +$$ + +where $\bar { D } ( t )$ is the average tumour diameter over the last 15 days, $D _ { \mathrm { m a x } } = 1 3 \mathrm { c m }$ is the maximum tumour diameter and $\sigma ( \cdot )$ is the sigmoid activation function. The parameters $\delta _ { c }$ and $\delta _ { r }$ are set to $\delta _ { c } = \delta _ { r } = D _ { \operatorname* { m a x } } / 2$ such that there is 0.5 probability of receiving treatment when tumour is half of its maximum size. $\gamma _ { c } , \gamma _ { r }$ control the amount of time-dependent confounding; the higher $\gamma _ { \star }$ is, the more important the history of tumour diameter is in assigning treatments. Thus, at each timestep, there are four treatment options options: no treatment $( A _ { 1 } )$ , chemotherpy $\left( A _ { 2 } \right)$ , radiotherapy $\left( A _ { 3 } \right)$ , combined chemotherapy and radiotherapy $( A _ { 4 } )$ . + +Since the work most relevant to ours is the one of Lim et al. (2018) we used the same data simulation and same settings for $\gamma = \gamma _ { c } = \gamma _ { r }$ as in their case. When $\gamma = 0$ , there is no time-dependent confounding and the treatments are randomly assigned. By increasing $\gamma$ we increase the influence of the volume size history (encoded in $\bar { D } ( t ) ,$ ) on the treatment probability. For example, assume $\bar { D } ( t ) =$ $\frac { 3 D _ { m a x } } { 4 }$ n (7), t, where obability of chemotherapy in thiis the sigmoid function. When i, $\begin{array} { r } { p _ { c } ( t ) = \sigma ( \frac { \gamma _ { c } } { D _ { m a x } } ( \bar { D } ( t ) - } \end{array}$ $\textstyle \frac { D _ { m a x } } { 2 } ) ) = \sigma ( 0 . 2 5 \gamma _ { c } )$ $\sigma ( \cdot )$ $\gamma = 1$ $p _ { c } ( t ) = 0 . 5 6$ $\gamma = 5$ $p _ { c } ( t ) = 0 . 7 7$ and when $\gamma = 1 0$ , $\dot { p } _ { c } ( t ) = 0 . 9 2$ in this example. $\gamma$ can be increased further to increase the bias. However, the values used in the experiments evaluate the model on a wide range of settings for the time-dependent confounding bias. + +# G MARGINAL STRUCTURAL MODELS + +Marginal Structural Models (Robins et al., 2000; Hernán et al., 2001) have been widely used in epidemiology and as part of follow up studies. In our case, we would like to estimate the effects of a sequence of treatments in the future given the current patient history: + +$$ +\mathbb { E } ( \mathbf { Y } _ { t + \tau } \mid \bar { \mathbf { A } } ( t , t + \tau - 1 ) = \bar { \mathbf { a } } ( t , t + \tau - 1 ) , \bar { \mathbf { H } } _ { t } ) = g ( \tau , a ( t , t + \tau - 1 ) , \bar { \mathbf { H } } _ { t } ) , +$$ + +where $g$ is a generic function and $\bar { \mathbf { a } } ( t , t + \tau - 1 ) = [ \mathbf { a } _ { t } , \dots \mathbf { a } _ { t + \tau - 1 } ]$ represents a possible sequence of treatments from timestep $t$ just until before the potential outcome $\mathbf { Y } _ { t + \tau }$ is observed. After removing the bias form time-dependent confounders, $\begin{array} { r } { \mathbb { E } \mathbf { \hat { ( } } \mathbf { Y } _ { t + \tau } \mid \bar { \mathbf { A } } ( t , t + \tau - 1 ) = \bar { \mathbf { a } } ( t , t + \tau - 1 ) , \bar { \mathbf { H } } _ { t } ) = } \end{array}$ $\mathbb { E } ( \mathbf { Y } _ { t + \tau } [ \bar { \mathbf { a } } ( t , t + \tau - 1 ) ]$ . + +Note that for implementing MSMs, we encode the treatments at timestep $t$ in the model of tumour growth as $\mathbf { A } _ { t } = [ A _ { t , c } , A _ { t , d } ]$ to indicate the binary application of chemotherapy and radiotherapy. In order to remove the time-dependent confounding bias and estimate future outcomes, we use the stabilized weights of MSMs to weight each patient in the dataset: + +$$ +S W ( t , \tau ) = \prod _ { n = t } ^ { t + \tau } \frac { f ( \mathbf { A } _ { n } \mid \bar { \mathbf { A } } _ { n - 1 } ) } { f ( \mathbf { A } _ { n } \mid \bar { \mathbf { A } } _ { n - 1 } , \bar { \mathbf { X } } _ { n } , \mathbf { V } ) } = \prod _ { n = t } ^ { t + \tau } \frac { \prod _ { k \in \{ c , d \} } f ( A _ { n , k } \mid \bar { \mathbf { A } } _ { n - 1 } ) } { \prod _ { k \in \{ c , d \} } f ( A _ { n , k } \mid \bar { \mathbf { A } } _ { n - 1 } , \bar { \mathbf { X } } _ { n } , \mathbf { V } ) } , +$$ + +where $f ( \cdot )$ represents the conditional probability mass function for discrete treatments. + +We adopt the implementation in (Hernán et al., 2001; Howe et al., 2012; Lim et al., 2018) for MSMs and use logistic regression for estimating the propensity weights as follows: + +$$ +f ( A _ { t , k } \mid \bar { \mathbf { A } } _ { t - 1 } ) = \sigma \Big ( \sum _ { j = 1 } ^ { k } \omega _ { k } ( \sum _ { i = 1 } ^ { t - 1 } A _ { t , j } ) \Big ) +$$ + +$$ +f ( A _ { t , k } \mid { \bar { \mathbf { H } } } _ { t } ) = \sigma { \Big ( } \sum _ { k \in \{ c , d \} } \phi _ { k } ( \sum _ { i = 1 } ^ { t - 1 } A _ { t , k } ) + \mathbf { w } _ { 1 } \mathbf { X } _ { t } + \mathbf { w } _ { 2 } \mathbf { X } _ { t - 1 } + \mathbf { w } _ { 3 } \mathbf { V } { \Big ) } +$$ + +where $\omega _ { \star } , \phi _ { \star }$ and $\mathbf { w } _ { \star }$ are regression coefficients, $k \in \{ c , d \}$ indicates the chemotherapy or radiotherapy treatments and $\sigma ( \cdot )$ is the sigmoid function. + +For predicting the outcome, the following regression model is used, where each individual patient is weighted by its propensity score: + +$$ +g ( \tau , a ( t , t + \tau - 1 ) , \bar { \mathbf { H } } _ { t } ) = \sum _ { k \in \{ c , d \} } \beta _ { k } ( \sum _ { n = t } ^ { t + \tau - 1 } A _ { n , k } ) + 1 _ { 1 } \mathbf { X } _ { t } + 1 _ { 2 } \mathbf { X } _ { t - 1 } + 1 _ { 3 } \mathbf { V } +$$ + +where $\beta _ { \star }$ and ${ \bf l } _ { \star }$ are regression coefficients. + +MSMs do not require hyperparameter tuning so we use the patients from both the train and validation sets for training. + +# H RECURRENT MARGINAL STRUCTURAL NETWORKS + +MSMs are very sensitive to model mis-specification in computing the propensity weights and estimating the outcomes. Recurrent Marginal Structural Models (RMSNs) (Lim et al., 2018) overcome this problem by using recurrent neural networks to estimate the propensity scores and to build the outcome model. RNNs are more robust to changes in the treatment assignment policy. RMSNs were implemented as descried in Lim et al. $( 2 0 1 8 ) ^ { 2 }$ . + +For implementing RMSNs, we also encode the treatments at timestep $t$ in the model of tumour growth as $\mathbf { A } _ { t } = [ A _ { t , c } , A _ { t , d } ]$ to indicate the binary application of chemotherapy and radiotherapy. The propensity weights are estimated using recurrent neural networks as follows: + +$$ +\begin{array} { r l } { f ( A _ { t , k } \ \vert \ \bar { \mathbf { A } } _ { t - 1 } ) = \mathrm { R N N } _ { S W _ { n } } ( \bar { \mathbf { A } } _ { t - 1 } ) } & { { } ~ f ( A _ { t , k } \ \vert \ \bar { \mathbf { X } } _ { t } , \bar { \mathbf { A } } _ { t - 1 } ) = \mathrm { R N N } _ { S W _ { d } } ( \bar { \mathbf { A } } _ { t - 1 } , \bar { \mathbf { X } } _ { t } , { \mathbf { V } } ) } \end{array} +$$ + +For predicting one-step-ahed outcome, R-MSNs use an encoder network: + +$$ +g ( 1 , a ( t , t ) , \bar { { \mathbf { H } } } _ { t } ) = { \mathbf { R } } { \mathbf { N } } { \mathbf { N } } _ { E } ( { \mathbf { a } } _ { t } , \bar { { \mathbf { A } } } _ { t - 1 } , \bar { { \mathbf { X } } } _ { t } , { \mathbf { V } } ) , +$$ + +where in the loss function, each patient is weighted by their stabilized IPTW. + +For estimating the treatment responses for a sequence of treatments in the future, RMSNs use a decoder network: + +$$ +g ( \tau , a ( t , t + \tau - 1 ) , \bar { \mathbf { H } } _ { t } ) = \mathrm { R N N } _ { D } ( \mathbf { a } _ { t } , \ldots , \mathbf { a } _ { t + \tau - 1 } , \bar { \mathbf { A } } _ { t - 1 } , \bar { \mathbf { X } } _ { t } , \mathbf { V } ) . +$$ + +See Lim et al. (2018) for more details about the R-MSNs model architecture and training procedure of the propensity weights, encoder and decoder networks. Tables 2 and 3 show the hyperparameter search ranges used to optimize this model for evaluation in our paper. The hyperparameters were selected in the same way as proposed by Lim et al. (2018), based on the error on the factual outcomes in the validation dataset. All of the models are trained using Adam optimizer for 100 epochs. + +Table 2: Hyperparameter search range for propensity networks and encoder (same as in Lim et al. (2018)). C is the size of the input. + +
HyperparameterSearch range
Iterations of Hyperparameter Search50
Learning rate0.01,0.005,0.001
Minibatch size64,128,256
RNN state size0.5C,1C,2C, 3C,4C
Dropout rate0.1, 0.2,0.3,0.4, 0.5
Max Gradient Norm0.5, 1.0, 2.0
+ +Table 3: Hyperparameter search range for decoder (same as in Lim et al. (2018)). C is the input size. + +
HyperparameterSearch range
Iterations of Hyperparameter Search20
Learning rate0.01,0.001,0.0001
Minibatch size256,512,1024
RNN state size1C,2C,4C,8C,16C
Dropout Rate0.1, 0.2, 0.3, 0.4, 0.5
Max Gradient Norm0.5, 1.0, 2.0, 4.0
+ +# I BASELINE RNN AND LINEAR MODEL + +For the baseline linear model, we fit the same regression model used for Marginal Structural Networks, but without using the IPTW. The baseline RNN uses an LSTM unit and, at each timestep, receives as input the current treatment, the patient covariates and the patient static features to perform one-stepahead prediction. To have a model of similar capacity to the CRN (similar number of parameters), we add a fully connected layer on top of the output of the LSTM unit in order to obtain the outcomes. Table 4 shows the hyperparameter search range used to optimize this model. The hyperparameters were selecting according to the error on the factual outcomes in the validation set. We train the baseline RNN using the Adam optimizer for 100 epochs. + +Table 4: Hyperparameter search range for baseline RNN model. C is the size of the input. + +
HyperparameterSearch range
Iterations of Hyperparameter Search50
Learning rate0.01,0.001,0.0001
Minibatch size64,128,256
RNNhidden units0.5C,1C,2C, 3C,4C
FC hidden units0.5C, 1C, 2C, 3C, 4C
RNN dropout probability0.1, 0.2, 0.3, 0.4, 0.5
+ +# J HYPERPARAMETER OPTIMIZATION FOR CRN + +As described in Appendix C, the dataset for training the decoder are used by splitting the sequences of the patients in the training set. This creates a larger dataset for training (where each patient $( i )$ contributes $T ^ { ( i ) } - \tau _ { \operatorname* { m a x } }$ times to the dataset) which requires a different hyperparameter search range. Moreover, the balancing representations computed by the encoder are used to initialize the state of the RNN for the decoder. Thus, the decoder RNN size is equal to the size of the balancing representation size of the encoder. Table 5 shows the hyperparameter search ranges for the encoder and decoder networks in CRN. We selected hyperparameters based on the error of the model on the factual outcomes in the validation dataset. All models are trained for 100 epochs. + +In addition, Tables 6 and 7 illustrate the optimal hyperparameters chosen. + +Table 5: Hyperparameter search range for CRN encoder. C is the size of the input and R is the size of the balancing representation. + +
HyperparameterSearch range encoderSearch range decoder
Iterations ofHyperparameter Search5030
Learning rate0.01,0.001, 0.00010.01,0.001, 0.0001
Minibatch size64,128,256256,512,1024
RNN hidden units0.5C,1C, 2C, 3C, 4CBalancing representation size of encoder
Balancing representation size0.5C,1C,2C,3C,4C0.5C,1C,2C,3C,4C
FC hidden units0.5R,1R,2R,3R,4R0.5R,1R,2R,3R,4R
RNN dropout probability0.1, 0.2, 0.3,0.4, 0.50.1,0.2, 0.3, 0.4, 0.5
+ +Table 6: Optimal hyperparameters for the CRN encoder when different degrees of time-dependent confounding are applied in the model of tumour growth. The parameters $\gamma _ { c }$ and $\gamma _ { r }$ measure the degree of time-dependent confounding applied. When $\gamma _ { c }$ and $\gamma _ { r }$ are set to the same value, we denote this with $\gamma _ { \star }$ . + +
2=02=12=22=3Y=42=5
Learning rate0.0010.10.0010.010.010.001
Minibatch size64646412864128
RNN hidden units121824182424
Balancing representation size18181218612
FC hidden units181836542448
RNN dropout probability0.10.10.10.20.20.1
=62=7Y=82=9Y=10
Learning rate0.0010.0010.010.0010.01
Minibatch size6464128128128
RNN hidden units2418122424
Balancing representation size1218241812
FC hidden units4872123612
RNN dropout probability0.10.20.10.10.1
Yc=0,γr=5 Yc=5,γr=0
Learning rate0.010.001
Minibatch size12864
RNN hidden units1212
Balancing representation size1824
FC hidden units3696
RNN dropout probability0.10.1
+ +Table 7: Optimal hyperparameters for the CRN decoder when different degrees of time-dependent confounding are applied in the model of tumour growth. The parameters $\gamma _ { c }$ and $\gamma _ { r }$ measure the degree of time-dependent confounding applied. When $\gamma _ { c }$ and $\gamma _ { r }$ are set to the same value, we denote this with $\gamma _ { \star }$ + +
y=12=2=32=4=5
Learning rate0.0010.0010.0010.0010.001
Minibatch size1024102451210241024
RNN hidden units181218612
Balancing representation size18186183
FC hidden units183618726
RNN dropout probability0.10.20.30.10.1
Yc =0,γr=5 Yc=5,γr=0
Learning rate0.010.001
Minibatch size5121024
RNN hidden units1824
Balancing representation size1812
FC hidden units3624
RNN dropout probability0.10.03
+ +# K FULL RESULTS FOR COUNTERFACTUAL PREDICTION + +# K.1 MULTI-STEP AHEAD PREDICTION OF COUNTERFACTUALS + +Figure 7 shows the normalized RMSE for multiple step-ahead prediction of counterfactuals. The RMSE is normalized by the maximum tumour volume: $\bar { V _ { m a x } } = 1 \bar { 1 } 5 0 \mathrm { c m ^ { 3 } }$ . The counterfactuals in this case are generated as described in Section 6.3 and Appendix I. We notice that performance gains of CRN compared to RMSN increase with the number of future timesteps for which the counterfactuals are estimated. + +![](images/995945365917721cfe59800f6c40ff0276b71b09419710e573d0a6203278d1ac.jpg) +Figure 7: Results for prediction of patient counterfactuals for multiple steps ahead. + +# K.2 DETAILED RESULTS FOR THE COUNTERFACTUAL PREDICTIONS + +Tables 8 and 9 show detailed results for the counterfactual predictions. + +Table 8: Normalized RMSE for one-step-ahead prediction of counterfactuals. The parameter $\gamma$ measures the degree of time-dependent confounding applied. + +
γ=0γ=1γ=2γ=3γ=4γ=5
Linear (no IPTW)0.99%1.08%1.36%1.68%2.11%2.77%
MSM0.99%1.08%1.34%1.63%2.02%2.61%
RNN CRN(入=0)0.70%0.70%0.84%1.05%1.24%1.69%
RMSN0.66% 0.60%0.77% 0.61%0.92% 0.72%0.95% 0.81%1.24% 0.94%1.54% 1.23%
CRN0.56%0.57%0.62%0.67%0.87%1.20%
γ=6Y=7Y=8Y=9γ=10
Linear (no IPTW)3.55%4.15%4.80%5.09%5.22%
MSM3.30%3.79%4.30%4.47%4.47%
RNN2.03%2.52%2.88%3.79%4.01%
CRN (λ= 0)1.98%2.42%2.73%3.17%
RMSN3.57%
1.70%2.18%2.37%2.77%2.83%
CRN1.48%1.56%2.05%2.36%2.41%
+ +Table 9: Normalized RMSE for $\tau$ -step-ahead prediction of counterfactuals. The parameter $\gamma$ measures the degree of time-dependent confounding applied. + +
γ=1γ=2γ=3γ=4γ=5
T=2RMSN0.90%1.15%1.53%2.14%2.91%
CRN0.84%0.96%1.21%1.46%2.45%
T=3RMSN0.97%1.36%1.87%2.44%3.47%
CRN0.86%0.96%1.47 %1.51%2.84%
T=4RMSN1.24%1.79%2.60%3.33%3.88%
CRN0.91%1.08%1.74%1.76%2.82%
T=5RMSN1.51%2.13%3.06%4.07%4.58%
CRN0.85%1.10%1.73%2.00%3.43%
T=6RMSN2.10%2.89%3.06%4.16%6.32%
CRN1.16%1.52%2.29%2.66%4.91%
+ +In order to evaluate how well the models select the correct treatment and timing of treatment we simulate counterfactual outcomes as follows. We generate 1000 test samples using the model of tumour growth described in Section 6. Let $\bar { \mathbf { H } } _ { t }$ be the current history of the patient and let $\tau$ be a future time horizon. For each timestep in the future, we have 4 treatment options at: no treatment $( A _ { 0 } )$ , chemotherapy $( A _ { 1 } )$ , radiotherapy $\left( A _ { 2 } \right)$ , chemotherapy and radiotherapy. $\left( A _ { 3 } \right)$ . + +Using the model of tumour growth where the outcome $\mathbf { Y } _ { t + \tau }$ is given by the volume of the tumour, we generate the following $2 \tau$ counterfactuals: + +Chemotherapy application + +$$ +\begin{array} { r l } { { \bf Y } _ { t + \tau } } & { { } | \mathrm { \bf ~ a } _ { t } = A _ { 1 } , \mathbf { a } _ { t + 1 } = A _ { 0 } , \ldots \mathbf { a } _ { t + \tau - 1 } = A _ { 0 } , \bar { \mathbf { H } } _ { t } } \\ { { \bf Y } _ { t + \tau } } & { { } | \mathrm { \bf ~ a } _ { t } = A _ { 0 } , \mathbf { a } _ { t + 1 } = A _ { 1 } , \ldots \mathbf { a } _ { t + \tau - 1 } = A _ { 0 } , \bar { \mathbf { H } } _ { t } } \\ { \mathrm { ~ \cdot ~ } \cdot \cdot } \\ { { \bf Y } _ { t + \tau } } & { { } | \mathrm { \bf \textup { \bf ~ a } } _ { t } = A _ { 0 } , \mathbf { a } _ { t + 1 } = A _ { 0 } , \ldots \mathbf { a } _ { t + \tau - 1 } = A _ { 1 } , \bar { \mathbf { H } } _ { t } } \end{array} +$$ + +Radiotherapy application + +$$ +\begin{array} { r l } { { \bf Y } _ { t + \tau } } & { { } | \quad { \bf a } _ { t } = A _ { 2 } , { \bf a } _ { t + 1 } = A _ { 0 } , \ldots { \bf a } _ { t + \tau - 1 } = A _ { 0 } , \bar { \bf H } _ { t } } \\ { { \bf Y } _ { t + \tau } } & { { } | \quad { \bf a } _ { t } = A _ { 0 } , { \bf a } _ { t + 1 } = A _ { 2 } , \ldots { \bf a } _ { t + \tau - 1 } = A _ { 0 } , \bar { \bf H } _ { t } } \\ { \ldots } & { { } } \\ { { \bf Y } _ { t + \tau } } & { { } | \quad { \bf a } _ { t } = A _ { 0 } , { \bf a } _ { t + 1 } = A _ { 0 } , \ldots { \bf a } _ { t + \tau - 1 } = A _ { 2 } , \bar { \bf H } _ { t } } \end{array} +$$ + +We perform this for each patient in the test set and at each time $t$ in the history. For instance, for a patient with 50 timesteps in the model of tumour growth and for time horizon $\tau = 3$ , we generate $2 \cdot 3 \cdot 5 0 = 3 0 0$ counterfactuals. + +Using the true generated couterfactual data, we select the treatment that has the lowest $\mathbf { Y } _ { t + \tau }$ among the $\tau$ options generated for each treatment. Then, we select the time of applying treatment (among $t , t + 1 , \ldots t + \tau - 1 )$ that resulted in the lowest $\mathbf { Y } _ { t + \tau }$ . For each model, we generate the counterfactuals under the same treatment plans and patient histories. Then, we perform the selection of treatment and timing of treatment in the same way and we compare these with the true ones. Note that in order to account for numerical instability (two outcomes $Y _ { t + \tau }$ having very similar values), we consider two outcomes the same if they are within $\epsilon = 0 . 0 0 1$ of each other. + +# M RESULTS ON FACTUAL PREDICTION ON MIMIC III + +Using the Medical Information Mart for Intensive Care (MIMIC III) (Johnson et al., 2016) database consisting of electronic health records from patients in the ICU, we also show how the CRN can be used on a real medical dataset. From MIMIC III we extracted the patients on antibiotics, with trajectories up to 30 timesteps, thus obtaining a dataset with 3487 patients. For each patient, we extracted 25 patient covariates including lab tests and vital signs measured over time, as well as static patient features such as age and gender. + +We used a binary treatment at each timestep indicating whether the patient was administered antibiotics or not. Note that for the longitudinal covariates we used aggregate value for each day since the ICU admission. The reason for this is because antibiotic treatment is decided daily for the patient. We split the dataset into 2826/313/348 patients for training, validation and testing respectively. We performed hyperparameter optimization on the validation patient set, using the search ranges in Table 5 and we again selected hyperparameters based on the error on the factual outcomes. + +We estimate the individualized effect of antibiotics assigned over time on the patient’s white blood cell count. A high white blood cell count is associated with severe illness and poor outcome for ICU patients (Waheed et al., 2003). Antibiotic administration in the ICU aims to reduce the white blood cell count. However, the effectiveness of the antibiotics treatment in reducing the white blood cell count is highly dependent on the time they are administered with respect to the history of the patient covariates. In this context we again have time-dependent confounders: the patient features change over time and are affected by the previous administration of antibiotics. Moreover, the history of the patient features also determines antibiotics administration and affects future patient outcomes (De Bus et al., 2018; Ali et al., 2019). + +In Table 10 we report the root mean squared error for factual prediction of the patients’ white blood cell count for multiple prediction horizons $\tau$ . Note that for this dataset we do not have access to counterfactual data, which is why we report error on factual predictions. + +Table 10: RMSE for $\tau$ -step-ahead prediction of factual outcomes on MIMIC III. + +
T=1T=2T=3 T=4
RMSN2.843.874.464.79
CRN2.683.544.074.67
+ +We notice that CRN also achieves better performance than RMSN in estimating factual outcomes in a real-world dataset containing electronic health records. In this context, where couterfactual data is not available, domain expert knowledge is required to validate the model’s counterfactual predictions under other antibiotic treatment alternatives. This further medical validation is outside the scope of this paper. \ No newline at end of file diff --git a/parse/train/BJg866NFvB/BJg866NFvB_content_list.json b/parse/train/BJg866NFvB/BJg866NFvB_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d04b6a784365f10668ef84c99447a478778ae541 --- /dev/null +++ b/parse/train/BJg866NFvB/BJg866NFvB_content_list.json @@ -0,0 +1,3392 @@ +[ + { + "type": "text", + "text": "ESTIMATING COUNTERFACTUAL TREATMENT OUTCOMES OVER TIME THROUGH ADVERSARIALLY BALANCED REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 101, + 821, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ioana Bica ", + "text_level": 1, + "bbox": [ + 184, + 195, + 259, + 208 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Department of Engineering Science University of Oxford, Oxford, UK The Alan Turing Institute, London, UK ioana.bica@eng.ox.ac.uk ", + "bbox": [ + 184, + 210, + 442, + 265 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ahmed M. Alaa Department of Electrical Engineering University of California, Los Angeles, USA ahmedmalaa@ucla.edu ", + "bbox": [ + 524, + 195, + 813, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "James Jordon ", + "text_level": 1, + "bbox": [ + 184, + 287, + 282, + 300 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Mihaela van der Schaar ", + "text_level": 1, + "bbox": [ + 506, + 286, + 673, + 300 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Department of Engineering Science University of Oxford, Oxford, UK james.jordon@wolfson.ox.ac.uk ", + "bbox": [ + 184, + 301, + 467, + 342 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "University of Cambridge, Cambridge, UK University of California, Los Angeles, USA The Alan Turing Institute, London, UK mv472@cam.ac.uk ", + "bbox": [ + 506, + 301, + 794, + 354 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 392, + 544, + 407 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Identifying when to give treatments to patients and how to select among multiple treatments over time are important medical problems with a few existing solutions. In this paper, we introduce the Counterfactual Recurrent Network (CRN), a novel sequence-to-sequence model that leverages the increasingly available patient observational data to estimate treatment effects over time and answer such medical questions. To handle the bias from time-varying confounders, covariates affecting the treatment assignment policy in the observational data, CRN uses domain adversarial training to build balancing representations of the patient history. At each timestep, CRN constructs a treatment invariant representation which removes the association between patient history and treatment assignments and thus can be reliably used for making counterfactual predictions. On a simulated model of tumour growth, with varying degree of time-dependent confounding, we show how our model achieves lower error in estimating counterfactuals and in choosing the correct treatment and timing of treatment than current state-of-the-art methods. ", + "bbox": [ + 233, + 422, + 766, + 617 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 642, + 336, + 657 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As clinical decision-makers are often faced with the problem of choosing between treatment alternatives for patients, reliably estimating their effects is paramount. While clinical trials represent the gold standard for causal inference, they are expensive, have a few patients and narrow inclusion criteria (Booth & Tannock, 2014). Leveraging the increasingly available observational data about patients, such as electronic health records, represents a more viable alternative for estimating treatment effects. ", + "bbox": [ + 174, + 672, + 825, + 742 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A large number of methods have been proposed for performing causal inference using observational data in the static setting (Johansson et al., 2016; Shalit et al., 2017; Alaa & van der Schaar, 2017; Li & Fu, 2017; Yoon et al., 2018; Alaa & van der Schaar, 2018; Yao et al., 2018) and only a few methods address the longitudinal setting (Xu et al., 2016; Roy et al., 2016; Soleimani et al., 2017; Schulam & Saria, 2017; Lim et al., 2018). However, estimating the effects of treatments over time poses unique opportunities such as understanding how diseases evolve under different treatment plans, how individual patients respond to medication over time, but also which are optimal timings for assigning treatments, thus providing new tools to improve clinical decision support systems. ", + "bbox": [ + 174, + 750, + 825, + 861 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The biggest challenge when estimating the effects of time-dependent treatments from observational data involves correctly handling the time-dependent confounders: patient covariates that are affected by past treatments which then influence future treatments and outcomes (Platt et al., 2009). For instance, consider that treatment A is given when a certain patient covariate (e.g. white blood cell count) has been outside of normal range values for several consecutive timesteps. Suppose also that this patient covariate was itself affected by the past administration of treatment B. If these patients are more likely to die, without adjusting for the time-dependent confounding (e.g. the changes in the white blood cell count over time), we will incorrectly conclude that treatment A is harmful to patients. Moreover, estimating the effect of a different sequence of treatments on the patient outcome would require not only adjusting for the bias at the current step (in treatment A), but also for the bias introduced by the previous application of treatment B. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/48102b092b83fa34ae96204e15fa6bcdf8c5bf203ea2fe1f45443282de05bbd2.jpg", + "image_caption": [ + "Figure 1: Applicability of CRN in cancer treatment planning. We illustrate 3 patients with different covariate and treatment histories $\\bar { \\mathbf { H } } _ { t }$ . For a current time $t$ , CRN can predict counterfactual trajectories (the coloured dashed branches) for planned treatments in the future. Through the counterfactual predictions, we can decide which treatment plan results in the best patient outcome (in this case, the lowest tumour volume). This way, CRN can be used to perform all of the following: choose optimal treatments (a), find timing when treatment is most effective (b) decide when to stop treatment (c). " + ], + "image_footnote": [], + "bbox": [ + 186, + 103, + 813, + 262 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 372, + 825, + 470 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Existing methods for causal inference in the static setting cannot be applied in this longitudinal setting since they are designed to handle the cross-sectional set-up, where the treatment and outcome depend only on a static value of the patient covariates. If we consider again the above example, these methods would not be able to model how the changes in patient covariates over time affect the assignment of treatments and they would also not be able to estimate the effect of a sequence of treatments on the patient outcome (e.g. sequential application of treatment A followed by treatment B). Different models that can handle these temporal dependencies in the observational data and varying-length patient histories are needed for estimating treatment effects over time. ", + "bbox": [ + 174, + 478, + 825, + 589 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Time-dependent confounders are present in observational data because doctors follow policies: the history of the patients’ covariates and the patients’ response to past treatments are used to decide future treatments (Mansournia et al., 2012). The direct use of supervised learning methods will be biased by the treatment policies present in the observational data and will not be able to correctly estimate counterfactuals for different treatment assignment policies. ", + "bbox": [ + 174, + 595, + 825, + 665 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Standard methods for adjusting for time-varying confounding and estimating the effects of timevarying exposures are based on ideas from epidemiology. The most widely used among these are Marginal Structural Models (MSMs) (Robins et al., 2000; Mansournia et al., 2012) which use the inverse probability of treatment weighting (IPTW) to adjust for the time-dependent confounding bias. Through IPTW, MSMs create a pseudo-population where the probability of treatment does not depend on the time-varying confounders. However, MSMs are not robust to model misspecification in computing the IPTWs. MSMs can also give high-variance estimates due to extreme weights; computing the IPTW involves dividing by probability of assigning a treatment conditional on patient history which can be numerically unstable if the probability is small. ", + "bbox": [ + 174, + 672, + 825, + 797 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We introduce the Counterfactual Recurrent Network (CRN), a novel sequence-to-sequence architecture for estimating treatment effects over time. CRN leverages recent advances in representation learning (Bengio et al., 2012) and domain adversarial training (Ganin et al., 2016) to overcome the problems of existing methods for causal inference over time. Our main contributions are as follows. ", + "bbox": [ + 174, + 805, + 825, + 861 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Treatment invariant representations over time. CRN constructs treatment invariant representations at each timestep in order to break the association between patient history and treatment assignment and thus removes the bias from time-dependent confounders. For this, CRN uses domain adversarial training (Ganin et al., 2016; Li et al., 2018; Sebag et al., 2019) to trade-off between building this balancing representation and predicting patient outcomes. We show that these representations remove the bias from time-varying confounders and can be reliably used for estimating counterfactual outcomes. This represents the first work that introduces ideas from domain adaptation to the area of estimating treatment effects over time. In addition, by building balancing representations, we propose a novel way of removing the bias introduced by time-varying confounders. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Counterfactual estimation of future outcomes. To estimate counterfactual outcomes for treatment plans (and not just single treatments), we integrate the domain adversarial training procedure as part of a sequence-to-sequence architecture. CRN consists of an encoder network which builds treatment invariant representations of the patient history that are used to initialize the decoder. The decoder network estimates outcomes under an intended sequence of future treatments, while also updating the balanced representation. By performing counterfactual estimation of future treatment outcomes, CRN can be used to answer critical medical questions such as deciding when to give treatments to patients, when to start and stop treatment regimes, and also how to select from multiple treatments over time. We illustrate in Figure 1 the applicability of our method in choosing optimal cancer treatments. ", + "bbox": [ + 174, + 180, + 825, + 306 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In our experiments, we evaluate CRN in a realistic set-up using a model of tumour growth (Geng et al., 2017). We show that CRN achieves better performance in predicting counterfactual outcomes, but also in choosing the right treatment and timing of treatment than current state-of-the-art methods. ", + "bbox": [ + 176, + 311, + 825, + 354 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 376, + 341, + 392 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We focus on methods for estimating treatment effects over time and for building balancing representations for causal inference. A more in-depth review of related work is in Appendix A. ", + "bbox": [ + 176, + 409, + 823, + 436 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Treatment effects over time. Standard methods for estimating the effects of time-varying exposures were first developed in the epidemiology literature and include the $\\mathrm { g }$ -computation formula, Structural Nested Models and Marginal Structural Models (MSMs) (Robins, 1986; 1994; Robins et al., 2000; Robins & Hernán, 2008). Originally, these methods have used predictors performing logistic/linear regression which makes them unsuitable for handling complex time-dependencies (Hernán et al., 2001; Mansournia et al., 2012; Mortimer et al., 2005). To address these limitations, methods that use Bayesian non-parametrics or recurrent neural networks as part of these frameworks have been proposed. (Xu et al., 2016; Roy et al., 2016; Lim et al., 2018). ", + "bbox": [ + 174, + 443, + 825, + 555 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To begin with, Xu et al. (2016) use Gaussian processes to model discrete patient outcomes as a generalized mixed-effects model and uses the $g$ -computation method to handle time-varying confounders. Soleimani et al. (2017) extend the approach in $\\mathrm { X u }$ et al. (2016) to the continuous time-setting and model treatment responses using linear time-invariant dynamical systems. Roy et al. (2016) use Dirichlet and Gaussian processes to model the observational data and estimate the IPTW in Marginal Structural Models. Schulam & Saria (2017) build upon work from Lok et al. (2008); Arjas & Parner (2004) and use marked point processes and Gaussian processes to learn causal effects in continuous-time data. These Bayesian non-parametric methods make strong assumptions about model structure and consequently cannot handle well heterogeneous treatment effects arising from baseline variables (Soleimani et al., 2017; Schulam & Saria, 2017) and multiple treatment outcomes (Xu et al., 2016; Schulam & Saria, 2017). ", + "bbox": [ + 174, + 563, + 825, + 714 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The work most related to ours is the one of Lim et al. (2018) which improves on the standard MSMs by using recurrent neural networks to estimate the inverse probability of treatment weights (IPTWs). Lim et al. (2018) introduces Recurrent Marginal Structural Networks (RMSNs) which also use a sequence-to-sequence deep learning architecture to forecast treatment responses in a similar fashion to our model. However, RMSNs require training additional RNNs to estimate the propensity weights and does not overcome the fundamental problems with IPTWs, such as the high-variance of the weights. Conversely, CRN takes advantage of the recent advances in machine learning, in particular, representation learning to propose a novel way of handling time-varying confounders. ", + "bbox": [ + 174, + 722, + 825, + 833 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Balancing representations for treatment effect estimation. Balancing the distribution of control and treated groups has been used for counterfactual estimation in the static setting. The methods proposed in the static setting for balancing representations are based on using discrepancy measures in the representation space between treated and untreated patients, which do not generalize to multiple treatments (Johansson et al., 2016; Shalit et al., 2017; Li & Fu, 2017; Yao et al., 2018). Moreover, due to the sequential assignment of treatments in the longitudinal setting, and due to the change of patient covariates over time according to previous treatments, the methods for the static setting are not directly applicable to the time-varying setting (Hernán et al., 2000; Mansournia et al., 2012). ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 PROBLEM FORMULATION ", + "text_level": 1, + "bbox": [ + 176, + 152, + 415, + 169 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Consider an observational dataset $\\mathcal { D } = \\left\\{ \\{ \\mathbf { x } _ { t } ^ { ( i ) } , \\mathbf { a } _ { t } ^ { ( i ) } , \\mathbf { y } _ { t + 1 } ^ { ( i ) } \\} _ { t = 1 } ^ { T ^ { ( i ) } } \\cup \\{ \\mathbf { v } ^ { ( i ) } \\} \\right\\} _ { i = 1 } ^ { N }$ consisting of information about $N$ independent patients. For each patient $( i )$ , we observe time-dependent covariates $\\mathbf { X } _ { t } ^ { ( i ) } \\in \\mathcal { X } _ { t }$ , treatment received $\\mathbf { A } _ { t } ^ { ( i ) } \\in \\{ A _ { 1 } , . . . A _ { K } \\} = \\mathcal { A }$ and outcomes $\\mathbf { Y } _ { t + 1 } ^ { ( i ) } \\in \\mathcal { V } _ { t + 1 }$ for $T ^ { ( i ) }$ discrete timesteps. The patient can also have baseline covariates $\\mathbf { V } ^ { ( i ) } \\in \\mathcal { V }$ such as gender and genetic information. Note that the ou me $\\mathbf { Y } _ { t + 1 } ^ { ( i ) }$ will be part of the observed covariates $\\mathbf { X } _ { t + 1 } ^ { ( i ) }$ . For simplicity, the patient $( i )$ ", + "bbox": [ + 173, + 179, + 826, + 284 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We adopt the potential outcomes framework proposed by (Neyman, 1923; Rubin, 1978) and extended by (Robins & Hernán, 2008) to account for time-varying treatments. Let $\\mathbf { Y } [ \\bar { \\mathbf { a } } ]$ be the potential outcomes, either factual or counterfactual, for each possible course of treatment a¯. Let $\\mathbf { \\bar { H } } _ { t } = ( \\bar { \\mathbf { X } } _ { t } , \\bar { \\mathbf { A } } _ { \\underline { { t } } - 1 } , \\mathbf { V } )$ represent the history of the patient covariates $\\bar { \\mathbf X } _ { t } = ( \\mathbf X _ { 1 } , \\dots , \\mathbf X _ { t } )$ , treatment assignments $\\bar { \\mathbf { A } } _ { t } = ( \\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { t } )$ and static features $\\mathbf { V }$ . We want to estimate: ", + "bbox": [ + 173, + 290, + 825, + 361 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ee54142209a0418282b4ea570e9bc262b47f0ee3678397ea7b1da9248f4bd3f6.jpg", + "text": "$$\n\\mathbb { E } ( \\mathbf { Y } _ { t + \\tau } [ \\bar { \\mathbf { a } } ( t , t + \\tau - 1 ) ] | \\bar { \\mathbf { H } } _ { t } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 397, + 366, + 598, + 383 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\bar { \\mathbf { a } } ( t , t + \\tau - 1 ) = [ \\mathbf { a } _ { t } , \\dots \\mathbf { a } _ { t + \\tau - 1 } ]$ represents a possible sequence of treatments from timestep $t$ just until before the potential outcome $\\mathbf { Y } _ { t + \\tau }$ is observed. We make the standard assumptions (Robins et al., 2000; Lim et al., 2018) needed to identify the treatment effects: consistency, positivity and no hidden confounders (sequential strong ignorability). See Appendix B for more more details. ", + "bbox": [ + 173, + 390, + 825, + 448 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 COUNTERFACTUAL RECURRENT NETWORK", + "text_level": 1, + "bbox": [ + 174, + 467, + 566, + 483 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The observational data can be used to train a supervised learning model to forecast: $\\mathbb { E } ( \\mathbf { Y } _ { t + \\tau } \\ )$ $\\bar { \\mathbf { A } } ( t , t + \\tau - 1 ) = \\bar { \\mathbf { a } } ( t , t + \\tau - 1 ) , \\bar { \\mathbf { H } } _ { t } )$ . However, without adjusting for the bias introduced by timevarying confounders, this model cannot be reliably used for making causal predictions (Robins et al., 2000; Robins & Hernán, 2008; Schulam $\\&$ Saria, 2017). The Counterfactual Recurrent Network (CRN) removes this bias through domain adversarial training and estimates the counterfactual outcomes $\\mathbb { E } ( \\mathbf { Y } _ { t + \\tau } [ \\bar { \\mathbf { a } } ( t , t + \\tau - 1 \\bar { ) } ] | \\bar { \\mathbf { H } } _ { t } )$ , for any intended future treatment assignment $\\bar { \\mathbf { a } } ( t , t + \\tau - 1 )$ . ", + "bbox": [ + 173, + 498, + 826, + 583 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Balancing representations. The history $\\bar { \\mathbf { H } } _ { t } = ( \\bar { \\mathbf { X } } _ { t } , \\bar { \\mathbf { A } } _ { t - 1 } , \\mathbf { V } )$ of the patient contains the timevarying confounders $\\bar { \\mathbf { X } } _ { t }$ which bias the treatment assignment $\\mathbf { A } _ { t } \\in \\{ A _ { 1 } , \\dotsc . . . A _ { K } \\}$ in the observational dataset. Inverse probability of treatment weighting, as performed by MSMs, creates a pseudopopulation where the probability of treatment ${ \\bf A } _ { t }$ does not depend on the time-varying confounders (Robins et al., 2000). In this paper, we propose instead building a representation of the history $\\bar { \\mathbf { H } } _ { t }$ that is not predictive of the treatment ${ \\bf A } _ { t }$ . This way, we remove the association between history, containing the time-varying confounders $\\bar { \\mathbf { X } } _ { t }$ , and current treatment ${ \\bf A } _ { t }$ . Robins (1999) shows that in this case, the estimation of counterfactual treatment outcomes is unbiased. See Appendix $\\textrm { C }$ for details and for an example of a causal graph with time-dependent confounders. ", + "bbox": [ + 173, + 588, + 826, + 715 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Let $\\Phi$ be the representation function that maps the patient history $\\bar { \\mathbf { H } } _ { t }$ to a representation space $\\mathcal { R }$ . To obtain unbiased treatment effects, $\\Phi$ needs to construct treatment invariant representations such that $P ( \\Phi ( { \\bar { \\mathbf { H } } } _ { t } ) \\mid \\mathbf { A } _ { t } = A _ { 1 } ) = \\cdots = P ( \\Phi ( { \\bar { \\mathbf { H } } } _ { t } ) \\mid \\mathbf { A } _ { t } = A _ { K } )$ . To achieve this and to estimate counterfactual outcomes under a planned sequence of treatments, we integrate the domain adversarial training framework proposed by Ganin et al. (2016) and extended by Sebag et al. (2019) to the multi-domain learning setting, into a sequence-to-sequence architecture. In our case, the different treatments at each timestep are considered the different domains. Note that the novelty here comes from the use of domain adversarial training to handle the bias from the time-dependent confounders, rather than the use of sequence-to-sequence models, which have already been applied to forecast treatment responses (Lim et al., 2018). Figure 2 illustrates our model architecture. ", + "bbox": [ + 173, + 720, + 825, + 861 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Encoder. The encoder network uses an RNN, with LSTM unit (Hochreiter & Schmidhuber, 1997), to process the history of treatments $\\bar { \\mathbf { A } } _ { t - 1 }$ , covariates $\\bar { \\mathbf { X } } _ { t }$ and baseline features $\\mathbf { V }$ to build a treatment invariant representation $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ , but also to predict one-step-ahead outcomes $\\mathbf { Y } _ { t + 1 }$ . To achieve this, the encoder network aims to maximize the loss of the treatment classifier $G _ { a }$ and minimize the loss of the outcome predictor network $G _ { y }$ . This way, the balanced representation $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ is not predictive of the assigned treatment ${ \\bf A } _ { t }$ , but is discriminative enough to estimate the outcome $\\mathbf { Y } _ { t + 1 }$ . To train this model using gradient descent, we use the Gradient Reversal Layer (Ganin et al., 2016). ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/39de32791ca33f62a2ed333d3baf6dd543573abf1582505564a41dc63718a59d.jpg", + "image_caption": [ + "Figure 2: CRN architecture. Encoder builds representation $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ that maximizes loss of treatment classifier $G _ { a }$ and minimizes loss of outcome predictor $G _ { y }$ . $\\bar { \\Phi ( \\bar { \\mathbf { H } } _ { t } ) }$ is used to initialize the decoder, which continues to update it to predict counterfactual outcomes of a sequence of future treatments. " + ], + "image_footnote": [], + "bbox": [ + 176, + 95, + 821, + 270 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 335, + 825, + 377 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Decoder. The decoder network uses the balanced representation computed by the encoder to initialize the state of an RNN that predicts the counterfactual outcomes for a sequence of future treatments. During training, the decoder uses as input the outcomes from the observational data $( \\mathbf { Y } _ { t + 1 } , . . . \\mathbf { Y } _ { t + \\tau - 1 } )$ , the static patient features $\\mathbf { V }$ and the intended sequence of treatments $\\bar { \\mathbf { a } } ( t , t + \\tau - 1 )$ . The decoder is trained in a similar way to the encoder to update the balanced representation and to estimate the outcomes. During testing, we do not have access to ground-truth outcomes; thus, the outcomes predicted by the decoder $( \\mathsf { \\bar { Y } } _ { t + 1 } , \\dots , \\hat { \\mathbf { Y } } _ { t + \\tau - 1 } )$ are auto-regressively used instead as inputs. By running the decoder with different treatment settings, and by auto-regressively feeding back the outcomes, we can determine when to start and end different treatments, which is the optimal time to give the treatment and which treatments to give over time to obtain the best patient outcomes. ", + "bbox": [ + 174, + 385, + 825, + 526 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The representation $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ is built by applying a fully connected layer, with Exponential Linear Unit (ELU) activation to the output of the LSTM. The treatment classifier $G _ { a }$ and the predictor network $G _ { y }$ consist of a hidden layer each, also with ELU activation. The output layer of $G _ { a }$ uses softmax activation, while the output layer of $G _ { y }$ uses linear activation for continuous predictions. For categorical outcomes, softmax activation can be used. We follow an approach similar to Lim et al. (2018) and we split the encoder and decoder training into separate steps. See Appendix E for details. ", + "bbox": [ + 174, + 532, + 825, + 617 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The encoder and decoder networks use variational dropout (Gal & Ghahramani, 2016) such that the CRN can also give uncertainty intervals for the treatment outcomes. This is particularity important in the estimation of treatment effects, since the model predictions should only be used when they have high confidence. Our model can also be modified to allow for irregular samplings of observations by using a PhasedLSTM (Neil et al., 2016). ", + "bbox": [ + 174, + 623, + 825, + 693 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 ADVERSARIALLY BALANCED REPRESENTATION OVER TIME ", + "text_level": 1, + "bbox": [ + 176, + 718, + 696, + 732 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "At each timestep $t$ , let the $K$ different possible treatments $\\mathbf { A } _ { t } \\in \\{ A _ { 1 } , \\dotsc . . . A _ { K } \\}$ represent our domains. As described in Section 4, to remove the bias from time-dependent confounders, we build a representation of history $\\bar { \\mathbf { H } } _ { t }$ that is invariant across treatments: $P ( \\mathbf { \\bar { \\Phi } } ( \\mathbf { \\bar { H } } _ { t } ) \\mid A _ { 1 } ) = \\cdots = P ( \\Phi ( \\mathbf { \\bar { H } } _ { t } ) \\mid \\mathbf { \\bar { \\Phi } } _ { X } )$ . ", + "bbox": [ + 174, + 748, + 828, + 792 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This requirement can be enforced by minimizing the distance in the distribution of $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ between any two pairs of treatments. Kifer et al. (2004); Ben-David et al. (2007), propose measuring the disparity between distributions based on their separability by a discriminatively-trained classifier. Let the symmetric hypothesis class $\\mathcal { H }$ consist of the set of symmetric multiclass classifiers, such as neural network architectures. The $\\mathcal { H }$ -divergence between all pairs of two distributions is defined in terms of the capacity of the hypothesis class $\\mathcal { H }$ to discriminate between examples from the multiple distributions. Empirically, minimizing the $\\varkappa -$ divergence involves building a representation where examples from the multiple domains are as indistinguishable as possible (Ben-David et al., 2007; Li et al., 2018; Sebag et al., 2019). Ganin et al. (2016) use this idea to propose an adversarial framework for domain adaptation involving building a representation which achieves maximum error on a domain classifier and minimum error on an outcome predictor. Similarly, in our case, we use domain adversarial training to build a representation of the patient history $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ that is both invariant to the treatment given at timestep $t$ , ${ \\bf A } _ { t }$ and that achieves low error in estimating the outcome $\\mathbf { Y } _ { t + 1 }$ . ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 160 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Let $G _ { a } \\big ( \\Phi ( \\bar { \\mathbf { H } } _ { t } ) ; \\theta _ { a } \\big )$ be the treatment classifier with parameters $\\theta _ { a }$ and let $G _ { a } ^ { j } ( \\Phi ( { \\bar { \\mathbf { H } } } _ { t } ) ; \\theta _ { a } )$ be the output corresponding to treatment $A _ { j }$ . Let $\\hat { G } _ { y } ( \\dot { \\Phi } ( \\bar { \\mathbf { H } } _ { t } ) ; \\theta _ { y } )$ be the predictor network with parameters $\\theta _ { y }$ . The representation function $\\Phi$ is parameterized by the parameters $\\theta _ { r }$ in the RNN: $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } ; \\theta _ { r } )$ . Figure 3 shows the adversarial training procedure used. ", + "bbox": [ + 173, + 165, + 552, + 265 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For timestep $t$ and patient $( i )$ , let $\\mathcal { L } _ { t , a } ^ { ( i ) } ( \\theta _ { r } , \\theta _ { a } )$ be the treatment (domain) loss and let $\\mathcal { L } _ { t , y } ^ { ( i ) } ( \\theta _ { r } , \\theta _ { y } )$ the outcome loss, defined as follows: ", + "bbox": [ + 173, + 268, + 550, + 319 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/f7a6761b284070fba3d17ff94f113542ed403489c6cb7b08242728a01d5e4917.jpg", + "text": "$$\n\\mathcal { L } _ { t , a } ^ { ( i ) } ( \\theta _ { r } , \\theta _ { a } ) = - \\sum _ { j = 1 } ^ { K } \\mathbb { I } _ { \\{ \\mathbf { a } _ { t } ^ { ( i ) } = a _ { j } \\} } \\log ( G _ { a } ^ { j } ( \\Phi ( \\bar { \\mathbf { H } } _ { t } ; \\theta _ { r } ) ; \\theta _ { a } ) )\n$$", + "text_format": "latex", + "bbox": [ + 176, + 324, + 544, + 369 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/8345f039c6d1a6ebfdbc510bd16403a83dd82aaf99fccb630b194552f5ccdb7a.jpg", + "text": "$$\n\\mathcal { L } _ { t , y } ^ { ( i ) } ( \\theta _ { r } , \\theta _ { y } ) = \\| \\mathbf { Y } _ { t + 1 } ^ { ( i ) } - \\left( G _ { y } ( \\Phi ( \\bar { \\mathbf { H } } _ { t } ; \\theta _ { r } ) , \\theta _ { y } ) \\right) \\| ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 176, + 383, + 500, + 407 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "If the outcome is binary, the cross-entropy loss can be used instead for $\\mathcal { L } _ { t , y }$ . To build treatment invariant representations and to also estimate patient outcomes, we aim to maximize treatment loss and minimize outcome loss. ", + "bbox": [ + 173, + 420, + 549, + 477 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/d393ec25b88d42f7d5cc610cfeb256d83fddebdab392f202255cb7d1638d9cd5.jpg", + "image_caption": [ + "Figure 3: Training procedure for building balancing representation. " + ], + "image_footnote": [], + "bbox": [ + 566, + 178, + 820, + 428 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$\\mathcal { L } _ { t , y } ^ { ( i ) }$ at timestep $t$ is given by: ", + "bbox": [ + 174, + 481, + 511, + 500 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ba271ea6cd93d58b5413ddb23d93ec1e598d1ad96d899586e36caf569446fe67.jpg", + "text": "$$\n\\mathcal { L } _ { t } ^ { ( i ) } ( \\theta _ { r } , \\theta _ { y } , \\theta _ { a } ) = \\sum _ { i = 1 } ^ { N } \\mathcal { L } _ { t , y } ^ { ( i ) } ( \\theta _ { r } , \\theta _ { y } ) - \\lambda \\mathcal { L } _ { t , a } ^ { ( i ) } ( \\theta _ { r } , \\theta _ { a } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 326, + 507, + 669, + 551 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where the hyperparameter $\\lambda$ controls this trade-off between domain discrimination and outcome prediction. We use the standard procedure for training domain adversarial networks from Ganin et al. (2016) and we start off with an initial value for $\\lambda$ and use an exponentially increasing schedule during training. To train the model using backpropagation, we use the Gradient Reversal Layer (GRL) (Ganin et al., 2016). For more details about the training procedure, see Appendix $\\mathrm { E }$ . ", + "bbox": [ + 173, + 558, + 825, + 630 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "By using the objective $\\mathcal { L } _ { t } ^ { ( i ) } ( \\theta _ { r } , \\theta _ { y } , \\theta _ { a } )$ , we reach the saddle point $( \\hat { \\theta } _ { r } , \\hat { \\theta } _ { y } , \\hat { \\theta } _ { a } )$ that achieves the equilibrium between domain discrimination and outcome estimation. ", + "bbox": [ + 171, + 635, + 823, + 667 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/2011bd68e3f56b4b4ebe390cd9b38fa98d81dc6505b0fa09bc569a3fefea196a.jpg", + "text": "$$\n( \\hat { \\theta } _ { r } , \\hat { \\theta } _ { y } ) = \\arg \\operatorname* { m i n } _ { \\theta _ { r } , \\theta _ { y } } \\mathcal { L } _ { t } ^ { ( i ) } ( \\theta _ { r } , \\theta _ { y } , \\hat { \\theta } _ { a } ) \\qquad \\hat { \\theta } _ { a } = \\arg \\operatorname* { m a x } _ { \\theta _ { a } } \\mathcal { L } _ { t } ^ { ( i ) } ( \\hat { \\theta } _ { r } , \\hat { \\theta } _ { y } , \\theta _ { a } ) .\n$$", + "text_format": "latex", + "bbox": [ + 245, + 674, + 753, + 703 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The result stated in Theorem 1 proves that the treatment (domain) loss part of our objective (from equation 2) aims to remove the time-dependent confounding bias. ", + "bbox": [ + 171, + 718, + 825, + 747 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 1. Let $t \\in \\{ 1 , 2 , \\dots \\}$ . For each $j = 1 , . . . , K$ , let $P _ { j }$ denote the distribution of $\\bar { \\mathbf { H } } _ { t }$ conditional on $\\mathbf { A } _ { t } = A _ { j }$ and let $P _ { j } ^ { \\Phi }$ denote the distribution of $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ conditional on $\\mathbf { A } _ { t } = A _ { j }$ . Let $G _ { a } ^ { j }$ denote the output of $G _ { a }$ corresponding to treatment $A _ { j }$ . Then the minimax game defined by ", + "bbox": [ + 174, + 751, + 825, + 800 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/7ce58ece0f668c44fcef24ad8527166ae87117ed618283aed5e99fe1b802935b.jpg", + "text": "$$\n\\underset { \\Phi } { \\mathop { \\operatorname* { m i n } } } \\underset { G _ { a } } { \\mathop { \\operatorname* { m a x } } } \\sum _ { j = 1 } ^ { K } \\mathbb { E } _ { \\bar { \\mathbf { H } } _ { t } \\sim P _ { j } } \\left[ \\log ( G _ { a } ^ { j } ( \\Phi ( \\bar { \\mathbf { H } } _ { t } ) ; \\theta _ { a } ) ) \\right] \\qquad \\quad s u b j e c t t o \\sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( \\Phi ( \\bar { \\mathbf { H } } _ { t } ) ) = 1\n$$", + "text_format": "latex", + "bbox": [ + 222, + 806, + 772, + 852 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "has a global minimum which is attained if and only if $P _ { 1 } ^ { \\Phi } = P _ { 2 } ^ { \\Phi } = \\ldots = P _ { K } ^ { \\Phi }$ , i.e. when the learned representations are invariant across all treatments. ", + "bbox": [ + 173, + 859, + 826, + 890 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proof. This result is a restatement of the one in Li et al. (2018). For details, see the Appendix D. ", + "bbox": [ + 168, + 907, + 816, + 925 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "A good representation allows us to obtain a low error in estimating counterfactuals for all treatments, while at the same time to minimize the $\\mathcal { H }$ -divergence between induced marginal distributions of all the domains. We use an algorithm that directly minimizes a combination of the $\\varkappa -$ divergence and the empirical training margin. ", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 181, + 326, + 196 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In real datasets, counterfactual outcomes and the degree of time-dependent confounding are not known (Schulam & Saria, 2017; Lim et al., 2018). To validate the $\\mathrm { C R N ^ { 1 } }$ , we evaluate it on a Pharmacokinetic-Pharmacodynamic model of tumour growth (Geng et al., 2017), which uses a stateof-the-art bio-mathematical model to simulate the combined effects of chemotherapy and radiotherapy in lung cancer patients. The same model was used by Lim et al. (2018) to evaluate RMSNs. ", + "bbox": [ + 174, + 212, + 825, + 284 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Model of tumour growth The volume of tumour $t$ days after diagnosis is modelled as follows: ", + "bbox": [ + 169, + 289, + 797, + 304 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/8455e5f469a601e403f6480a206839150f42db4f3a488f061732968fa8537010.jpg", + "text": "$$\nV ( t + 1 ) = \\Big ( 1 + \\underbrace { \\rho \\mathrm { l o g } \\big ( \\frac { K } { V ( t ) } \\big ) } _ { \\mathrm { T u m o r ~ g r o w t h } } - \\underbrace { \\beta _ { c } C ( t ) } _ { \\mathrm { C h e m o t h e r a p y } } - \\underbrace { \\big ( \\alpha _ { r } d ( t ) + \\beta _ { r } d ( t ) ^ { 2 } \\big ) } _ { \\mathrm { R a d i o t h e r a p y } } + \\underbrace { e _ { t } } _ { \\mathrm { N o i s e } } \\Big ) V ( t )\n$$", + "text_format": "latex", + "bbox": [ + 235, + 310, + 764, + 364 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $K , \\rho , \\beta _ { c } , \\alpha _ { r } , \\beta _ { r } , e _ { t }$ are sampled as described in Geng et al. (2017). To incorporate heterogeneity in patient responses, the prior means for $\\beta _ { c }$ and $\\alpha _ { r }$ are adjusted to create patient subgroups, which are used as baseline features. The chemotherapy concentration $C ( t )$ and radiotherapy dose $d ( t )$ are modelled as described in Appendix F. Time-varying confounding is introduced by modelling emotherapy and radiotherapy assigndepending on the tumour diameter: ables, and $p _ { c }$ $p _ { r }$ $\\begin{array} { r } { p _ { c } ( t ) = \\sigma \\big ( \\frac { \\gamma _ { c } } { D _ { \\mathrm { m a x } } } ( \\bar { D } ( t ) - \\delta _ { c } ) \\big ) } \\end{array}$ $\\begin{array} { r } { p _ { r } ( t ) \\dot { = } \\sigma \\big ( \\frac { \\gamma _ { r } } { D _ { \\mathrm { m a x } } } ( \\hat { \\bar { D } } ( t ) - } \\end{array}$ $\\delta _ { r } )$ \u0001 where is the average diameter over the last 15 days, $D _ { \\mathrm { m a x } } = 1 3 \\mathrm { c m } , \\sigma ( \\cdot )$ is the sigmoid and $\\delta _ { c } = \\delta _ { r } = D _ { \\operatorname* { m a x } } / 2$ . The amount of time-dependent confounding is controlled through $\\gamma _ { c } , \\gamma _ { r }$ ; the higher $\\gamma _ { \\star }$ is, the more important the history is in assigning treatments. At each timestep, there are four treatment options: no treatment, chemotherapy, radiotherapy, combined chemotherapy and radiotherapy. For details about data simulation, see Appendix F. ", + "bbox": [ + 173, + 371, + 825, + 530 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Benchmarks We used the following benchmarks for performance comparison: Marginal Structural Models (MSMs) (Robins et al., 2000), which use logistic regression for estimating the IPTWs and linear regression for prediction (see Appendix G for details). We also compare against the Recurrent Marginal Structural Networks (RMSNs) Lim et al. (2018), which is the current state-of-the-art model in estimating treatment responses. RMSNs use RNNs to estimate the IPTWs and the patient outcomes (details in Appendix H). To show that standard supervised learning models do not handle the timevarying confounders we compare against an RNN and a linear regression model, which receive as input treatments and covariates to predict the outcome (see Appendix I for details). Our model architecture follows the description in Sections 4 and 5, with full training details and hyperparameter optimization in Appendix J. To show the importance of adversarial training, we also benchmark against CRN $\\lambda = 0$ ) a model with the same architecture, but with $\\lambda = 0$ , i.e our model architecture without adversarial training. ", + "bbox": [ + 173, + 536, + 825, + 703 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.1 EVALUATE MODELS ON COUNTERFACTUAL PREDICTIONS ", + "text_level": 1, + "bbox": [ + 174, + 722, + 607, + 736 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Previous methods focused on evaluating the error only for factual outcomes (observed patient outcomes) (Lim et al., 2018). However, to build decision support systems, we need to evaluate how well the models estimate the counterfactual outcomes, i.e patient outcomes under alternative treatment options. The parameters $\\gamma _ { c }$ and $\\gamma _ { r }$ control the treatment assignment policy, i.e. the degree of time-dependent confounding present in the data. We evaluate the benchmarks under different degrees of time-dependent confounding by setting $\\gamma = \\gamma _ { c } = \\gamma _ { r }$ . For each $\\gamma$ we simulate a 10000 patients for training, 1000 for validation (hyperparameter tuning) and 1000 for out-of-sample testing. For the patients in the test set, for each time $t$ , we also simulate counterfactuals $\\mathbf { Y } _ { t + 1 }$ , represented by tumour volume $V ( t + 1 )$ , under all possible treatment options. ", + "bbox": [ + 173, + 747, + 825, + 873 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/cfd2e5bc6015b41a94296fbf8b32467dc117f38d981d224e897f47516e68addb.jpg", + "image_caption": [ + "Figure 4: Results for prediction of patient counterfactuals. " + ], + "image_footnote": [], + "bbox": [ + 197, + 102, + 812, + 272 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 4 (a) shows the normalized root mean squared error (RMSE) for one-step ahead estimation of counterfactuals with varying degree of time-dependent confounding $\\gamma$ . The RMSE is normalized by the maximum tumour volume: $V _ { m a x } = \\mathrm { 1 \\bar { 1 } 5 0 c m ^ { 3 } }$ . The linear and MSM models provide a baseline for performance as they achieve the highest RMSE. While the use of IPTW in MSMs helps when $\\gamma$ increases, using linear modelling has severe limitations. When there is no time-dependent confounding, the machine learning methods achieve similar performance, close to $0 . 6 \\%$ RMSE. As the bias in the dataset increases, the harder it becomes for the RNN and the CRN $\\lambda = 0 )$ ) to generalize to estimate outcomes of treatments not matching the training policy. When $\\gamma = 1 0$ , CRN improves by $4 8 . 1 \\%$ on the same model architecture without domain adversarial training CRN $\\lambda = 0$ ). ", + "bbox": [ + 173, + 313, + 825, + 439 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Our proposed model achieves the lowest RMSE across all values of $\\gamma$ . Compared to RMSNs, CRN improves by $\\sim 1 7 \\%$ when $\\gamma > 6$ . To highlight the gains of our method even for smaller $\\gamma$ , Figure 4 (b) shows the RMSE for five-step ahead prediction (with counterfactuals generated as described in Section 6.2 and Appendix L). RMSNs also use a decoder for sequence prediction. However, RMSNs require training additional RNNs to estimate the IPTW, which are used to weight each sample during the decoder training. For $\\tau$ -step ahead prediction, IPTW involves multiplying $\\tau$ weights which can result in high variance. The results in Figure 4 (b) show the problems with using IPTW to handle the time-dependent confounding bias. See Appendix K for more results on multi-step ahead prediction. ", + "bbox": [ + 173, + 445, + 825, + 558 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Balancing representation: To evaluate whether the CRN has indeed learnt treatment invariant represenations, for $\\gamma = 5$ , we illustrate in Figure 5 the T-SNE embeddings of the balancing representations $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ built by the CRN encoder for test patients. We color each point by the treatment $\\mathbf { A } _ { t } \\in \\{ \\mathrm { n o } \\}$ treatment, chemotherapy, radiotherapy, combined chemotherapy and radiotherapy} received at timestep $t$ to highlight the invariance of $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ across the different treatments. In Figure 5(b), we show $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ only for chemotherapy and radiotherapy for better understanding. ", + "bbox": [ + 174, + 564, + 825, + 647 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/81db339d71547cdef7a4a6bfc60803c8c22dc7a97cbc2b111eb3ea319e11801f.jpg", + "image_caption": [ + "Figure 5: TSNE embedding of the balancing representation $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ learnt by the CRN encoder at different timesteps $t$ . Notice that $\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )$ is not predictive of the treatment ${ \\bf A } _ { t }$ given at timestep $t$ . " + ], + "image_footnote": [], + "bbox": [ + 181, + 654, + 812, + 717 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.2 EVALUATE RECOMMENDING THE RIGHT TREATMENT AND TIMING OF TREATMENT", + "text_level": 1, + "bbox": [ + 169, + 773, + 781, + 787 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Evaluating the models just in terms of the RMSE on counterfactual estimation is also not enough for assessing their reliability when used as part of decision support systems. In this section we assess how well the models can select the correct treatment and timing of treatment for several forecasting horizons $\\tau$ . We generate test sets consisting of 1000 patients where for each horizon $\\tau$ and for each time $t$ in a patient’s trajectory, there are $\\tau$ options for giving chemotherapy at one of $t , \\dots t + \\tau - 1$ and $\\tau$ options for giving radiotherapy at one of $t , \\dots t + \\tau - 1$ . At the rest of the future timesteps, no treatment is applied. These $2 \\tau$ treatment plans are assessed in terms of the tumour volume outcome $\\mathbf { Y } _ { t + \\tau }$ . We select the treatment (chemotherapy or radiotherapy) that achieves lowest $\\mathbf { Y } _ { t + \\tau }$ , and within the correct treatment the timing with lowest $\\mathbf { Y } _ { t + \\tau }$ . We also compute the normalized RMSE for predicting $\\mathbf { Y } _ { t + \\tau }$ . See Appendix L for more details about the test set. The models are evaluated for 3 settings of $\\gamma _ { c }$ and $\\gamma _ { r }$ . ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/e8c20a276d8d98067f0238283f13ca2afea3e0c0dbb97025c67f09d8280e2526.jpg", + "table_caption": [ + "Table 1: Results for recommending the correct treatment and timing of treatment. " + ], + "table_footnote": [], + "table_body": "
%c=5,γr=5% =5,γr=01 Yc =0,γr =5
TCRNRMSNMSMCRN RMSNMSMCRNRMSNMSM
Normalized RMSE2.43%3.16%6.75%1.08% 1.35%3.68%1.54%1.59%3.23%
2.83%3.95%7.65% 1.21%1.81%3.84%1.81%2.25%3.52%
3４56７3.18%4.37% 7.95%1.33%2.13%3.91%2.03%2.71%3.63%
3.51%5.61%8.19%1.42% 2.41%3.97%2.23%2.73%3.71%
3.93%6.21% 8.52%1.53%2.43%4.04%2.43%2.88%3.79%
Treatment Accuracy83.1%75.3%73.9%83.2% 78.6%77.1%92.9%87.3%74.9%
82.5%74.1%68.5%81.3%77.7% 73.9%85.7%83.8%74.1%
3４56７73.5%72.7%63.2%78.3% 77.2%72.3%83.8%82.1%72.8%
69.4%66.7%62.7%79.5% 76.3%71.8%78.6%69.7%64.5%
71.2%68.8%62.4%72.7% 71.8%71.6%71.9%69.3%61.2%
Treatment Timing Accuracy79.6%78.1%67.6%80.5%76.8% 77.5%79.8%75.7%60.6%
73.9%70.3%63.1%79.0%77.2%73.4% 75.4%71.4%58.2%
69.8%68.6%62.4%78.3%73.3%63.6%66.9%31.3%29.5%
3456７66.9%66.2%62.6%73.5%72.1%63.9%65.8%24.2%15.5%
64.5%63.6%62.2%70.6%57.4%44.2%63.9%25.6%12.5%
", + "bbox": [ + 174, + 116, + 825, + 387 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 404, + 823, + 431 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 1 shows the results for this evaluation set-up. The treatment accuracy denotes the percentage of patients for which the correct treatment was selected, while the treatment timing accuracy is the percentage for which the correct timing was selected. Note that when $\\gamma _ { c } = 0$ and $\\gamma _ { r } = 5$ , RMSN and MSM select the wrong treatment timing for projection horizons $\\tau > 4$ . CRN performs similarly among the different policies present in the observational data and achieve the lowest RMSE and highest accuracy in selecting the correct treatment and timing of treatment. ", + "bbox": [ + 173, + 438, + 825, + 522 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In Appendix M we also show the applicability of the CRN in more complex medical scenarios involving real data. We provide experimental results based on the Medical Information Mart for Intensive Care (MIMIC III) database (Johnson et al., 2016) consisting of electronic health records from patients in the ICU. ", + "bbox": [ + 174, + 529, + 825, + 585 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 604, + 318, + 621 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Despite its wide applicability, the problem of causal inference for time-dependent treatments has been relatively less studied compared to problem of causal inference in the static setting. Both new methods and theory are necessary to be able to harness the full potential of observational data for learning individualized effects of complex treatment scenarios. Further work in this direction is needed for proposing alternative methods for handling time-dependent confounders, for modelling combinations of treatments assigned over time or for estimating the individualized effects of time-dependent treatments with associated dosage. ", + "bbox": [ + 174, + 637, + 825, + 734 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we introduced the Counterfactual Recurrent Network (CRN), a model that estimates individualized effects of treatments over time using a novel way of handling the bias from timedependent confounders through adversarial training. Using a model of tumour growth, we validated CRN in realistic medical scenarios and we showed improvements over existing state-of-the-art methods. We also showed the applicability of the CRN a real dataset consiting of patient electronic health records. The counterfactual predictions of CRN have the potential to be used as part of clinical decision support systems to address relevant medical challenges involving selecting the best treatments for patients over time, identify optimal treatment timings but also when the treatment is no longer needed. In future work, we will aim to build better balancing representations and to provide theoretical guarantees for the expected error on the counterfactuals. ", + "bbox": [ + 173, + 741, + 825, + 881 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 356, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We would like to thank the reviewers for their valuable feedback. The research presented in this paper was supported by The Alan Turing Institute, under the EPSRC grant EP/N510129/1 and by the US Office of Naval Research (ONR). ", + "bbox": [ + 174, + 133, + 825, + 175 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 195, + 287, + 210 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Alberto Abadie and Guido W Imbens. Matching on the estimated propensity score. Econometrica, 84(2):781–807, 2016. ", + "bbox": [ + 174, + 218, + 825, + 247 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Ahmed Alaa and Mihaela van der Schaar. Limits of estimating heterogeneous treatment effects: Guidelines for practical algorithm design. In International Conference on Machine Learning, pp. 129–138, 2018. ", + "bbox": [ + 174, + 255, + 825, + 297 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Ahmed M Alaa and Mihaela van der Schaar. 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", + "bbox": [ + 173, + 474, + 826, + 517 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 103, + 248, + 117 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A EXTENDED RELATED WORK ", + "text_level": 1, + "bbox": [ + 178, + 132, + 441, + 147 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Causal inference in the static setting: A large number of methods have been proposed to learn treatment effects from observational data in the static setting. In this case, it is needed to adjust for the selection bias; bias caused by the fact that, in the observational dataset, the treatment assignments depend on the patient features. Several ways of handling the selection bias involve using propensity matching (Austin, 2011; Imai & Ratkovic, 2014; Abadie & Imbens, 2016), building representations where treated and un-treated populations had similar distributions (Johansson et al., 2016; Shalit et al., 2017; Li & Fu, 2017; Yao et al., 2018) or performing propensity-aware hyperparameter tuning (Alaa & van der Schaar, 2017; 2018). However, these methods for the static setting cannot be extended directly to time-varying treatments (Hernán et al., 2000; Schisterman et al., 2009). ", + "bbox": [ + 174, + 162, + 825, + 289 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Learning optimal policies: A related problem to ours involves learning the optimal treatment policies from logged data (Swaminathan & Joachims, 2015a;b; Atan et al., 2018). That is, learning the treatment option that would give the best reward. Note the difference to the causal inference setting considered in this paper, where the aim is to learn the counterfactual patient outcomes under all possible treatment options. Learning all of the counterfactual outcomes is a harder problem and can also be used for finding the optimal treatment. ", + "bbox": [ + 174, + 295, + 825, + 378 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A method for learning optimal policies, proposed by Atan et al. (2018) uses domain adversarial training to build a representation that is invariant to the following two domains: observational data and simulated randomized clinical trial data, where the treatments have equal probabilities. Atan et al. (2018) only considers the static setting and aims to choose the optimal treatment instead of estimating all of the counterfactual outcomes. In our paper the aim is to eliminate the bias from the time-dependent confounders and reliably estimate all of the potential outcomes; thus, at each timestep $t$ we build a representation that is invariant to the treatment. ", + "bbox": [ + 174, + 386, + 825, + 483 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Off-policy evaluation in reinforcement learning: In reinforcement learning, a similar problem to ours is off-policy evaluation, which uses retrospective observational data, also known as logged bandit feedback (Hoiles & Van Der Schaar, 2016; Paduraru et al., 2013; Doroudi et al., 2017). In this case, ˘ the retrospective observational data consists of sequences of states, actions and rewards which were generated by an agent operating under an unknown policy. The off-policy evaluation methods aim to use this data to estimate the expected reward of a target policy. These methods use algorithms based on importance sampling (Precup, 2000; Thomas et al., 2015; Guo et al., 2017), action-value function approximation (model based) (Hallak et al., 2015) or doubly robust combination of both approaches (Jiang & Li, 2015). Nevertheless, these methods focus on obtaining average rewards of policies, while in our case the aim is to estimate individualized patient outcomes for future treatments. ", + "bbox": [ + 174, + 491, + 825, + 630 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B ASSUMPTIONS ", + "text_level": 1, + "bbox": [ + 176, + 650, + 331, + 665 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The standard assumptions needed for identifying the treatment effects are (Robins & Hernán, 2008; Lim et al., 2018; Schulam & Saria, 2017): ", + "bbox": [ + 174, + 681, + 825, + 709 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Assumption 1: Consistency. If ${ \\bf A } _ { t } = { \\bf a } _ { t }$ for a given patient, then the potential outcome for treatment $\\mathbf { a } _ { t }$ is the same as the observed (factual) outcome: $\\mathbf Y _ { t + 1 } [ \\mathbf a _ { t } ] = \\mathbf Y _ { t + 1 }$ . ", + "bbox": [ + 173, + 715, + 823, + 746 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Assumption 2: Positivity (Overlap) (Imai & Van Dyk, 2004): If $P ( \\bar { \\mathbf { A } } _ { t - 1 } = \\bar { \\mathbf { a } } _ { t - 1 } , \\bar { \\mathbf { X } } _ { t } = \\bar { \\mathbf { x } } _ { t } ) \\neq 0$ then $P ( \\mathbf { A } _ { t } = \\mathbf { a } _ { t } \\mid \\bar { \\mathbf { A } } _ { t - 1 } = \\bar { \\mathbf { a } } _ { t - 1 } , \\bar { \\mathbf { X } } _ { t } = \\bar { \\mathbf { x } } _ { t } ) > 0$ for all $\\bar { \\mathbf { a } } _ { t }$ . ", + "bbox": [ + 171, + 751, + 825, + 780 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Assumption 3: Sequential strong ignorability. $\\mathbf { Y } _ { t + 1 } [ \\mathbf { a } _ { t } ] \\perp \\perp \\mathbf { A } _ { t } \\mid \\bar { \\mathbf { A } } _ { t - 1 } , \\bar { \\mathbf { X } } _ { t } , \\forall \\mathbf { a } _ { t } \\in \\mathcal { A } , \\forall t .$ ", + "bbox": [ + 171, + 785, + 772, + 803 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Assumption 2 means that, for each timestep, each treatment has non-zero probability of being assigned. Assumption 3 means that there are no hidden confounders, that is, all of covariates affecting both the treatment assignment and the outcomes are present in the the observational dataset. Note that while assumption 3 is standard across all methods for estimating treatment effects, it is not testable in practice (Robins et al., 2000; Pearl et al., 2009). ", + "bbox": [ + 174, + 808, + 825, + 878 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C TIME-DEPENDENT CONFOUNDING ", + "text_level": 1, + "bbox": [ + 174, + 102, + 495, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Figure 6 illustrates the causal graphs for a time-varying exposures with 2-steps (Robins et al., 2000). In Figure 6 (a), the covariate $X$ is a time-dependent confounder because it affects the treatment assignments and at the same time, its value is changed by past treatments (Mansournia et al., 2017), as illustrated by the red arrows. Thus, the treatment probabilities at each time $t$ depend on the history of covariate $X$ and past treatments. Note that $U _ { 0 }$ and $U _ { 1 }$ are hidden variables which only affect the covariates, i.e. they do not have arrows into the treatments. Thus, the no hidden confounders assumption (Assumption 3) is satisfied. ", + "bbox": [ + 173, + 133, + 826, + 232 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Figure 6 (a) and (b) illustrate the two cases when there is no bias from time-dependent confounding. In Figure 6 (a) the treatment probabilities are independent, while in Figure 6 (b) they depend on past treatments. ", + "bbox": [ + 174, + 238, + 826, + 280 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/36d7077a848a6b126878a2fc9fc2c440777cd99b9c6aaab7a41d4644439ac523.jpg", + "image_caption": [ + "Figure 6: Causal graphs for 2-step time-varying exposures (Robins et al., 2000). $X _ { 0 } , X _ { 1 }$ are patient covariates, $A _ { 0 } , A _ { 1 }$ are treatments, $U _ { 0 }$ , $U _ { 1 }$ are unobserved variable and $Y _ { 2 }$ is the outcome. " + ], + "image_footnote": [], + "bbox": [ + 174, + 295, + 823, + 625 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Marginal Structural Models Robins et al. (2000). To remove the association between timedependent confounders and time-varying treatments, Marginal Structural Models propose using inverse probability of treatment weighting (IPTW). Without loss of generality, consider the use of MSMs with univariate treatments, baseline variables and outcomes. The outcome after $t$ timesteps is parametrized as follows: $\\mathbf { E } [ Y _ { t + 1 } \\mid \\mathbf { a } _ { 1 } , \\dots \\mathbf { a } _ { t } , V ] = g ( \\mathbf { a } _ { 1 } , \\dots \\mathbf { a } _ { n } , V ; \\theta )$ , where $g ( \\cdot )$ is usually a linear function with parameters $\\theta$ . To remove the bias from the time-dependent confounders present in the observational dataset, in the regression model $g ( \\cdot )$ MSMs weights each patients using either stabilized weights: ", + "bbox": [ + 173, + 690, + 825, + 803 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/be646c540e38b18432c6a9dde10788452a0b283c3451bb2a75243c36617bd11c.jpg", + "text": "$$\nS W ( t ) = \\prod _ { l = 1 } ^ { t } \\frac { f ( \\mathbf { A } _ { l } \\mid \\bar { \\mathbf { A } } _ { l - 1 } ) } { f ( \\mathbf { A } _ { l } \\mid \\bar { \\mathbf { X } } _ { l } , \\bar { \\mathbf { A } } _ { l - 1 } , \\mathbf { V } ) }\n$$", + "text_format": "latex", + "bbox": [ + 379, + 801, + 619, + 844 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "or unstabilized weights: ", + "bbox": [ + 174, + 848, + 331, + 863 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/1252f242c461c5051d6b991b41aa2e7d2c81119e5a06eb89ad0e4ee358dd23f2.jpg", + "text": "$$\nW ( t ) = \\prod _ { l = 1 } ^ { t } { \\frac { 1 } { f ( \\mathbf { A } _ { l } \\mid \\bar { \\mathbf { X } } _ { l } , \\bar { \\mathbf { A } } _ { l - 1 } , \\mathbf { V } ) } } ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 862, + 614, + 905 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $f ( \\cdot )$ represents the conditional probability mass function for discrete treatments. ", + "bbox": [ + 171, + 909, + 740, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Inverse probability of treatment weighting (IPTW) creates a pseudo-population where each member consists of themselves and $W - 1$ (or $S W - 1 )$ copies added though weighting. In this pseudopopulation, Robins Robins (1999) shows that $\\bar { \\mathbf X } _ { t }$ does not predict treatment ${ \\bf A } _ { t }$ , thus removing the bias from time-dependent confounders. ", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "When using unstabilized weights $W$ , the causal graph in the pseudo-population is the one in Figure 6 (a) where $\\mathbf { \\bar { \\nabla } } P ( \\mathbf { A } _ { t } \\mid \\bar { \\mathbf { X } } _ { t } , \\bar { \\mathbf { A } } _ { t - 1 } , \\mathbf { \\bar { \\xi } } V ) = P ( \\mathbf { A } _ { t } )$ . On the other hand, when using stabilized weights $S W$ , causal graph in the pseudo-population is the one in Figure 6 (b) where $\\mathbf { \\bar { \\nabla } } P ( \\mathbf { A } _ { t } \\mid \\bar { \\mathbf { X } } _ { t } , \\bar { \\mathbf { A } } _ { t - 1 } , V ) =$ $P ( \\mathbf { A } _ { t } \\mid \\bar { \\mathbf { A } } _ { t - 1 } )$ . ", + "bbox": [ + 173, + 166, + 825, + 223 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Counterfactual Recurrent Networks. Instead of using IPTW, we proposed building a representation of $\\bar { \\mathbf { X } } _ { t } , \\bar { \\mathbf { A } } _ { t - 1 } , V$ that is not predictive of treatment ${ \\bf A } _ { t }$ . At timestep $t$ , we have $k$ different possible treatments $\\mathbf { A } _ { t } \\in \\{ A _ { 1 } , \\dotsc . . . A _ { K } \\}$ . We build a representation of the history and covariates and treatments that has the same distribution across the different possible treatments: $P ( \\Phi ( { \\bar { \\mathbf { X } } } _ { t } , { \\bar { \\mathbf { A } } } _ { t - 1 } , \\mathbf { V } ) \\mid \\mathbf { A } _ { t } =$ $A _ { 1 } ) = \\cdots = P ( \\Phi ( { \\bar { \\mathbf { X } } } _ { t } , { \\bar { \\mathbf { A } } } _ { t - 1 } , \\mathbf { V } ) \\mid \\mathbf { A } _ { t } = A _ { K } )$ . By breaking the association between past exposure and current treatments ${ \\bf A } _ { t }$ , we satisfy the causal graph in Figure 6 (a) and thus we remove the bias from time-dependent confounders. ", + "bbox": [ + 173, + 229, + 825, + 327 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "D PROOF OF THEOREM 1 ", + "text_level": 1, + "bbox": [ + 174, + 101, + 398, + 118 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We first prove the following proposition. ", + "bbox": [ + 174, + 133, + 441, + 148 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proposition 1. For fixed $\\Phi$ , let $x ^ { \\prime } = \\Phi ( \\bar { \\mathbf { h } } _ { t } )$ . Then the optimal prediction probabilities of $G _ { a }$ are given by ", + "bbox": [ + 174, + 151, + 823, + 179 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/3bbac7548f876e8f96ac7e5eefec823656a640d60cd989ccd295c2ad6e76c27a.jpg", + "text": "$$\n{ G _ { a } ^ { j } } ^ { * } ( x ^ { \\prime } ) = \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 406, + 178, + 591, + 219 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. For fixed $\\Phi$ , the optimal prediction probabilities are given by ", + "bbox": [ + 173, + 231, + 620, + 247 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/e8a5a3b6df443b833d937b582e4b9710c1ec7517262e3eeeaa5a22ed16ddb79e.jpg", + "text": "$$\nG _ { a } ^ { * } = \\arg \\operatorname* { m a x } _ { G _ { a } } \\sum _ { j = 1 } ^ { K } \\int _ { x ^ { \\prime } } \\log ( G _ { a } ^ { j } ( x ^ { \\prime } ) ) P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) d x ^ { \\prime } \\qquad \\mathrm { s u b j e c t ~ t o } \\sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( x ^ { \\prime } ) = 1 .\n$$", + "text_format": "latex", + "bbox": [ + 227, + 252, + 769, + 299 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Maximising the value function pointwise and applying Lagrange multiplies, we get ", + "bbox": [ + 171, + 303, + 722, + 319 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/2d5de6eb3aeede2e67d256d7fb0332c254121b4c257c1e223e9983d4722caaba.jpg", + "text": "$$\nG _ { a } ^ { * } = \\arg \\operatorname* { m a x } _ { G _ { a } } \\sum _ { j = 1 } ^ { K } \\log ( G _ { a } ^ { j } ( x ^ { \\prime } ) ) P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) + \\lambda \\Bigg ( \\sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( x ^ { \\prime } ) - 1 \\Bigg ) .\n$$", + "text_format": "latex", + "bbox": [ + 284, + 324, + 714, + 369 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Setting the derivative (w.r.t. $G _ { a } ^ { j } { } ^ { * } ( x ^ { \\prime } ) )$ to 0 and solving for $G _ { a } ^ { j } { } ^ { * } ( x ^ { \\prime } )$ we get ", + "bbox": [ + 174, + 377, + 660, + 395 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/d535bf772a4383484cdcc75726bbd8b4493254c057261184bd8bec9529017ce8.jpg", + "text": "$$\n{ G _ { a } ^ { j } } ^ { * } ( x ^ { \\prime } ) = - \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\lambda }\n$$", + "text_format": "latex", + "bbox": [ + 426, + 400, + 571, + 434 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $\\lambda$ can now be solved for using the constraint to be $\\begin{array} { r } { \\lambda = - \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } \\end{array}$ . This gives the result. ", + "bbox": [ + 173, + 440, + 825, + 472 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. (of Theorem 1) By substituting the expression from Proposition 1 into the minimax game defined in Eq. 6, the objective for $\\Phi$ becomes ", + "bbox": [ + 174, + 486, + 823, + 516 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/9b2144766465cf6c74716cf0fb154657afb69f660116b20151d4635b2948070a.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\Phi } \\sum _ { j = 1 } ^ { K } \\mathbb { E } _ { x ^ { \\prime } \\sim P _ { j } ^ { \\Phi } } \\left[ \\log \\left( \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } \\right) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 522, + 640, + 568 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We then note that ", + "bbox": [ + 173, + 574, + 290, + 588 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/6ad792127c3b6818733137894cdbe6437d1186236cdce3a017ad3e32210bf22e.jpg", + "text": "$$\n\\sum _ { j = 1 } ^ { K } \\mathbb { E } _ { x ^ { \\prime } \\sim P _ { j } ^ { \\Phi } } \\bigg [ \\log \\bigg ( \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } \\bigg ) \\bigg ] + K \\log K = \\sum _ { j = 1 } ^ { K } \\Bigg ( \\mathbb { E } _ { x ^ { \\prime } \\sim P _ { j } ^ { \\Phi } } \\bigg [ \\log \\bigg ( \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } \\bigg ) \\bigg ] + \\log K \\Bigg )\n$$", + "text_format": "latex", + "bbox": [ + 181, + 592, + 839, + 637 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/53014b3501172eb19a3041fd05388c2ee49749d22d5baf8b5c3db947be95da45.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle = \\sum _ { j = 1 } ^ { K } \\mathbb { E } _ { x ^ { \\prime } \\sim P _ { j } ^ { \\Phi } } \\left[ \\log \\left( \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } \\right) \\right] } } \\\\ { { \\displaystyle = \\sum _ { j = 1 } ^ { K } K L \\biggl ( P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) \\biggl \\| \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) \\biggr ) } } \\\\ { { \\displaystyle = K \\cdot J S D ( P _ { 1 } ^ { \\Phi } , . . . , P _ { K } ^ { \\Phi } ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 503, + 654, + 782, + 765 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where $K L ( \\cdot | | \\cdot )$ is the Kullback-Leibler divergence and $J S D ( \\cdot , . . . , \\cdot )$ is the multi-distribution Jensennon-negative and 0 if and only if all distributions are equal, we have that Shannon Divergence (Li et al., 2018). Since $K \\log K$ is a constant and the multi-distribution JSD is $P _ { 1 } ^ { \\Phi } = \\ldots = P _ { K } ^ { \\Phi }$ . □ ", + "bbox": [ + 173, + 770, + 825, + 814 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "E TRAINING PROCEDURE FOR CRN ", + "text_level": 1, + "bbox": [ + 173, + 101, + 486, + 118 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Let $\\mathcal { D } = \\left\\{ \\{ \\mathbf { x } _ { t } ^ { ( i ) } , \\mathbf { a } _ { t } ^ { ( i ) } , \\mathbf { y } _ { t + 1 } ^ { ( i ) } \\} _ { t = 1 } ^ { T ^ { ( i ) } } \\cup \\{ \\mathbf { v } ^ { ( i ) } \\} \\right\\} _ { i = 1 } ^ { N }$ be an observational dataset consisting of information about $N$ independent patients that we use to train CRN. The encoder and decoder networks part of CRN are trained into two separate steps. ", + "bbox": [ + 173, + 131, + 826, + 185 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "To begin with, the encoder is trained to built treatment invariant representations of the patient history and to perform one-step ahead prediction. After the encoder is optimized, we use it to compute the balancing representation $\\mathbf { b r } _ { t } ^ { ( i ) }$ for each timestep in the trajectory of patient $( i )$ . To train the decoder, we modify the training dataset as follows. For each patient $( i )$ , we split their trajectory into shorter sequences of the $\\tau _ { \\mathrm { m a x } }$ timesteps of the form: ", + "bbox": [ + 173, + 191, + 825, + 265 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/7ef22625577cecaaeff6cdf74124f56ee9333453a80b144f6ef126a30bd61a58.jpg", + "text": "$$\n\\left\\{ \\mathbf { b r } _ { l } ^ { ( i ) } \\cup \\{ \\mathbf { y } _ { l + t } ^ { ( i ) } , \\mathbf { a } _ { l + t } ^ { ( i ) } , \\mathbf { y } _ { l + t + 1 } ^ { ( i ) } \\} _ { t = 1 } ^ { \\tau _ { m a x } } \\cup \\mathbf { v } ^ { ( i ) } \\right\\} ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 272, + 647, + 299 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "for $l = 1 , . . . T ^ { ( i ) } - \\tau _ { \\mathrm { m a x } }$ . Thus, each patients contributes with $T ^ { ( i ) } - \\tau _ { \\operatorname* { m a x } }$ examples in the dataset for training the decoder. The different sequences obtained for all patents are randomly grouped into minibatches and used for training. ", + "bbox": [ + 174, + 306, + 825, + 351 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The pseudocode in Algorithm 1 shows the training procedure used for the encoder and decoder networks part of CRN. The model was implemented in TensorFlow and trained on an NVIDIA Tesla K80 GPU. The Adam optimizer (Kingma & Ba, 2014) was used for training and both the encoder and the decoder are trained for 100 epochs. ", + "bbox": [ + 174, + 356, + 825, + 412 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Algorithm 1 Pseudo-code for training CRN ", + "text_level": 1, + "bbox": [ + 176, + 103, + 462, + 118 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Input: Training data: $\\mathcal { D } = \\left\\{ \\{ \\mathbf { x } _ { t } ^ { ( i ) } , \\mathbf { a } _ { t } ^ { ( i ) } , \\mathbf { y } _ { t + 1 } ^ { ( i ) } \\} _ { t = 1 } ^ { T ^ { ( i ) } } \\cup \\mathbf { v } ^ { ( i ) } \\right\\} _ { i = 1 } ^ { N }$ ", + "bbox": [ + 189, + 125, + 596, + 152 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(1) Encoder optimization: parameters $\\theta _ { E , r } , \\theta _ { E , a } , \\theta _ { E , y }$ ", + "bbox": [ + 191, + 159, + 558, + 174 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Learning rate: $\\mu$ ", + "bbox": [ + 191, + 174, + 299, + 186 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "for $p = 1 , \\ldots , \\operatorname* { m a x }$ epochs do $\\lambda _ { p } = \\frac { 2 } { 1 + \\exp ( - 1 0 \\cdot p ) } - 1$ ", + "bbox": [ + 189, + 188, + 393, + 232 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "for Batch $\\boldsymbol { B } = \\left\\{ \\{ \\mathbf { x } _ { t } ^ { ( i ) } , \\mathbf { a } _ { t } ^ { ( i ) } , \\mathbf { y } _ { t + 1 } ^ { ( i ) } \\} _ { t = 0 } ^ { T ^ { ( i ) } } \\cup \\mathbf { v } ^ { ( i ) } \\right\\} _ { i = 1 } ^ { | \\mathcal { B } | }$ in epoch do ", + "bbox": [ + 205, + 231, + 616, + 260 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "end for end for ", + "bbox": [ + 191, + 382, + 259, + 410 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(2) Compute the encoder balanced representation and use it to initialize the decoder hidden state. ", + "text_level": 1, + "bbox": [ + 187, + 422, + 825, + 450 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "for $i = 1 , . . . , N$ do for $t = 1 , \\ldots , T ^ { ( i ) }$ do 0 $\\mathbf { b r } _ { t } ^ { ( i ) } = \\operatorname { e n c o d e r } ( \\bar { \\mathbf { x } } _ { t } ^ { ( i ) } , \\bar { \\mathbf { a } } _ { t - 1 } ^ { ( i ) } , \\mathbf { v } ^ { ( i ) } ; \\boldsymbol { \\theta } _ { E , r } )$ end for \nend for ", + "bbox": [ + 187, + 450, + 480, + 523 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(3) Split dataset in sequences of $\\tau _ { \\mathrm { m a x } }$ timesteps: ", + "text_level": 1, + "bbox": [ + 191, + 536, + 519, + 551 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/7aaef18128b350701d8946743a5d0650bd8c537ee0f71f01e74e47db1d9eb8fb.jpg", + "text": "$$\n\\left\\{ \\left\\{ \\mathbf { b r } _ { l } ^ { ( i ) } \\cup \\{ \\mathbf { y } _ { l + t } ^ { ( i ) } , \\mathbf { a } _ { l + t } ^ { ( i ) } , \\mathbf { y } _ { l + t + 1 } ^ { ( i ) } \\} _ { t = 1 } ^ { \\tau _ { m a x } } \\cup \\mathbf { v } ^ { ( i ) } \\right\\} _ { l = 1 } ^ { T ^ { ( i ) } - \\tau _ { \\operatorname* { m a x } } } \\right\\} _ { i = 1 } ^ { N }\n$$", + "text_format": "latex", + "bbox": [ + 308, + 558, + 704, + 597 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(4) Optimize decoder: parameters $\\theta _ { D , r } , \\theta _ { D , a } , \\theta _ { D , y }$ Learning rate: $\\mu$ ", + "bbox": [ + 187, + 609, + 531, + 637 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "for $\\mathrm { p } = 1 , \\ldots$ , max epochs do ", + "bbox": [ + 189, + 637, + 395, + 681 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/ac852ab763eeff945a9bc6d1ff38554c2fd5e81feec5d26c4276ae8f3c311e87.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { ~ \\gamma = \\frac { 2 } { 1 + \\exp ( - 1 0 \\cdot p ) } - 1 } } \\\\ & { \\mathrm { ~ } } \\\\ & { \\mathrm { ~ \\gamma ~ R a t c h ~ } \\mathcal { B } = \\left\\{ \\mathbf { b r } _ { l } ^ { ( i ) } \\ \\bigcup \\left\\{ \\mathbf { y } _ { l + t } ^ { ( i ) } , \\ \\mathbf { a } _ { l + t } ^ { ( i ) } , \\mathbf { y } _ { l + t + 1 } ^ { ( i ) } \\right\\} _ { t = 0 } ^ { \\tau _ { m a x } } \\ | \\ \\mathbf { y } | ^ { ( i ) } \\right\\} _ { i = 1 } ^ { | B | } \\mathrm { ~ i n ~ e ~ } } \\\\ & { \\mathrm { ~ C o m p u t ~ } \\mathcal { L } _ { D , a } ^ { D } ( \\theta _ { D , r } , \\theta _ { D , a } ) = \\frac { 1 } { | B | } \\sum _ { i \\in B } \\sum _ { r = 1 } ^ { r _ { m a x } } \\mathcal { L } _ { t , a } ^ { ( i ) } ( \\theta _ { D , r } , \\theta _ { D , a } ) } \\\\ & { \\mathrm { ~ C o m p u t ~ } \\mathcal { L } _ { D , y } ^ { D } ( \\theta _ { D , r } , \\theta _ { D , y } ) = \\frac { 1 } { | B | } \\sum _ { i \\in B } \\sum _ { r = 1 } ^ { r _ { m a x } } \\mathcal { L } _ { t , y } ^ { ( i ) } ( \\theta _ { D , r } , \\theta _ { D , y } ) } \\\\ & { \\mathrm { ~ \\ } } \\\\ & { \\theta _ { D , r } \\cdot \\theta _ { D , r } - \\mu \\left( \\frac { \\partial \\mathcal { L } _ { D , y } ^ { D } ( \\theta _ { D , r } , \\theta _ { D , y } ) } { \\partial \\theta _ { D , r } } - \\lambda _ { p } \\frac { \\partial \\mathcal { L } _ { D , 0 } ^ { D } ( \\theta _ { D , r } , \\theta _ { D , a } ) } { \\partial \\theta _ { D , r } } \\right) } \\\\ & { \\theta _ { D , y } + \\theta _ { D , y } - \\mu \\frac { \\partial \\mathcal { L } _ { D , y } ^ { D } ( \\theta _ { D , r } , \\theta _ { D , y } ) } { \\partial \\theta _ { D , r } } } \\\\ & { \\theta _ { D , a } \\gets \\theta _ { D , \\theta } - \\mu \\frac { \\partial \\mathcal { L } _ { D , x } ^ { D } ( \\theta _ { D , r } , \\theta _ { D , x } ) } { \\partial \\theta _ { D , a } } } \\\\ & \\theta _ { D , a } \\gets \\theta _ { D , \\theta } - \\mu \\frac \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 654, + 710, + 832 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "end for ", + "bbox": [ + 205, + 829, + 258, + 840 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "end for ", + "bbox": [ + 191, + 842, + 243, + 854 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Output: Trained CRN encoder (parameters $\\theta _ { E , r } , \\theta _ { E , a } , \\theta _ { E , y } )$ and trained CRN decoder (parameters $\\theta _ { D , r } , \\theta _ { D , a } , \\theta _ { D , y } . ~ )$ ) ", + "bbox": [ + 186, + 868, + 823, + 898 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "F PHARMACOKINETIC-PHARMACODYNAMIC MODEL OF TUMOUR GROWTH ", + "text_level": 1, + "bbox": [ + 171, + 102, + 812, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "To evaluate the CRN on counterfactual estimation, we need access to the data generation mechanism to build a test set that consists of patient outcomes under all possible treatment options. For this purpose, we use the state-of-the-art pharmacokinetic-pharmacodynamic (PK-PD) model of tumour growth proposed by Geng et al. (2017) and also used by Lim et al. (2018) for evaluating RMSMs. The PK-PD model characterizes patients suffering from non-small cell lung cancer and models the evolution of their tumour under the combined effects of chemotherapy and radiotherapy. In addition, the model includes different distributions of tumour sizes based on the cancer stage at diagnosis. ", + "bbox": [ + 173, + 132, + 826, + 232 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Model of tumour growth The volume of tumour $t$ days after diagnosis is modelled as follows: ", + "bbox": [ + 169, + 237, + 797, + 252 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/e2557ef0023c3a665990cf4317f4ba76916ae9620ba84de4bb4caaaf65bac40f.jpg", + "text": "$$\nV ( t + 1 ) = \\Big ( 1 + \\underbrace { \\rho \\mathrm { l o g } ( \\frac { K } { V ( t ) } ) } _ { \\mathrm { T u m o r ~ g r o w t h } } - \\underbrace { \\beta _ { c } C ( t ) } _ { \\mathrm { C h e m o t h e r a p y } } - \\underbrace { \\big ( \\alpha _ { r } d ( t ) + \\beta _ { r } d ( t ) ^ { 2 } \\big ) } _ { \\mathrm { R a d i o t h e r a p y } } + \\underbrace { e _ { t } } _ { \\mathrm { N o i s e } } \\Big ) V ( t )\n$$", + "text_format": "latex", + "bbox": [ + 235, + 258, + 763, + 311 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where the parameters $K , \\rho , \\beta _ { c } , \\alpha _ { r } , \\beta _ { r }$ are sampled from the prior distributions described in (Geng et al., 2017) and $e _ { t } \\sim \\mathcal { N } ( 0 , 0 . 0 1 ^ { 2 } )$ is a noise term that accounts for randomness in the tumour growth. ", + "bbox": [ + 171, + 318, + 825, + 347 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "To incorporate heterogeneity among patient responses, due to, for instance, gender or genetic factors Bartsch et al. (2007), the prior means for $\\beta _ { c }$ and $\\alpha _ { r }$ are adjusted to create three patient subgroups $S ^ { ( i ) } \\in \\{ 1 , 2 , 3 \\}$ as described in Lim et al. (2018). This way, we incorporate in the model of tumour growth specific characteristics that affect the patient’s individualized response to treatments. Thus, the prior mean $\\mu _ { \\beta _ { c } }$ of $\\beta _ { c }$ and the prior mean $\\mu _ { \\alpha _ { r } }$ of $\\alpha _ { r }$ are augmented as follows. ", + "bbox": [ + 173, + 353, + 826, + 425 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/05e546345d7ffc1053caf0ba81224b78b4e995b12e03101af2bf1b1e35d119db.jpg", + "text": "$$\n\\begin{array} { r } { \\mu _ { \\beta _ { c } } ^ { \\prime } ( i ) = \\left\\{ \\begin{array} { l l } { 1 . 1 \\mu _ { \\beta _ { c } } , \\mathrm { i f } \\mathrm { S } ^ { ( i ) } = 3 } \\\\ { \\mu _ { \\beta _ { c } } , \\mathrm { o t h e r w i s e } } \\end{array} \\right. \\qquad \\mu _ { \\alpha _ { r } } ^ { \\prime } ( i ) = \\left\\{ \\begin{array} { l l } { 1 . 1 \\mu _ { \\alpha _ { r } } , \\mathrm { i f } \\mathrm { S } ^ { ( i ) } = 1 } \\\\ { \\mu _ { \\alpha _ { r } } , \\mathrm { o t h e r w i s e } } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 245, + 431, + 735, + 468 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\mu _ { \\beta _ { c } }$ and $\\mu _ { \\alpha _ { r } }$ are the mean parameters from Geng et al. (2017) and $\\mu _ { \\beta _ { c } } ^ { \\prime } ( i )$ and $\\mu _ { \\alpha _ { r } } ^ { \\prime } ( i )$ are the parameters used in the data simulation. The patient subgroup $S ^ { ( i ) } \\in \\{ 1 , 2 , 3 \\}$ is used as baseline features. ", + "bbox": [ + 173, + 474, + 825, + 522 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The chemotherapy drug concentration follows an exponential decay with half life of 1 day: ", + "bbox": [ + 173, + 527, + 767, + 542 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/1c4201090bb9934d95d2db562dfb063997c31c260912886b5f4e49b471e2709e.jpg", + "text": "$$\nC ( t ) = \\tilde { C } ( t ) + C ( t - 1 ) / 2 ,\n$$", + "text_format": "latex", + "bbox": [ + 403, + 549, + 593, + 569 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\tilde { C } ( t ) = 5 . 0 m g / m ^ { 3 }$ of Vinblastine if chemotherapy is given at time $t$ . $d ( t ) = 2 . 0 G y$ fractions of radiotherapy if the radiotherapy treatment is applied at timestep $t$ . ", + "bbox": [ + 171, + 577, + 823, + 606 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Time-varying confounding is introduced by modelling chemotherapy and radiotherapy assignment as Bernoulli random variables, with probabilities $p _ { c }$ and $p _ { r }$ depending on the tumour diameter: ", + "bbox": [ + 173, + 611, + 825, + 641 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/5fdbff385364eeb0a9503f2f851555ed99ef98105cd93892f77cd5e230072271.jpg", + "text": "$$\np _ { c } ( t ) = \\sigma \\left( \\frac { \\gamma _ { c } } { D _ { \\mathrm { m a x } } } ( \\bar { D } ( t ) - \\delta _ { c } ) \\right) \\qquad p _ { r } ( t ) = \\sigma \\left( \\frac { \\gamma _ { r } } { D _ { \\mathrm { m a x } } } ( \\bar { D } ( t ) - \\delta _ { r } ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 243, + 646, + 753, + 681 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "where $\\bar { D } ( t )$ is the average tumour diameter over the last 15 days, $D _ { \\mathrm { m a x } } = 1 3 \\mathrm { c m }$ is the maximum tumour diameter and $\\sigma ( \\cdot )$ is the sigmoid activation function. The parameters $\\delta _ { c }$ and $\\delta _ { r }$ are set to $\\delta _ { c } = \\delta _ { r } = D _ { \\operatorname* { m a x } } / 2$ such that there is 0.5 probability of receiving treatment when tumour is half of its maximum size. $\\gamma _ { c } , \\gamma _ { r }$ control the amount of time-dependent confounding; the higher $\\gamma _ { \\star }$ is, the more important the history of tumour diameter is in assigning treatments. Thus, at each timestep, there are four treatment options options: no treatment $( A _ { 1 } )$ , chemotherpy $\\left( A _ { 2 } \\right)$ , radiotherapy $\\left( A _ { 3 } \\right)$ , combined chemotherapy and radiotherapy $( A _ { 4 } )$ . ", + "bbox": [ + 173, + 688, + 826, + 786 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Since the work most relevant to ours is the one of Lim et al. (2018) we used the same data simulation and same settings for $\\gamma = \\gamma _ { c } = \\gamma _ { r }$ as in their case. When $\\gamma = 0$ , there is no time-dependent confounding and the treatments are randomly assigned. By increasing $\\gamma$ we increase the influence of the volume size history (encoded in $\\bar { D } ( t ) ,$ ) on the treatment probability. For example, assume $\\bar { D } ( t ) =$ $\\frac { 3 D _ { m a x } } { 4 }$ n (7), t, where obability of chemotherapy in thiis the sigmoid function. When i, $\\begin{array} { r } { p _ { c } ( t ) = \\sigma ( \\frac { \\gamma _ { c } } { D _ { m a x } } ( \\bar { D } ( t ) - } \\end{array}$ $\\textstyle \\frac { D _ { m a x } } { 2 } ) ) = \\sigma ( 0 . 2 5 \\gamma _ { c } )$ $\\sigma ( \\cdot )$ $\\gamma = 1$ $p _ { c } ( t ) = 0 . 5 6$ $\\gamma = 5$ $p _ { c } ( t ) = 0 . 7 7$ and when $\\gamma = 1 0$ , $\\dot { p } _ { c } ( t ) = 0 . 9 2$ in this example. $\\gamma$ can be increased further to increase the bias. However, the values used in the experiments evaluate the model on a wide range of settings for the time-dependent confounding bias. ", + "bbox": [ + 173, + 791, + 826, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "G MARGINAL STRUCTURAL MODELS ", + "text_level": 1, + "bbox": [ + 173, + 102, + 503, + 118 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Marginal Structural Models (Robins et al., 2000; Hernán et al., 2001) have been widely used in epidemiology and as part of follow up studies. In our case, we would like to estimate the effects of a sequence of treatments in the future given the current patient history: ", + "bbox": [ + 173, + 132, + 825, + 176 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/9444ac0857824582bc73ef18ac71a61e8192c86ef7507c2e839e8a675b4261cd.jpg", + "text": "$$\n\\mathbb { E } ( \\mathbf { Y } _ { t + \\tau } \\mid \\bar { \\mathbf { A } } ( t , t + \\tau - 1 ) = \\bar { \\mathbf { a } } ( t , t + \\tau - 1 ) , \\bar { \\mathbf { H } } _ { t } ) = g ( \\tau , a ( t , t + \\tau - 1 ) , \\bar { \\mathbf { H } } _ { t } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 240, + 179, + 754, + 198 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $g$ is a generic function and $\\bar { \\mathbf { a } } ( t , t + \\tau - 1 ) = [ \\mathbf { a } _ { t } , \\dots \\mathbf { a } _ { t + \\tau - 1 } ]$ represents a possible sequence of treatments from timestep $t$ just until before the potential outcome $\\mathbf { Y } _ { t + \\tau }$ is observed. After removing the bias form time-dependent confounders, $\\begin{array} { r } { \\mathbb { E } \\mathbf { \\hat { ( } } \\mathbf { Y } _ { t + \\tau } \\mid \\bar { \\mathbf { A } } ( t , t + \\tau - 1 ) = \\bar { \\mathbf { a } } ( t , t + \\tau - 1 ) , \\bar { \\mathbf { H } } _ { t } ) = } \\end{array}$ $\\mathbb { E } ( \\mathbf { Y } _ { t + \\tau } [ \\bar { \\mathbf { a } } ( t , t + \\tau - 1 ) ]$ . ", + "bbox": [ + 173, + 202, + 825, + 261 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Note that for implementing MSMs, we encode the treatments at timestep $t$ in the model of tumour growth as $\\mathbf { A } _ { t } = [ A _ { t , c } , A _ { t , d } ]$ to indicate the binary application of chemotherapy and radiotherapy. In order to remove the time-dependent confounding bias and estimate future outcomes, we use the stabilized weights of MSMs to weight each patient in the dataset: ", + "bbox": [ + 173, + 265, + 825, + 321 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/a5bb5d4f77ece4e8cc83cb4646a2ef2f441964da9acf66c5beaac9645edf1954.jpg", + "text": "$$\nS W ( t , \\tau ) = \\prod _ { n = t } ^ { t + \\tau } \\frac { f ( \\mathbf { A } _ { n } \\mid \\bar { \\mathbf { A } } _ { n - 1 } ) } { f ( \\mathbf { A } _ { n } \\mid \\bar { \\mathbf { A } } _ { n - 1 } , \\bar { \\mathbf { X } } _ { n } , \\mathbf { V } ) } = \\prod _ { n = t } ^ { t + \\tau } \\frac { \\prod _ { k \\in \\{ c , d \\} } f ( A _ { n , k } \\mid \\bar { \\mathbf { A } } _ { n - 1 } ) } { \\prod _ { k \\in \\{ c , d \\} } f ( A _ { n , k } \\mid \\bar { \\mathbf { A } } _ { n - 1 } , \\bar { \\mathbf { X } } _ { n } , \\mathbf { V } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 210, + 325, + 759, + 369 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $f ( \\cdot )$ represents the conditional probability mass function for discrete treatments. ", + "bbox": [ + 173, + 375, + 741, + 388 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We adopt the implementation in (Hernán et al., 2001; Howe et al., 2012; Lim et al., 2018) for MSMs and use logistic regression for estimating the propensity weights as follows: ", + "bbox": [ + 174, + 395, + 823, + 424 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/2c28f5e80be1e62c1cdbc04bb7df7ad9f5d86fcdc192f37ca20dce5bd42fa1d4.jpg", + "text": "$$\nf ( A _ { t , k } \\mid \\bar { \\mathbf { A } } _ { t - 1 } ) = \\sigma \\Big ( \\sum _ { j = 1 } ^ { k } \\omega _ { k } ( \\sum _ { i = 1 } ^ { t - 1 } A _ { t , j } ) \\Big )\n$$", + "text_format": "latex", + "bbox": [ + 362, + 429, + 633, + 474 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/856def0c59a05cc36c4b5801e07513800a9553b9082256112054c6628527284e.jpg", + "text": "$$\nf ( A _ { t , k } \\mid { \\bar { \\mathbf { H } } } _ { t } ) = \\sigma { \\Big ( } \\sum _ { k \\in \\{ c , d \\} } \\phi _ { k } ( \\sum _ { i = 1 } ^ { t - 1 } A _ { t , k } ) + \\mathbf { w } _ { 1 } \\mathbf { X } _ { t } + \\mathbf { w } _ { 2 } \\mathbf { X } _ { t - 1 } + \\mathbf { w } _ { 3 } \\mathbf { V } { \\Big ) }\n$$", + "text_format": "latex", + "bbox": [ + 263, + 478, + 735, + 523 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $\\omega _ { \\star } , \\phi _ { \\star }$ and $\\mathbf { w } _ { \\star }$ are regression coefficients, $k \\in \\{ c , d \\}$ indicates the chemotherapy or radiotherapy treatments and $\\sigma ( \\cdot )$ is the sigmoid function. ", + "bbox": [ + 173, + 526, + 825, + 555 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For predicting the outcome, the following regression model is used, where each individual patient is weighted by its propensity score: ", + "bbox": [ + 174, + 561, + 825, + 590 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/fddcb4ac03c21e50eb7c668628e43c469f08c0036e911f4a1f32d43aa1266dd2.jpg", + "text": "$$\ng ( \\tau , a ( t , t + \\tau - 1 ) , \\bar { \\mathbf { H } } _ { t } ) = \\sum _ { k \\in \\{ c , d \\} } \\beta _ { k } ( \\sum _ { n = t } ^ { t + \\tau - 1 } A _ { n , k } ) + 1 _ { 1 } \\mathbf { X } _ { t } + 1 _ { 2 } \\mathbf { X } _ { t - 1 } + 1 _ { 3 } \\mathbf { V }\n$$", + "text_format": "latex", + "bbox": [ + 245, + 594, + 753, + 641 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "where $\\beta _ { \\star }$ and ${ \\bf l } _ { \\star }$ are regression coefficients. ", + "bbox": [ + 176, + 645, + 457, + 660 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "MSMs do not require hyperparameter tuning so we use the patients from both the train and validation sets for training. ", + "bbox": [ + 173, + 666, + 825, + 695 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "H RECURRENT MARGINAL STRUCTURAL NETWORKS", + "text_level": 1, + "bbox": [ + 174, + 714, + 635, + 732 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "MSMs are very sensitive to model mis-specification in computing the propensity weights and estimating the outcomes. Recurrent Marginal Structural Models (RMSNs) (Lim et al., 2018) overcome this problem by using recurrent neural networks to estimate the propensity scores and to build the outcome model. RNNs are more robust to changes in the treatment assignment policy. RMSNs were implemented as descried in Lim et al. $( 2 0 1 8 ) ^ { 2 }$ . ", + "bbox": [ + 173, + 746, + 825, + 816 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For implementing RMSNs, we also encode the treatments at timestep $t$ in the model of tumour growth as $\\mathbf { A } _ { t } = [ A _ { t , c } , A _ { t , d } ]$ to indicate the binary application of chemotherapy and radiotherapy. The propensity weights are estimated using recurrent neural networks as follows: ", + "bbox": [ + 173, + 823, + 826, + 866 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/dbe8a53299b880de732f2f912d47b082a5925b8f40af189f767c2cd073366c9c.jpg", + "text": "$$\n\\begin{array} { r l } { f ( A _ { t , k } \\ \\vert \\ \\bar { \\mathbf { A } } _ { t - 1 } ) = \\mathrm { R N N } _ { S W _ { n } } ( \\bar { \\mathbf { A } } _ { t - 1 } ) } & { { } ~ f ( A _ { t , k } \\ \\vert \\ \\bar { \\mathbf { X } } _ { t } , \\bar { \\mathbf { A } } _ { t - 1 } ) = \\mathrm { R N N } _ { S W _ { d } } ( \\bar { \\mathbf { A } } _ { t - 1 } , \\bar { \\mathbf { X } } _ { t } , { \\mathbf { V } } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 869, + 774, + 888 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For predicting one-step-ahed outcome, R-MSNs use an encoder network: ", + "bbox": [ + 173, + 103, + 655, + 118 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/511e15653c3075d0e2841707cd4c67da4ac11c434a6f2dfbe5c9ef04f55dcb32.jpg", + "text": "$$\ng ( 1 , a ( t , t ) , \\bar { { \\mathbf { H } } } _ { t } ) = { \\mathbf { R } } { \\mathbf { N } } { \\mathbf { N } } _ { E } ( { \\mathbf { a } } _ { t } , \\bar { { \\mathbf { A } } } _ { t - 1 } , \\bar { { \\mathbf { X } } } _ { t } , { \\mathbf { V } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 333, + 125, + 632, + 142 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where in the loss function, each patient is weighted by their stabilized IPTW. ", + "bbox": [ + 173, + 148, + 678, + 164 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For estimating the treatment responses for a sequence of treatments in the future, RMSNs use a decoder network: ", + "bbox": [ + 173, + 170, + 825, + 198 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/68c68cc97c61597a66d6be1e97b0834a2864ee0ac9e05a5885b1e535a1b5a34f.jpg", + "text": "$$\ng ( \\tau , a ( t , t + \\tau - 1 ) , \\bar { \\mathbf { H } } _ { t } ) = \\mathrm { R N N } _ { D } ( \\mathbf { a } _ { t } , \\ldots , \\mathbf { a } _ { t + \\tau - 1 } , \\bar { \\mathbf { A } } _ { t - 1 } , \\bar { \\mathbf { X } } _ { t } , \\mathbf { V } ) .\n$$", + "text_format": "latex", + "bbox": [ + 261, + 204, + 704, + 223 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "See Lim et al. (2018) for more details about the R-MSNs model architecture and training procedure of the propensity weights, encoder and decoder networks. Tables 2 and 3 show the hyperparameter search ranges used to optimize this model for evaluation in our paper. The hyperparameters were selected in the same way as proposed by Lim et al. (2018), based on the error on the factual outcomes in the validation dataset. All of the models are trained using Adam optimizer for 100 epochs. ", + "bbox": [ + 173, + 228, + 825, + 300 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/8f44ae79d1c64e6aed47c9612de89063cbccff1d590c72e48b0dabe5583559a0.jpg", + "table_caption": [ + "Table 2: Hyperparameter search range for propensity networks and encoder (same as in Lim et al. (2018)). C is the size of the input. " + ], + "table_footnote": [], + "table_body": "
HyperparameterSearch range
Iterations of Hyperparameter Search50
Learning rate0.01,0.005,0.001
Minibatch size64,128,256
RNN state size0.5C,1C,2C, 3C,4C
Dropout rate0.1, 0.2,0.3,0.4, 0.5
Max Gradient Norm0.5, 1.0, 2.0
", + "bbox": [ + 308, + 366, + 691, + 468 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/a3810b4945de140730cedd3c45d707685692fe99033c7fe0ef5176094d4e5d6a.jpg", + "table_caption": [ + "Table 3: Hyperparameter search range for decoder (same as in Lim et al. (2018)). C is the input size. " + ], + "table_footnote": [], + "table_body": "
HyperparameterSearch range
Iterations of Hyperparameter Search20
Learning rate0.01,0.001,0.0001
Minibatch size256,512,1024
RNN state size1C,2C,4C,8C,16C
Dropout Rate0.1, 0.2, 0.3, 0.4, 0.5
Max Gradient Norm0.5, 1.0, 2.0, 4.0
", + "bbox": [ + 310, + 525, + 687, + 627 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "I BASELINE RNN AND LINEAR MODEL ", + "text_level": 1, + "bbox": [ + 173, + 651, + 511, + 667 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "For the baseline linear model, we fit the same regression model used for Marginal Structural Networks, but without using the IPTW. The baseline RNN uses an LSTM unit and, at each timestep, receives as input the current treatment, the patient covariates and the patient static features to perform one-stepahead prediction. To have a model of similar capacity to the CRN (similar number of parameters), we add a fully connected layer on top of the output of the LSTM unit in order to obtain the outcomes. Table 4 shows the hyperparameter search range used to optimize this model. The hyperparameters were selecting according to the error on the factual outcomes in the validation set. We train the baseline RNN using the Adam optimizer for 100 epochs. ", + "bbox": [ + 173, + 683, + 826, + 795 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/fc6cd61931eae4ab97ee384c8e288cf4b018a39a80938a279147f70ca04a66f6.jpg", + "table_caption": [ + "Table 4: Hyperparameter search range for baseline RNN model. C is the size of the input. " + ], + "table_footnote": [], + "table_body": "
HyperparameterSearch range
Iterations of Hyperparameter Search50
Learning rate0.01,0.001,0.0001
Minibatch size64,128,256
RNNhidden units0.5C,1C,2C, 3C,4C
FC hidden units0.5C, 1C, 2C, 3C, 4C
RNN dropout probability0.1, 0.2, 0.3, 0.4, 0.5
", + "bbox": [ + 307, + 140, + 691, + 243 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "J HYPERPARAMETER OPTIMIZATION FOR CRN ", + "text_level": 1, + "bbox": [ + 173, + 265, + 578, + 280 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "As described in Appendix C, the dataset for training the decoder are used by splitting the sequences of the patients in the training set. This creates a larger dataset for training (where each patient $( i )$ contributes $T ^ { ( i ) } - \\tau _ { \\operatorname* { m a x } }$ times to the dataset) which requires a different hyperparameter search range. Moreover, the balancing representations computed by the encoder are used to initialize the state of the RNN for the decoder. Thus, the decoder RNN size is equal to the size of the balancing representation size of the encoder. Table 5 shows the hyperparameter search ranges for the encoder and decoder networks in CRN. We selected hyperparameters based on the error of the model on the factual outcomes in the validation dataset. All models are trained for 100 epochs. ", + "bbox": [ + 173, + 295, + 825, + 407 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "In addition, Tables 6 and 7 illustrate the optimal hyperparameters chosen. ", + "bbox": [ + 173, + 415, + 653, + 429 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/054c905b8d5128439b29b2a2f1a8c4646f038ad9ffa20a7fb83fe9fc07ca060e.jpg", + "table_caption": [ + "Table 5: Hyperparameter search range for CRN encoder. C is the size of the input and R is the size of the balancing representation. " + ], + "table_footnote": [], + "table_body": "
HyperparameterSearch range encoderSearch range decoder
Iterations ofHyperparameter Search5030
Learning rate0.01,0.001, 0.00010.01,0.001, 0.0001
Minibatch size64,128,256256,512,1024
RNN hidden units0.5C,1C, 2C, 3C, 4CBalancing representation size of encoder
Balancing representation size0.5C,1C,2C,3C,4C0.5C,1C,2C,3C,4C
FC hidden units0.5R,1R,2R,3R,4R0.5R,1R,2R,3R,4R
RNN dropout probability0.1, 0.2, 0.3,0.4, 0.50.1,0.2, 0.3, 0.4, 0.5
", + "bbox": [ + 179, + 496, + 818, + 611 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/be26ad2392900c92c00d97f50a7b7a974456bbfd5ea5c87a8d4289102a64946d.jpg", + "table_caption": [ + "Table 6: Optimal hyperparameters for the CRN encoder when different degrees of time-dependent confounding are applied in the model of tumour growth. The parameters $\\gamma _ { c }$ and $\\gamma _ { r }$ measure the degree of time-dependent confounding applied. When $\\gamma _ { c }$ and $\\gamma _ { r }$ are set to the same value, we denote this with $\\gamma _ { \\star }$ . " + ], + "table_footnote": [], + "table_body": "
2=02=12=22=3Y=42=5
Learning rate0.0010.10.0010.010.010.001
Minibatch size64646412864128
RNN hidden units121824182424
Balancing representation size18181218612
FC hidden units181836542448
RNN dropout probability0.10.10.10.20.20.1
=62=7Y=82=9Y=10
Learning rate0.0010.0010.010.0010.01
Minibatch size6464128128128
RNN hidden units2418122424
Balancing representation size1218241812
FC hidden units4872123612
RNN dropout probability0.10.20.10.10.1
Yc=0,γr=5 Yc=5,γr=0
Learning rate0.010.001
Minibatch size12864
RNN hidden units1212
Balancing representation size1824
FC hidden units3696
RNN dropout probability0.10.1
", + "bbox": [ + 176, + 220, + 823, + 525 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/687b2d7bcb6a2bbd6521ae0156444e723158ae510c99bb50c69ba95277e250b5.jpg", + "table_caption": [ + "Table 7: Optimal hyperparameters for the CRN decoder when different degrees of time-dependent confounding are applied in the model of tumour growth. The parameters $\\gamma _ { c }$ and $\\gamma _ { r }$ measure the degree of time-dependent confounding applied. When $\\gamma _ { c }$ and $\\gamma _ { r }$ are set to the same value, we denote this with $\\gamma _ { \\star }$ " + ], + "table_footnote": [], + "table_body": "
y=12=2=32=4=5
Learning rate0.0010.0010.0010.0010.001
Minibatch size1024102451210241024
RNN hidden units181218612
Balancing representation size18186183
FC hidden units183618726
RNN dropout probability0.10.20.30.10.1
Yc =0,γr=5 Yc=5,γr=0
Learning rate0.010.001
Minibatch size5121024
RNN hidden units1824
Balancing representation size1812
FC hidden units3624
RNN dropout probability0.10.03
", + "bbox": [ + 196, + 683, + 802, + 886 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "K FULL RESULTS FOR COUNTERFACTUAL PREDICTION ", + "text_level": 1, + "bbox": [ + 174, + 103, + 642, + 118 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "K.1 MULTI-STEP AHEAD PREDICTION OF COUNTERFACTUALS", + "text_level": 1, + "bbox": [ + 174, + 133, + 614, + 148 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Figure 7 shows the normalized RMSE for multiple step-ahead prediction of counterfactuals. The RMSE is normalized by the maximum tumour volume: $\\bar { V _ { m a x } } = 1 \\bar { 1 } 5 0 \\mathrm { c m ^ { 3 } }$ . The counterfactuals in this case are generated as described in Section 6.3 and Appendix I. We notice that performance gains of CRN compared to RMSN increase with the number of future timesteps for which the counterfactuals are estimated. ", + "bbox": [ + 173, + 159, + 826, + 229 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/995945365917721cfe59800f6c40ff0276b71b09419710e573d0a6203278d1ac.jpg", + "image_caption": [ + "Figure 7: Results for prediction of patient counterfactuals for multiple steps ahead. " + ], + "image_footnote": [], + "bbox": [ + 174, + 272, + 802, + 696 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "K.2 DETAILED RESULTS FOR THE COUNTERFACTUAL PREDICTIONS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 650, + 117 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Tables 8 and 9 show detailed results for the counterfactual predictions. ", + "bbox": [ + 176, + 128, + 635, + 143 + ], + "page_idx": 25 + }, + { + "type": "table", + "img_path": "images/5e4b6ce9dd70503d985ef7e7dd194266135bcc5bd2d8a2e33f29980dcb1269f5.jpg", + "table_caption": [ + "Table 8: Normalized RMSE for one-step-ahead prediction of counterfactuals. The parameter $\\gamma$ measures the degree of time-dependent confounding applied. " + ], + "table_footnote": [], + "table_body": "
γ=0γ=1γ=2γ=3γ=4γ=5
Linear (no IPTW)0.99%1.08%1.36%1.68%2.11%2.77%
MSM0.99%1.08%1.34%1.63%2.02%2.61%
RNN CRN(入=0)0.70%0.70%0.84%1.05%1.24%1.69%
RMSN0.66% 0.60%0.77% 0.61%0.92% 0.72%0.95% 0.81%1.24% 0.94%1.54% 1.23%
CRN0.56%0.57%0.62%0.67%0.87%1.20%
γ=6Y=7Y=8Y=9γ=10
Linear (no IPTW)3.55%4.15%4.80%5.09%5.22%
MSM3.30%3.79%4.30%4.47%4.47%
RNN2.03%2.52%2.88%3.79%4.01%
CRN (λ= 0)1.98%2.42%2.73%3.17%
RMSN3.57%
1.70%2.18%2.37%2.77%2.83%
CRN1.48%1.56%2.05%2.36%2.41%
", + "bbox": [ + 250, + 210, + 746, + 419 + ], + "page_idx": 25 + }, + { + "type": "table", + "img_path": "images/9023ec73868e46c7abeeffa770036089d4b550bbe2fc469bed1429cbd594ab2f.jpg", + "table_caption": [ + "Table 9: Normalized RMSE for $\\tau$ -step-ahead prediction of counterfactuals. The parameter $\\gamma$ measures the degree of time-dependent confounding applied. " + ], + "table_footnote": [], + "table_body": "
γ=1γ=2γ=3γ=4γ=5
T=2RMSN0.90%1.15%1.53%2.14%2.91%
CRN0.84%0.96%1.21%1.46%2.45%
T=3RMSN0.97%1.36%1.87%2.44%3.47%
CRN0.86%0.96%1.47 %1.51%2.84%
T=4RMSN1.24%1.79%2.60%3.33%3.88%
CRN0.91%1.08%1.74%1.76%2.82%
T=5RMSN1.51%2.13%3.06%4.07%4.58%
CRN0.85%1.10%1.73%2.00%3.43%
T=6RMSN2.10%2.89%3.06%4.16%6.32%
CRN1.16%1.52%2.29%2.66%4.91%
", + "bbox": [ + 284, + 489, + 712, + 667 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "In order to evaluate how well the models select the correct treatment and timing of treatment we simulate counterfactual outcomes as follows. We generate 1000 test samples using the model of tumour growth described in Section 6. Let $\\bar { \\mathbf { H } } _ { t }$ be the current history of the patient and let $\\tau$ be a future time horizon. For each timestep in the future, we have 4 treatment options at: no treatment $( A _ { 0 } )$ , chemotherapy $( A _ { 1 } )$ , radiotherapy $\\left( A _ { 2 } \\right)$ , chemotherapy and radiotherapy. $\\left( A _ { 3 } \\right)$ . ", + "bbox": [ + 173, + 132, + 825, + 204 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Using the model of tumour growth where the outcome $\\mathbf { Y } _ { t + \\tau }$ is given by the volume of the tumour, we generate the following $2 \\tau$ counterfactuals: ", + "bbox": [ + 173, + 210, + 825, + 238 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Chemotherapy application ", + "bbox": [ + 253, + 247, + 426, + 262 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/c1af7f88e3d092438d75d3c04a0d91e1122e9b724b25de734172def022302208.jpg", + "text": "$$\n\\begin{array} { r l } { { \\bf Y } _ { t + \\tau } } & { { } | \\mathrm { \\bf ~ a } _ { t } = A _ { 1 } , \\mathbf { a } _ { t + 1 } = A _ { 0 } , \\ldots \\mathbf { a } _ { t + \\tau - 1 } = A _ { 0 } , \\bar { \\mathbf { H } } _ { t } } \\\\ { { \\bf Y } _ { t + \\tau } } & { { } | \\mathrm { \\bf ~ a } _ { t } = A _ { 0 } , \\mathbf { a } _ { t + 1 } = A _ { 1 } , \\ldots \\mathbf { a } _ { t + \\tau - 1 } = A _ { 0 } , \\bar { \\mathbf { H } } _ { t } } \\\\ { \\mathrm { ~ \\cdot ~ } \\cdot \\cdot } \\\\ { { \\bf Y } _ { t + \\tau } } & { { } | \\mathrm { \\bf \\textup { \\bf ~ a } } _ { t } = A _ { 0 } , \\mathbf { a } _ { t + 1 } = A _ { 0 } , \\ldots \\mathbf { a } _ { t + \\tau - 1 } = A _ { 1 } , \\bar { \\mathbf { H } } _ { t } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 383, + 261, + 748, + 338 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Radiotherapy application ", + "bbox": [ + 259, + 335, + 428, + 349 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/c5556e18245143eb2e8682cc985a6453ab423123221e0413e2a9dc3ba68f645b.jpg", + "text": "$$\n\\begin{array} { r l } { { \\bf Y } _ { t + \\tau } } & { { } | \\quad { \\bf a } _ { t } = A _ { 2 } , { \\bf a } _ { t + 1 } = A _ { 0 } , \\ldots { \\bf a } _ { t + \\tau - 1 } = A _ { 0 } , \\bar { \\bf H } _ { t } } \\\\ { { \\bf Y } _ { t + \\tau } } & { { } | \\quad { \\bf a } _ { t } = A _ { 0 } , { \\bf a } _ { t + 1 } = A _ { 2 } , \\ldots { \\bf a } _ { t + \\tau - 1 } = A _ { 0 } , \\bar { \\bf H } _ { t } } \\\\ { \\ldots } & { { } } \\\\ { { \\bf Y } _ { t + \\tau } } & { { } | \\quad { \\bf a } _ { t } = A _ { 0 } , { \\bf a } _ { t + 1 } = A _ { 0 } , \\ldots { \\bf a } _ { t + \\tau - 1 } = A _ { 2 } , \\bar { \\bf H } _ { t } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 382, + 349, + 748, + 424 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "We perform this for each patient in the test set and at each time $t$ in the history. For instance, for a patient with 50 timesteps in the model of tumour growth and for time horizon $\\tau = 3$ , we generate $2 \\cdot 3 \\cdot 5 0 = 3 0 0$ counterfactuals. ", + "bbox": [ + 176, + 428, + 821, + 469 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Using the true generated couterfactual data, we select the treatment that has the lowest $\\mathbf { Y } _ { t + \\tau }$ among the $\\tau$ options generated for each treatment. Then, we select the time of applying treatment (among $t , t + 1 , \\ldots t + \\tau - 1 )$ that resulted in the lowest $\\mathbf { Y } _ { t + \\tau }$ . For each model, we generate the counterfactuals under the same treatment plans and patient histories. Then, we perform the selection of treatment and timing of treatment in the same way and we compare these with the true ones. Note that in order to account for numerical instability (two outcomes $Y _ { t + \\tau }$ having very similar values), we consider two outcomes the same if they are within $\\epsilon = 0 . 0 0 1$ of each other. ", + "bbox": [ + 173, + 477, + 825, + 575 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "M RESULTS ON FACTUAL PREDICTION ON MIMIC III ", + "text_level": 1, + "bbox": [ + 174, + 102, + 635, + 118 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Using the Medical Information Mart for Intensive Care (MIMIC III) (Johnson et al., 2016) database consisting of electronic health records from patients in the ICU, we also show how the CRN can be used on a real medical dataset. From MIMIC III we extracted the patients on antibiotics, with trajectories up to 30 timesteps, thus obtaining a dataset with 3487 patients. For each patient, we extracted 25 patient covariates including lab tests and vital signs measured over time, as well as static patient features such as age and gender. ", + "bbox": [ + 174, + 132, + 825, + 217 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "We used a binary treatment at each timestep indicating whether the patient was administered antibiotics or not. Note that for the longitudinal covariates we used aggregate value for each day since the ICU admission. The reason for this is because antibiotic treatment is decided daily for the patient. We split the dataset into 2826/313/348 patients for training, validation and testing respectively. We performed hyperparameter optimization on the validation patient set, using the search ranges in Table 5 and we again selected hyperparameters based on the error on the factual outcomes. ", + "bbox": [ + 173, + 223, + 825, + 308 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "We estimate the individualized effect of antibiotics assigned over time on the patient’s white blood cell count. A high white blood cell count is associated with severe illness and poor outcome for ICU patients (Waheed et al., 2003). Antibiotic administration in the ICU aims to reduce the white blood cell count. However, the effectiveness of the antibiotics treatment in reducing the white blood cell count is highly dependent on the time they are administered with respect to the history of the patient covariates. In this context we again have time-dependent confounders: the patient features change over time and are affected by the previous administration of antibiotics. Moreover, the history of the patient features also determines antibiotics administration and affects future patient outcomes (De Bus et al., 2018; Ali et al., 2019). ", + "bbox": [ + 174, + 314, + 825, + 440 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "In Table 10 we report the root mean squared error for factual prediction of the patients’ white blood cell count for multiple prediction horizons $\\tau$ . Note that for this dataset we do not have access to counterfactual data, which is why we report error on factual predictions. ", + "bbox": [ + 174, + 446, + 825, + 489 + ], + "page_idx": 27 + }, + { + "type": "table", + "img_path": "images/d48466a4154c491a71a2f92d63c525ecb9af8c07d5657d4890cde36604e7d756.jpg", + "table_caption": [ + "Table 10: RMSE for $\\tau$ -step-ahead prediction of factual outcomes on MIMIC III. " + ], + "table_footnote": [], + "table_body": "
T=1T=2T=3 T=4
RMSN2.843.874.464.79
CRN2.683.544.074.67
", + "bbox": [ + 348, + 534, + 650, + 587 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "We notice that CRN also achieves better performance than RMSN in estimating factual outcomes in a real-world dataset containing electronic health records. In this context, where couterfactual data is not available, domain expert knowledge is required to validate the model’s counterfactual predictions under other antibiotic treatment alternatives. This further medical validation is outside the scope of this paper. 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To handle the bias from time-varying confounders, covariates affect-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 401, + 470, + 413 + ], + "spans": [ + { + "bbox": [ + 141, + 401, + 470, + 413 + ], + "score": 1.0, + "content": "ing the treatment assignment policy in the observational data, CRN uses domain", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 412, + 470, + 425 + ], + "spans": [ + { + "bbox": [ + 141, + 412, + 470, + 425 + ], + "score": 1.0, + "content": "adversarial training to build balancing representations of the patient history. At", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 423, + 470, + 436 + ], + "spans": [ + { + "bbox": [ + 141, + 423, + 470, + 436 + ], + "score": 1.0, + "content": "each timestep, CRN constructs a treatment invariant representation which removes", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 434, + 469, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 434, + 469, + 446 + ], + "score": 1.0, + "content": "the association between patient history and treatment assignments and thus can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 445, + 470, + 456 + ], + "spans": [ + { + "bbox": [ + 141, + 445, + 470, + 456 + ], + "score": 1.0, + "content": "be reliably used for making counterfactual predictions. On a simulated model of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 456, + 469, + 468 + ], + "spans": [ + { + "bbox": [ + 141, + 456, + 469, + 468 + ], + "score": 1.0, + "content": "tumour growth, with varying degree of time-dependent confounding, we show how", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 467, + 469, + 479 + ], + "spans": [ + { + "bbox": [ + 141, + 467, + 469, + 479 + ], + "score": 1.0, + "content": "our model achieves lower error in estimating counterfactuals and in choosing the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 478, + 459, + 489 + ], + "spans": [ + { + "bbox": [ + 141, + 478, + 459, + 489 + ], + "score": 1.0, + "content": "correct treatment and timing of treatment than current state-of-the-art methods.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 509, + 206, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 208, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 208, + 524 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 545 + ], + "score": 1.0, + "content": "As clinical decision-makers are often faced with the problem of choosing between treatment alterna-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "tives for patients, reliably estimating their effects is paramount. While clinical trials represent the gold", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "standard for causal inference, they are expensive, have a few patients and narrow inclusion criteria", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 566, + 507, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 507, + 579 + ], + "score": 1.0, + "content": "(Booth & Tannock, 2014). 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+ "content": "treatments over time are important medical problems with a few existing solutions.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 357, + 470, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 470, + 369 + ], + "score": 1.0, + "content": "In this paper, we introduce the Counterfactual Recurrent Network (CRN), a novel", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 369, + 471, + 380 + ], + "spans": [ + { + "bbox": [ + 141, + 369, + 471, + 380 + ], + "score": 1.0, + "content": "sequence-to-sequence model that leverages the increasingly available patient ob-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 380, + 469, + 390 + ], + "spans": [ + { + "bbox": [ + 141, + 380, + 469, + 390 + ], + "score": 1.0, + "content": "servational data to estimate treatment effects over time and answer such medical", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 390, + 471, + 402 + ], + 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To handle the bias from time-varying confounders, covariates affect-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 401, + 470, + 413 + ], + "spans": [ + { + "bbox": [ + 141, + 401, + 470, + 413 + ], + "score": 1.0, + "content": "ing the treatment assignment policy in the observational data, CRN uses domain", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 412, + 470, + 425 + ], + "spans": [ + { + "bbox": [ + 141, + 412, + 470, + 425 + ], + "score": 1.0, + "content": "adversarial training to build balancing representations of the patient history. At", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 423, + 470, + 436 + ], + "spans": [ + { + "bbox": [ + 141, + 423, + 470, + 436 + ], + "score": 1.0, + "content": "each timestep, CRN constructs a treatment invariant representation which removes", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 434, + 469, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 434, + 469, + 446 + ], + "score": 1.0, + "content": "the association between patient history and treatment assignments and thus can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 445, + 470, + 456 + ], + "spans": [ + { + "bbox": [ + 141, + 445, + 470, + 456 + ], + "score": 1.0, + "content": "be reliably used for making counterfactual predictions. On a simulated model of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 456, + 469, + 468 + ], + "spans": [ + { + "bbox": [ + 141, + 456, + 469, + 468 + ], + "score": 1.0, + "content": "tumour growth, with varying degree of time-dependent confounding, we show how", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 467, + 469, + 479 + ], + "spans": [ + { + "bbox": [ + 141, + 467, + 469, + 479 + ], + "score": 1.0, + "content": "our model achieves lower error in estimating counterfactuals and in choosing the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 478, + 459, + 489 + ], + "spans": [ + { + "bbox": [ + 141, + 478, + 459, + 489 + ], + "score": 1.0, + "content": "correct treatment and timing of treatment than current state-of-the-art methods.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 28.5, + "bbox_fs": [ + 141, + 335, + 471, + 489 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 509, + 206, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 208, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 208, + 524 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 545 + ], + "score": 1.0, + "content": "As clinical decision-makers are often faced with the problem of choosing between treatment alterna-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "tives for patients, reliably estimating their effects is paramount. While clinical trials represent the gold", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "standard for causal inference, they are expensive, have a few patients and narrow inclusion criteria", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 566, + 507, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 507, + 579 + ], + "score": 1.0, + "content": "(Booth & Tannock, 2014). 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For", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "score": 1.0, + "content": "instance, consider that treatment A is given when a certain patient covariate (e.g. white blood cell", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "count) has been outside of normal range values for several consecutive timesteps. Suppose also that", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "this patient covariate was itself affected by the past administration of treatment B. If these patients", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "are more likely to die, without adjusting for the time-dependent confounding (e.g. the changes in", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "the white blood cell count over time), we will incorrectly conclude that treatment A is harmful to", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "patients. 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We illustrate 3 patients with different", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 228, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 235, + 242 + ], + "score": 1.0, + "content": "covariate and treatment histories", + "type": "text" + }, + { + "bbox": [ + 235, + 228, + 249, + 240 + ], + "score": 0.89, + "content": "\\bar { \\mathbf { H } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 229, + 325, + 242 + ], + "score": 1.0, + "content": ". For a current time", + "type": "text" + }, + { + "bbox": [ + 325, + 231, + 330, + 239 + ], + "score": 0.63, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 229, + 506, + 242 + ], + "score": 1.0, + "content": ", CRN can predict counterfactual trajectories", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "(the coloured dashed branches) for planned treatments in the future. Through the counterfactual", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "predictions, we can decide which treatment plan results in the best patient outcome (in this case, the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 263, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 274 + ], + "score": 1.0, + "content": "lowest tumour volume). This way, CRN can be used to perform all of the following: choose optimal", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 274, + 497, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 497, + 285 + ], + "score": 1.0, + "content": "treatments (a), find timing when treatment is most effective (b) decide when to stop treatment (c).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 505, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "count) has been outside of normal range values for several consecutive timesteps. Suppose also that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "this patient covariate was itself affected by the past administration of treatment B. If these patients", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "are more likely to die, without adjusting for the time-dependent confounding (e.g. the changes in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "the white blood cell count over time), we will incorrectly conclude that treatment A is harmful to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "patients. Moreover, estimating the effect of a different sequence of treatments on the patient outcome", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "would require not only adjusting for the bias at the current step (in treatment A), but also for the bias", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 363, + 324, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 324, + 374 + ], + "score": 1.0, + "content": "introduced by the previous application of treatment B.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "score": 1.0, + "content": "Existing methods for causal inference in the static setting cannot be applied in this longitudinal setting", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "since they are designed to handle the cross-sectional set-up, where the treatment and outcome depend", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "only on a static value of the patient covariates. 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Different", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "models that can handle these temporal dependencies in the observational data and varying-length", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 456, + 387, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 387, + 468 + ], + "score": 1.0, + "content": "patient histories are needed for estimating treatment effects over time.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 472, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "Time-dependent confounders are present in observational data because doctors follow policies: the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "history of the patients’ covariates and the patients’ response to past treatments are used to decide", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 506 + ], + "score": 1.0, + "content": "future treatments (Mansournia et al., 2012). The direct use of supervised learning methods will be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "biased by the treatment policies present in the observational data and will not be able to correctly", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 517, + 379, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 379, + 529 + ], + "score": 1.0, + "content": "estimate counterfactuals for different treatment assignment policies.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 507, + 546 + ], + "score": 1.0, + "content": "Standard methods for adjusting for time-varying confounding and estimating the effects of time-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "varying exposures are based on ideas from epidemiology. The most widely used among these are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "Marginal Structural Models (MSMs) (Robins et al., 2000; Mansournia et al., 2012) which use the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "score": 1.0, + "content": "inverse probability of treatment weighting (IPTW) to adjust for the time-dependent confounding", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 576, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 591 + ], + "score": 1.0, + "content": "bias. Through IPTW, MSMs create a pseudo-population where the probability of treatment does not", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "depend on the time-varying confounders. However, MSMs are not robust to model misspecification", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "in computing the IPTWs. MSMs can also give high-variance estimates due to extreme weights;", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "computing the IPTW involves dividing by probability of assigning a treatment conditional on patient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 621, + 383, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 383, + 634 + ], + "score": 1.0, + "content": "history which can be numerically unstable if the probability is small.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "We introduce the Counterfactual Recurrent Network (CRN), a novel sequence-to-sequence archi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "tecture for estimating treatment effects over time. CRN leverages recent advances in representation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "learning (Bengio et al., 2012) and domain adversarial training (Ganin et al., 2016) to overcome the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 671, + 507, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 507, + 683 + ], + "score": 1.0, + "content": "problems of existing methods for causal inference over time. Our main contributions are as follows.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Treatment invariant representations over time. CRN constructs treatment invariant represen-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "tations at each timestep in order to break the association between patient history and treatment", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "assignment and thus removes the bias from time-dependent confounders. 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We illustrate 3 patients with different", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 228, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 235, + 242 + ], + "score": 1.0, + "content": "covariate and treatment histories", + "type": "text" + }, + { + "bbox": [ + 235, + 228, + 249, + 240 + ], + "score": 0.89, + "content": "\\bar { \\mathbf { H } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 229, + 325, + 242 + ], + "score": 1.0, + "content": ". For a current time", + "type": "text" + }, + { + "bbox": [ + 325, + 231, + 330, + 239 + ], + "score": 0.63, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 229, + 506, + 242 + ], + "score": 1.0, + "content": ", CRN can predict counterfactual trajectories", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "(the coloured dashed branches) for planned treatments in the future. Through the counterfactual", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "predictions, we can decide which treatment plan results in the best patient outcome (in this case, the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 263, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 274 + ], + "score": 1.0, + "content": "lowest tumour volume). This way, CRN can be used to perform all of the following: choose optimal", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 274, + 497, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 497, + 285 + ], + "score": 1.0, + "content": "treatments (a), find timing when treatment is most effective (b) decide when to stop treatment (c).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 505, + 373 + ], + "lines": [], + "index": 12, + "bbox_fs": [ + 105, + 296, + 506, + 374 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "score": 1.0, + "content": "Existing methods for causal inference in the static setting cannot be applied in this longitudinal setting", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "since they are designed to handle the cross-sectional set-up, where the treatment and outcome depend", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "only on a static value of the patient covariates. 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The most widely used among these are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "Marginal Structural Models (MSMs) (Robins et al., 2000; Mansournia et al., 2012) which use the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "score": 1.0, + "content": "inverse probability of treatment weighting (IPTW) to adjust for the time-dependent confounding", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 576, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 506, + 591 + ], + "score": 1.0, + "content": "bias. 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MSMs can also give high-variance estimates due to extreme weights;", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "computing the IPTW involves dividing by probability of assigning a treatment conditional on patient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 621, + 383, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 383, + 634 + ], + "score": 1.0, + "content": "history which can be numerically unstable if the probability is small.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 533, + 507, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "We introduce the Counterfactual Recurrent Network (CRN), a novel sequence-to-sequence archi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "tecture for estimating treatment effects over time. CRN leverages recent advances in representation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "learning (Bengio et al., 2012) and domain adversarial training (Ganin et al., 2016) to overcome the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 671, + 507, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 507, + 683 + ], + "score": 1.0, + "content": "problems of existing methods for causal inference over time. Our main contributions are as follows.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 638, + 507, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "Treatment invariant representations over time. CRN constructs treatment invariant represen-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "tations at each timestep in order to break the association between patient history and treatment", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 721 + ], + "score": 1.0, + "content": "assignment and thus removes the bias from time-dependent confounders. For this, CRN uses domain", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "adversarial training (Ganin et al., 2016; Li et al., 2018; Sebag et al., 2019) to trade-off between build-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "ing this balancing representation and predicting patient outcomes. We show that these representations", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "remove the bias from time-varying confounders and can be reliably used for estimating counterfactual", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "outcomes. This represents the first work that introduces ideas from domain adaptation to the area of", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "estimating treatment effects over time. In addition, by building balancing representations, we propose", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 406, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 406, + 139 + ], + "score": 1.0, + "content": "a novel way of removing the bias introduced by time-varying confounders.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "ing this balancing representation and predicting patient outcomes. We show that these representations", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "remove the bias from time-varying confounders and can be reliably used for estimating counterfactual", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "outcomes. This represents the first work that introduces ideas from domain adaptation to the area of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 129 + ], + "score": 1.0, + "content": "estimating treatment effects over time. In addition, by building balancing representations, we propose", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 406, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 406, + 139 + ], + "score": 1.0, + "content": "a novel way of removing the bias introduced by time-varying confounders.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 143, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 505, + 155 + ], + "score": 1.0, + "content": "Counterfactual estimation of future outcomes. To estimate counterfactual outcomes for treatment", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "plans (and not just single treatments), we integrate the domain adversarial training procedure as part", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "of a sequence-to-sequence architecture. CRN consists of an encoder network which builds treatment", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "invariant representations of the patient history that are used to initialize the decoder. The decoder", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "network estimates outcomes under an intended sequence of future treatments, while also updating the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "balanced representation. By performing counterfactual estimation of future treatment outcomes, CRN", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 507, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 507, + 222 + ], + "score": 1.0, + "content": "can be used to answer critical medical questions such as deciding when to give treatments to patients,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 506, + 232 + ], + "score": 1.0, + "content": "when to start and stop treatment regimes, and also how to select from multiple treatments over time.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 488, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 488, + 244 + ], + "score": 1.0, + "content": "We illustrate in Figure 1 the applicability of our method in choosing optimal cancer treatments.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 108, + 247, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 246, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 506, + 262 + ], + "score": 1.0, + "content": "In our experiments, we evaluate CRN in a realistic set-up using a model of tumour growth (Geng", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 507, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 507, + 272 + ], + "score": 1.0, + "content": "et al., 2017). We show that CRN achieves better performance in predicting counterfactual outcomes,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "but also in choosing the right treatment and timing of treatment than current state-of-the-art methods.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 298, + 209, + 311 + ], + "lines": [ + { + "bbox": [ + 104, + 297, + 210, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 297, + 210, + 313 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 108, + 324, + 504, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "We focus on methods for estimating treatment effects over time and for building balancing represen-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 335, + 453, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 453, + 347 + ], + "score": 1.0, + "content": "tations for causal inference. A more in-depth review of related work is in Appendix A.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "Treatment effects over time. Standard methods for estimating the effects of time-varying exposures", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 368, + 375 + ], + "score": 1.0, + "content": "were first developed in the epidemiology literature and include the", + "type": "text" + }, + { + "bbox": [ + 368, + 365, + 375, + 375 + ], + "score": 0.26, + "content": "\\mathrm { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "-computation formula, Structural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 374, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 506, + 385 + ], + "score": 1.0, + "content": "Nested Models and Marginal Structural Models (MSMs) (Robins, 1986; 1994; Robins et al., 2000;", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "Robins & Hernán, 2008). Originally, these methods have used predictors performing logistic/linear", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "score": 1.0, + "content": "regression which makes them unsuitable for handling complex time-dependencies (Hernán et al.,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "2001; Mansournia et al., 2012; Mortimer et al., 2005). To address these limitations, methods that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "use Bayesian non-parametrics or recurrent neural networks as part of these frameworks have been", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 429, + 356, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 356, + 441 + ], + "score": 1.0, + "content": "proposed. (Xu et al., 2016; Roy et al., 2016; Lim et al., 2018).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "To begin with, Xu et al. (2016) use Gaussian processes to model discrete patient outcomes as", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 456, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 309, + 470 + ], + "score": 1.0, + "content": "a generalized mixed-effects model and uses the", + "type": "text" + }, + { + "bbox": [ + 310, + 459, + 316, + 468 + ], + "score": 0.8, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 456, + 506, + 470 + ], + "score": 1.0, + "content": "-computation method to handle time-varying", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 360, + 479 + ], + "score": 1.0, + "content": "confounders. Soleimani et al. (2017) extend the approach in", + "type": "text" + }, + { + "bbox": [ + 361, + 468, + 375, + 478 + ], + "score": 0.28, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "et al. (2016) to the continuous", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "time-setting and model treatment responses using linear time-invariant dynamical systems. Roy et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "(2016) use Dirichlet and Gaussian processes to model the observational data and estimate the IPTW", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "in Marginal Structural Models. Schulam & Saria (2017) build upon work from Lok et al. (2008);", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "Arjas & Parner (2004) and use marked point processes and Gaussian processes to learn causal effects", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "in continuous-time data. These Bayesian non-parametric methods make strong assumptions about", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "model structure and consequently cannot handle well heterogeneous treatment effects arising from", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "baseline variables (Soleimani et al., 2017; Schulam & Saria, 2017) and multiple treatment outcomes", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 556, + 274, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 274, + 567 + ], + "score": 1.0, + "content": "(Xu et al., 2016; Schulam & Saria, 2017).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "score": 1.0, + "content": "The work most related to ours is the one of Lim et al. (2018) which improves on the standard MSMs", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "by using recurrent neural networks to estimate the inverse probability of treatment weights (IPTWs).", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "score": 1.0, + "content": "Lim et al. (2018) introduces Recurrent Marginal Structural Networks (RMSNs) which also use a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 606, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 617 + ], + "score": 1.0, + "content": "sequence-to-sequence deep learning architecture to forecast treatment responses in a similar fashion", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "to our model. However, RMSNs require training additional RNNs to estimate the propensity weights", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "and does not overcome the fundamental problems with IPTWs, such as the high-variance of the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "weights. Conversely, CRN takes advantage of the recent advances in machine learning, in particular,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 649, + 451, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 451, + 661 + ], + "score": 1.0, + "content": "representation learning to propose a novel way of handling time-varying confounders.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "Balancing representations for treatment effect estimation. Balancing the distribution of control", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "and treated groups has been used for counterfactual estimation in the static setting. The methods", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "proposed in the static setting for balancing representations are based on using discrepancy measures", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "in the representation space between treated and untreated patients, which do not generalize to multiple", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 708, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 507, + 723 + ], + "score": 1.0, + "content": "treatments (Johansson et al., 2016; Shalit et al., 2017; Li & Fu, 2017; Yao et al., 2018). 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To estimate counterfactual outcomes for treatment", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "plans (and not just single treatments), we integrate the domain adversarial training procedure as part", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "of a sequence-to-sequence architecture. CRN consists of an encoder network which builds treatment", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "invariant representations of the patient history that are used to initialize the decoder. The decoder", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "network estimates outcomes under an intended sequence of future treatments, while also updating the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "balanced representation. 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We show that CRN achieves better performance in predicting counterfactual outcomes,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "but also in choosing the right treatment and timing of treatment than current state-of-the-art methods.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 246, + 507, + 282 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 298, + 209, + 311 + ], + "lines": [ + { + "bbox": [ + 104, + 297, + 210, + 313 + ], + "spans": [ + { + "bbox": [ + 104, + 297, + 210, + 313 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 108, + 324, + 504, + 346 + ], + "lines": [ + { + "bbox": [ + 106, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "We focus on methods for estimating treatment effects over time and for building balancing represen-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 335, + 453, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 453, + 347 + ], + "score": 1.0, + "content": "tations for causal inference. A more in-depth review of related work is in Appendix A.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 106, + 323, + 506, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "Treatment effects over time. Standard methods for estimating the effects of time-varying exposures", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 368, + 375 + ], + "score": 1.0, + "content": "were first developed in the epidemiology literature and include the", + "type": "text" + }, + { + "bbox": [ + 368, + 365, + 375, + 375 + ], + "score": 0.26, + "content": "\\mathrm { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "-computation formula, Structural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 374, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 506, + 385 + ], + "score": 1.0, + "content": "Nested Models and Marginal Structural Models (MSMs) (Robins, 1986; 1994; Robins et al., 2000;", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "Robins & Hernán, 2008). Originally, these methods have used predictors performing logistic/linear", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "score": 1.0, + "content": "regression which makes them unsuitable for handling complex time-dependencies (Hernán et al.,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "2001; Mansournia et al., 2012; Mortimer et al., 2005). To address these limitations, methods that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "use Bayesian non-parametrics or recurrent neural networks as part of these frameworks have been", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 429, + 356, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 356, + 441 + ], + "score": 1.0, + "content": "proposed. (Xu et al., 2016; Roy et al., 2016; Lim et al., 2018).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 351, + 506, + 441 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "To begin with, Xu et al. 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Roy et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "(2016) use Dirichlet and Gaussian processes to model the observational data and estimate the IPTW", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "in Marginal Structural Models. Schulam & Saria (2017) build upon work from Lok et al. (2008);", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "Arjas & Parner (2004) and use marked point processes and Gaussian processes to learn causal effects", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "in continuous-time data. 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See Appendix B for more more details.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 107, + 370, + 347, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 348, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 348, + 385 + ], + "score": 1.0, + "content": "4 COUNTERFACTUAL RECURRENT NETWORK", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 506, + 462 + ], + "lines": [ + { + "bbox": [ + 104, + 393, + 504, + 409 + ], + "spans": [ + { + "bbox": [ + 104, + 393, + 461, + 409 + ], + "score": 1.0, + "content": "The observational data can be used to train a supervised learning model to forecast:", + "type": "text" + }, + { + "bbox": [ + 462, + 396, + 504, + 408 + ], + "score": 0.91, + "content": "\\mathbb { E } ( \\mathbf { Y } _ { t + \\tau } \\ )", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 107, + 406, + 264, + 419 + ], + "score": 0.9, + "content": "\\bar { \\mathbf { A } } ( t , t + \\tau - 1 ) = \\bar { \\mathbf { a } } ( t , t + \\tau - 1 ) , \\bar { \\mathbf { H } } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 406, + 506, + 420 + ], + "score": 1.0, + "content": ". 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The Counterfactual Recurrent Network", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "(CRN) removes this bias through domain adversarial training and estimates the counterfactual", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 146, + 463 + ], + "score": 1.0, + "content": "outcomes", + "type": "text" + }, + { + "bbox": [ + 146, + 450, + 263, + 463 + ], + "score": 0.9, + "content": "\\mathbb { E } ( \\mathbf { Y } _ { t + \\tau } [ \\bar { \\mathbf { a } } ( t , t + \\tau - 1 \\bar { ) } ] | \\bar { \\mathbf { H } } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 450, + 444, + 463 + ], + "score": 1.0, + "content": ", for any intended future treatment assignment", + "type": "text" + }, + { + "bbox": [ + 444, + 450, + 503, + 462 + ], + "score": 0.91, + "content": "\\bar { \\mathbf { a } } ( t , t + \\tau - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 450, + 506, + 463 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 506, + 567 + ], + "lines": [ + { + "bbox": [ + 104, + 465, + 508, + 482 + ], + "spans": [ + { + "bbox": [ + 104, + 465, + 278, + 482 + ], + "score": 1.0, + "content": "Balancing representations. The history", + "type": "text" + }, + { + "bbox": [ + 278, + 466, + 368, + 479 + ], + "score": 0.93, + "content": "\\bar { \\mathbf { H } } _ { t } = ( \\bar { \\mathbf { X } } _ { t } , \\bar { \\mathbf { A } } _ { t - 1 } , \\mathbf { V } )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 465, + 508, + 482 + ], + "score": 1.0, + "content": "of the patient contains the time-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 188, + 491 + ], + "score": 1.0, + "content": "varying confounders", + "type": "text" + }, + { + "bbox": [ + 188, + 478, + 201, + 489 + ], + "score": 0.9, + "content": "\\bar { \\mathbf { X } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 477, + 345, + 491 + ], + "score": 1.0, + "content": "which bias the treatment assignment", + "type": "text" + }, + { + "bbox": [ + 345, + 479, + 426, + 491 + ], + "score": 0.92, + "content": "\\mathbf { A } _ { t } \\in \\{ A _ { 1 } , \\dotsc . . . A _ { K } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "in the observational", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "dataset. Inverse probability of treatment weighting, as performed by MSMs, creates a pseudo-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 290, + 513 + ], + "score": 1.0, + "content": "population where the probability of treatment", + "type": "text" + }, + { + "bbox": [ + 291, + 501, + 304, + 511 + ], + "score": 0.89, + "content": "{ \\bf A } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "does not depend on the time-varying confounders", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 510, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 490, + 524 + ], + "score": 1.0, + "content": "(Robins et al., 2000). 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Robins (1999) shows that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 481, + 557 + ], + "score": 1.0, + "content": "in this case, the estimation of counterfactual treatment outcomes is unbiased. See Appendix", + "type": "text" + }, + { + "bbox": [ + 481, + 545, + 490, + 554 + ], + "score": 0.29, + "content": "\\textrm { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 554, + 423, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 423, + 568 + ], + "score": 1.0, + "content": "details and for an example of a causal graph with time-dependent confounders.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 122, + 585 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 573, + 131, + 582 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 572, + 375, + 585 + ], + "score": 1.0, + "content": "be the representation function that maps the patient history", + "type": "text" + }, + { + "bbox": [ + 375, + 571, + 389, + 583 + ], + "score": 0.9, + "content": "\\bar { \\mathbf { H } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 572, + 493, + 585 + ], + "score": 1.0, + "content": "to a representation space", + "type": "text" + }, + { + "bbox": [ + 493, + 573, + 503, + 582 + ], + "score": 0.78, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 572, + 506, + 585 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 257, + 595 + ], + "score": 1.0, + "content": "To obtain unbiased treatment effects,", + "type": "text" + }, + { + "bbox": [ + 257, + 583, + 266, + 593 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "needs to construct treatment invariant representations such", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 125, + 608 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 126, + 594, + 366, + 606 + ], + "score": 0.88, + "content": "P ( \\Phi ( { \\bar { \\mathbf { H } } } _ { t } ) \\mid \\mathbf { A } _ { t } = A _ { 1 } ) = \\cdots = P ( \\Phi ( { \\bar { \\mathbf { H } } } _ { t } ) \\mid \\mathbf { A } _ { t } = A _ { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 593, + 506, + 608 + ], + "score": 1.0, + "content": ". 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We make the standard assumptions (Robins", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "et al., 2000; Lim et al., 2018) needed to identify the treatment effects: consistency, positivity and no", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 342, + 475, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 475, + 356 + ], + "score": 1.0, + "content": "hidden confounders (sequential strong ignorability). See Appendix B for more more details.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 309, + 506, + 356 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 370, + 347, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 348, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 348, + 385 + ], + "score": 1.0, + "content": "4 COUNTERFACTUAL RECURRENT NETWORK", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 506, + 462 + ], + "lines": [ + { + "bbox": [ + 104, + 393, + 504, + 409 + ], + "spans": [ + { + "bbox": [ + 104, + 393, + 461, + 409 + ], + "score": 1.0, + "content": "The observational data can be used to train a supervised learning model to forecast:", + "type": "text" + }, + { + "bbox": [ + 462, + 396, + 504, + 408 + ], + "score": 0.91, + "content": "\\mathbb { E } ( \\mathbf { Y } _ { t + \\tau } \\ )", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 107, + 406, + 264, + 419 + ], + "score": 0.9, + "content": "\\bar { \\mathbf { A } } ( t , t + \\tau - 1 ) = \\bar { \\mathbf { a } } ( t , t + \\tau - 1 ) , \\bar { \\mathbf { H } } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 406, + 506, + 420 + ], + "score": 1.0, + "content": ". However, without adjusting for the bias introduced by time-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "varying confounders, this model cannot be reliably used for making causal predictions (Robins et al.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 275, + 441 + ], + "score": 1.0, + "content": "2000; Robins & Hernán, 2008; Schulam", + "type": "text" + }, + { + "bbox": [ + 275, + 429, + 284, + 439 + ], + "score": 0.26, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "Saria, 2017). The Counterfactual Recurrent Network", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "(CRN) removes this bias through domain adversarial training and estimates the counterfactual", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 450, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 146, + 463 + ], + "score": 1.0, + "content": "outcomes", + "type": "text" + }, + { + "bbox": [ + 146, + 450, + 263, + 463 + ], + "score": 0.9, + "content": "\\mathbb { E } ( \\mathbf { Y } _ { t + \\tau } [ \\bar { \\mathbf { a } } ( t , t + \\tau - 1 \\bar { ) } ] | \\bar { \\mathbf { H } } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 450, + 444, + 463 + ], + "score": 1.0, + "content": ", for any intended future treatment assignment", + "type": "text" + }, + { + "bbox": [ + 444, + 450, + 503, + 462 + ], + "score": 0.91, + "content": "\\bar { \\mathbf { a } } ( t , t + \\tau - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 450, + 506, + 463 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 393, + 506, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 506, + 567 + ], + "lines": [ + { + "bbox": [ + 104, + 465, + 508, + 482 + ], + "spans": [ + { + "bbox": [ + 104, + 465, + 278, + 482 + ], + "score": 1.0, + "content": "Balancing representations. The history", + "type": "text" + }, + { + "bbox": [ + 278, + 466, + 368, + 479 + ], + "score": 0.93, + "content": "\\bar { \\mathbf { H } } _ { t } = ( \\bar { \\mathbf { X } } _ { t } , \\bar { \\mathbf { A } } _ { t - 1 } , \\mathbf { V } )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 465, + 508, + 482 + ], + "score": 1.0, + "content": "of the patient contains the time-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 188, + 491 + ], + "score": 1.0, + "content": "varying confounders", + "type": "text" + }, + { + "bbox": [ + 188, + 478, + 201, + 489 + ], + "score": 0.9, + "content": "\\bar { \\mathbf { X } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 477, + 345, + 491 + ], + "score": 1.0, + "content": "which bias the treatment assignment", + "type": "text" + }, + { + "bbox": [ + 345, + 479, + 426, + 491 + ], + "score": 0.92, + "content": "\\mathbf { A } _ { t } \\in \\{ A _ { 1 } , \\dotsc . . . A _ { K } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "in the observational", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "dataset. Inverse probability of treatment weighting, as performed by MSMs, creates a pseudo-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 290, + 513 + ], + "score": 1.0, + "content": "population where the probability of treatment", + "type": "text" + }, + { + "bbox": [ + 291, + 501, + 304, + 511 + ], + "score": 0.89, + "content": "{ \\bf A } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "does not depend on the time-varying confounders", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 510, + 504, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 490, + 524 + ], + "score": 1.0, + "content": "(Robins et al., 2000). In this paper, we propose instead building a representation of the history", + "type": "text" + }, + { + "bbox": [ + 490, + 510, + 504, + 522 + ], + "score": 0.89, + "content": "\\bar { \\mathbf { H } } _ { t }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 520, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 262, + 536 + ], + "score": 1.0, + "content": "that is not predictive of the treatment", + "type": "text" + }, + { + "bbox": [ + 262, + 523, + 276, + 533 + ], + "score": 0.86, + "content": "{ \\bf A } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 520, + 506, + 536 + ], + "score": 1.0, + "content": ". This way, we remove the association between history,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 274, + 546 + ], + "score": 1.0, + "content": "containing the time-varying confounders", + "type": "text" + }, + { + "bbox": [ + 275, + 533, + 288, + 544 + ], + "score": 0.89, + "content": "\\bar { \\mathbf { X } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 533, + 381, + 546 + ], + "score": 1.0, + "content": ", and current treatment", + "type": "text" + }, + { + "bbox": [ + 381, + 534, + 394, + 544 + ], + "score": 0.87, + "content": "{ \\bf A } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 533, + 506, + 546 + ], + "score": 1.0, + "content": ". Robins (1999) shows that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 481, + 557 + ], + "score": 1.0, + "content": "in this case, the estimation of counterfactual treatment outcomes is unbiased. See Appendix", + "type": "text" + }, + { + "bbox": [ + 481, + 545, + 490, + 554 + ], + "score": 0.29, + "content": "\\textrm { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 554, + 423, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 423, + 568 + ], + "score": 1.0, + "content": "details and for an example of a causal graph with time-dependent confounders.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 465, + 508, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 122, + 585 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 573, + 131, + 582 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 572, + 375, + 585 + ], + "score": 1.0, + "content": "be the representation function that maps the patient history", + "type": "text" + }, + { + "bbox": [ + 375, + 571, + 389, + 583 + ], + "score": 0.9, + "content": "\\bar { \\mathbf { H } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 572, + 493, + 585 + ], + "score": 1.0, + "content": "to a representation space", + "type": "text" + }, + { + "bbox": [ + 493, + 573, + 503, + 582 + ], + "score": 0.78, + "content": "\\mathcal { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 572, + 506, + 585 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 257, + 595 + ], + "score": 1.0, + "content": "To obtain unbiased treatment effects,", + "type": "text" + }, + { + "bbox": [ + 257, + 583, + 266, + 593 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "needs to construct treatment invariant representations such", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 125, + 608 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 126, + 594, + 366, + 606 + ], + "score": 0.88, + "content": "P ( \\Phi ( { \\bar { \\mathbf { H } } } _ { t } ) \\mid \\mathbf { A } _ { t } = A _ { 1 } ) = \\cdots = P ( \\Phi ( { \\bar { \\mathbf { H } } } _ { t } ) \\mid \\mathbf { A } _ { t } = A _ { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 593, + 506, + 608 + ], + "score": 1.0, + "content": ". To achieve this and to estimate", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "counterfactual outcomes under a planned sequence of treatments, we integrate the domain adversarial", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "training framework proposed by Ganin et al. (2016) and extended by Sebag et al. (2019) to the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "multi-domain learning setting, into a sequence-to-sequence architecture. In our case, the different", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "treatments at each timestep are considered the different domains. Note that the novelty here comes", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "from the use of domain adversarial training to handle the bias from the time-dependent confounders,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "rather than the use of sequence-to-sequence models, which have already been applied to forecast", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 671, + 436, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 436, + 684 + ], + "score": 1.0, + "content": "treatment responses (Lim et al., 2018). Figure 2 illustrates our model architecture.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 571, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 699 + ], + "score": 1.0, + "content": "Encoder. The encoder network uses an RNN, with LSTM unit (Hochreiter & Schmidhuber, 1997),", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 246, + 710 + ], + "score": 1.0, + "content": "to process the history of treatments", + "type": "text" + }, + { + "bbox": [ + 247, + 698, + 270, + 711 + ], + "score": 0.91, + "content": "\\bar { \\mathbf { A } } _ { t - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 699, + 316, + 710 + ], + "score": 1.0, + "content": ", covariates", + "type": "text" + }, + { + "bbox": [ + 316, + 698, + 329, + 710 + ], + "score": 0.89, + "content": "\\bar { \\mathbf { X } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 699, + 415, + 710 + ], + "score": 1.0, + "content": "and baseline features", + "type": "text" + }, + { + "bbox": [ + 415, + 699, + 425, + 709 + ], + "score": 0.41, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 699, + 506, + 710 + ], + "score": 1.0, + "content": "to build a treatment", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 507, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 203, + 722 + ], + "score": 1.0, + "content": "invariant representation", + "type": "text" + }, + { + "bbox": [ + 203, + 709, + 231, + 722 + ], + "score": 0.94, + "content": "\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 709, + 413, + 722 + ], + "score": 1.0, + "content": ", but also to predict one-step-ahead outcomes", + "type": "text" + }, + { + "bbox": [ + 414, + 710, + 437, + 721 + ], + "score": 0.9, + "content": "\\mathbf { Y } _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 709, + 507, + 722 + ], + "score": 1.0, + "content": ". 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For more details about the training procedure, see Appendix", + "type": "text" + }, + { + "bbox": [ + 432, + 487, + 439, + 497 + ], + "score": 0.29, + "content": "\\mathrm { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 486, + 443, + 500 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 105, + 503, + 504, + 529 + ], + "lines": [ + { + "bbox": [ + 102, + 500, + 508, + 522 + ], + "spans": [ + { + "bbox": [ + 102, + 500, + 205, + 522 + ], + "score": 1.0, + "content": "By using the objective", + "type": "text" + }, + { + "bbox": [ + 205, + 503, + 267, + 518 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { t } ^ { ( i ) } ( \\theta _ { r } , \\theta _ { y } , \\theta _ { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 500, + 385, + 522 + ], + "score": 1.0, + "content": ", we reach the saddle point", + "type": "text" + }, + { + "bbox": [ + 385, + 504, + 430, + 519 + ], + "score": 0.93, + "content": "( \\hat { \\theta } _ { r } , \\hat { \\theta } _ { y } , \\hat { \\theta } _ { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 500, + 508, + 522 + ], + "score": 1.0, + "content": "that achieves the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 516, + 384, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 384, + 530 + ], + "score": 1.0, + "content": "equilibrium between domain discrimination and outcome estimation.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 534, + 461, + 557 + ], + "lines": [ + { + "bbox": [ + 150, + 534, + 461, + 557 + ], + "spans": [ + { + "bbox": [ + 150, + 534, + 461, + 557 + ], + "score": 0.91, + "content": "( \\hat { \\theta } _ { r } , \\hat { \\theta } _ { y } ) = \\arg \\operatorname* { m i n } _ { \\theta _ { r } , \\theta _ { y } } \\mathcal { L } _ { t } ^ { ( i ) } ( \\theta _ { r } , \\theta _ { y } , \\hat { \\theta } _ { a } ) \\qquad \\hat { \\theta } _ { a } = \\arg \\operatorname* { m a x } _ { \\theta _ { a } } \\mathcal { L } _ { t } ^ { ( i ) } ( \\hat { \\theta } _ { r } , \\hat { \\theta } _ { y } , \\theta _ { a } ) .", + "type": "interline_equation", + "image_path": "2011bd68e3f56b4b4ebe390cd9b38fa98d81dc6505b0fa09bc569a3fefea196a.jpg" + } + ] + } + ], + "index": 48, + "virtual_lines": [ + { + "bbox": [ + 150, + 534, + 461, + 557 + ], + "spans": [], + "index": 48 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 569, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 583 + ], + "score": 1.0, + "content": "The result stated in Theorem 1 proves that the treatment (domain) loss part of our objective (from", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 581, + 371, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 371, + 594 + ], + "score": 1.0, + "content": "equation 2) aims to remove the time-dependent confounding bias.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5 + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 504, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 178, + 609 + ], + "score": 1.0, + "content": "Theorem 1. Let", + "type": "text" + }, + { + "bbox": [ + 178, + 596, + 242, + 608 + ], + "score": 0.92, + "content": "t \\in \\{ 1 , 2 , \\dots \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 594, + 291, + 609 + ], + "score": 1.0, + "content": ". For each", + "type": "text" + }, + { + "bbox": [ + 291, + 596, + 346, + 608 + ], + "score": 0.91, + "content": "j = 1 , . . . , K", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 594, + 366, + 609 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 366, + 596, + 378, + 609 + ], + "score": 0.81, + "content": "P _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 594, + 490, + 609 + ], + "score": 1.0, + "content": "denote the distribution of", + "type": "text" + }, + { + "bbox": [ + 490, + 595, + 504, + 608 + ], + "score": 0.88, + "content": "\\bar { \\mathbf { H } } _ { t }", + "type": "inline_equation" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 607, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 167, + 622 + ], + "score": 1.0, + "content": "conditional on", + "type": "text" + }, + { + "bbox": [ + 167, + 609, + 205, + 621 + ], + "score": 0.92, + "content": "\\mathbf { A } _ { t } = A _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 607, + 237, + 622 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 237, + 608, + 252, + 622 + ], + "score": 0.9, + "content": "P _ { j } ^ { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 607, + 357, + 622 + ], + "score": 1.0, + "content": "denote the distribution of", + "type": "text" + }, + { + "bbox": [ + 357, + 609, + 385, + 621 + ], + "score": 0.9, + "content": "\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 607, + 447, + 622 + ], + "score": 1.0, + "content": "conditional on", + "type": "text" + }, + { + "bbox": [ + 447, + 609, + 484, + 621 + ], + "score": 0.91, + "content": "\\mathbf { A } _ { t } = A _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 607, + 506, + 622 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 107, + 621, + 488, + 635 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 120, + 633 + ], + "score": 0.89, + "content": "G _ { a } ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 621, + 204, + 635 + ], + "score": 1.0, + "content": "denote the output of", + "type": "text" + }, + { + "bbox": [ + 204, + 622, + 217, + 632 + ], + "score": 0.88, + "content": "G _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 621, + 330, + 635 + ], + "score": 1.0, + "content": "corresponding to treatment", + "type": "text" + }, + { + "bbox": [ + 330, + 622, + 342, + 633 + ], + "score": 0.89, + "content": "A _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 621, + 488, + 635 + ], + "score": 1.0, + "content": ". Then the minimax game defined by", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52 + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 639, + 473, + 675 + ], + "lines": [ + { + "bbox": [ + 136, + 639, + 473, + 675 + ], + "spans": [ + { + "bbox": [ + 136, + 639, + 473, + 675 + ], + "score": 0.93, + "content": "\\underset { \\Phi } { \\mathop { \\operatorname* { m i n } } } \\underset { G _ { a } } { \\mathop { \\operatorname* { m a x } } } \\sum _ { j = 1 } ^ { K } \\mathbb { E } _ { \\bar { \\mathbf { H } } _ { t } \\sim P _ { j } } \\left[ \\log ( G _ { a } ^ { j } ( \\Phi ( \\bar { \\mathbf { H } } _ { t } ) ; \\theta _ { a } ) ) \\right] \\qquad \\quad s u b j e c t t o \\sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( \\Phi ( \\bar { \\mathbf { H } } _ { t } ) ) = 1", + "type": "interline_equation", + "image_path": "7ce58ece0f668c44fcef24ad8527166ae87117ed618283aed5e99fe1b802935b.jpg" + } + ] + } + ], + "index": 55, + "virtual_lines": [ + { + "bbox": [ + 136, + 639, + 473, + 651.0 + ], + "spans": [], + "index": 54 + }, + { + "bbox": [ + 136, + 651.0, + 473, + 663.0 + ], + "spans": [], + "index": 55 + }, + { + "bbox": [ + 136, + 663.0, + 473, + 675.0 + ], + "spans": [], + "index": 56 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 681, + 506, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 510, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 323, + 699 + ], + "score": 1.0, + "content": "has a global minimum which is attained if and only if", + "type": "text" + }, + { + "bbox": [ + 323, + 682, + 414, + 695 + ], + "score": 0.92, + "content": "P _ { 1 } ^ { \\Phi } = P _ { 2 } ^ { \\Phi } = \\ldots = P _ { K } ^ { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 676, + 510, + 702 + ], + "score": 1.0, + "content": ", i.e. when the learned", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 694, + 312, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 312, + 705 + ], + "score": 1.0, + "content": "representations are invariant across all treatments.", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 57.5 + }, + { + "type": "text", + "bbox": [ + 103, + 719, + 500, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 719, + 502, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 502, + 733 + ], + "score": 1.0, + "content": "Proof. This result is a restatement of the one in Li et al. (2018). 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Then the minimax game defined by", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52, + "bbox_fs": [ + 105, + 594, + 506, + 635 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 639, + 473, + 675 + ], + "lines": [ + { + "bbox": [ + 136, + 639, + 473, + 675 + ], + "spans": [ + { + "bbox": [ + 136, + 639, + 473, + 675 + ], + "score": 0.93, + "content": "\\underset { \\Phi } { \\mathop { \\operatorname* { m i n } } } \\underset { G _ { a } } { \\mathop { \\operatorname* { m a x } } } \\sum _ { j = 1 } ^ { K } \\mathbb { E } _ { \\bar { \\mathbf { H } } _ { t } \\sim P _ { j } } \\left[ \\log ( G _ { a } ^ { j } ( \\Phi ( \\bar { \\mathbf { H } } _ { t } ) ; \\theta _ { a } ) ) \\right] \\qquad \\quad s u b j e c t t o \\sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( \\Phi ( \\bar { \\mathbf { H } } _ { t } ) ) = 1", + "type": "interline_equation", + "image_path": "7ce58ece0f668c44fcef24ad8527166ae87117ed618283aed5e99fe1b802935b.jpg" + } + ] + } + ], + "index": 55, + "virtual_lines": [ + { + "bbox": [ + 136, + 639, + 473, + 651.0 + ], + "spans": [], + "index": 54 + }, + { + "bbox": [ + 136, + 651.0, + 473, + 663.0 + ], + "spans": [], + "index": 55 + }, + { + "bbox": [ + 136, + 663.0, + 473, + 675.0 + ], + "spans": [], + "index": 56 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 681, + 506, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 510, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 323, + 699 + ], + "score": 1.0, + "content": "has a global minimum which is attained if and only if", + "type": "text" + }, + { + "bbox": [ + 323, + 682, + 414, + 695 + ], + "score": 0.92, + "content": "P _ { 1 } ^ { \\Phi } = P _ { 2 } ^ { \\Phi } = \\ldots = P _ { K } ^ { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 676, + 510, + 702 + ], + "score": 1.0, + "content": ", i.e. when the learned", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 694, + 312, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 312, + 705 + ], + "score": 1.0, + "content": "representations are invariant across all treatments.", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 57.5, + "bbox_fs": [ + 105, + 676, + 510, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 103, + 719, + 500, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 719, + 502, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 502, + 733 + ], + "score": 1.0, + "content": "Proof. This result is a restatement of the one in Li et al. (2018). For details, see the Appendix D.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 59, + "bbox_fs": [ + 106, + 719, + 502, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 506, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 94 + ], + "score": 1.0, + "content": "A good representation allows us to obtain a low error in estimating counterfactuals for all treatments,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 263, + 105 + ], + "score": 1.0, + "content": "while at the same time to minimize the", + "type": "text" + }, + { + "bbox": [ + 264, + 94, + 273, + 104 + ], + "score": 0.68, + "content": "\\mathcal { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "-divergence between induced marginal distributions of all", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 425, + 117 + ], + "score": 1.0, + "content": "the domains. We use an algorithm that directly minimizes a combination of the", + "type": "text" + }, + { + "bbox": [ + 426, + 105, + 443, + 115 + ], + "score": 0.78, + "content": "\\varkappa -", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "divergence and", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 227, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 227, + 128 + ], + "score": 1.0, + "content": "the empirical training margin.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 144, + 200, + 156 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 201, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 201, + 158 + ], + "score": 1.0, + "content": "6 EXPERIMENTS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 225 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 181 + ], + "score": 1.0, + "content": "In real datasets, counterfactual outcomes and the degree of time-dependent confounding are not", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 179, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 391, + 192 + ], + "score": 1.0, + "content": "known (Schulam & Saria, 2017; Lim et al., 2018). To validate the", + "type": "text" + }, + { + "bbox": [ + 391, + 180, + 417, + 190 + ], + "score": 0.73, + "content": "\\mathrm { C R N ^ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 179, + 506, + 192 + ], + "score": 1.0, + "content": ", we evaluate it on a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 190, + 507, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 507, + 204 + ], + "score": 1.0, + "content": "Pharmacokinetic-Pharmacodynamic model of tumour growth (Geng et al., 2017), which uses a state-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 199, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 217 + ], + "score": 1.0, + "content": "of-the-art bio-mathematical model to simulate the combined effects of chemotherapy and radiotherapy", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 213, + 474, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 474, + 225 + ], + "score": 1.0, + "content": "in lung cancer patients. The same model was used by Lim et al. 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Time-varying confounding is introduced by modelling", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 117, + 338, + 487, + 371 + ], + "spans": [ + { + "bbox": [ + 117, + 338, + 264, + 371 + ], + "score": 1.0, + "content": "emotherapy and radiotherapy assigndepending on the tumour diameter:", + "type": "text" + }, + { + "bbox": [ + 383, + 338, + 402, + 371 + ], + "score": 1.0, + "content": "ables, and", + "type": "text" + }, + { + "bbox": [ + 477, + 340, + 487, + 350 + ], + "score": 0.79, + "content": "p _ { c }", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 349, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 117, + 361 + ], + "score": 0.82, + "content": "p _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 349, + 383, + 364 + ], + "score": 0.92, + "content": "\\begin{array} { r } { p _ { c } ( t ) = \\sigma \\big ( \\frac { \\gamma _ { c } } { D _ { \\mathrm { m a x } } } ( \\bar { D } ( t ) - \\delta _ { c } ) \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 349, + 505, + 364 + ], + "score": 0.9, + "content": "\\begin{array} { r } { p _ { r } ( t ) \\dot { = } \\sigma \\big ( \\frac { \\gamma _ { r } } { D _ { \\mathrm { m a x } } } ( \\hat { \\bar { D } } ( t ) - } \\end{array}", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 363, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 119, + 375 + ], + "score": 0.72, + "content": "\\delta _ { r } )", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 363, + 154, + 377 + ], + "score": 1.0, + "content": "\u0001 where", + "type": "text" + }, + { + "bbox": [ + 175, + 363, + 361, + 377 + ], + "score": 1.0, + "content": "is the average diameter over the last 15 days,", + "type": "text" + }, + { + "bbox": [ + 362, + 364, + 442, + 375 + ], + "score": 0.89, + "content": "D _ { \\mathrm { m a x } } = 1 3 \\mathrm { c m } , \\sigma ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 363, + 506, + 377 + ], + "score": 1.0, + "content": "is the sigmoid", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 374, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 124, + 390 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 376, + 203, + 387 + ], + "score": 0.91, + "content": "\\delta _ { c } = \\delta _ { r } = D _ { \\operatorname* { m a x } } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 374, + 477, + 390 + ], + "score": 1.0, + "content": ". The amount of time-dependent confounding is controlled through", + "type": "text" + }, + { + "bbox": [ + 477, + 377, + 502, + 387 + ], + "score": 0.59, + "content": "\\gamma _ { c } , \\gamma _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 374, + 506, + 390 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 149, + 400 + ], + "score": 1.0, + "content": "the higher", + "type": "text" + }, + { + "bbox": [ + 150, + 389, + 161, + 398 + ], + "score": 0.86, + "content": "\\gamma _ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 386, + 505, + 400 + ], + "score": 1.0, + "content": "is, the more important the history is in assigning treatments. At each timestep, there", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "are four treatment options: no treatment, chemotherapy, radiotherapy, combined chemotherapy and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 408, + 365, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 365, + 421 + ], + "score": 1.0, + "content": "radiotherapy. For details about data simulation, see Appendix F.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "Benchmarks We used the following benchmarks for performance comparison: Marginal Structural", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "score": 1.0, + "content": "Models (MSMs) (Robins et al., 2000), which use logistic regression for estimating the IPTWs and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "linear regression for prediction (see Appendix G for details). We also compare against the Recurrent", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "score": 1.0, + "content": "Marginal Structural Networks (RMSNs) Lim et al. (2018), which is the current state-of-the-art model", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 104, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "in estimating treatment responses. RMSNs use RNNs to estimate the IPTWs and the patient outcomes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "(details in Appendix H). To show that standard supervised learning models do not handle the time-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 492, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 504 + ], + "score": 1.0, + "content": "varying confounders we compare against an RNN and a linear regression model, which receive", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "as input treatments and covariates to predict the outcome (see Appendix I for details). Our model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "architecture follows the description in Sections 4 and 5, with full training details and hyperparameter", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "optimization in Appendix J. 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To validate the", + "type": "text" + }, + { + "bbox": [ + 391, + 180, + 417, + 190 + ], + "score": 0.73, + "content": "\\mathrm { C R N ^ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 179, + 506, + 192 + ], + "score": 1.0, + "content": ", we evaluate it on a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 190, + 507, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 507, + 204 + ], + "score": 1.0, + "content": "Pharmacokinetic-Pharmacodynamic model of tumour growth (Geng et al., 2017), which uses a state-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 199, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 217 + ], + "score": 1.0, + "content": "of-the-art bio-mathematical model to simulate the combined effects of chemotherapy and radiotherapy", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 213, + 474, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 474, + 225 + ], + "score": 1.0, + "content": "in lung cancer patients. 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The amount of time-dependent confounding is controlled through", + "type": "text" + }, + { + "bbox": [ + 477, + 377, + 502, + 387 + ], + "score": 0.59, + "content": "\\gamma _ { c } , \\gamma _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 374, + 506, + 390 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 386, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 149, + 400 + ], + "score": 1.0, + "content": "the higher", + "type": "text" + }, + { + "bbox": [ + 150, + 389, + 161, + 398 + ], + "score": 0.86, + "content": "\\gamma _ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 386, + 505, + 400 + ], + "score": 1.0, + "content": "is, the more important the history is in assigning treatments. At each timestep, there", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "are four treatment options: no treatment, chemotherapy, radiotherapy, combined chemotherapy and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 408, + 365, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 365, + 421 + ], + "score": 1.0, + "content": "radiotherapy. For details about data simulation, see Appendix F.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 294, + 506, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "Benchmarks We used the following benchmarks for performance comparison: Marginal Structural", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 450 + ], + "score": 1.0, + "content": "Models (MSMs) (Robins et al., 2000), which use logistic regression for estimating the IPTWs and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "linear regression for prediction (see Appendix G for details). We also compare against the Recurrent", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "score": 1.0, + "content": "Marginal Structural Networks (RMSNs) Lim et al. (2018), which is the current state-of-the-art model", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 104, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "in estimating treatment responses. RMSNs use RNNs to estimate the IPTWs and the patient outcomes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "(details in Appendix H). To show that standard supervised learning models do not handle the time-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 492, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 504 + ], + "score": 1.0, + "content": "varying confounders we compare against an RNN and a linear regression model, which receive", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "as input treatments and covariates to predict the outcome (see Appendix I for details). Our model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "architecture follows the description in Sections 4 and 5, with full training details and hyperparameter", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 537 + ], + "score": 1.0, + "content": "optimization in Appendix J. To show the importance of adversarial training, we also benchmark", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 535, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 163, + 547 + ], + "score": 1.0, + "content": "against CRN", + "type": "text" + }, + { + "bbox": [ + 163, + 536, + 190, + 546 + ], + "score": 0.84, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 535, + 371, + 547 + ], + "score": 1.0, + "content": ") a model with the same architecture, but with", + "type": "text" + }, + { + "bbox": [ + 372, + 535, + 397, + 545 + ], + "score": 0.9, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 535, + 506, + 547 + ], + "score": 1.0, + "content": ", i.e our model architecture", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 545, + 221, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 221, + 559 + ], + "score": 1.0, + "content": "without adversarial training.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 425, + 506, + 559 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 572, + 372, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 571, + 374, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 374, + 584 + ], + "score": 1.0, + "content": "6.1 EVALUATE MODELS ON COUNTERFACTUAL PREDICTIONS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 592, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 505, + 605 + ], + "score": 1.0, + "content": "Previous methods focused on evaluating the error only for factual outcomes (observed patient", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 617 + ], + "score": 1.0, + "content": "outcomes) (Lim et al., 2018). However, to build decision support systems, we need to evaluate", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 627 + ], + "score": 1.0, + "content": "how well the models estimate the counterfactual outcomes, i.e patient outcomes under alternative", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 625, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 244, + 638 + ], + "score": 1.0, + "content": "treatment options. 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Notice that", + "type": "text" + }, + { + "bbox": [ + 239, + 588, + 267, + 601 + ], + "score": 0.92, + "content": "\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 588, + 400, + 602 + ], + "score": 1.0, + "content": "is not predictive of the treatment", + "type": "text" + }, + { + "bbox": [ + 401, + 590, + 414, + 600 + ], + "score": 0.86, + "content": "{ \\bf A } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 588, + 485, + 602 + ], + "score": 1.0, + "content": "given at timestep", + "type": "text" + }, + { + "bbox": [ + 485, + 590, + 490, + 599 + ], + "score": 0.65, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 588, + 495, + 602 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + } + ], + "index": 29.25 + }, + { + "type": "title", + "bbox": [ + 104, + 613, + 478, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 479, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 479, + 624 + ], + "score": 1.0, + "content": "6.2 EVALUATE RECOMMENDING THE RIGHT TREATMENT AND TIMING OF TREATMENT", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Evaluating the models just in terms of the RMSE on counterfactual estimation is also not enough for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "assessing their reliability when used as part of decision support systems. In this section we assess", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 653, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 653, + 506, + 669 + ], + "score": 1.0, + "content": "how well the models can select the correct treatment and timing of treatment for several forecasting", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 143, + 678 + ], + "score": 1.0, + "content": "horizons", + "type": "text" + }, + { + "bbox": [ + 144, + 668, + 150, + 676 + ], + "score": 0.75, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 665, + 445, + 678 + ], + "score": 1.0, + "content": ". 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Notice that", + "type": "text" + }, + { + "bbox": [ + 239, + 588, + 267, + 601 + ], + "score": 0.92, + "content": "\\Phi ( { \\bar { \\mathbf { H } } } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 588, + 400, + 602 + ], + "score": 1.0, + "content": "is not predictive of the treatment", + "type": "text" + }, + { + "bbox": [ + 401, + 590, + 414, + 600 + ], + "score": 0.86, + "content": "{ \\bf A } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 588, + 485, + 602 + ], + "score": 1.0, + "content": "given at timestep", + "type": "text" + }, + { + "bbox": [ + 485, + 590, + 490, + 599 + ], + "score": 0.65, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 588, + 495, + 602 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + } + ], + "index": 29.25 + }, + { + "type": "title", + "bbox": [ + 104, + 613, + 478, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 612, + 479, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 479, + 624 + ], + "score": 1.0, + "content": "6.2 EVALUATE RECOMMENDING THE RIGHT TREATMENT AND TIMING OF TREATMENT", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Evaluating the models just in terms of the RMSE on counterfactual estimation is also not enough for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "assessing their reliability when used as part of decision support systems. In this section we assess", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 653, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 653, + 506, + 669 + ], + "score": 1.0, + "content": "how well the models can select the correct treatment and timing of treatment for several forecasting", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 143, + 678 + ], + "score": 1.0, + "content": "horizons", + "type": "text" + }, + { + "bbox": [ + 144, + 668, + 150, + 676 + ], + "score": 0.75, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 665, + 445, + 678 + ], + "score": 1.0, + "content": ". 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%c=5,γr=5% =5,γr=01 Yc =0,γr =5
TCRNRMSNMSMCRN RMSNMSMCRNRMSNMSM
Normalized RMSE2.43%3.16%6.75%1.08% 1.35%3.68%1.54%1.59%3.23%
2.83%3.95%7.65% 1.21%1.81%3.84%1.81%2.25%3.52%
3４56７3.18%4.37% 7.95%1.33%2.13%3.91%2.03%2.71%3.63%
3.51%5.61%8.19%1.42% 2.41%3.97%2.23%2.73%3.71%
3.93%6.21% 8.52%1.53%2.43%4.04%2.43%2.88%3.79%
Treatment Accuracy83.1%75.3%73.9%83.2% 78.6%77.1%92.9%87.3%74.9%
82.5%74.1%68.5%81.3%77.7% 73.9%85.7%83.8%74.1%
3４56７73.5%72.7%63.2%78.3% 77.2%72.3%83.8%82.1%72.8%
69.4%66.7%62.7%79.5% 76.3%71.8%78.6%69.7%64.5%
71.2%68.8%62.4%72.7% 71.8%71.6%71.9%69.3%61.2%
Treatment Timing Accuracy79.6%78.1%67.6%80.5%76.8% 77.5%79.8%75.7%60.6%
73.9%70.3%63.1%79.0%77.2%73.4% 75.4%71.4%58.2%
69.8%68.6%62.4%78.3%73.3%63.6%66.9%31.3%29.5%
3456７66.9%66.2%62.6%73.5%72.1%63.9%65.8%24.2%15.5%
64.5%63.6%62.2%70.6%57.4%44.2%63.9%25.6%12.5%
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TCRNRMSNMSMCRN RMSNMSMCRNRMSNMSM
Normalized RMSE2.43%3.16%6.75%1.08% 1.35%3.68%1.54%1.59%3.23%
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Treatment Accuracy83.1%75.3%73.9%83.2% 78.6%77.1%92.9%87.3%74.9%
82.5%74.1%68.5%81.3%77.7% 73.9%85.7%83.8%74.1%
3４56７73.5%72.7%63.2%78.3% 77.2%72.3%83.8%82.1%72.8%
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71.2%68.8%62.4%72.7% 71.8%71.6%71.9%69.3%61.2%
Treatment Timing Accuracy79.6%78.1%67.6%80.5%76.8% 77.5%79.8%75.7%60.6%
73.9%70.3%63.1%79.0%77.2%73.4% 75.4%71.4%58.2%
69.8%68.6%62.4%78.3%73.3%63.6%66.9%31.3%29.5%
3456７66.9%66.2%62.6%73.5%72.1%63.9%65.8%24.2%15.5%
64.5%63.6%62.2%70.6%57.4%44.2%63.9%25.6%12.5%
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In this case, it is needed to adjust for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 152, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 505, + 164 + ], + "score": 1.0, + "content": "the selection bias; bias caused by the fact that, in the observational dataset, the treatment assignments", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "score": 1.0, + "content": "depend on the patient features. Several ways of handling the selection bias involve using propensity", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "score": 1.0, + "content": "matching (Austin, 2011; Imai & Ratkovic, 2014; Abadie & Imbens, 2016), building representations", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "score": 1.0, + "content": "where treated and un-treated populations had similar distributions (Johansson et al., 2016; Shalit et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "score": 1.0, + "content": "2017; Li & Fu, 2017; Yao et al., 2018) or performing propensity-aware hyperparameter tuning (Alaa", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "score": 1.0, + "content": "& van der Schaar, 2017; 2018). However, these methods for the static setting cannot be extended", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 436, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 436, + 230 + ], + "score": 1.0, + "content": "directly to time-varying treatments (Hernán et al., 2000; Schisterman et al., 2009).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 234, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "score": 1.0, + "content": "Learning optimal policies: A related problem to ours involves learning the optimal treatment", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 505, + 259 + ], + "score": 1.0, + "content": "policies from logged data (Swaminathan & Joachims, 2015a;b; Atan et al., 2018). That is, learning", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 256, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 269 + ], + "score": 1.0, + "content": "the treatment option that would give the best reward. Note the difference to the causal inference", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "score": 1.0, + "content": "setting considered in this paper, where the aim is to learn the counterfactual patient outcomes under", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "all possible treatment options. 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Nevertheless, these methods focus on obtaining average rewards of policies,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 488, + 479, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 479, + 500 + ], + "score": 1.0, + "content": "while in our case the aim is to estimate individualized patient outcomes for future treatments.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 515, + 203, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 205, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 205, + 530 + ], + "score": 1.0, + "content": "B ASSUMPTIONS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "The standard assumptions needed for identifying the treatment effects are (Robins & Hernán, 2008;", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 551, + 276, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 276, + 562 + ], + "score": 1.0, + "content": "Lim et al., 2018; Schulam & Saria, 2017):", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 106, + 567, + 504, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 568, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 235, + 579 + ], + "score": 1.0, + "content": "Assumption 1: Consistency. If", + "type": "text" + }, + { + "bbox": [ + 236, + 568, + 271, + 579 + ], + "score": 0.92, + "content": "{ \\bf A } _ { t } = { \\bf a } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 568, + 505, + 579 + ], + "score": 1.0, + "content": "for a given patient, then the potential outcome for treatment", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 578, + 382, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 117, + 590 + ], + "score": 0.83, + "content": "\\mathbf { a } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 578, + 305, + 592 + ], + "score": 1.0, + "content": "is the same as the observed (factual) outcome:", + "type": "text" + }, + { + "bbox": [ + 306, + 579, + 378, + 591 + ], + "score": 0.93, + "content": "\\mathbf Y _ { t + 1 } [ \\mathbf a _ { t } ] = \\mathbf Y _ { t + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 578, + 382, + 592 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 105, + 595, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 373, + 609 + ], + "score": 1.0, + "content": "Assumption 2: Positivity (Overlap) (Imai & Van Dyk, 2004): If", + "type": "text" + }, + { + "bbox": [ + 373, + 594, + 505, + 608 + ], + "score": 0.91, + "content": "P ( \\bar { \\mathbf { A } } _ { t - 1 } = \\bar { \\mathbf { a } } _ { t - 1 } , \\bar { \\mathbf { X } } _ { t } = \\bar { \\mathbf { x } } _ { t } ) \\neq 0", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 606, + 340, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 126, + 619 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 126, + 606, + 299, + 619 + ], + "score": 0.9, + "content": "P ( \\mathbf { A } _ { t } = \\mathbf { a } _ { t } \\mid \\bar { \\mathbf { A } } _ { t - 1 } = \\bar { \\mathbf { a } } _ { t - 1 } , \\bar { \\mathbf { X } } _ { t } = \\bar { \\mathbf { x } } _ { t } ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 606, + 326, + 619 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 326, + 608, + 336, + 618 + ], + "score": 0.87, + "content": "\\bar { \\mathbf { a } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 606, + 340, + 619 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 105, + 622, + 473, + 636 + ], + "lines": [ + { + "bbox": [ + 104, + 622, + 471, + 637 + ], + "spans": [ + { + "bbox": [ + 104, + 622, + 302, + 637 + ], + "score": 1.0, + "content": "Assumption 3: Sequential strong ignorability.", + "type": "text" + }, + { + "bbox": [ + 302, + 622, + 471, + 636 + ], + "score": 0.85, + "content": "\\mathbf { Y } _ { t + 1 } [ \\mathbf { a } _ { t } ] \\perp \\perp \\mathbf { A } _ { t } \\mid \\bar { \\mathbf { A } } _ { t - 1 } , \\bar { \\mathbf { X } } _ { t } , \\forall \\mathbf { a } _ { t } \\in \\mathcal { A } , \\forall t .", + "type": "inline_equation" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "Assumption 2 means that, for each timestep, each treatment has non-zero probability of being", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 650, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 665 + ], + "score": 1.0, + "content": "assigned. 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Note that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "while assumption 3 is standard across all methods for estimating treatment effects, it is not testable in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 684, + 299, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 299, + 696 + ], + "score": 1.0, + "content": "practice (Robins et al., 2000; Pearl et al., 2009).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 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": "title", + "bbox": [ + 108, + 82, + 152, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 154, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 154, + 94 + ], + "score": 1.0, + "content": "APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 109, + 105, + 270, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 272, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 272, + 119 + ], + "score": 1.0, + "content": "A EXTENDED RELATED WORK", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 129, + 505, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 130, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 130, + 505, + 142 + ], + "score": 1.0, + "content": "Causal inference in the static setting: A large number of methods have been proposed to learn", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "treatment effects from observational data in the static setting. In this case, it is needed to adjust for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 152, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 505, + 164 + ], + "score": 1.0, + "content": "the selection bias; bias caused by the fact that, in the observational dataset, the treatment assignments", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 176 + ], + "score": 1.0, + "content": "depend on the patient features. Several ways of handling the selection bias involve using propensity", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "score": 1.0, + "content": "matching (Austin, 2011; Imai & Ratkovic, 2014; Abadie & Imbens, 2016), building representations", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "score": 1.0, + "content": "where treated and un-treated populations had similar distributions (Johansson et al., 2016; Shalit et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 506, + 209 + ], + "score": 1.0, + "content": "2017; Li & Fu, 2017; Yao et al., 2018) or performing propensity-aware hyperparameter tuning (Alaa", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "score": 1.0, + "content": "& van der Schaar, 2017; 2018). However, these methods for the static setting cannot be extended", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 436, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 436, + 230 + ], + "score": 1.0, + "content": "directly to time-varying treatments (Hernán et al., 2000; Schisterman et al., 2009).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 130, + 506, + 230 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 234, + 505, + 300 + ], + "lines": [ + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 505, + 246 + ], + "score": 1.0, + "content": "Learning optimal policies: A related problem to ours involves learning the optimal treatment", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 244, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 505, + 259 + ], + "score": 1.0, + "content": "policies from logged data (Swaminathan & Joachims, 2015a;b; Atan et al., 2018). That is, learning", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 256, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 269 + ], + "score": 1.0, + "content": "the treatment option that would give the best reward. Note the difference to the causal inference", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "score": 1.0, + "content": "setting considered in this paper, where the aim is to learn the counterfactual patient outcomes under", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "all possible treatment options. 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At timestep", + "type": "text" + }, + { + "bbox": [ + 379, + 194, + 384, + 203 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 192, + 423, + 205 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 424, + 194, + 430, + 203 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "different possible", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 149, + 216 + ], + "score": 1.0, + "content": "treatments", + "type": "text" + }, + { + "bbox": [ + 150, + 204, + 230, + 216 + ], + "score": 0.92, + "content": "\\mathbf { A } _ { t } \\in \\{ A _ { 1 } , \\dotsc . . . A _ { K } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 204, + 505, + 216 + ], + "score": 1.0, + "content": ". We build a representation of the history and covariates and treatments", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 392, + 227 + ], + "score": 1.0, + "content": "that has the same distribution across the different possible treatments:", + "type": "text" + }, + { + "bbox": [ + 393, + 214, + 505, + 227 + ], + "score": 0.88, + "content": "P ( \\Phi ( { \\bar { \\mathbf { X } } } _ { t } , { \\bar { \\mathbf { A } } } _ { t - 1 } , \\mathbf { V } ) \\mid \\mathbf { A } _ { t } =", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 225, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 293, + 238 + ], + "score": 0.9, + "content": "A _ { 1 } ) = \\cdots = P ( \\Phi ( { \\bar { \\mathbf { X } } } _ { t } , { \\bar { \\mathbf { A } } } _ { t - 1 } , \\mathbf { V } ) \\mid \\mathbf { A } _ { t } = A _ { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 225, + 505, + 239 + ], + "score": 1.0, + "content": ". By breaking the association between past exposure", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 199, + 250 + ], + "score": 1.0, + "content": "and current treatments", + "type": "text" + }, + { + "bbox": [ + 199, + 238, + 212, + 248 + ], + "score": 0.87, + "content": "{ \\bf A } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 237, + 505, + 250 + ], + "score": 1.0, + "content": ", we satisfy the causal graph in Figure 6 (a) and thus we remove the bias", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 246, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 246, + 260 + ], + "score": 1.0, + "content": "from time-dependent confounders.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 181, + 505, + 260 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 80, + 244, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 245, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 245, + 96 + ], + "score": 1.0, + "content": "D PROOF OF THEOREM 1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 270, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 270, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 270, + 120 + ], + "score": 1.0, + "content": "We first prove the following proposition.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 120, + 504, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 210, + 135 + ], + "score": 1.0, + "content": "Proposition 1. For fixed", + "type": "text" + }, + { + "bbox": [ + 211, + 121, + 219, + 131 + ], + "score": 0.73, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 119, + 236, + 135 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 237, + 120, + 285, + 133 + ], + "score": 0.93, + "content": "x ^ { \\prime } = \\Phi ( \\bar { \\mathbf { h } } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 119, + 474, + 135 + ], + "score": 1.0, + "content": ". Then the optimal prediction probabilities of", + "type": "text" + }, + { + "bbox": [ + 474, + 123, + 487, + 132 + ], + "score": 0.9, + "content": "G _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 119, + 506, + 135 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 131, + 144, + 146 + ], + "spans": [ + { + "bbox": [ + 104, + 131, + 144, + 146 + ], + "score": 1.0, + "content": "given by", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 141, + 362, + 174 + ], + "lines": [ + { + "bbox": [ + 249, + 141, + 362, + 174 + ], + "spans": [ + { + "bbox": [ + 249, + 141, + 362, + 174 + ], + "score": 0.94, + "content": "{ G _ { a } ^ { j } } ^ { * } ( x ^ { \\prime } ) = \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } .", + "type": "interline_equation", + "image_path": "3bbac7548f876e8f96ac7e5eefec823656a640d60cd989ccd295c2ad6e76c27a.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 249, + 141, + 362, + 157.5 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 249, + 157.5, + 362, + 174.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 183, + 380, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 380, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 174, + 198 + ], + "score": 1.0, + "content": "Proof. For fixed", + "type": "text" + }, + { + "bbox": [ + 175, + 184, + 183, + 194 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 181, + 380, + 198 + ], + "score": 1.0, + "content": ", the optimal prediction probabilities are given by", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 139, + 200, + 471, + 237 + ], + "lines": [ + { + "bbox": [ + 139, + 200, + 471, + 237 + ], + "spans": [ + { + "bbox": [ + 139, + 200, + 471, + 237 + ], + "score": 0.93, + "content": "G _ { a } ^ { * } = \\arg \\operatorname* { m a x } _ { G _ { a } } \\sum _ { j = 1 } ^ { K } \\int _ { x ^ { \\prime } } \\log ( G _ { a } ^ { j } ( x ^ { \\prime } ) ) P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) d x ^ { \\prime } \\qquad \\mathrm { s u b j e c t ~ t o } \\sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( x ^ { \\prime } ) = 1 .", + "type": "interline_equation", + "image_path": "e8a5a3b6df443b833d937b582e4b9710c1ec7517262e3eeeaa5a22ed16ddb79e.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 139, + 200, + 471, + 212.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 139, + 212.33333333333334, + 471, + 224.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 139, + 224.66666666666669, + 471, + 237.00000000000003 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 240, + 442, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 239, + 441, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 441, + 255 + ], + "score": 1.0, + "content": "Maximising the value function pointwise and applying Lagrange multiplies, we get", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 257, + 437, + 293 + ], + "lines": [ + { + "bbox": [ + 174, + 257, + 437, + 293 + ], + "spans": [ + { + "bbox": [ + 174, + 257, + 437, + 293 + ], + "score": 0.94, + "content": "G _ { a } ^ { * } = \\arg \\operatorname* { m a x } _ { G _ { a } } \\sum _ { j = 1 } ^ { K } \\log ( G _ { a } ^ { j } ( x ^ { \\prime } ) ) P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) + \\lambda \\Bigg ( \\sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( x ^ { \\prime } ) - 1 \\Bigg ) .", + "type": "interline_equation", + "image_path": "2d5de6eb3aeede2e67d256d7fb0332c254121b4c257c1e223e9983d4722caaba.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 174, + 257, + 437, + 269.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 174, + 269.0, + 437, + 281.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 174, + 281.0, + 437, + 293.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 404, + 313 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 405, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 219, + 315 + ], + "score": 1.0, + "content": "Setting the derivative (w.r.t.", + "type": "text" + }, + { + "bbox": [ + 219, + 299, + 257, + 312 + ], + "score": 0.92, + "content": "G _ { a } ^ { j } { } ^ { * } ( x ^ { \\prime } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 296, + 339, + 315 + ], + "score": 1.0, + "content": "to 0 and solving for", + "type": "text" + }, + { + "bbox": [ + 339, + 299, + 374, + 312 + ], + "score": 0.95, + "content": "G _ { a } ^ { j } { } ^ { * } ( x ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 296, + 405, + 315 + ], + "score": 1.0, + "content": "we get", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 317, + 350, + 344 + ], + "lines": [ + { + "bbox": [ + 261, + 317, + 350, + 344 + ], + "spans": [ + { + "bbox": [ + 261, + 317, + 350, + 344 + ], + "score": 0.95, + "content": "{ G _ { a } ^ { j } } ^ { * } ( x ^ { \\prime } ) = - \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\lambda }", + "type": "interline_equation", + "image_path": "d535bf772a4383484cdcc75726bbd8b4493254c057261184bd8bec9529017ce8.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 261, + 317, + 350, + 344 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 103, + 346, + 507, + 369 + ], + "spans": [ + { + "bbox": [ + 103, + 346, + 134, + 369 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 352, + 142, + 362 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 346, + 348, + 369 + ], + "score": 1.0, + "content": "can now be solved for using the constraint to be", + "type": "text" + }, + { + "bbox": [ + 348, + 349, + 437, + 365 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\lambda = - \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 346, + 507, + 369 + ], + "score": 1.0, + "content": ". This gives the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 361, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 135, + 376 + ], + "score": 1.0, + "content": "result.", + "type": "text" + }, + { + "bbox": [ + 496, + 366, + 504, + 371 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 504, + 409 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "score": 1.0, + "content": "Proof. (of Theorem 1) By substituting the expression from Proposition 1 into the minimax game", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 397, + 290, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 243, + 410 + ], + "score": 1.0, + "content": "defined in Eq. 6, the objective for", + "type": "text" + }, + { + "bbox": [ + 243, + 398, + 251, + 407 + ], + "score": 0.85, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 397, + 290, + 410 + ], + "score": 1.0, + "content": "becomes", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 414, + 392, + 450 + ], + "lines": [ + { + "bbox": [ + 219, + 414, + 392, + 450 + ], + "spans": [ + { + "bbox": [ + 219, + 414, + 392, + 450 + ], + "score": 0.94, + "content": "\\operatorname* { m i n } _ { \\Phi } \\sum _ { j = 1 } ^ { K } \\mathbb { E } _ { x ^ { \\prime } \\sim P _ { j } ^ { \\Phi } } \\left[ \\log \\left( \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } \\right) \\right] .", + "type": "interline_equation", + "image_path": "9b2144766465cf6c74716cf0fb154657afb69f660116b20151d4635b2948070a.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 414, + 392, + 432.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 219, + 432.0, + 392, + 450.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 178, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 178, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 178, + 466 + ], + "score": 1.0, + "content": "We then note that", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 469, + 514, + 505 + ], + "lines": [ + { + "bbox": [ + 111, + 469, + 514, + 505 + ], + "spans": [ + { + "bbox": [ + 111, + 469, + 514, + 505 + ], + "score": 0.88, + "content": "\\sum _ { j = 1 } ^ { K } \\mathbb { E } _ { x ^ { \\prime } \\sim P _ { j } ^ { \\Phi } } \\bigg [ \\log \\bigg ( \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } \\bigg ) \\bigg ] + K \\log K = \\sum _ { j = 1 } ^ { K } \\Bigg ( \\mathbb { E } _ { x ^ { \\prime } \\sim P _ { j } ^ { \\Phi } } \\bigg [ \\log \\bigg ( \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } \\bigg ) \\bigg ] + \\log K \\Bigg )", + "type": "interline_equation", + "image_path": "6ad792127c3b6818733137894cdbe6437d1186236cdce3a017ad3e32210bf22e.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 111, + 469, + 514, + 481.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 111, + 481.0, + 514, + 493.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 111, + 493.0, + 514, + 505.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 308, + 518, + 479, + 606 + ], + "lines": [ + { + "bbox": [ + 308, + 518, + 479, + 606 + ], + "spans": [ + { + "bbox": [ + 308, + 518, + 479, + 606 + ], + "score": 0.58, + "content": "\\begin{array} { l } { { \\displaystyle = \\sum _ { j = 1 } ^ { K } \\mathbb { E } _ { x ^ { \\prime } \\sim P _ { j } ^ { \\Phi } } \\left[ \\log \\left( \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } \\right) \\right] } } \\\\ { { \\displaystyle = \\sum _ { j = 1 } ^ { K } K L \\biggl ( P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) \\biggl \\| \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) \\biggr ) } } \\\\ { { \\displaystyle = K \\cdot J S D ( P _ { 1 } ^ { \\Phi } , . . . , P _ { K } ^ { \\Phi } ) } } \\end{array}", + "type": "interline_equation", + "image_path": 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"inline_equation" + }, + { + "bbox": [ + 169, + 609, + 325, + 623 + ], + "score": 1.0, + "content": "is the Kullback-Leibler divergence and", + "type": "text" + }, + { + "bbox": [ + 326, + 610, + 378, + 622 + ], + "score": 0.93, + "content": "J S D ( \\cdot , . . . , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 609, + 507, + 623 + ], + "score": 1.0, + "content": "is the multi-distribution Jensen-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 621, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 397, + 650 + ], + "score": 1.0, + "content": "non-negative and 0 if and only if all distributions are equal, we have that", + "type": "text" + }, + { + "bbox": [ + 106, + 621, + 284, + 633 + ], + "score": 1.0, + "content": "Shannon Divergence (Li et al., 2018). Since", + "type": "text" + }, + { + "bbox": [ + 284, + 622, + 320, + 633 + ], + "score": 0.91, + "content": "K \\log K", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "is a constant and the multi-distribution JSD is", + "type": "text" + }, + { + "bbox": [ + 398, + 632, + 461, + 645 + ], + "score": 0.93, + "content": "P _ { 1 } ^ { \\Phi } = \\ldots = P _ { K } ^ { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 626, + 469, + 650 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 632, + 505, + 643 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "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": "title", + "bbox": [ + 107, + 80, + 244, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 245, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 245, + 96 + ], + "score": 1.0, + "content": "D PROOF OF THEOREM 1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 270, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 270, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 270, + 120 + ], + "score": 1.0, + "content": "We first prove the following proposition.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 105, + 270, + 120 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 120, + 504, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 210, + 135 + ], + "score": 1.0, + "content": "Proposition 1. For fixed", + "type": "text" + }, + { + "bbox": [ + 211, + 121, + 219, + 131 + ], + "score": 0.73, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 119, + 236, + 135 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 237, + 120, + 285, + 133 + ], + "score": 0.93, + "content": "x ^ { \\prime } = \\Phi ( \\bar { \\mathbf { h } } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 119, + 474, + 135 + ], + "score": 1.0, + "content": ". Then the optimal prediction probabilities of", + "type": "text" + }, + { + "bbox": [ + 474, + 123, + 487, + 132 + ], + "score": 0.9, + "content": "G _ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 119, + 506, + 135 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 131, + 144, + 146 + ], + "spans": [ + { + "bbox": [ + 104, + 131, + 144, + 146 + ], + "score": 1.0, + "content": "given by", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 104, + 119, + 506, + 146 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 249, + 141, + 362, + 174 + ], + "lines": [ + { + "bbox": [ + 249, + 141, + 362, + 174 + ], + "spans": [ + { + "bbox": [ + 249, + 141, + 362, + 174 + ], + "score": 0.94, + "content": "{ G _ { a } ^ { j } } ^ { * } ( x ^ { \\prime } ) = \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } .", + "type": "interline_equation", + "image_path": "3bbac7548f876e8f96ac7e5eefec823656a640d60cd989ccd295c2ad6e76c27a.jpg" + } + ] + } + ], + "index": 4.5, + "virtual_lines": [ + { + "bbox": [ + 249, + 141, + 362, + 157.5 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 249, + 157.5, + 362, + 174.0 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 183, + 380, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 181, + 380, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 174, + 198 + ], + "score": 1.0, + "content": "Proof. For fixed", + "type": "text" + }, + { + "bbox": [ + 175, + 184, + 183, + 194 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 181, + 380, + 198 + ], + "score": 1.0, + "content": ", the optimal prediction probabilities are given by", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 181, + 380, + 198 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 139, + 200, + 471, + 237 + ], + "lines": [ + { + "bbox": [ + 139, + 200, + 471, + 237 + ], + "spans": [ + { + "bbox": [ + 139, + 200, + 471, + 237 + ], + "score": 0.93, + "content": "G _ { a } ^ { * } = \\arg \\operatorname* { m a x } _ { G _ { a } } \\sum _ { j = 1 } ^ { K } \\int _ { x ^ { \\prime } } \\log ( G _ { a } ^ { j } ( x ^ { \\prime } ) ) P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) d x ^ { \\prime } \\qquad \\mathrm { s u b j e c t ~ t o } \\sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( x ^ { \\prime } ) = 1 .", + "type": "interline_equation", + "image_path": "e8a5a3b6df443b833d937b582e4b9710c1ec7517262e3eeeaa5a22ed16ddb79e.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 139, + 200, + 471, + 212.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 139, + 212.33333333333334, + 471, + 224.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 139, + 224.66666666666669, + 471, + 237.00000000000003 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 240, + 442, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 239, + 441, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 441, + 255 + ], + "score": 1.0, + "content": "Maximising the value function pointwise and applying Lagrange multiplies, we get", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 239, + 441, + 255 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 257, + 437, + 293 + ], + "lines": [ + { + "bbox": [ + 174, + 257, + 437, + 293 + ], + "spans": [ + { + "bbox": [ + 174, + 257, + 437, + 293 + ], + "score": 0.94, + "content": "G _ { a } ^ { * } = \\arg \\operatorname* { m a x } _ { G _ { a } } \\sum _ { j = 1 } ^ { K } \\log ( G _ { a } ^ { j } ( x ^ { \\prime } ) ) P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) + \\lambda \\Bigg ( \\sum _ { j = 1 } ^ { K } G _ { a } ^ { j } ( x ^ { \\prime } ) - 1 \\Bigg ) .", + "type": "interline_equation", + "image_path": "2d5de6eb3aeede2e67d256d7fb0332c254121b4c257c1e223e9983d4722caaba.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 174, + 257, + 437, + 269.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 174, + 269.0, + 437, + 281.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 174, + 281.0, + 437, + 293.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 299, + 404, + 313 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 405, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 219, + 315 + ], + "score": 1.0, + "content": "Setting the derivative (w.r.t.", + "type": "text" + }, + { + "bbox": [ + 219, + 299, + 257, + 312 + ], + "score": 0.92, + "content": "G _ { a } ^ { j } { } ^ { * } ( x ^ { \\prime } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 296, + 339, + 315 + ], + "score": 1.0, + "content": "to 0 and solving for", + "type": "text" + }, + { + "bbox": [ + 339, + 299, + 374, + 312 + ], + "score": 0.95, + "content": "G _ { a } ^ { j } { } ^ { * } ( x ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 296, + 405, + 315 + ], + "score": 1.0, + "content": "we get", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 296, + 405, + 315 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 317, + 350, + 344 + ], + "lines": [ + { + "bbox": [ + 261, + 317, + 350, + 344 + ], + "spans": [ + { + "bbox": [ + 261, + 317, + 350, + 344 + ], + "score": 0.95, + "content": "{ G _ { a } ^ { j } } ^ { * } ( x ^ { \\prime } ) = - 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This gives the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 361, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 135, + 376 + ], + "score": 1.0, + "content": "result.", + "type": "text" + }, + { + "bbox": [ + 496, + 366, + 504, + 371 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 103, + 346, + 507, + 376 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 504, + 409 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 400 + ], + "score": 1.0, + "content": "Proof. (of Theorem 1) By substituting the expression from Proposition 1 into the minimax game", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 397, + 290, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 243, + 410 + ], + "score": 1.0, + "content": "defined in Eq. 6, the objective for", + "type": "text" + }, + { + "bbox": [ + 243, + 398, + 251, + 407 + ], + "score": 0.85, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 397, + 290, + 410 + ], + "score": 1.0, + "content": "becomes", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 385, + 505, + 410 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 219, + 414, + 392, + 450 + ], + "lines": [ + { + "bbox": [ + 219, + 414, + 392, + 450 + ], + "spans": [ + { + "bbox": [ + 219, + 414, + 392, + 450 + ], + "score": 0.94, + "content": "\\operatorname* { m i n } _ { \\Phi } \\sum _ { j = 1 } ^ { K } \\mathbb { E } _ { x ^ { \\prime } \\sim P _ { j } ^ { \\Phi } } \\left[ \\log \\left( \\frac { P _ { j } ^ { \\Phi } ( x ^ { \\prime } ) } { \\sum _ { i = 1 } ^ { K } P _ { i } ^ { \\Phi } ( x ^ { \\prime } ) } \\right) \\right] .", + "type": "interline_equation", + "image_path": "9b2144766465cf6c74716cf0fb154657afb69f660116b20151d4635b2948070a.jpg" + } + ] + } + ], + "index": 20.5, + "virtual_lines": [ + { + "bbox": [ + 219, + 414, + 392, + 432.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 219, + 432.0, + 392, + 450.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 178, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 178, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 178, + 466 + ], + "score": 1.0, + "content": "We then note that", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 455, + 178, + 466 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 469, + 514, + 505 + ], + "lines": [ + { + "bbox": 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HyperparameterSearch range
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HyperparameterSearch range
Iterations of Hyperparameter Search20
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Minibatch size256,512,1024
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Dropout Rate0.1, 0.2, 0.3, 0.4, 0.5
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HyperparameterSearch range
Iterations of Hyperparameter Search50
Learning rate0.01,0.005,0.001
Minibatch size64,128,256
RNN state size0.5C,1C,2C, 3C,4C
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HyperparameterSearch range
Iterations of Hyperparameter Search20
Learning rate0.01,0.001,0.0001
Minibatch size256,512,1024
RNN state size1C,2C,4C,8C,16C
Dropout Rate0.1, 0.2, 0.3, 0.4, 0.5
Max Gradient Norm0.5, 1.0, 2.0, 4.0
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HyperparameterSearch range
Iterations of Hyperparameter Search50
Learning rate0.01,0.001,0.0001
Minibatch size64,128,256
RNNhidden units0.5C,1C,2C, 3C,4C
FC hidden units0.5C, 1C, 2C, 3C, 4C
RNN dropout probability0.1, 0.2, 0.3, 0.4, 0.5
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HyperparameterSearch range encoderSearch range decoder
Iterations ofHyperparameter Search5030
Learning rate0.01,0.001, 0.00010.01,0.001, 0.0001
Minibatch size64,128,256256,512,1024
RNN hidden units0.5C,1C, 2C, 3C, 4CBalancing representation size of encoder
Balancing representation size0.5C,1C,2C,3C,4C0.5C,1C,2C,3C,4C
FC hidden units0.5R,1R,2R,3R,4R0.5R,1R,2R,3R,4R
RNN dropout probability0.1, 0.2, 0.3,0.4, 0.50.1,0.2, 0.3, 0.4, 0.5
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HyperparameterSearch range
Iterations of Hyperparameter Search50
Learning rate0.01,0.001,0.0001
Minibatch size64,128,256
RNNhidden units0.5C,1C,2C, 3C,4C
FC hidden units0.5C, 1C, 2C, 3C, 4C
RNN dropout probability0.1, 0.2, 0.3, 0.4, 0.5
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C is the size of the input and R is the size of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 361, + 223, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 223, + 374 + ], + "score": 1.0, + "content": "the balancing representation.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "table_body", + "bbox": [ + 110, + 393, + 501, + 484 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 110, + 393, + 501, + 484 + ], + "spans": [ + { + "bbox": [ + 110, + 393, + 501, + 484 + ], + "score": 0.98, + "html": "
HyperparameterSearch range encoderSearch range decoder
Iterations ofHyperparameter Search5030
Learning rate0.01,0.001, 0.00010.01,0.001, 0.0001
Minibatch size64,128,256256,512,1024
RNN hidden units0.5C,1C, 2C, 3C, 4CBalancing representation size of encoder
Balancing representation size0.5C,1C,2C,3C,4C0.5C,1C,2C,3C,4C
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Note that in order to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 301, + 446 + ], + "score": 1.0, + "content": "account for numerical instability (two outcomes", + "type": "text" + }, + { + "bbox": [ + 301, + 433, + 322, + 445 + ], + "score": 0.91, + "content": "Y _ { t + \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "having very similar values), we consider two", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 444, + 354, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 255, + 456 + ], + "score": 1.0, + "content": "outcomes the same if they are within", + "type": "text" + }, + { + "bbox": [ + 255, + 444, + 297, + 455 + ], + "score": 0.89, + "content": "\\epsilon = 0 . 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 444, + 354, + 456 + ], + "score": 1.0, + "content": "of each other.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 376, + 506, + 456 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 389, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 390, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 390, + 96 + ], + "score": 1.0, + "content": "M RESULTS ON FACTUAL PREDICTION ON MIMIC III", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "Using the Medical Information Mart for Intensive Care (MIMIC III) (Johnson et al., 2016) database", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "consisting of electronic health records from patients in the ICU, we also show how the CRN can", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 504, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 504, + 140 + ], + "score": 1.0, + "content": "be used on a real medical dataset. From MIMIC III we extracted the patients on antibiotics, with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "score": 1.0, + "content": "trajectories up to 30 timesteps, thus obtaining a dataset with 3487 patients. For each patient, we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 505, + 162 + ], + "score": 1.0, + "content": "extracted 25 patient covariates including lab tests and vital signs measured over time, as well as static", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 267, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 267, + 174 + ], + "score": 1.0, + "content": "patient features such as age and gender.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 177, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "We used a binary treatment at each timestep indicating whether the patient was administered antibi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "otics or not. Note that for the longitudinal covariates we used aggregate value for each day since the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "score": 1.0, + "content": "ICU admission. The reason for this is because antibiotic treatment is decided daily for the patient.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "We split the dataset into 2826/313/348 patients for training, validation and testing respectively. We", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "performed hyperparameter optimization on the validation patient set, using the search ranges in Table", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 446, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 446, + 245 + ], + "score": 1.0, + "content": "5 and we again selected hyperparameters based on the error on the factual outcomes.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "We estimate the individualized effect of antibiotics assigned over time on the patient’s white blood", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "score": 1.0, + "content": "cell count. A high white blood cell count is associated with severe illness and poor outcome for ICU", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "patients (Waheed et al., 2003). Antibiotic administration in the ICU aims to reduce the white blood", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "cell count. However, the effectiveness of the antibiotics treatment in reducing the white blood cell", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "count is highly dependent on the time they are administered with respect to the history of the patient", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "score": 1.0, + "content": "covariates. In this context we again have time-dependent confounders: the patient features change", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "over time and are affected by the previous administration of antibiotics. Moreover, the history of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "the patient features also determines antibiotics administration and affects future patient outcomes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 338, + 258, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 258, + 349 + ], + "score": 1.0, + "content": "(De Bus et al., 2018; Ali et al., 2019).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "In Table 10 we report the root mean squared error for factual prediction of the patients’ white blood", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 283, + 378 + ], + "score": 1.0, + "content": "cell count for multiple prediction horizons", + "type": "text" + }, + { + "bbox": [ + 283, + 367, + 290, + 375 + ], + "score": 0.67, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 365, + 506, + 378 + ], + "score": 1.0, + "content": ". 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This further medical validation is outside the scope of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 517, + 150, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 150, + 532 + ], + "score": 1.0, + "content": "this paper.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + } + ], + "page_idx": 27, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "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 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 389, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 390, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 390, + 96 + ], + "score": 1.0, + "content": "M RESULTS ON FACTUAL PREDICTION ON MIMIC III", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "Using the Medical Information Mart for Intensive Care (MIMIC III) (Johnson et al., 2016) database", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "consisting of electronic health records from patients in the ICU, we also show how the CRN can", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 504, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 504, + 140 + ], + "score": 1.0, + "content": "be used on a real medical dataset. From MIMIC III we extracted the patients on antibiotics, with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "score": 1.0, + "content": "trajectories up to 30 timesteps, thus obtaining a dataset with 3487 patients. For each patient, we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 505, + 162 + ], + "score": 1.0, + "content": "extracted 25 patient covariates including lab tests and vital signs measured over time, as well as static", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 267, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 267, + 174 + ], + "score": 1.0, + "content": "patient features such as age and gender.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 106, + 506, + 174 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 177, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 190 + ], + "score": 1.0, + "content": "We used a binary treatment at each timestep indicating whether the patient was administered antibi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "otics or not. Note that for the longitudinal covariates we used aggregate value for each day since the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "score": 1.0, + "content": "ICU admission. The reason for this is because antibiotic treatment is decided daily for the patient.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 210, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 505, + 223 + ], + "score": 1.0, + "content": "We split the dataset into 2826/313/348 patients for training, validation and testing respectively. We", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "performed hyperparameter optimization on the validation patient set, using the search ranges in Table", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 233, + 446, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 233, + 446, + 245 + ], + "score": 1.0, + "content": "5 and we again selected hyperparameters based on the error on the factual outcomes.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 177, + 506, + 245 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "We estimate the individualized effect of antibiotics assigned over time on the patient’s white blood", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 272 + ], + "score": 1.0, + "content": "cell count. A high white blood cell count is associated with severe illness and poor outcome for ICU", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 271, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 506, + 284 + ], + "score": 1.0, + "content": "patients (Waheed et al., 2003). Antibiotic administration in the ICU aims to reduce the white blood", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "cell count. However, the effectiveness of the antibiotics treatment in reducing the white blood cell", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "count is highly dependent on the time they are administered with respect to the history of the patient", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 318 + ], + "score": 1.0, + "content": "covariates. In this context we again have time-dependent confounders: the patient features change", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "over time and are affected by the previous administration of antibiotics. Moreover, the history of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "the patient features also determines antibiotics administration and affects future patient outcomes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 338, + 258, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 258, + 349 + ], + "score": 1.0, + "content": "(De Bus et al., 2018; Ali et al., 2019).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 249, + 506, + 349 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "In Table 10 we report the root mean squared error for factual prediction of the patients’ white blood", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 283, + 378 + ], + "score": 1.0, + "content": "cell count for multiple prediction horizons", + "type": "text" + }, + { + "bbox": [ + 283, + 367, + 290, + 375 + ], + "score": 0.67, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 365, + 506, + 378 + ], + "score": 1.0, + "content": ". 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%c=5,γr=5% =5,γr=01 Yc =0,γr =5
TCRNRMSNMSMCRN RMSNMSMCRNRMSNMSM
Normalized RMSE2.43%3.16%6.75%1.08% 1.35%3.68%1.54%1.59%3.23%
2.83%3.95%7.65% 1.21%1.81%3.84%1.81%2.25%3.52%
3４56７3.18%4.37% 7.95%1.33%2.13%3.91%2.03%2.71%3.63%
3.51%5.61%8.19%1.42% 2.41%3.97%2.23%2.73%3.71%
3.93%6.21% 8.52%1.53%2.43%4.04%2.43%2.88%3.79%
Treatment Accuracy83.1%75.3%73.9%83.2% 78.6%77.1%92.9%87.3%74.9%
82.5%74.1%68.5%81.3%77.7% 73.9%85.7%83.8%74.1%
3４56７73.5%72.7%63.2%78.3% 77.2%72.3%83.8%82.1%72.8%
69.4%66.7%62.7%79.5% 76.3%71.8%78.6%69.7%64.5%
71.2%68.8%62.4%72.7% 71.8%71.6%71.9%69.3%61.2%
Treatment Timing Accuracy79.6%78.1%67.6%80.5%76.8% 77.5%79.8%75.7%60.6%
73.9%70.3%63.1%79.0%77.2%73.4% 75.4%71.4%58.2%
69.8%68.6%62.4%78.3%73.3%63.6%66.9%31.3%29.5%
3456７66.9%66.2%62.6%73.5%72.1%63.9%65.8%24.2%15.5%
64.5%63.6%62.2%70.6%57.4%44.2%63.9%25.6%12.5%
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HyperparameterSearch range encoderSearch range decoder
Iterations ofHyperparameter Search5030
Learning rate0.01,0.001, 0.00010.01,0.001, 0.0001
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RNN hidden units0.5C,1C, 2C, 3C, 4CBalancing representation size of encoder
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FC hidden units0.5R,1R,2R,3R,4R0.5R,1R,2R,3R,4R
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HyperparameterSearch range
Iterations of Hyperparameter Search50
Learning rate0.01,0.001,0.0001
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equivalent to special-purpose computers. Their inter-neuronal connection weights represent the learnt Neural Program that instructs the networks on how to compute the data. However, without storing Neural Programs, they are restricted to only one, overwriting learnt programs when trained on new data. Here we design Neurocoder, a new class of generalpurpose neural networks in which the neural network “codes” itself in a dataresponsive way by composing relevant programs from a set of shareable, modular programs stored in external memory. For the first time, a Neural Program is efficiently treated as a datum in memory. Integrating Neurocoder into current neural architectures, we demonstrate new capacity to learn modular programs, reuse simple programs to build complex ones, handle pattern shifts and remember old programs as new ones are learnt, and show substantial performance improvement in solving object recognition, playing video games and continual learning tasks. + +# 14 1 Introduction + +15 From its inception in 1943 until recently, the fundamental architectures of Artificial Neural Net +16 works remained largely unchanged - a program is executed by passing data through a network of +17 artificial neurons whose inter-neuronal connection weights are learnt through training with data. +18 These inter-neuronal connection weights, or Neural Programs, correspond to a program in modern +19 computers [32]. Memory Augmented Neural Networks (MANN) are an innovative solution allow +20 ing networks to access external memory for manipulating data [11, 12]. But they were still unable +21 to store Neural Programs in such external memory, and this severely limits machine learning. Stor +22 ing inter-neuronal connection weights only in their network does not permit modular separation of +23 Neural programs and is analogous to a computer with one fixed program. Recent works introduce +24 conditional computation via adjusting or activating parts of a network in an input-dependent manner +25 [39, 33, 4, 13, 28], but networks remain monolithic. Current networks forget when retrained, old +26 inter-neuronal connection weights are merged with new ones or erased. +27 The brain is modular, not a monolithic system [8, 6]. Neuroscience research indicates that the brain is +28 divided into functional modules [19, 7, 9]. If the neural program for each module is kept in separate +29 networks, networks proliferate. Modular neural networks, another form of conditional computation, +30 combine the output of multiple expert networks, but as the experts grow, the networks grow drasti +31 cally [20, 14, 35, 29]. This requires huge computational storage and introduces redundancy as these +32 experts do not share common basic programs. +33 A pathway out of this bind is to keep such basic programs in memory and combine them as required. +34 This brings neural networks towards modern general-purpose computers that use the stored-program +35 principle [37, 40] to efficiently access reusable programs in external memory. Here we show how +36 Neurocoder, a new neural framework, introduces a new class of general-purpose conditional compu +37 tation machines in which a neural network can be “coded” in an input-dependent manner. Efficient +38 decomposition of Neural Programs creates shareable modular components that can reconstruct the +39 whole program space. These components change their “shapes” based on training and are stored +40 in an external Program Memory. Then, in a data-responsive way, a Program Controller retrieves +41 relevant components to build the Neural Program. This is analogous to shape-shifting Lego bricks +42 that can be reused to build unlimited shapes and structures (See Appendix Fig. 4). +43 Using adaptive modular components vastly increases the learning capacity of the neural network +44 by allowing re-utilisation of parameters, effectively curbing network growth as programs increase. +45 More importantly, unlike pre-defined sub-networks or modules [20, 1] that combine at activation +46 level, the construction of our modular components is dynamic and performed on the weight space. +47 The Neural Program construction is learnt through training via traditional backpropagation [30] as +48 the architecture is end-to-end differentiable. + +![](images/e389c763e9b4533f0b062c87f24ebd5da2780f7e18d1c6818a55a57ce1880a0c.jpg) +Figure 1: Neurocoder (a) The Main Network uses a working program to compute the output for the input. Here only the final layer of the Main Network is adaptively loaded with the working program $( I )$ . Other layers use traditional Neural Programs as connection weights (fixed-after-training). (b) The Program Controller’s composition network controls access to the Program Memory, emitting queries and interpolating gate control signals in response to the input (2). It then performs recurrent multi-head program attention to the Program Status (3), triggering attention weights to the Singular Programs (4). The attended Singular Programs form an active program using low-rank approximation (5). Residual program produced by the Program Controller’s integration network (6) plus the active program derives the working program. (c) The Program Memory stores the representations (singular programs) required to reconstruct the active program to be used by the Program Controller. Access is controlled through the Program Status including keys $( k )$ , and slot usage $( m )$ that are updated during the training and computation (7). + +# 49 2 Methods + +# 2.1 System overview + +51 A Neurocoder is a neural network (Main Network) coupled to an external Program Memory through +52 a Program Controller. The working program of the Main Network processes the input data to pro +53 duce the output. This working program is “coded” by the Program Controller by creating an input +54 dependent active program from the Program Memory (Fig. 1). The following gives a high-level +55 description of the Neurocoder framework and then the details. + +# 56 Neurocoder stores Singular Value Decomposition of Neural Programs in Program Memory + +57 The Neural Program needs to be stored efficiently in Program Memory. This is challenging as there +58 may be millions of inter-neuronal connection weights, thus storing them directly ([22]) is grossly +59 inefficient. Instead, the Neurocoder forms the basis of a subspace spanned by Neural Programs and +60 stores the singular values and vectors of this subspace in memory slots of the Program Memory +61 (hereafter referred to as singular programs). Based on the input, relevant singular programs are +62 retrieved, a new program is reconstructed and then loaded in the Main Network to process the input. +63 This representational choice significantly reduces the number of stored elements and allows each +64 singular program to effectively represent a unitary function of the active program. + +65 The active program matrix $\mathbf { P }$ can be composed by standard low-rank approximation as + +$$ +\mathbf { P } = \mathbf { U S V } ^ { \mathbf { T } } = \sum _ { n } ^ { r _ { m } } \sigma _ { n } u _ { n } v _ { n } ^ { \top } +$$ + +66 where $\mathbf { U }$ and $\mathbf { V }$ are matrices of the left and right singular vectors, and S the matrix of singular values. +67 $r _ { m }$ is the total number of components we want to retrieve. $\left\{ \sigma _ { n } \right\} _ { n = 1 } ^ { r _ { m } }$ is the attended singular values, +68 {un} mn= and $\{ v _ { n } \} _ { n = 1 } ^ { r _ { m } }$ the attended singular vectors of S, U, and $\mathbf { V }$ , respectively. The Program +69 Memory is crafted as three singular program memories $\{ { \bf M } _ { U } , { \bf M } _ { V } , { \bf M } _ { S } \}$ –each of their memory +70 slot stores a singular component or singular program. The process “codes” the active program +71 using singular programs from the program memories. The coding is conditioned on input $x _ { t }$ , yet +72 we drop index $t$ for notation simplification and leave the details on the computation of $\sigma _ { n } , u _ { n } , v _ { n }$ in +73 Sec. 2.2. +74 The Program Memory also maintains the status for each singular program in terms of access and +75 usage. To access a singular program, program keys $( k )$ are used. These keys are low-dimensional +76 vectors that represent the singular program function and computed by a neural network that ef +77 fectively compresses the singular program. The program usage $( m )$ measures memory utilisation, +78 recording how much a memory slot is used in constructing a program. The components of the +79 Program Memory are summarised in Fig. 1 (c). +80 Recurrent multi-head program attention mechanisms for program storage and retrieval +81 Neural networks use the concept of differentiable attention to access memory [11, 2]. This de +82 fines a weighting distribution over the memory slots essentially weighting the degree to which each +83 memory slot participates in a read or write operation. This is unlike conventional computers that use +84 a unique address to access a single memory slot. +85 Here we use two kinds of attention. First is content-based attention [11, 12] to ensure that the singu +86 lar program is selected based on its functionality and the data input. This is achieved by producing +87 a query vector based on the input and comparing it to the program keys $( k )$ using cosine similarity. +88 Higher cosine similarity scores indicate higher attention weights to the singular programs associated +89 with those program keys. Second, to encourage better memory utilisation, higher attention weights +90 are assigned to slots with lower program usage $( m )$ through usage-based attention [12, 31]. The +91 attention weights from the two schemas are then combined using interpolating gates to compose the +92 final attention weights to the Program Memory. +93 We adapt multi-head attention [11, 38] that applies multiple attentions in parallel to retrieve $H$ singu +94 lar components. Besides, we introduce a recurrent attention mechanism, in which multi-head access +95 is performed recurrently in $J$ steps. The $j$ -th set of $H$ retrieved components is conditioned on the +96 previous ones. This recurrent, multi-head attention allows the composition network to incrementally +97 search for optimal components for building relevant active programs. + +# 98 Neurocoder learns to “code” a relevant working program via training + +99 The structure of the Program Memory and the role of the Program Controller facilitates the au +100 tomatic construction of working programs via training. The Program Controller controls memory +101 access through its composition network that creates the attention weight defining how to weight the +102 singular programs in the memories. A weighted summation of the singular programs results in the +103 attended singular program. Applying the recurrent multi-head attention described earlier, multiple +104 attended singular programs are retrieved to construct an active program (Eq. 1). Then the Program +105 Controller generates a residual program using its integration network, adding to the active program +106 to produce the working program of the Main Network. This addition enables creation of flexible +107 higher-rank working programs, which compensates for the low-rank coding process. The structure +108 of the Program Controller is illustrated in Fig. 1 (b). +109 The singular programs are trained to represent unitary functions necessary for any computation +110 whilst the composition and integration networks are trained to compose the relevant programs for +111 the considering task. As such, beside minimising the task loss, we enforce orthogonality of stored +112 singular vectors by minimising $\mathcal { L } _ { o } = \mathbf { M } _ { U } \mathbf { M } _ { U } ^ { \top } - \mathbf { \bar { I } } + \mathbf { M } _ { V } \mathbf { M } _ { V } ^ { \top } - \mathbf { I }$ . The parameters of the networks, +113 and the stored singular programs are adjusted using gradient training via minimising the total loss + +$$ +\mathscr { L } = \mathcal { L } _ { t a s k } + a \mathcal { L } _ { o } +$$ + +where $\mathcal { L } _ { t a s k }$ represents the supervised task loss and $\mathcal { L } _ { o }$ represents the orthogonal loss weighted by a hyper-parameter $a$ to enforce orthogonality of the singular vectors. + +# 2.2 Attention mechanisms for Program Memory + +117 Here we ddenoted as $w _ { i n } ^ { u , v , \sigma }$ progra)–the a attention mechanintion weight to the $i$ ms used in this paper. Given -th slot of the singular progra $w _ { i n } ^ { u }$ , e $w _ { i n } ^ { v }$ , i $w _ { i n } ^ { \sigma }$ $\mathbf { M } _ { U }$ o, $\mathbf { M } _ { V }$ +119 , we retrieve the $n$ -th singular vector as follows, + +$$ +u _ { n } = \sum _ { i = 1 } ^ { P _ { u } } w _ { i n } ^ { u } \mathbf { M } _ { U } \left( i \right) +$$ + +$$ +v _ { n } = \sum _ { i = 1 } ^ { P _ { v } } w _ { i n } ^ { v } \mathbf { M } _ { V } \left( i \right) +$$ + +120 For the singular values, we need to enforce $\sigma _ { 1 } > \sigma _ { 2 } > . . . > \sigma _ { r _ { m } } > 0$ , thus we retrieve using + +$$ +\sigma _ { n } = \left\{ \begin{array} { l l } { \mathrm { s o f t p l u s } \left( \sum _ { i = 1 } ^ { P _ { s } } w _ { i n } ^ { \sigma } \mathbf { M } _ { S } \left( i \right) \right) } & { n = r _ { m } } \\ { \sigma _ { n + 1 } + \mathrm { s o f t p l u s } \left( \sum _ { i = 1 } ^ { P _ { s } } w _ { i n } ^ { \sigma } \mathbf { M } _ { S } \left( i \right) \right) } & { n < r _ { m } } \end{array} \right. +$$ + +121 Here, $P _ { u } , P _ { v }$ and $P _ { s }$ are the number of memory slots of $\mathbf { M } _ { U }$ , $\mathbf { M } _ { V }$ and $\mathbf { M } _ { S }$ , respectively. In this +122 paper, we set $P = P _ { u } = P _ { v } = P _ { s }$ as the number of memory slots of the Program Memory. We note +123 that thead ese notations are speci, and an attention step ed for some data input , hence the full notatio $x _ { t }$ and the hould be $n$ later maps to an attention To simplify notations, we +$h$ $j$ $w _ { t i j h } ^ { u , v , \sigma }$ +125 will drop $u , v , \sigma$ from now and describe the computation of a representative $w _ { t i j h }$ for any of the +126 three program memories in the following parts. + +# 127 Recurrent Access to the Program Memory via the composition network + +128 To perform program attention, the Program Controller employs a composition network (denoted +129 as $f _ { \theta } )$ ), which takes the current input $x _ { t }$ and produce program composition control signals $( \pmb { \xi } _ { t } ^ { p } )$ . +130 If $f _ { \theta }$ performs all attentions concurrently via multi-head attention as in [11, 38], it may lead to +131 program collapse [22]. To have a better control of the component formation and alleviate program +132 collapse, we propose to recurrently attend to the program memory. To this end, we implement $f _ { \theta }$ as +133 $\pmb { \xi } _ { t } ^ { p } = \left\{ \pmb { \xi } _ { t j } ^ { p } \right\} _ { j = 1 } ^ { J }$ al network (LST. At access step $j$ M [16]) and let it access the program memory , the recurrent network updates its hidden sta $J$ times, resultin and generates $\xi _ { t j } ^ { p }$ +135 using recurrent dynamics as + +$$ +{ \pmb { \xi } } _ { t j } ^ { p } , h _ { j } = f _ { \theta } \left( x _ { t } , h _ { j - 1 } \right) +$$ + +136 where $h _ { 0 }$ is initialized as zeros and $\xi _ { t j } ^ { p }$ is the program composition control signal at step $j$ that +137 depends on both on the input data $x _ { t }$ and the the previous state $h _ { j - 1 }$ . Particularly, the control signal +138 contains the queries and the interpolation gates for each head to compute the program attention +139 weight: $\pmb { \xi } _ { t j } ^ { p } = \{ q _ { t j h } , g _ { t i j h } \} _ { h = 1 } ^ { H }$ . Here, at each attention step, we perform multi-head attention with +140 $H$ as the number of attention heads and thus, each $\xi _ { t j } ^ { p }$ consists of $H$ pairs of queries and gates. +141 Hence, the total number of retrieved components $\boldsymbol { r } _ { m } = \boldsymbol { \bar { J } } \times \boldsymbol { H }$ and the index $n = j \times H + h$ . + +# 142 Attending to Programs by “Name” + +143 Inspired by the content-based attention mechanism for data memory [11], we use the query to look +144 for the singular programs. In computer programming, to find the appropriate program for some +145 computation, we often refer to the program description or at least the name of the program. Here, we +146 create the “name” for our neural programs by compressing the program content to a low-dimensional +147 key vector. As such, we employ a neural network $( f _ { \varphi } )$ to compute the program memory keys as + +$$ +k _ { i } = f _ { \varphi } \left( \mathbf { M } \left( i \right) \right) +$$ + +148 where $\boldsymbol { k } _ { i } \in \mathbb { R } ^ { K }$ and $i$ is the row index of the program memory. Here, $f _ { \varphi }$ learns to compress each +149 memory slot into a $K$ -dimensional vector. As the singular programs evolve, their keys get updated. +150 In this paper, we update the program keys after each learning iteration during training. +151 Finally the content-based program memory attention $c _ { t i j h }$ is computed using cosine distance be +152 tween the program keys $k _ { i }$ and the queries $q _ { t j h }$ as + +$$ +c _ { t i j h } = \mathrm { s o f t m a x } ^ { ( i ) } \left( \frac { q _ { t j h } \cdot k _ { i } } { \left| \left| q _ { t j h } \right| \right| \cdot \left| \left| k _ { i } \right| \right| } \right) +$$ + +# 153 Making Every Program Count + +154 Similarly to [12, 31], in addition to the content-based attention, we employ a least-used reading +155 strategy to encourage the Program Controller to assign different singular programs to different com +156 ponents. In particular, we calculate the memory usage for each program slot across attentions as +157 + +$$ +m _ { t i j h } = \operatorname* { m a x } _ { \tilde { j } \leq j } \left( w _ { t i \tilde { j } h } \right) +$$ + +158 Since we want to consider only $l _ { I }$ amongst $P$ memory slots that have smallest usages, let $\hat { m } _ { t j h } ^ { l _ { I } }$ +159 denote the value of the $l _ { I }$ -th smallest usage, then the least-used attention is computed as + +$$ +l _ { t i j h } = \left\{ \begin{array} { l l } { \underset { i } { \operatorname* { m a x } } \left( m _ { t i j h } \right) - m _ { t i j h } } & { ; m _ { t i j h } \leq \hat { m } _ { t j h } ^ { l _ { I } } } \\ { 0 } & { ; m _ { t i j h } > \hat { m } _ { t j h } ^ { l _ { I } } } \end{array} \right. +$$ + +160 The final program memory attention is computed as + +$$ +w _ { t i j h } = \mathrm { s i g m o i d } \left( g _ { t i j h } \right) c _ { t i j h } + \left( 1 - \mathrm { s i g m o i d } \left( g _ { t i j h } \right) \right) l _ { t i j h } +$$ + +161 Since the usage record are computed along the memory accesses, the multi-step Neurocoder utilises +162 this attention mechanism better than the single-step Neurocoder, creating different attention styles +163 (see Sec. 3.2). The composition the active program $\mathbf { P } _ { t }$ is illustrated in Appendix’s Fig. 5. + +# 2.3 Program Integration via the integration network + +165 Since the working program $\mathbf { P } _ { t }$ only contains top $r _ { m }$ principal components, it is low-rank and may +166 be not flexible enough for sophisticated computation. We propose to enhance $\mathbf { P } _ { t }$ with a residual +167 program $\mathbf { R } -$ a traditional connection weight trained as the integration network’s parameters, which +168 is constant after training w.r.t $t$ . The residual program represents the sum of the remaining less +169 important components. To this end, we suppress $\mathbf { R }$ with a multiplier that is smaller than $\sigma _ { t r _ { m } } .$ – the +170 smallest singular value of the main components - resulting in the integration formula + +$$ +W _ { t } = \mathbf { P } _ { t } + w _ { t } ^ { r } \sigma _ { t r _ { m } } \mathbf { R } +$$ + +171 where $w _ { t } ^ { r } = \mathrm { s i g m o i d } \left( f _ { \phi } \left( x _ { t } \right) \right)$ is an adaptive gating value that controls the contribution of the +172 residual program. $f _ { \phi }$ is the integration network in the Program Controller and hence, in our imple +173 mentation, the integration control signal sent by the Program Controller is $\lambda _ { t } ^ { p } = \{ w _ { t } ^ { r } , \sigma _ { t r _ { m } } \}$ . We +174 note that in our experiments, the program integration can be disabled ( $W _ { t }$ is directly set to $\mathbf { P } _ { t }$ ) to +175 prove the contribution of $\mathbf { P } _ { t }$ or reduce the number of parameters. The working program $W _ { t }$ is then +176 used by the Main Network to execute the input data $x _ { t }$ (see (Fig. 1 (a))). For example, with linear +177 classifier Main Network, the execution is $y _ { t } = x _ { t } W _ { t }$ . Appendix’s Table 2 summarises the notations +178 used for important parameters of Neurocoder. + +![](images/f992f503335ad0a3f753e1c8f03e87de2fd352b9c040fd798ec6c09805c37c77.jpg) +Figure 2: (a) MNIST test set classification error vs the number of steps $( J )$ in Neurocoder (blue), compared with a linear classifier (red). (b) 1st column: Digit images; Middle column: Single-step attention weights for 30 slots in $\mathbf { M } _ { U }$ (vertical axis) for first 3 singular vectors (horizontal axis) for each digit; Last column: Multi-step attention weights for 10 slots in $\mathbf { M } _ { U }$ (vertical axis) for first 3 singular vectors (horizontal axis). Multi-step attention is able to produce far more diverse patterns with fewer slots - 10 slots compared to single-step 30 slots. (c) Two attention patterns of singlestep Neurocoder. The binary decision tree derived from single-step Neurocoder’s attention patterns. The two patterns across components represent the decisions going up and down across the binary tree. Visualisation for (d) multi-step $J = 5$ , 20 memory slots) and (e) single-step $J = 1$ , 10 memory slots) cases showing while processing a sequence of the polynomial auto-regression task. The Neurocoder’s attentions to $\mathbf { M } _ { U }$ that form the first component of the active program are shown over sequence timesteps (upper) with Neurocoder’s $y _ { t }$ prediction (orange) and ground truth (blue) (lower). The vertical dash green lines separate polynomial chunks. Each chuck represents a local pattern, and thus ideally requires a specific active program to compute the input $x _ { t }$ . Although both predict well, only the multi-step Neurocoder discovers the chunk boundaries, assigning program attention to the first component in accordance with sequence changes. + +# 179 3 Results + +180 To demonstrate the flexibility of Neurocoder framework, we consider different learning paradigms: +181 instance-based, sequential, multi-task and continual learning. We do not focus on breaking perfor +182 mance records by augmenting state-of-the-art models with Neurocoder. Rather our inquiry is on +183 re-coding feed-forward layers with the Neurocoder’s programs and testing on varied data types to +184 demonstrate its intrinsic properties. For some experiments, we include ablation studies. +185 We compare the performance of diverse Main Networks (MN) with and without Neurocoder. We +186 also augment the Main Networks with other recent conditional computing methods, either modular +187 (sparse Mixture of Experts, Neural Stored-program Memory) or monolithic (HyperNets, FiLM) to +188 form stronger baselines across our experiments. In our experiments, we always apply Neurocoder +189 to all layers of multi-layer perceptrons (MLP) or just the final feed-forward layer of deep CNN +190 networks (LeNet, DenseNet, ResNet), RNNs (GRU, LSTM), MANN (NTM). Other competitors +191 such as MOE, NSM, HyperNet and FiLM are applied to the Main Networks in the same manner. + +![](images/7e25b32d05c9ae7db2df1ea7bbd96b41284179be7b8a5c34b278f599cab1c811.jpg) +Figure 3: Learning curves (mean and std. over 5 runs) on representative Atari 2600 games. All baselines are applied to the actor/critic networks in the A3C agent. + +# 192 3.1 Instance-based learning - Object Recognition + +We tested Neurocoder on instance-based learning through classical image classification tasks using MNIST [24] and CIFAR [21] datasets. The first experiment interpreted Neurocoder’s behaviour in classifying digits into 10 classes $( 0 - 9 )$ using linear classifier Main Network. With equivalent model size, Neurocoder using the novel recurrent attention surpasses the performance of the linear classifier [24] by up to $5 \%$ (Fig. 2 (a)). + +To differentiate the input, Neurocoder attends to different components of the active program to guide the decision-making process. Fig. 2 (b) shows single-step and multi-step attention to the first 3 singular vectors for each digit across memory slots. Multi-step attention produces richer patterns compared to single-step Neurocoder that manages only 2 attention weight patterns. + +Fig. 2 (c) illustrates how Neurocoder performs modular learning by showing the attention assignment for top 3 singular vectors as a binary decision tree. Digits under the same parental node share similar attention paths, and thereby similar active programs. Some digits look unique (e.g. 7) resulting in active programs composed of unique attention paths, discriminating themselves early in the decision tree. Some digits (e.g. 0 and 9) share the same attention pattern for the first 3 components and are thus unclassifiable. They can only be distinguished by considering more singular vectors. + +We integrated Neurocoder with deep networks - 5-layer LeNet and 100-layer DenseNet - and tested on CIFAR datasets. Neurocoder significantly outperformed the original Main Networks with performance gain $1 - 5 \%$ . Compared with recent conditional computing models such as sparse Mixture of Experts (MOE [35]) and Neural Stored-program Memory (NSM [22]), Neurocoder required a tenth of the number of parameters and performed better by up to $8 - 1 0 \%$ (see Appendix’s Table 3). + +# 3.2 Sequential learning - Adaption to sequence changes and game playing using reinforcement learning + +Recurrent neural networks (RNN) can learn from sequential data by updating the hidden states of the networks. However, this does not suffice when local patterns shift, as is often the case. We now demonstrate that Neurocoder helps RNNs overcome this limitation by composing diverse programs to handle sequence changes. + +Synthetic polynomial auto-regression We created a simple auto-regression task in which data points are sampled from polynomial function chunks that change over time. The Main Network is a strong RNN–Gated Recurrent Unit (GRU [5]). We found that GRU integrated with a single-step or multi-step Neurocoder converged much faster than all other baselines. The other conditional computing counterparts (HyperNet [13], FiLM [28]) adapt by re-scaling weights or activation of the GRU, which were shown inferior to our modular approach (Appendix’s Fig. 6). + +225 Visualising the first singular vector attention weights in $\mathbf { M } _ { U }$ , we find that the multi-step attention +226 Neurocoder changes its attention following polynomial changes - it attends to the same singular pro +227 gram when processing data from the same polynomial and alters attention for data from a different +228 polynomial (Fig. 2(d)). In contrast, the single-step Neurocoder only changes its attention when +229 there is a remarkable change in $y$ -coordinate values (Fig. 2(e)). Although single-step Neurocoder +230 converges well, it did not discover the underlying structure of the data, and thus underperformed +231 the multi-step Neurocoder. We hypothesise that when recurrence is employed, usage-based atten +232 tion takes effect, stipulating better memory utilisation and diverse attentions over timesteps. We ran +233 multi-step Neurocoder without usage-based attention. The results were worse than the full multi +234 step Neurocoder, which confirms our hypothesis (Appendix’s Fig. 6). + +
MethodMN (MLP[17])MN (MLP ours)NSMNeurocoder
Adam55.16±1.3853.55±1.2754.85±2.8158.46±0.46
Adagrad58.08±1.0657.83±2.7458.42±1.8762.28±4.03
L266.00±3.7364.37±2.4062.83±7.2169.89±1.72
SI64.76±3.0964.41±3.3664.36±2.9967.96±3.22
EWC58.85±2.5958.41±2.3758.12±3.2465.66±1.25
O-EWC57.33±1.4457.78±1.8458.55±3.4073.97±1.50
+ +Table 1: Incremental domain continual learning with Split MNIST. Final test accuracy (mean and std.) over 10 runs. + +Atari game reinforcement learning We used reinforcement learning as a further testbed to show the ability to adapt to environmental changes. We performed experiments on several Atari 2600 games [3] wherein the agent was implemented as the Asynchronous Advantage Actor-Critic (A3C [26]). In the Atari platform, agents are allowed to observe the screen snapshot of the games and act to earn the highest score. We augmented the A3C by employing Neurocoder’s working programs for feed-forward layers of the actor and critic networks, aiming to decompose the policy and value function into singular programs that were selected depending on the game state. + +242 Frostbite and Montezuma’s Revenge. These games are known to be challenging for A3C and other +243 algorithms [26]. We trained A3C and HyperNet-based A3C for over 300 million steps, yet these +244 models did not show any sign of learning, performing equivalently to random agents. For such com +245 plicated environments with sparse rewards, both the monolithic neural networks and the HyperNet’s +246 unstored fast-weights fail to learn (almost zero scores). In contrast, Neurocoder enabled A3C to +247 achieve from 1, 500 to 3, 000 scores on these environments (Fig. 3), confirming the importance of +248 decomposing a complex solution to smaller, simple stored programs. + +# 49 3.3 Multi-task learning - Solving mutliple algorithms simultenously + +Here we explore the modular learning capability of Neurocoder in multi-task setting. Inspired by algorithmic sequencing tasks [22], we created a challenging sequential multi-task benchmark wherein the input sequence is a series of sub-sequences from 4 algorithms: Copy, Repeat Copy, Associative Recall and Priority Sort [11]. Each sub-sequence, following a task identification vector, represents the input for each task. In each input sequence, $n$ tasks were sampled from the set of 4 algorithms randomly with replacement and the output sequences were created correspondingly. + +We trained a MANN–Neural Turing Machine (NTM [11]) Main Network with FiLM, HyperNet and our Neurocoder augmentation on sequences of $n = 4$ tasks, and tested with sequences of $n = 4$ and $n = 8$ tasks. Appendix’s Fig. 7 demonstrates that Neurocoder was performant in both test settings, not only achieving lowest error on $n = 4$ , but also being the only one generalised well to $n = 8$ scenario, which was unseen during training. + +# 3.4 Continual learning $-$ Learning tasks sequentially without catastrophic forgetting + +In continual learning, standard neural networks often suffer from “catastrophic forgetting” in which they cannot retain knowledge acquired from old tasks upon learning new ones [10]. Our Neurocoder offers natural mitigation of such catastrophic forgetting in neural networks by attending to different singular programs whilst learning different tasks. + +In this case, in addition to the Main Network, we examine several continual learning algorithms with and without Neurocoder. These algorithms, including Elastic Weight Consolidation (EWC [41]) and Synaptic Intelligence (SI [41]), work by regularising the loss function and thus can be easily combined with Neurocoder by modifying the loss $\mathcal { L } _ { t a s k }$ . We demonstrate that Neurocoder + +70 can improve these continual learning algorithms without requiring additional assumptions as in other +1 approaches [25, 36, 34] that either utilise task embedding or replay memory. + +Split MNIST We first considered the split MNIST dataset–a standard continual learning benchmark wherein the original MNIST was split into a 5 2-way classification tasks, consecutively presented to a Multi-layer Perceptron Main Network (MLP). We followed the benchmarking as in [17] in which various optimisers and state-of-the-art continual learning methods were examined under incremental task and domain scenarios. We measured the performance of the MLP versus Neurocoder and NSM under each continual learning method. In both scenarios, Neurocoder was compatible with all continual leaning methods, demonstrating superior performance over MLP and NSM with performance gain between 1 to $1 6 \%$ (see Appendix’s Table 5 and 1). + +280 Split CIFAR We verified the scalability of Neurocoder to more challenging datasets. We split +281 CIFAR datasets as in the split MNIST, resulting in 5-task 2-way split CIFAR10 and a 20-task 5-way +282 split CIFAR100. We used Main Network ResNet [15]–a very deep CNN architecture. +283 When we stressed the orthogonal loss $a = 1 0$ ) and used bigger program memory (100 slots), Neu +284 rocoder improved ResNet classification by $1 5 \%$ and $1 0 \%$ on CIFAR10 and CIFAR100, respectively. +285 When we integrated Neurocoder with Synaptic Intelligence (SI [41]), the performance was further +286 improved, maintaining a stable performance above $8 0 \%$ accuracy for CIFAR10 and outperforming +287 using SI alone by $1 0 \%$ for CIFAR100 (see Appendix’s Fig. 8). + +# 288 4 Discussion + +Our experiments demonstrate that Neurocoder is capable of re-coding Neural Programs in distinctive neural networks, amplifying their capabilities in diverse learning scenarios: instance-based, sequential, multi-task and continual learning. This consistently results in significant performance increase, and further creates novel robustness to pattern shift and catastrophic forgetting. This ability for each architecture to re-code itself is made possible without changing the way it is trained, or majorly increasing the number of parameters it needs to learn (see Appendix Table 7). + +The MNIST problem illustrates the reasoning process of Neurocoder when classifying digit images wherein its singular program assignment resembles a binary tree decision-making process - it shows how some singular programs are shared, others are not. The polynomial auto-regression problem highlights the importance of efficient memory utilisation in re-constructing the working program enabling discovery of hidden structures in sequential data. Training our framework with reinforcement learning, we enable neural agents to solve complex games wherein traditional methods fail or learn slowly. Neurocoder also works well with multi-task setting, as shown in the challenging multi-algorithm benchmark. Finally, continual learning problems show that Neurocoder mitigates catastrophic forgetting efficiently under different learning settings/algorithms. + +304 Our solution offers a single framework that is scalable and adaptable to various problems and learn +305 ing paradigms. Unlike previous attempts to employ a bank of separate big programs [20, 35, 22], +306 Neurocoder maintains only shareable, smaller components that can reconstruct the whole program +307 space, thereby heavily utilising the parameters and preventing the model from proliferating. We +308 note that Neurocoder is orthogonal to approaches employing tensor decomposition to reduce the +309 number of parameters or hasten the computation [27, 23]. Neurocoder composes rather than decom +310 pose the neural weights. Our aim is not only to enable efficient parameter usage, but also achieve +311 general-purpose computing power, outperforming other methods in numerous learning problems. +312 One limitation of this work is the number of additional hyperparameters, which prevents us from +313 fully tuning Neurocoder. Our research aims to add new capabilities to current neural networks to +314 improve their performance and make them robust in different learning scenario. Hence, we do +315 not see any intermediate negative societal impact. In future work, we will extend Neurocoder’s +316 application beyond feed-forward layers. It would be interesting to efficiently replace all neural layers +317 including CNN or Transformer by Neurocoder’s programs. 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[N/A] \ No newline at end of file diff --git a/parse/train/O0ow9RQW1nP/O0ow9RQW1nP_content_list.json b/parse/train/O0ow9RQW1nP/O0ow9RQW1nP_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..f5cfb3cea0bbeb259c5cc7308f87957583e1e439 --- /dev/null +++ b/parse/train/O0ow9RQW1nP/O0ow9RQW1nP_content_list.json @@ -0,0 +1,1216 @@ +[ + { + "type": "text", + "text": "Neurocoder: Learning General-Purpose Computation Using Stored Neural Programs ", + "text_level": 1, + "bbox": [ + 228, + 122, + 769, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ", + "bbox": [ + 423, + 226, + 578, + 281 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 318, + 535, + 334 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Artificial Neural Networks are functionally equivalent to special-purpose computers. Their inter-neuronal connection weights represent the learnt Neural Program that instructs the networks on how to compute the data. However, without storing Neural Programs, they are restricted to only one, overwriting learnt programs when trained on new data. Here we design Neurocoder, a new class of generalpurpose neural networks in which the neural network “codes” itself in a dataresponsive way by composing relevant programs from a set of shareable, modular programs stored in external memory. For the first time, a Neural Program is efficiently treated as a datum in memory. Integrating Neurocoder into current neural architectures, we demonstrate new capacity to learn modular programs, reuse simple programs to build complex ones, handle pattern shifts and remember old programs as new ones are learnt, and show substantial performance improvement in solving object recognition, playing video games and continual learning tasks. ", + "bbox": [ + 148, + 348, + 766, + 529 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "14 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 553, + 310, + 570 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "15 From its inception in 1943 until recently, the fundamental architectures of Artificial Neural Net \n16 works remained largely unchanged - a program is executed by passing data through a network of \n17 artificial neurons whose inter-neuronal connection weights are learnt through training with data. \n18 These inter-neuronal connection weights, or Neural Programs, correspond to a program in modern \n19 computers [32]. Memory Augmented Neural Networks (MANN) are an innovative solution allow \n20 ing networks to access external memory for manipulating data [11, 12]. But they were still unable \n21 to store Neural Programs in such external memory, and this severely limits machine learning. Stor \n22 ing inter-neuronal connection weights only in their network does not permit modular separation of \n23 Neural programs and is analogous to a computer with one fixed program. Recent works introduce \n24 conditional computation via adjusting or activating parts of a network in an input-dependent manner \n25 [39, 33, 4, 13, 28], but networks remain monolithic. Current networks forget when retrained, old \n26 inter-neuronal connection weights are merged with new ones or erased. \n27 The brain is modular, not a monolithic system [8, 6]. Neuroscience research indicates that the brain is \n28 divided into functional modules [19, 7, 9]. If the neural program for each module is kept in separate \n29 networks, networks proliferate. Modular neural networks, another form of conditional computation, \n30 combine the output of multiple expert networks, but as the experts grow, the networks grow drasti \n31 cally [20, 14, 35, 29]. This requires huge computational storage and introduces redundancy as these \n32 experts do not share common basic programs. \n33 A pathway out of this bind is to keep such basic programs in memory and combine them as required. \n34 This brings neural networks towards modern general-purpose computers that use the stored-program \n35 principle [37, 40] to efficiently access reusable programs in external memory. Here we show how \n36 Neurocoder, a new neural framework, introduces a new class of general-purpose conditional compu \n37 tation machines in which a neural network can be “coded” in an input-dependent manner. Efficient \n38 decomposition of Neural Programs creates shareable modular components that can reconstruct the \n39 whole program space. These components change their “shapes” based on training and are stored \n40 in an external Program Memory. Then, in a data-responsive way, a Program Controller retrieves \n41 relevant components to build the Neural Program. This is analogous to shape-shifting Lego bricks \n42 that can be reused to build unlimited shapes and structures (See Appendix Fig. 4). \n43 Using adaptive modular components vastly increases the learning capacity of the neural network \n44 by allowing re-utilisation of parameters, effectively curbing network growth as programs increase. \n45 More importantly, unlike pre-defined sub-networks or modules [20, 1] that combine at activation \n46 level, the construction of our modular components is dynamic and performed on the weight space. \n47 The Neural Program construction is learnt through training via traditional backpropagation [30] as \n48 the architecture is end-to-end differentiable. ", + "bbox": [ + 147, + 584, + 825, + 751 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 756, + 825, + 840 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 845, + 823, + 901 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/e389c763e9b4533f0b062c87f24ebd5da2780f7e18d1c6818a55a57ce1880a0c.jpg", + "image_caption": [ + "Figure 1: Neurocoder (a) The Main Network uses a working program to compute the output for the input. Here only the final layer of the Main Network is adaptively loaded with the working program $( I )$ . Other layers use traditional Neural Programs as connection weights (fixed-after-training). (b) The Program Controller’s composition network controls access to the Program Memory, emitting queries and interpolating gate control signals in response to the input (2). It then performs recurrent multi-head program attention to the Program Status (3), triggering attention weights to the Singular Programs (4). The attended Singular Programs form an active program using low-rank approximation (5). Residual program produced by the Program Controller’s integration network (6) plus the active program derives the working program. (c) The Program Memory stores the representations (singular programs) required to reconstruct the active program to be used by the Program Controller. Access is controlled through the Program Status including keys $( k )$ , and slot usage $( m )$ that are updated during the training and computation (7). " + ], + "image_footnote": [], + "bbox": [ + 186, + 88, + 818, + 367 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 578, + 825, + 661 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 667, + 825, + 750 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "49 2 Methods ", + "text_level": 1, + "bbox": [ + 155, + 777, + 277, + 794 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 System overview ", + "text_level": 1, + "bbox": [ + 174, + 814, + 326, + 828 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "51 A Neurocoder is a neural network (Main Network) coupled to an external Program Memory through \n52 a Program Controller. The working program of the Main Network processes the input data to pro \n53 duce the output. This working program is “coded” by the Program Controller by creating an input \n54 dependent active program from the Program Memory (Fig. 1). The following gives a high-level \n55 description of the Neurocoder framework and then the details. ", + "bbox": [ + 147, + 842, + 823, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "56 Neurocoder stores Singular Value Decomposition of Neural Programs in Program Memory ", + "text_level": 1, + "bbox": [ + 150, + 90, + 805, + 106 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "57 The Neural Program needs to be stored efficiently in Program Memory. This is challenging as there \n58 may be millions of inter-neuronal connection weights, thus storing them directly ([22]) is grossly \n59 inefficient. Instead, the Neurocoder forms the basis of a subspace spanned by Neural Programs and \n60 stores the singular values and vectors of this subspace in memory slots of the Program Memory \n61 (hereafter referred to as singular programs). Based on the input, relevant singular programs are \n62 retrieved, a new program is reconstructed and then loaded in the Main Network to process the input. \n63 This representational choice significantly reduces the number of stored elements and allows each \n64 singular program to effectively represent a unitary function of the active program. ", + "bbox": [ + 145, + 114, + 825, + 227 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "65 The active program matrix $\\mathbf { P }$ can be composed by standard low-rank approximation as ", + "bbox": [ + 147, + 231, + 743, + 247 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9f81cb11dec9c11b78cca6d7f7e9e1a93d9f3d74864311254010fb6362f61e22.jpg", + "text": "$$\n\\mathbf { P } = \\mathbf { U S V } ^ { \\mathbf { T } } = \\sum _ { n } ^ { r _ { m } } \\sigma _ { n } u _ { n } v _ { n } ^ { \\top }\n$$", + "text_format": "latex", + "bbox": [ + 401, + 252, + 594, + 292 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "66 where $\\mathbf { U }$ and $\\mathbf { V }$ are matrices of the left and right singular vectors, and S the matrix of singular values. \n67 $r _ { m }$ is the total number of components we want to retrieve. $\\left\\{ \\sigma _ { n } \\right\\} _ { n = 1 } ^ { r _ { m } }$ is the attended singular values, \n68 {un} mn= and $\\{ v _ { n } \\} _ { n = 1 } ^ { r _ { m } }$ the attended singular vectors of S, U, and $\\mathbf { V }$ , respectively. The Program \n69 Memory is crafted as three singular program memories $\\{ { \\bf M } _ { U } , { \\bf M } _ { V } , { \\bf M } _ { S } \\}$ –each of their memory \n70 slot stores a singular component or singular program. The process “codes” the active program \n71 using singular programs from the program memories. The coding is conditioned on input $x _ { t }$ , yet \n72 we drop index $t$ for notation simplification and leave the details on the computation of $\\sigma _ { n } , u _ { n } , v _ { n }$ in \n73 Sec. 2.2. \n74 The Program Memory also maintains the status for each singular program in terms of access and \n75 usage. To access a singular program, program keys $( k )$ are used. These keys are low-dimensional \n76 vectors that represent the singular program function and computed by a neural network that ef \n77 fectively compresses the singular program. The program usage $( m )$ measures memory utilisation, \n78 recording how much a memory slot is used in constructing a program. The components of the \n79 Program Memory are summarised in Fig. 1 (c). \n80 Recurrent multi-head program attention mechanisms for program storage and retrieval \n81 Neural networks use the concept of differentiable attention to access memory [11, 2]. This de \n82 fines a weighting distribution over the memory slots essentially weighting the degree to which each \n83 memory slot participates in a read or write operation. This is unlike conventional computers that use \n84 a unique address to access a single memory slot. \n85 Here we use two kinds of attention. First is content-based attention [11, 12] to ensure that the singu \n86 lar program is selected based on its functionality and the data input. This is achieved by producing \n87 a query vector based on the input and comparing it to the program keys $( k )$ using cosine similarity. \n88 Higher cosine similarity scores indicate higher attention weights to the singular programs associated \n89 with those program keys. Second, to encourage better memory utilisation, higher attention weights \n90 are assigned to slots with lower program usage $( m )$ through usage-based attention [12, 31]. The \n91 attention weights from the two schemas are then combined using interpolating gates to compose the \n92 final attention weights to the Program Memory. \n93 We adapt multi-head attention [11, 38] that applies multiple attentions in parallel to retrieve $H$ singu \n94 lar components. Besides, we introduce a recurrent attention mechanism, in which multi-head access \n95 is performed recurrently in $J$ steps. The $j$ -th set of $H$ retrieved components is conditioned on the \n96 previous ones. This recurrent, multi-head attention allows the composition network to incrementally \n97 search for optimal components for building relevant active programs. ", + "bbox": [ + 147, + 296, + 825, + 410 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 416, + 825, + 500 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 513, + 825, + 583 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 589, + 825, + 700 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 707, + 823, + 776 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "98 Neurocoder learns to “code” a relevant working program via training ", + "text_level": 1, + "bbox": [ + 150, + 790, + 658, + 805 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "99 The structure of the Program Memory and the role of the Program Controller facilitates the au \n100 tomatic construction of working programs via training. The Program Controller controls memory \n101 access through its composition network that creates the attention weight defining how to weight the \n102 singular programs in the memories. A weighted summation of the singular programs results in the \n103 attended singular program. Applying the recurrent multi-head attention described earlier, multiple \n104 attended singular programs are retrieved to construct an active program (Eq. 1). Then the Program \n105 Controller generates a residual program using its integration network, adding to the active program \n106 to produce the working program of the Main Network. This addition enables creation of flexible \n107 higher-rank working programs, which compensates for the low-rank coding process. The structure \n108 of the Program Controller is illustrated in Fig. 1 (b). \n109 The singular programs are trained to represent unitary functions necessary for any computation \n110 whilst the composition and integration networks are trained to compose the relevant programs for \n111 the considering task. As such, beside minimising the task loss, we enforce orthogonality of stored \n112 singular vectors by minimising $\\mathcal { L } _ { o } = \\mathbf { M } _ { U } \\mathbf { M } _ { U } ^ { \\top } - \\mathbf { \\bar { I } } + \\mathbf { M } _ { V } \\mathbf { M } _ { V } ^ { \\top } - \\mathbf { I }$ . The parameters of the networks, \n113 and the stored singular programs are adjusted using gradient training via minimising the total loss ", + "bbox": [ + 143, + 814, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 90, + 826, + 133 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 138, + 825, + 210 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/73ae3d1e18ef6e591429508b2ed4e92a9f313315a3a72f46d0c305e7d5f3e144.jpg", + "text": "$$\n\\mathscr { L } = \\mathcal { L } _ { t a s k } + a \\mathcal { L } _ { o }\n$$", + "text_format": "latex", + "bbox": [ + 437, + 228, + 560, + 243 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\mathcal { L } _ { t a s k }$ represents the supervised task loss and $\\mathcal { L } _ { o }$ represents the orthogonal loss weighted by a hyper-parameter $a$ to enforce orthogonality of the singular vectors. ", + "bbox": [ + 161, + 244, + 825, + 273 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 Attention mechanisms for Program Memory ", + "text_level": 1, + "bbox": [ + 160, + 287, + 522, + 303 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "117 Here we ddenoted as $w _ { i n } ^ { u , v , \\sigma }$ progra)–the a attention mechanintion weight to the $i$ ms used in this paper. Given -th slot of the singular progra $w _ { i n } ^ { u }$ , e $w _ { i n } ^ { v }$ , i $w _ { i n } ^ { \\sigma }$ $\\mathbf { M } _ { U }$ o, $\\mathbf { M } _ { V }$ \n119 , we retrieve the $n$ -th singular vector as follows, ", + "bbox": [ + 142, + 313, + 825, + 356 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7f593b14fbe5fa2db83bfae351abab39af3ebcf4aa3fc552641badb79ceb64dc.jpg", + "text": "$$\nu _ { n } = \\sum _ { i = 1 } ^ { P _ { u } } w _ { i n } ^ { u } \\mathbf { M } _ { U } \\left( i \\right)\n$$", + "text_format": "latex", + "bbox": [ + 426, + 371, + 571, + 415 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/22ef38c2bf214494039065f6b7c509007ff039d38b1ab448d686f393e65d8af4.jpg", + "text": "$$\nv _ { n } = \\sum _ { i = 1 } ^ { P _ { v } } w _ { i n } ^ { v } \\mathbf { M } _ { V } \\left( i \\right)\n$$", + "text_format": "latex", + "bbox": [ + 426, + 421, + 570, + 464 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "120 For the singular values, we need to enforce $\\sigma _ { 1 } > \\sigma _ { 2 } > . . . > \\sigma _ { r _ { m } } > 0$ , thus we retrieve using ", + "bbox": [ + 138, + 463, + 781, + 479 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/189a9f5c9f12697e6d17a3e2309cbe37090458ece1e4757b8e6d2895b27d7fbc.jpg", + "text": "$$\n\\sigma _ { n } = \\left\\{ \\begin{array} { l l } { \\mathrm { s o f t p l u s } \\left( \\sum _ { i = 1 } ^ { P _ { s } } w _ { i n } ^ { \\sigma } \\mathbf { M } _ { S } \\left( i \\right) \\right) } & { n = r _ { m } } \\\\ { \\sigma _ { n + 1 } + \\mathrm { s o f t p l u s } \\left( \\sum _ { i = 1 } ^ { P _ { s } } w _ { i n } ^ { \\sigma } \\mathbf { M } _ { S } \\left( i \\right) \\right) } & { n < r _ { m } } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 310, + 493, + 686, + 542 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "121 Here, $P _ { u } , P _ { v }$ and $P _ { s }$ are the number of memory slots of $\\mathbf { M } _ { U }$ , $\\mathbf { M } _ { V }$ and $\\mathbf { M } _ { S }$ , respectively. In this \n122 paper, we set $P = P _ { u } = P _ { v } = P _ { s }$ as the number of memory slots of the Program Memory. We note \n123 that thead ese notations are speci, and an attention step ed for some data input , hence the full notatio $x _ { t }$ and the hould be $n$ later maps to an attention To simplify notations, we \n$h$ $j$ $w _ { t i j h } ^ { u , v , \\sigma }$ \n125 will drop $u , v , \\sigma$ from now and describe the computation of a representative $w _ { t i j h }$ for any of the \n126 three program memories in the following parts. ", + "bbox": [ + 142, + 542, + 826, + 627 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "127 Recurrent Access to the Program Memory via the composition network ", + "text_level": 1, + "bbox": [ + 150, + 640, + 671, + 655 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "128 To perform program attention, the Program Controller employs a composition network (denoted \n129 as $f _ { \\theta } )$ ), which takes the current input $x _ { t }$ and produce program composition control signals $( \\pmb { \\xi } _ { t } ^ { p } )$ . \n130 If $f _ { \\theta }$ performs all attentions concurrently via multi-head attention as in [11, 38], it may lead to \n131 program collapse [22]. To have a better control of the component formation and alleviate program \n132 collapse, we propose to recurrently attend to the program memory. To this end, we implement $f _ { \\theta }$ as \n133 $\\pmb { \\xi } _ { t } ^ { p } = \\left\\{ \\pmb { \\xi } _ { t j } ^ { p } \\right\\} _ { j = 1 } ^ { J }$ al network (LST. At access step $j$ M [16]) and let it access the program memory , the recurrent network updates its hidden sta $J$ times, resultin and generates $\\xi _ { t j } ^ { p }$ \n135 using recurrent dynamics as ", + "bbox": [ + 140, + 662, + 825, + 782 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8d5ff96f4903eab24618ce065adce652211fa542cb32b96df1282f89efe4a773.jpg", + "text": "$$\n{ \\pmb { \\xi } } _ { t j } ^ { p } , h _ { j } = f _ { \\theta } \\left( x _ { t } , h _ { j - 1 } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 419, + 801, + 576, + 819 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "136 where $h _ { 0 }$ is initialized as zeros and $\\xi _ { t j } ^ { p }$ is the program composition control signal at step $j$ that \n137 depends on both on the input data $x _ { t }$ and the the previous state $h _ { j - 1 }$ . Particularly, the control signal \n138 contains the queries and the interpolation gates for each head to compute the program attention \n139 weight: $\\pmb { \\xi } _ { t j } ^ { p } = \\{ q _ { t j h } , g _ { t i j h } \\} _ { h = 1 } ^ { H }$ . Here, at each attention step, we perform multi-head attention with \n140 $H$ as the number of attention heads and thus, each $\\xi _ { t j } ^ { p }$ consists of $H$ pairs of queries and gates. \n141 Hence, the total number of retrieved components $\\boldsymbol { r } _ { m } = \\boldsymbol { \\bar { J } } \\times \\boldsymbol { H }$ and the index $n = j \\times H + h$ . ", + "bbox": [ + 140, + 820, + 825, + 912 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "142 Attending to Programs by “Name” ", + "text_level": 1, + "bbox": [ + 143, + 92, + 416, + 106 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "143 Inspired by the content-based attention mechanism for data memory [11], we use the query to look \n144 for the singular programs. In computer programming, to find the appropriate program for some \n145 computation, we often refer to the program description or at least the name of the program. Here, we \n146 create the “name” for our neural programs by compressing the program content to a low-dimensional \n147 key vector. As such, we employ a neural network $( f _ { \\varphi } )$ to compute the program memory keys as ", + "bbox": [ + 140, + 114, + 825, + 185 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/fbbd4d53165389137c65107dbb9b892e43aaf951aa496a542b7ddfbe0a58479f.jpg", + "text": "$$\nk _ { i } = f _ { \\varphi } \\left( \\mathbf { M } \\left( i \\right) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 442, + 203, + 553, + 220 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "148 where $\\boldsymbol { k } _ { i } \\in \\mathbb { R } ^ { K }$ and $i$ is the row index of the program memory. Here, $f _ { \\varphi }$ learns to compress each \n149 memory slot into a $K$ -dimensional vector. As the singular programs evolve, their keys get updated. \n150 In this paper, we update the program keys after each learning iteration during training. \n151 Finally the content-based program memory attention $c _ { t i j h }$ is computed using cosine distance be \n152 tween the program keys $k _ { i }$ and the queries $q _ { t j h }$ as ", + "bbox": [ + 140, + 224, + 826, + 267 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 273, + 826, + 303 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1bb5f46999968c3a1c30f85f187deb3f1106cda326578faa900fb8408b86eb63.jpg", + "text": "$$\nc _ { t i j h } = \\mathrm { s o f t m a x } ^ { ( i ) } \\left( \\frac { q _ { t j h } \\cdot k _ { i } } { \\left| \\left| q _ { t j h } \\right| \\right| \\cdot \\left| \\left| k _ { i } \\right| \\right| } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 375, + 318, + 622, + 353 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "153 Making Every Program Count ", + "text_level": 1, + "bbox": [ + 143, + 364, + 390, + 381 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "154 Similarly to [12, 31], in addition to the content-based attention, we employ a least-used reading \n155 strategy to encourage the Program Controller to assign different singular programs to different com \n156 ponents. In particular, we calculate the memory usage for each program slot across attentions as \n157 ", + "bbox": [ + 142, + 388, + 825, + 440 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/20d2e1b3e95cc99e60d768bc0080c3996ae302e11b21cde534c96b944ec0386e.jpg", + "text": "$$\nm _ { t i j h } = \\operatorname* { m a x } _ { \\tilde { j } \\leq j } \\left( w _ { t i \\tilde { j } h } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 423, + 440, + 575, + 470 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "158 Since we want to consider only $l _ { I }$ amongst $P$ memory slots that have smallest usages, let $\\hat { m } _ { t j h } ^ { l _ { I } }$ \n159 denote the value of the $l _ { I }$ -th smallest usage, then the least-used attention is computed as ", + "bbox": [ + 143, + 477, + 825, + 508 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/3eff87705475f2c68a1c4480bd1c57076406de4a285ac39dac6df397bafd8964.jpg", + "text": "$$\nl _ { t i j h } = \\left\\{ \\begin{array} { l l } { \\underset { i } { \\operatorname* { m a x } } \\left( m _ { t i j h } \\right) - m _ { t i j h } } & { ; m _ { t i j h } \\leq \\hat { m } _ { t j h } ^ { l _ { I } } } \\\\ { 0 } & { ; m _ { t i j h } > \\hat { m } _ { t j h } ^ { l _ { I } } } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 333, + 522, + 663, + 566 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "160 The final program memory attention is computed as ", + "bbox": [ + 142, + 568, + 516, + 583 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7f5851a3247de278761e154705908476d92d30012c85ff80f22df974bfe7e41f.jpg", + "text": "$$\nw _ { t i j h } = \\mathrm { s i g m o i d } \\left( g _ { t i j h } \\right) c _ { t i j h } + \\left( 1 - \\mathrm { s i g m o i d } \\left( g _ { t i j h } \\right) \\right) l _ { t i j h }\n$$", + "text_format": "latex", + "bbox": [ + 303, + 602, + 694, + 619 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "161 Since the usage record are computed along the memory accesses, the multi-step Neurocoder utilises \n162 this attention mechanism better than the single-step Neurocoder, creating different attention styles \n163 (see Sec. 3.2). The composition the active program $\\mathbf { P } _ { t }$ is illustrated in Appendix’s Fig. 5. ", + "bbox": [ + 142, + 622, + 825, + 665 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.3 Program Integration via the integration network ", + "text_level": 1, + "bbox": [ + 163, + 679, + 552, + 695 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "165 Since the working program $\\mathbf { P } _ { t }$ only contains top $r _ { m }$ principal components, it is low-rank and may \n166 be not flexible enough for sophisticated computation. We propose to enhance $\\mathbf { P } _ { t }$ with a residual \n167 program $\\mathbf { R } -$ a traditional connection weight trained as the integration network’s parameters, which \n168 is constant after training w.r.t $t$ . The residual program represents the sum of the remaining less \n169 important components. To this end, we suppress $\\mathbf { R }$ with a multiplier that is smaller than $\\sigma _ { t r _ { m } } .$ – the \n170 smallest singular value of the main components - resulting in the integration formula ", + "bbox": [ + 140, + 704, + 825, + 790 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/5a1ea2eaac36e14e3d2dca9766e69d0bd941880cfbe050d8a72f9f696febc4cf.jpg", + "text": "$$\nW _ { t } = \\mathbf { P } _ { t } + w _ { t } ^ { r } \\sigma _ { t r _ { m } } \\mathbf { R }\n$$", + "text_format": "latex", + "bbox": [ + 423, + 808, + 575, + 825 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "171 where $w _ { t } ^ { r } = \\mathrm { s i g m o i d } \\left( f _ { \\phi } \\left( x _ { t } \\right) \\right)$ is an adaptive gating value that controls the contribution of the \n172 residual program. $f _ { \\phi }$ is the integration network in the Program Controller and hence, in our imple \n173 mentation, the integration control signal sent by the Program Controller is $\\lambda _ { t } ^ { p } = \\{ w _ { t } ^ { r } , \\sigma _ { t r _ { m } } \\}$ . We \n174 note that in our experiments, the program integration can be disabled ( $W _ { t }$ is directly set to $\\mathbf { P } _ { t }$ ) to \n175 prove the contribution of $\\mathbf { P } _ { t }$ or reduce the number of parameters. The working program $W _ { t }$ is then \n176 used by the Main Network to execute the input data $x _ { t }$ (see (Fig. 1 (a))). For example, with linear \n177 classifier Main Network, the execution is $y _ { t } = x _ { t } W _ { t }$ . Appendix’s Table 2 summarises the notations \n178 used for important parameters of Neurocoder. ", + "bbox": [ + 140, + 827, + 825, + 912 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/f992f503335ad0a3f753e1c8f03e87de2fd352b9c040fd798ec6c09805c37c77.jpg", + "image_caption": [ + "Figure 2: (a) MNIST test set classification error vs the number of steps $( J )$ in Neurocoder (blue), compared with a linear classifier (red). (b) 1st column: Digit images; Middle column: Single-step attention weights for 30 slots in $\\mathbf { M } _ { U }$ (vertical axis) for first 3 singular vectors (horizontal axis) for each digit; Last column: Multi-step attention weights for 10 slots in $\\mathbf { M } _ { U }$ (vertical axis) for first 3 singular vectors (horizontal axis). Multi-step attention is able to produce far more diverse patterns with fewer slots - 10 slots compared to single-step 30 slots. (c) Two attention patterns of singlestep Neurocoder. The binary decision tree derived from single-step Neurocoder’s attention patterns. The two patterns across components represent the decisions going up and down across the binary tree. Visualisation for (d) multi-step $J = 5$ , 20 memory slots) and (e) single-step $J = 1$ , 10 memory slots) cases showing while processing a sequence of the polynomial auto-regression task. The Neurocoder’s attentions to $\\mathbf { M } _ { U }$ that form the first component of the active program are shown over sequence timesteps (upper) with Neurocoder’s $y _ { t }$ prediction (orange) and ground truth (blue) (lower). The vertical dash green lines separate polynomial chunks. Each chuck represents a local pattern, and thus ideally requires a specific active program to compute the input $x _ { t }$ . Although both predict well, only the multi-step Neurocoder discovers the chunk boundaries, assigning program attention to the first component in accordance with sequence changes. " + ], + "image_footnote": [], + "bbox": [ + 173, + 87, + 826, + 337 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 618, + 823, + 646 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "179 3 Results ", + "text_level": 1, + "bbox": [ + 142, + 691, + 266, + 708 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "180 To demonstrate the flexibility of Neurocoder framework, we consider different learning paradigms: \n181 instance-based, sequential, multi-task and continual learning. We do not focus on breaking perfor \n182 mance records by augmenting state-of-the-art models with Neurocoder. Rather our inquiry is on \n183 re-coding feed-forward layers with the Neurocoder’s programs and testing on varied data types to \n184 demonstrate its intrinsic properties. For some experiments, we include ablation studies. \n185 We compare the performance of diverse Main Networks (MN) with and without Neurocoder. We \n186 also augment the Main Networks with other recent conditional computing methods, either modular \n187 (sparse Mixture of Experts, Neural Stored-program Memory) or monolithic (HyperNets, FiLM) to \n188 form stronger baselines across our experiments. In our experiments, we always apply Neurocoder \n189 to all layers of multi-layer perceptrons (MLP) or just the final feed-forward layer of deep CNN \n190 networks (LeNet, DenseNet, ResNet), RNNs (GRU, LSTM), MANN (NTM). Other competitors \n191 such as MOE, NSM, HyperNet and FiLM are applied to the Main Networks in the same manner. ", + "bbox": [ + 142, + 738, + 823, + 808 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 814, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/7e25b32d05c9ae7db2df1ea7bbd96b41284179be7b8a5c34b278f599cab1c811.jpg", + "image_caption": [ + "Figure 3: Learning curves (mean and std. over 5 runs) on representative Atari 2600 games. All baselines are applied to the actor/critic networks in the A3C agent. " + ], + "image_footnote": [], + "bbox": [ + 173, + 88, + 825, + 236 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "192 3.1 Instance-based learning - Object Recognition ", + "text_level": 1, + "bbox": [ + 142, + 300, + 526, + 315 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We tested Neurocoder on instance-based learning through classical image classification tasks using MNIST [24] and CIFAR [21] datasets. The first experiment interpreted Neurocoder’s behaviour in classifying digits into 10 classes $( 0 - 9 )$ using linear classifier Main Network. With equivalent model size, Neurocoder using the novel recurrent attention surpasses the performance of the linear classifier [24] by up to $5 \\%$ (Fig. 2 (a)). ", + "bbox": [ + 165, + 327, + 823, + 396 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To differentiate the input, Neurocoder attends to different components of the active program to guide the decision-making process. Fig. 2 (b) shows single-step and multi-step attention to the first 3 singular vectors for each digit across memory slots. Multi-step attention produces richer patterns compared to single-step Neurocoder that manages only 2 attention weight patterns. ", + "bbox": [ + 165, + 402, + 823, + 458 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Fig. 2 (c) illustrates how Neurocoder performs modular learning by showing the attention assignment for top 3 singular vectors as a binary decision tree. Digits under the same parental node share similar attention paths, and thereby similar active programs. Some digits look unique (e.g. 7) resulting in active programs composed of unique attention paths, discriminating themselves early in the decision tree. Some digits (e.g. 0 and 9) share the same attention pattern for the first 3 components and are thus unclassifiable. They can only be distinguished by considering more singular vectors. ", + "bbox": [ + 173, + 464, + 823, + 547 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We integrated Neurocoder with deep networks - 5-layer LeNet and 100-layer DenseNet - and tested on CIFAR datasets. Neurocoder significantly outperformed the original Main Networks with performance gain $1 - 5 \\%$ . Compared with recent conditional computing models such as sparse Mixture of Experts (MOE [35]) and Neural Stored-program Memory (NSM [22]), Neurocoder required a tenth of the number of parameters and performed better by up to $8 - 1 0 \\%$ (see Appendix’s Table 3). ", + "bbox": [ + 165, + 554, + 825, + 623 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 Sequential learning - Adaption to sequence changes and game playing using reinforcement learning ", + "text_level": 1, + "bbox": [ + 173, + 641, + 738, + 670 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Recurrent neural networks (RNN) can learn from sequential data by updating the hidden states of the networks. However, this does not suffice when local patterns shift, as is often the case. We now demonstrate that Neurocoder helps RNNs overcome this limitation by composing diverse programs to handle sequence changes. ", + "bbox": [ + 168, + 680, + 825, + 736 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Synthetic polynomial auto-regression We created a simple auto-regression task in which data points are sampled from polynomial function chunks that change over time. The Main Network is a strong RNN–Gated Recurrent Unit (GRU [5]). We found that GRU integrated with a single-step or multi-step Neurocoder converged much faster than all other baselines. The other conditional computing counterparts (HyperNet [13], FiLM [28]) adapt by re-scaling weights or activation of the GRU, which were shown inferior to our modular approach (Appendix’s Fig. 6). ", + "bbox": [ + 171, + 752, + 823, + 835 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "225 Visualising the first singular vector attention weights in $\\mathbf { M } _ { U }$ , we find that the multi-step attention \n226 Neurocoder changes its attention following polynomial changes - it attends to the same singular pro \n227 gram when processing data from the same polynomial and alters attention for data from a different \n228 polynomial (Fig. 2(d)). In contrast, the single-step Neurocoder only changes its attention when \n229 there is a remarkable change in $y$ -coordinate values (Fig. 2(e)). Although single-step Neurocoder \n230 converges well, it did not discover the underlying structure of the data, and thus underperformed \n231 the multi-step Neurocoder. We hypothesise that when recurrence is employed, usage-based atten \n232 tion takes effect, stipulating better memory utilisation and diverse attentions over timesteps. We ran \n233 multi-step Neurocoder without usage-based attention. The results were worse than the full multi \n234 step Neurocoder, which confirms our hypothesis (Appendix’s Fig. 6). ", + "bbox": [ + 142, + 842, + 823, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/06ee98d5e15fb67549b39c75f337a7184d9d269eefcccfe677f8248a1273b0b6.jpg", + "table_caption": [], + "table_footnote": [ + "Table 1: Incremental domain continual learning with Split MNIST. Final test accuracy (mean and std.) over 10 runs. " + ], + "table_body": "
MethodMN (MLP[17])MN (MLP ours)NSMNeurocoder
Adam55.16±1.3853.55±1.2754.85±2.8158.46±0.46
Adagrad58.08±1.0657.83±2.7458.42±1.8762.28±4.03
L266.00±3.7364.37±2.4062.83±7.2169.89±1.72
SI64.76±3.0964.41±3.3664.36±2.9967.96±3.22
EWC58.85±2.5958.41±2.3758.12±3.2465.66±1.25
O-EWC57.33±1.4457.78±1.8458.55±3.4073.97±1.50
", + "bbox": [ + 236, + 88, + 761, + 189 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 256, + 823, + 327 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Atari game reinforcement learning We used reinforcement learning as a further testbed to show the ability to adapt to environmental changes. We performed experiments on several Atari 2600 games [3] wherein the agent was implemented as the Asynchronous Advantage Actor-Critic (A3C [26]). In the Atari platform, agents are allowed to observe the screen snapshot of the games and act to earn the highest score. We augmented the A3C by employing Neurocoder’s working programs for feed-forward layers of the actor and critic networks, aiming to decompose the policy and value function into singular programs that were selected depending on the game state. ", + "bbox": [ + 169, + 343, + 825, + 440 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "242 Frostbite and Montezuma’s Revenge. These games are known to be challenging for A3C and other \n243 algorithms [26]. We trained A3C and HyperNet-based A3C for over 300 million steps, yet these \n244 models did not show any sign of learning, performing equivalently to random agents. For such com \n245 plicated environments with sparse rewards, both the monolithic neural networks and the HyperNet’s \n246 unstored fast-weights fail to learn (almost zero scores). In contrast, Neurocoder enabled A3C to \n247 achieve from 1, 500 to 3, 000 scores on these environments (Fig. 3), confirming the importance of \n248 decomposing a complex solution to smaller, simple stored programs. ", + "bbox": [ + 142, + 446, + 825, + 545 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "49 3.3 Multi-task learning - Solving mutliple algorithms simultenously ", + "text_level": 1, + "bbox": [ + 156, + 563, + 653, + 578 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Here we explore the modular learning capability of Neurocoder in multi-task setting. Inspired by algorithmic sequencing tasks [22], we created a challenging sequential multi-task benchmark wherein the input sequence is a series of sub-sequences from 4 algorithms: Copy, Repeat Copy, Associative Recall and Priority Sort [11]. Each sub-sequence, following a task identification vector, represents the input for each task. In each input sequence, $n$ tasks were sampled from the set of 4 algorithms randomly with replacement and the output sequences were created correspondingly. ", + "bbox": [ + 171, + 589, + 823, + 671 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We trained a MANN–Neural Turing Machine (NTM [11]) Main Network with FiLM, HyperNet and our Neurocoder augmentation on sequences of $n = 4$ tasks, and tested with sequences of $n = 4$ and $n = 8$ tasks. Appendix’s Fig. 7 demonstrates that Neurocoder was performant in both test settings, not only achieving lowest error on $n = 4$ , but also being the only one generalised well to $n = 8$ scenario, which was unseen during training. ", + "bbox": [ + 171, + 679, + 825, + 748 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.4 Continual learning $-$ Learning tasks sequentially without catastrophic forgetting ", + "text_level": 1, + "bbox": [ + 161, + 767, + 767, + 782 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In continual learning, standard neural networks often suffer from “catastrophic forgetting” in which they cannot retain knowledge acquired from old tasks upon learning new ones [10]. Our Neurocoder offers natural mitigation of such catastrophic forgetting in neural networks by attending to different singular programs whilst learning different tasks. ", + "bbox": [ + 174, + 794, + 823, + 849 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this case, in addition to the Main Network, we examine several continual learning algorithms with and without Neurocoder. These algorithms, including Elastic Weight Consolidation (EWC [41]) and Synaptic Intelligence (SI [41]), work by regularising the loss function and thus can be easily combined with Neurocoder by modifying the loss $\\mathcal { L } _ { t a s k }$ . We demonstrate that Neurocoder ", + "bbox": [ + 169, + 856, + 823, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "70 can improve these continual learning algorithms without requiring additional assumptions as in other \n1 approaches [25, 36, 34] that either utilise task embedding or replay memory. ", + "bbox": [ + 153, + 90, + 825, + 119 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Split MNIST We first considered the split MNIST dataset–a standard continual learning benchmark wherein the original MNIST was split into a 5 2-way classification tasks, consecutively presented to a Multi-layer Perceptron Main Network (MLP). We followed the benchmarking as in [17] in which various optimisers and state-of-the-art continual learning methods were examined under incremental task and domain scenarios. We measured the performance of the MLP versus Neurocoder and NSM under each continual learning method. In both scenarios, Neurocoder was compatible with all continual leaning methods, demonstrating superior performance over MLP and NSM with performance gain between 1 to $1 6 \\%$ (see Appendix’s Table 5 and 1). ", + "bbox": [ + 163, + 136, + 823, + 247 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "280 Split CIFAR We verified the scalability of Neurocoder to more challenging datasets. We split \n281 CIFAR datasets as in the split MNIST, resulting in 5-task 2-way split CIFAR10 and a 20-task 5-way \n282 split CIFAR100. We used Main Network ResNet [15]–a very deep CNN architecture. \n283 When we stressed the orthogonal loss $a = 1 0$ ) and used bigger program memory (100 slots), Neu \n284 rocoder improved ResNet classification by $1 5 \\%$ and $1 0 \\%$ on CIFAR10 and CIFAR100, respectively. \n285 When we integrated Neurocoder with Synaptic Intelligence (SI [41]), the performance was further \n286 improved, maintaining a stable performance above $8 0 \\%$ accuracy for CIFAR10 and outperforming \n287 using SI alone by $1 0 \\%$ for CIFAR100 (see Appendix’s Fig. 8). ", + "bbox": [ + 143, + 263, + 823, + 306 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 311, + 823, + 382 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "288 4 Discussion ", + "text_level": 1, + "bbox": [ + 143, + 402, + 294, + 420 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Our experiments demonstrate that Neurocoder is capable of re-coding Neural Programs in distinctive neural networks, amplifying their capabilities in diverse learning scenarios: instance-based, sequential, multi-task and continual learning. This consistently results in significant performance increase, and further creates novel robustness to pattern shift and catastrophic forgetting. This ability for each architecture to re-code itself is made possible without changing the way it is trained, or majorly increasing the number of parameters it needs to learn (see Appendix Table 7). ", + "bbox": [ + 171, + 435, + 823, + 518 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The MNIST problem illustrates the reasoning process of Neurocoder when classifying digit images wherein its singular program assignment resembles a binary tree decision-making process - it shows how some singular programs are shared, others are not. The polynomial auto-regression problem highlights the importance of efficient memory utilisation in re-constructing the working program enabling discovery of hidden structures in sequential data. Training our framework with reinforcement learning, we enable neural agents to solve complex games wherein traditional methods fail or learn slowly. Neurocoder also works well with multi-task setting, as shown in the challenging multi-algorithm benchmark. Finally, continual learning problems show that Neurocoder mitigates catastrophic forgetting efficiently under different learning settings/algorithms. ", + "bbox": [ + 165, + 525, + 825, + 650 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "304 Our solution offers a single framework that is scalable and adaptable to various problems and learn \n305 ing paradigms. Unlike previous attempts to employ a bank of separate big programs [20, 35, 22], \n306 Neurocoder maintains only shareable, smaller components that can reconstruct the whole program \n307 space, thereby heavily utilising the parameters and preventing the model from proliferating. We \n308 note that Neurocoder is orthogonal to approaches employing tensor decomposition to reduce the \n309 number of parameters or hasten the computation [27, 23]. Neurocoder composes rather than decom \n310 pose the neural weights. Our aim is not only to enable efficient parameter usage, but also achieve \n311 general-purpose computing power, outperforming other methods in numerous learning problems. \n312 One limitation of this work is the number of additional hyperparameters, which prevents us from \n313 fully tuning Neurocoder. Our research aims to add new capabilities to current neural networks to \n314 improve their performance and make them robust in different learning scenario. Hence, we do \n315 not see any intermediate negative societal impact. In future work, we will extend Neurocoder’s \n316 application beyond feed-forward layers. It would be interesting to efficiently replace all neural layers \n317 including CNN or Transformer by Neurocoder’s programs. We can also further extend Neurocoder’s \n318 ability by allowing a growing Program Memory, in which the model decides to add or erase memory \n319 slots as the number of data patterns grows or shrinks beyond the current program space’s capacity. \n320 Such a system represents a more flexible general-purpose computer that can dynamically allocate \n321 computing resources by itself without human pre-specification. ", + "bbox": [ + 140, + 656, + 825, + 767 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 772, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "References \n[1] Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Dan Klein. Neural module networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 39–48, 2016. \n[2] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. 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In Advances in neural information processing systems, pages 5998–6008, 2017. \n[39] Christoph von der Malsburg. The correlation theory of brain function, 1981. \n[40] John Von Neumann. First draft of a report on the edvac. IEEE Annals of the History of Computing, 15(4):27–75, 1993. \n[41] Friedemann Zenke, Ben Poole, and Surya Ganguli. Continual learning through synaptic intelligence. Proceedings of machine learning research, 70:3987, 2017. ", + "bbox": [ + 151, + 88, + 828, + 915 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 158, + 47, + 828, + 921 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 151, + 89, + 828, + 323 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 116, + 339, + 130 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] See Discussion and Appendix’s \"Training procedure and hyper-parameter selections.\" \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Discussion. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 238, + 135, + 825, + 253 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 214, + 257, + 493, + 272 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ", + "bbox": [ + 235, + 276, + 736, + 308 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "3. If you ran experiments... 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", + "bbox": [ + 238, + 330, + 826, + 489 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 215, + 494, + 823, + 510 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] \n(b) Did you mention the license of the assets? [No] All assets are public. We will mention the license detail after the paper is published. \n(c) Did you include any new assets either in the supplemental material or as a URL? [No] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] ", + "bbox": [ + 238, + 513, + 825, + 635 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 214, + 638, + 707, + 654 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? 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Recent works introduce", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 88, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 88, + 564, + 100, + 573 + ], + "score": 1.0, + "content": "24", + "type": "text" + }, + { + "bbox": [ + 104, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "conditional computation via adjusting or activating parts of a network in an input-dependent manner", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 89, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 89, + 574, + 100, + 584 + ], + "score": 1.0, + "content": "25", + "type": "text" + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "[39, 33, 4, 13, 28], but networks remain monolithic. Current networks forget when retrained, old", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 89, + 583, + 392, + 595 + ], + "spans": [ + { + "bbox": [ + 89, + 586, + 100, + 595 + ], + "score": 1.0, + "content": "26", + "type": "text" + }, + { + "bbox": [ + 105, + 583, + 392, + 595 + ], + "score": 1.0, + "content": "inter-neuronal connection weights are merged with new ones or erased.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 90, + 599, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 89, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 89, + 601, + 99, + 610 + ], + "score": 1.0, + "content": "27", + "type": "text" + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "The brain is modular, not a monolithic system [8, 6]. 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This requires huge computational storage and introduces redundancy as these", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 88, + 655, + 292, + 667 + ], + "spans": [ + { + "bbox": [ + 88, + 656, + 100, + 667 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 105, + 655, + 292, + 667 + ], + "score": 1.0, + "content": "experts do not share common basic programs.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 90, + 670, + 504, + 714 + ], + "lines": [ + { + "bbox": [ + 89, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 89, + 672, + 100, + 682 + ], + "score": 1.0, + "content": "33", + "type": "text" + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "A pathway out of this bind is to keep such basic programs in memory and combine them as required.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 89, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 89, + 683, + 100, + 693 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "This brings neural networks towards modern general-purpose computers that use the stored-program", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 89, + 692, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 89, + 694, + 100, + 704 + ], + "score": 1.0, + "content": "35", + "type": "text" + }, + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "score": 1.0, + "content": "principle [37, 40] to efficiently access reusable programs in external memory. 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Current networks forget when retrained, old", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 583, + 392, + 595 + ], + "spans": [ + { + "bbox": [ + 89, + 586, + 100, + 595 + ], + "score": 1.0, + "content": "26", + "type": "text" + }, + { + "bbox": [ + 105, + 583, + 392, + 595 + ], + "score": 1.0, + "content": "inter-neuronal connection weights are merged with new ones or erased.", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 89, + 601, + 99, + 610 + ], + "score": 1.0, + "content": "27", + "type": "text" + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "The brain is modular, not a monolithic system [8, 6]. Neuroscience research indicates that the brain is", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 89, + 612, + 100, + 622 + ], + "score": 1.0, + "content": "28", + "type": "text" + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "divided into functional modules [19, 7, 9]. If the neural program for each module is kept in separate", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 89, + 624, + 99, + 633 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "networks, networks proliferate. Modular neural networks, another form of conditional computation,", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 89, + 634, + 100, + 644 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "combine the output of multiple expert networks, but as the experts grow, the networks grow drasti-", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 642, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 88, + 645, + 99, + 654 + ], + "score": 1.0, + "content": "31", + "type": "text" + }, + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "score": 1.0, + "content": "cally [20, 14, 35, 29]. This requires huge computational storage and introduces redundancy as these", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 655, + 292, + 667 + ], + "spans": [ + { + "bbox": [ + 88, + 656, + 100, + 667 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 105, + 655, + 292, + 667 + ], + "score": 1.0, + "content": "experts do not share common basic programs.", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 89, + 672, + 100, + 682 + ], + "score": 1.0, + "content": "33", + "type": "text" + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "A pathway out of this bind is to keep such basic programs in memory and combine them as required.", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 680, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 89, + 683, + 100, + 693 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 105, + 680, + 505, + 694 + ], + "score": 1.0, + "content": "This brings neural networks towards modern general-purpose computers that use the stored-program", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 692, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 89, + 694, + 100, + 704 + ], + "score": 1.0, + "content": "35", + "type": "text" + }, + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "score": 1.0, + "content": "principle [37, 40] to efficiently access reusable programs in external memory. Here we show how", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 702, + 505, + 716 + ], + "spans": [ + { + "bbox": [ + 88, + 705, + 100, + 715 + ], + "score": 1.0, + "content": "36", + "type": "text" + }, + { + "bbox": [ + 105, + 702, + 505, + 716 + ], + "score": 1.0, + "content": "Neurocoder, a new neural framework, introduces a new class of general-purpose conditional compu-", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 89, + 460, + 99, + 469 + ], + "score": 1.0, + "content": "37", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "tation machines in which a neural network can be “coded” in an input-dependent manner. Efficient", + "type": "text", + "cross_page": true + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 469, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 89, + 471, + 99, + 480 + ], + "score": 1.0, + "content": "38", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 469, + 506, + 481 + ], + "score": 1.0, + "content": "decomposition of Neural Programs creates shareable modular components that can reconstruct the", + "type": "text", + "cross_page": true + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 480, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 88, + 482, + 100, + 492 + ], + "score": 1.0, + "content": "39", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 480, + 506, + 493 + ], + "score": 1.0, + "content": "whole program space. These components change their “shapes” based on training and are stored", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 491, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 89, + 493, + 100, + 502 + ], + "score": 1.0, + "content": "40", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "in an external Program Memory. Then, in a data-responsive way, a Program Controller retrieves", + "type": "text", + "cross_page": true + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 89, + 504, + 99, + 513 + ], + "score": 1.0, + "content": "41", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "relevant components to build the Neural Program. This is analogous to shape-shifting Lego bricks", + "type": "text", + "cross_page": true + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 512, + 435, + 524 + ], + "spans": [ + { + "bbox": [ + 89, + 514, + 100, + 523 + ], + "score": 1.0, + "content": "42", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 512, + 435, + 524 + ], + "score": 1.0, + "content": "that can be reused to build unlimited shapes and structures (See Appendix Fig. 4).", + "type": "text", + "cross_page": true + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 89, + 531, + 100, + 540 + ], + "score": 1.0, + "content": "43", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "Using adaptive modular components vastly increases the learning capacity of the neural network", + "type": "text", + "cross_page": true + } + 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(b)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 332, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 347 + ], + "score": 1.0, + "content": "The Program Controller’s composition network controls access to the Program Memory, emitting", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "queries and interpolating gate control signals in response to the input (2). 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(b)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 332, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 347 + ], + "score": 1.0, + "content": "The Program Controller’s composition network controls access to the Program Memory, emitting", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "queries and interpolating gate control signals in response to the input (2). 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Then the Program", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 86, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 100, + 723 + ], + "score": 1.0, + "content": "105", + "type": "text" + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "Controller generates a residual program using its integration network, adding to the active program", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 48 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 92, + 72, + 493, + 84 + ], + "lines": [ + { + "bbox": [ + 87, + 69, + 495, + 88 + ], + "spans": [ + { + "bbox": [ + 87, + 69, + 495, + 88 + ], + "score": 1.0, + "content": "56 Neurocoder stores Singular Value Decomposition of Neural Programs in Program Memory", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "index", + "bbox": [ + 89, + 91, + 505, + 180 + ], + "lines": [ + { + "bbox": [ + 89, + 91, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 89, + 93, + 99, + 103 + ], + "score": 1.0, + "content": "57", + "type": "text" + }, + { + "bbox": [ + 105, + 91, + 506, + 104 + ], + "score": 1.0, + "content": "The Neural Program needs to be stored efficiently in Program Memory. This is challenging as there", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 101, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 88, + 104, + 99, + 114 + ], + "score": 1.0, + "content": "58", + "type": "text" + }, + { + "bbox": [ + 105, + 101, + 505, + 115 + ], + "score": 1.0, + "content": "may be millions of inter-neuronal connection weights, thus storing them directly ([22]) is grossly", + "type": "text" + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 113, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 88, + 115, + 99, + 125 + ], + "score": 1.0, + "content": "59", + "type": "text" + }, + { + "bbox": [ + 105, + 113, + 505, + 126 + ], + "score": 1.0, + "content": "inefficient. Instead, the Neurocoder forms the basis of a subspace spanned by Neural Programs and", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 121, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 88, + 126, + 99, + 136 + ], + "score": 1.0, + "content": "60", + "type": "text" + }, + { + "bbox": [ + 104, + 121, + 506, + 138 + ], + "score": 1.0, + "content": "stores the singular values and vectors of this subspace in memory slots of the Program Memory", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 134, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 88, + 137, + 99, + 147 + ], + "score": 1.0, + "content": "61", + "type": "text" + }, + { + "bbox": [ + 105, + 134, + 506, + 149 + ], + "score": 1.0, + "content": "(hereafter referred to as singular programs). 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This de-", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 89, + 431, + 99, + 441 + ], + "score": 1.0, + "content": "82", + "type": "text" + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "fines a weighting distribution over the memory slots essentially weighting the degree to which each", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 89, + 442, + 100, + 452 + ], + "score": 1.0, + "content": "83", + "type": "text" + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "memory slot participates in a read or write operation. This is unlike conventional computers that use", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 451, + 302, + 464 + ], + "spans": [ + { + "bbox": [ + 89, + 453, + 100, + 462 + ], + "score": 1.0, + "content": "84", + "type": "text" + }, + { + "bbox": [ + 104, + 451, + 302, + 464 + ], + "score": 1.0, + "content": "a unique address to access a single memory slot.", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 89, + 470, + 100, + 479 + ], + "score": 1.0, + "content": "85", + "type": "text" + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "Here we use two kinds of attention. First is content-based attention [11, 12] to ensure that the singu-", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 476, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 88, + 480, + 100, + 490 + ], + "score": 1.0, + "content": "86", + "type": "text" + }, + { + "bbox": [ + 104, + 476, + 506, + 492 + ], + "score": 1.0, + "content": "lar program is selected based on its functionality and the data input. 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Second, to encourage better memory utilisation, higher attention weights", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 521, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 88, + 524, + 99, + 533 + ], + "score": 1.0, + "content": "90", + "type": "text" + }, + { + "bbox": [ + 105, + 521, + 302, + 534 + ], + "score": 1.0, + "content": "are assigned to slots with lower program usage", + "type": "text" + }, + { + "bbox": [ + 302, + 523, + 318, + 533 + ], + "score": 0.6, + "content": "( m )", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 521, + 506, + 534 + ], + "score": 1.0, + "content": "through usage-based attention [12, 31]. The", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 88, + 535, + 99, + 545 + ], + "score": 1.0, + "content": "91", + "type": "text" + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "attention weights from the two schemas are then combined using interpolating gates to compose the", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 542, + 298, + 558 + ], + "spans": [ + { + "bbox": [ + 89, + 546, + 100, + 555 + ], + "score": 1.0, + "content": "92", + "type": "text" + }, + { + "bbox": [ + 105, + 542, + 298, + 558 + ], + "score": 1.0, + "content": "final attention weights to the Program Memory.", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 89, + 562, + 100, + 572 + ], + "score": 1.0, + "content": "93", + "type": "text" + }, + { + "bbox": [ + 106, + 559, + 466, + 573 + ], + "score": 1.0, + "content": "We adapt multi-head attention [11, 38] that applies multiple attentions in parallel to retrieve", + "type": "text" + }, + { + "bbox": [ + 467, + 561, + 477, + 570 + ], + "score": 0.76, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "singu-", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 571, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 89, + 573, + 100, + 582 + ], + "score": 1.0, + "content": "94", + "type": "text" + }, + { + "bbox": [ + 106, + 571, + 505, + 583 + ], + "score": 1.0, + "content": "lar components. 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Then the Program", + "type": "text" + } + ], + "index": 50, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 100, + 723 + ], + "score": 1.0, + "content": "105", + "type": "text" + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "Controller generates a residual program using its integration network, adding to the active program", + "type": "text" + } + ], + "index": 51, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 86, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "106 to produce the working program of the Main Network. This addition enables creation of flexible", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 87, + 86, + 100, + 95 + ], + "score": 1.0, + "content": "107", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "higher-rank working programs, which compensates for the low-rank coding process. The structure", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 95, + 316, + 107 + ], + "spans": [ + { + "bbox": [ + 87, + 97, + 100, + 105 + ], + "score": 1.0, + "content": "108", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 95, + 316, + 107 + ], + "score": 1.0, + "content": "of the Program Controller is illustrated in Fig. 1 (b).", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 86, + 113, + 99, + 122 + ], + "score": 1.0, + "content": "109", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "The singular programs are trained to represent unitary functions necessary for any computation", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 86, + 123, + 100, + 134 + ], + "score": 1.0, + "content": "110", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "whilst the composition and integration networks are trained to compose the relevant programs for", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 133, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 86, + 135, + 99, + 144 + ], + "score": 1.0, + "content": "111", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "score": 1.0, + "content": "the considering task. 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The parameters of the networks,", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 155, + 498, + 167 + ], + "spans": [ + { + "bbox": [ + 86, + 157, + 100, + 166 + ], + "score": 1.0, + "content": "113", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 155, + 498, + 167 + ], + "score": 1.0, + "content": "and the stored singular programs are adjusted using gradient training via minimising the total loss", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + } + ], + "index": 48, + "bbox_fs": [ + 86, + 645, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 87, + 72, + 506, + 106 + ], + "lines": [ + { + "bbox": [ + 86, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 86, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "106 to produce the working program of the Main Network. This addition enables creation of flexible", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 87, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 87, + 86, + 100, + 95 + ], + "score": 1.0, + "content": "107", + "type": "text" + }, + { + "bbox": [ + 104, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "higher-rank working programs, which compensates for the low-rank coding process. 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In computer programming, to find the appropriate program for some", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 86, + 113, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 86, + 115, + 100, + 124 + ], + "score": 1.0, + "content": "145", + "type": "text" + }, + { + "bbox": [ + 105, + 113, + 505, + 125 + ], + "score": 1.0, + "content": "computation, we often refer to the program description or at least the name of the program. 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As such, we employ a neural network", + "type": "text" + }, + { + "bbox": [ + 306, + 135, + 324, + 147 + ], + "score": 0.86, + "content": "( f _ { \\varphi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 135, + 491, + 148 + ], + "score": 1.0, + "content": "to compute the program memory keys as", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 271, + 161, + 339, + 175 + ], + "lines": [ + { + "bbox": [ + 271, + 161, + 339, + 175 + ], + "spans": [ + { + "bbox": [ + 271, + 161, + 339, + 175 + ], + "score": 0.92, + "content": "k _ { i } = f _ { \\varphi } \\left( \\mathbf { M } \\left( i \\right) \\right)", + "type": "interline_equation", + "image_path": "fbbd4d53165389137c65107dbb9b892e43aaf951aa496a542b7ddfbe0a58479f.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 271, + 161, + 339, + 175 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 178, + 506, + 212 + ], + "lines": [ + { + "bbox": [ + 86, + 177, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 86, + 180, + 101, + 190 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 102, + 177, + 133, + 191 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 178, + 172, + 190 + ], + "score": 0.93, + "content": "\\boldsymbol { k } _ { i } \\in \\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 177, + 191, + 191 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 179, + 196, + 189 + ], + "score": 0.81, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 177, + 393, + 191 + ], + "score": 1.0, + "content": "is the row index of the program memory. 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As the singular programs evolve, their keys get updated.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 87, + 200, + 452, + 214 + ], + "spans": [ + { + "bbox": [ + 87, + 203, + 100, + 212 + ], + "score": 1.0, + "content": "150", + "type": "text" + }, + { + "bbox": [ + 104, + 200, + 452, + 214 + ], + "score": 1.0, + "content": "In this paper, we update the program keys after each learning iteration during training.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 88, + 217, + 506, + 240 + ], + "lines": [ + { + "bbox": [ + 84, + 216, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 84, + 216, + 324, + 231 + ], + "score": 1.0, + "content": "151 Finally the content-based program memory attention", + "type": "text" + }, + { + "bbox": [ + 325, + 219, + 345, + 230 + ], + "score": 0.89, + "content": "c _ { t i j h }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 216, + 505, + 231 + ], + "score": 1.0, + "content": "is computed using cosine distance be-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 85, + 228, + 308, + 241 + ], + "spans": [ + { + "bbox": [ + 85, + 228, + 204, + 241 + ], + "score": 1.0, + "content": "152 tween the program keys", + "type": "text" + }, + { + "bbox": [ + 204, + 228, + 214, + 239 + ], + "score": 0.87, + "content": "k _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 228, + 278, + 241 + ], + "score": 1.0, + "content": "and the queries", + "type": "text" + }, + { + "bbox": [ + 278, + 230, + 295, + 240 + ], + "score": 0.87, + "content": "q _ { t j h }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 228, + 308, + 241 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 252, + 381, + 280 + ], + "lines": [ + { + "bbox": [ + 230, + 252, + 381, + 280 + ], + "spans": [ + { + "bbox": [ + 230, + 252, + 381, + 280 + ], + "score": 0.94, + "content": "c _ { t i j h } = \\mathrm { s o f t m a x } ^ { ( i ) } \\left( \\frac { q _ { t j h } \\cdot k _ { i } } { \\left| \\left| q _ { t j h } \\right| \\right| \\cdot \\left| \\left| k _ { i } \\right| \\right| } \\right)", + "type": "interline_equation", + "image_path": "1bb5f46999968c3a1c30f85f187deb3f1106cda326578faa900fb8408b86eb63.jpg" + } + ] + } + ], + "index": 12.5, + "virtual_lines": [ + { + "bbox": [ + 230, + 252, + 381, + 266.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 230, + 266.0, + 381, + 280.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "title", + "bbox": [ + 88, + 289, + 239, + 302 + ], + "lines": [ + { + "bbox": [ + 84, + 288, + 240, + 304 + ], + "spans": [ + { + "bbox": [ + 84, + 288, + 240, + 304 + ], + "score": 1.0, + "content": "153 Making Every Program Count", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 87, + 308, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 87, + 307, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 87, + 311, + 100, + 320 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 105, + 307, + 505, + 322 + ], + "score": 1.0, + "content": "Similarly to [12, 31], in addition to the content-based attention, we employ a least-used reading", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 319, + 504, + 333 + ], + "spans": [ + { + "bbox": [ + 86, + 321, + 100, + 331 + ], + "score": 1.0, + "content": "155", + "type": "text" + }, + { + "bbox": [ + 105, + 319, + 504, + 333 + ], + "score": 1.0, + "content": "strategy to encourage the Program Controller to assign different singular programs to different com-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 330, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 86, + 333, + 99, + 342 + ], + "score": 1.0, + "content": "156", + "type": "text" + }, + { + "bbox": [ + 105, + 330, + 505, + 344 + ], + "score": 1.0, + "content": "ponents. In particular, we calculate the memory usage for each program slot across attentions as", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 85, + 342, + 100, + 353 + ], + "spans": [ + { + "bbox": [ + 85, + 342, + 100, + 353 + ], + "score": 1.0, + "content": "157", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 259, + 349, + 352, + 373 + ], + "lines": [ + { + "bbox": [ + 259, + 349, + 352, + 373 + ], + "spans": [ + { + "bbox": [ + 259, + 349, + 352, + 373 + ], + "score": 0.95, + "content": "m _ { t i j h } = \\operatorname* { m a x } _ { \\tilde { j } \\leq j } \\left( w _ { t i \\tilde { j } h } \\right)", + "type": "interline_equation", + "image_path": "20d2e1b3e95cc99e60d768bc0080c3996ae302e11b21cde534c96b944ec0386e.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 259, + 349, + 352, + 373 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 88, + 378, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 87, + 368, + 504, + 398 + ], + "spans": [ + { + "bbox": [ + 87, + 380, + 101, + 389 + ], + "score": 1.0, + "content": "158", + "type": "text" + }, + { + "bbox": [ + 100, + 368, + 240, + 398 + ], + "score": 1.0, + "content": "Since we want to consider only", + "type": "text" + }, + { + "bbox": [ + 240, + 379, + 249, + 390 + ], + "score": 0.86, + "content": "l _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 368, + 288, + 398 + ], + "score": 1.0, + "content": "amongst", + "type": "text" + }, + { + "bbox": [ + 289, + 379, + 298, + 388 + ], + "score": 0.83, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 368, + 482, + 398 + ], + "score": 1.0, + "content": "memory slots that have smallest usages, let", + "type": "text" + }, + { + "bbox": [ + 482, + 377, + 504, + 392 + ], + "score": 0.92, + "content": "\\hat { m } _ { t j h } ^ { l _ { I } }", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 84, + 390, + 460, + 403 + ], + "spans": [ + { + "bbox": [ + 84, + 390, + 199, + 403 + ], + "score": 1.0, + "content": "159 denote the value of the", + "type": "text" + }, + { + "bbox": [ + 199, + 391, + 208, + 402 + ], + "score": 0.86, + "content": "l _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 390, + 460, + 403 + ], + "score": 1.0, + "content": "-th smallest usage, then the least-used attention is computed as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 414, + 406, + 449 + ], + "lines": [ + { + "bbox": [ + 204, + 414, + 406, + 449 + ], + "spans": [ + { + "bbox": [ + 204, + 414, + 406, + 449 + ], + "score": 0.93, + "content": "l _ { t i j h } = \\left\\{ \\begin{array} { l l } { \\underset { i } { \\operatorname* { m a x } } \\left( m _ { t i j h } \\right) - m _ { t i j h } } & { ; m _ { t i j h } \\leq \\hat { m } _ { t j h } ^ { l _ { I } } } \\\\ { 0 } & { ; m _ { t i j h } > \\hat { m } _ { t j h } ^ { l _ { I } } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "3eff87705475f2c68a1c4480bd1c57076406de4a285ac39dac6df397bafd8964.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 414, + 406, + 431.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 204, + 431.5, + 406, + 449.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 450, + 316, + 462 + ], + "lines": [ + { + "bbox": [ + 85, + 449, + 316, + 464 + ], + "spans": [ + { + "bbox": [ + 85, + 449, + 316, + 464 + ], + "score": 1.0, + "content": "160 The final program memory attention is computed as", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 477, + 425, + 491 + ], + "lines": [ + { + "bbox": [ + 186, + 477, + 425, + 491 + ], + "spans": [ + { + "bbox": [ + 186, + 477, + 425, + 491 + ], + "score": 0.9, + "content": "w _ { t i j h } = \\mathrm { s i g m o i d } \\left( g _ { t i j h } \\right) c _ { t i j h } + \\left( 1 - \\mathrm { s i g m o i d } \\left( g _ { t i j h } \\right) \\right) l _ { t i j h }", + "type": "interline_equation", + "image_path": "7f5851a3247de278761e154705908476d92d30012c85ff80f22df974bfe7e41f.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 186, + 477, + 425, + 491 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 493, + 505, + 527 + ], + "lines": [ + { + "bbox": [ + 86, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 86, + 495, + 100, + 504 + ], + "score": 1.0, + "content": "161", + "type": "text" + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "Since the usage record are computed along the memory accesses, the multi-step Neurocoder utilises", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 503, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 86, + 505, + 100, + 515 + ], + "score": 1.0, + "content": "162", + "type": "text" + }, + { + "bbox": [ + 105, + 503, + 506, + 517 + ], + "score": 1.0, + "content": "this attention mechanism better than the single-step Neurocoder, creating different attention styles", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 514, + 466, + 527 + ], + "spans": [ + { + "bbox": [ + 86, + 516, + 100, + 526 + ], + "score": 1.0, + "content": "163", + "type": "text" + }, + { + "bbox": [ + 105, + 514, + 313, + 527 + ], + "score": 1.0, + "content": "(see Sec. 3.2). 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In particular, we calculate the memory usage for each program slot across attentions as", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 342, + 100, + 353 + ], + "spans": [ + { + "bbox": [ + 85, + 342, + 100, + 353 + ], + "score": 1.0, + "content": "157", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + } + ], + "index": 16.5, + "bbox_fs": [ + 85, + 307, + 505, + 353 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 259, + 349, + 352, + 373 + ], + "lines": [ + { + "bbox": [ + 259, + 349, + 352, + 373 + ], + "spans": [ + { + "bbox": [ + 259, + 349, + 352, + 373 + ], + "score": 0.95, + "content": "m _ { t i j h } = \\operatorname* { m a x } _ { \\tilde { j } \\leq j } \\left( w _ { t i \\tilde { j } h } \\right)", + "type": "interline_equation", + "image_path": "20d2e1b3e95cc99e60d768bc0080c3996ae302e11b21cde534c96b944ec0386e.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 259, + 349, + 352, + 373 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "index", + "bbox": [ + 88, + 378, + 505, + 403 + ], + "lines": [ + { + "bbox": [ + 87, + 368, + 504, + 398 + ], + "spans": [ + { + "bbox": [ + 87, + 380, + 101, + 389 + ], + "score": 1.0, + "content": "158", + "type": "text" + }, + { + "bbox": [ + 100, + 368, + 240, + 398 + ], + "score": 1.0, + "content": "Since we want to consider only", + "type": "text" + }, + { + "bbox": [ + 240, + 379, + 249, + 390 + ], + "score": 0.86, + "content": "l _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 368, + 288, + 398 + ], + "score": 1.0, + "content": "amongst", + "type": "text" + }, + { + "bbox": [ + 289, + 379, + 298, + 388 + ], + "score": 0.83, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 368, + 482, + 398 + ], + "score": 1.0, + "content": "memory slots that have smallest usages, let", + "type": "text" + }, + { + "bbox": [ + 482, + 377, + 504, + 392 + ], + "score": 0.92, + "content": "\\hat { m } _ { t j h } ^ { l _ { I } }", + "type": "inline_equation" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 390, + 460, + 403 + ], + "spans": [ + { + "bbox": [ + 84, + 390, + 199, + 403 + ], + "score": 1.0, + "content": "159 denote the value of the", + "type": "text" + }, + { + "bbox": [ + 199, + 391, + 208, + 402 + ], + "score": 0.86, + "content": "l _ { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 390, + 460, + 403 + ], + "score": 1.0, + "content": "-th smallest usage, then the least-used attention is computed as", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + } + ], + "index": 20.5, + "bbox_fs": [ + 84, + 368, + 504, + 403 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 414, + 406, + 449 + ], + "lines": [ + { + "bbox": [ + 204, + 414, + 406, + 449 + ], + "spans": [ + { + "bbox": [ + 204, + 414, + 406, + 449 + ], + "score": 0.93, + "content": "l _ { t i j h } = \\left\\{ \\begin{array} { l l } { \\underset { i } { \\operatorname* { m a x } } \\left( m _ { t i j h } \\right) - m _ { t i j h } } & { ; 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Multi-step attention produces richer patterns", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 352, + 439, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 439, + 365 + ], + "score": 1.0, + "content": "compared to single-step Neurocoder that manages only 2 attention weight patterns.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 368, + 504, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "Fig. 2 (c) illustrates how Neurocoder performs modular learning by showing the attention assign-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "ment for top 3 singular vectors as a binary decision tree. Digits under the same parental node share", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "similar attention paths, and thereby similar active programs. Some digits look unique (e.g. 7) result-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "ing in active programs composed of unique attention paths, discriminating themselves early in the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "decision tree. Some digits (e.g. 0 and 9) share the same attention pattern for the first 3 components", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 422, + 496, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 496, + 435 + ], + "score": 1.0, + "content": "and are thus unclassifiable. They can only be distinguished by considering more singular vectors.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 101, + 439, + 505, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "We integrated Neurocoder with deep networks - 5-layer LeNet and 100-layer DenseNet - and tested", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "on CIFAR datasets. Neurocoder significantly outperformed the original Main Networks with perfor-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 460, + 154, + 474 + ], + "score": 1.0, + "content": "mance gain", + "type": "text" + }, + { + "bbox": [ + 154, + 461, + 184, + 471 + ], + "score": 0.91, + "content": "1 - 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 460, + 506, + 474 + ], + "score": 1.0, + "content": ". Compared with recent conditional computing models such as sparse Mixture of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "Experts (MOE [35]) and Neural Stored-program Memory (NSM [22]), Neurocoder required a tenth", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 482, + 486, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 343, + 496 + ], + "score": 1.0, + "content": "of the number of parameters and performed better by up to", + "type": "text" + }, + { + "bbox": [ + 343, + 483, + 380, + 493 + ], + "score": 0.9, + "content": "8 - 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 482, + 486, + 496 + ], + "score": 1.0, + "content": "(see Appendix’s Table 3).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 106, + 508, + 452, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 453, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 453, + 522 + ], + "score": 1.0, + "content": "3.2 Sequential learning - Adaption to sequence changes and game playing using", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 128, + 518, + 230, + 532 + ], + "spans": [ + { + "bbox": [ + 128, + 518, + 230, + 532 + ], + "score": 1.0, + "content": "reinforcement learning", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 103, + 539, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "score": 1.0, + "content": "Recurrent neural networks (RNN) can learn from sequential data by updating the hidden states of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "the networks. However, this does not suffice when local patterns shift, as is often the case. We now", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "demonstrate that Neurocoder helps RNNs overcome this limitation by composing diverse programs", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 572, + 222, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 222, + 585 + ], + "score": 1.0, + "content": "to handle sequence changes.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 105, + 596, + 504, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "Synthetic polynomial auto-regression We created a simple auto-regression task in which data", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 607, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 619 + ], + "score": 1.0, + "content": "points are sampled from polynomial function chunks that change over time. The Main Network is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "a strong RNN–Gated Recurrent Unit (GRU [5]). We found that GRU integrated with a single-step", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "or multi-step Neurocoder converged much faster than all other baselines. 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In contrast, the single-step Neurocoder only changes its attention when", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 86, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "229", + "type": "text" + }, + { + "bbox": [ + 105, + 710, + 236, + 724 + ], + "score": 1.0, + "content": "there is a remarkable change in", + "type": "text" + }, + { + "bbox": [ + 236, + 713, + 243, + 722 + ], + "score": 0.81, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "-coordinate values (Fig. 2(e)). 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All", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 205, + 374, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 374, + 218 + ], + "score": 1.0, + "content": "baselines are applied to the actor/critic networks in the A3C agent.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 87, + 238, + 322, + 250 + ], + "lines": [ + { + "bbox": [ + 84, + 236, + 324, + 253 + ], + "spans": [ + { + "bbox": [ + 84, + 236, + 324, + 253 + ], + "score": 1.0, + "content": "192 3.1 Instance-based learning - Object Recognition", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 101, + 259, + 504, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 257, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 506, + 272 + ], + "score": 1.0, + "content": "We tested Neurocoder on instance-based learning through classical image classification tasks using", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "MNIST [24] and CIFAR [21] datasets. The first experiment interpreted Neurocoder’s behaviour", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 251, + 293 + ], + "score": 1.0, + "content": "in classifying digits into 10 classes", + "type": "text" + }, + { + "bbox": [ + 252, + 281, + 282, + 292 + ], + "score": 0.84, + "content": "( 0 - 9 )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "using linear classifier Main Network. With equivalent", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 505, + 305 + ], + "score": 1.0, + "content": "model size, Neurocoder using the novel recurrent attention surpasses the performance of the linear", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 302, + 263, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 198, + 315 + ], + "score": 1.0, + "content": "classifier [24] by up to", + "type": "text" + }, + { + "bbox": [ + 199, + 302, + 214, + 313 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 302, + 263, + 315 + ], + "score": 1.0, + "content": "(Fig. 2 (a)).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 257, + 506, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 101, + 319, + 504, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 505, + 331 + ], + "score": 1.0, + "content": "To differentiate the input, Neurocoder attends to different components of the active program to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 329, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 506, + 343 + ], + "score": 1.0, + "content": "guide the decision-making process. Fig. 2 (b) shows single-step and multi-step attention to the first", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "3 singular vectors for each digit across memory slots. Multi-step attention produces richer patterns", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 352, + 439, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 439, + 365 + ], + "score": 1.0, + "content": "compared to single-step Neurocoder that manages only 2 attention weight patterns.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 319, + 506, + 365 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 368, + 504, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "Fig. 2 (c) illustrates how Neurocoder performs modular learning by showing the attention assign-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "ment for top 3 singular vectors as a binary decision tree. Digits under the same parental node share", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "similar attention paths, and thereby similar active programs. Some digits look unique (e.g. 7) result-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "ing in active programs composed of unique attention paths, discriminating themselves early in the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "decision tree. Some digits (e.g. 0 and 9) share the same attention pattern for the first 3 components", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 422, + 496, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 496, + 435 + ], + "score": 1.0, + "content": "and are thus unclassifiable. They can only be distinguished by considering more singular vectors.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 367, + 505, + 435 + ] + }, + { + "type": "text", + "bbox": [ + 101, + 439, + 505, + 494 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "We integrated Neurocoder with deep networks - 5-layer LeNet and 100-layer DenseNet - and tested", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 449, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 506, + 462 + ], + "score": 1.0, + "content": "on CIFAR datasets. Neurocoder significantly outperformed the original Main Networks with perfor-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 460, + 154, + 474 + ], + "score": 1.0, + "content": "mance gain", + "type": "text" + }, + { + "bbox": [ + 154, + 461, + 184, + 471 + ], + "score": 0.91, + "content": "1 - 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 460, + 506, + 474 + ], + "score": 1.0, + "content": ". Compared with recent conditional computing models such as sparse Mixture of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "Experts (MOE [35]) and Neural Stored-program Memory (NSM [22]), Neurocoder required a tenth", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 482, + 486, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 343, + 496 + ], + "score": 1.0, + "content": "of the number of parameters and performed better by up to", + "type": "text" + }, + { + "bbox": [ + 343, + 483, + 380, + 493 + ], + "score": 0.9, + "content": "8 - 1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 482, + 486, + 496 + ], + "score": 1.0, + "content": "(see Appendix’s Table 3).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 439, + 506, + 496 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 508, + 452, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 453, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 453, + 522 + ], + "score": 1.0, + "content": "3.2 Sequential learning - Adaption to sequence changes and game playing using", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 128, + 518, + 230, + 532 + ], + "spans": [ + { + "bbox": [ + 128, + 518, + 230, + 532 + ], + "score": 1.0, + "content": "reinforcement learning", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 103, + 539, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 551 + ], + "score": 1.0, + "content": "Recurrent neural networks (RNN) can learn from sequential data by updating the hidden states of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "the networks. However, this does not suffice when local patterns shift, as is often the case. We now", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "demonstrate that Neurocoder helps RNNs overcome this limitation by composing diverse programs", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 572, + 222, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 222, + 585 + ], + "score": 1.0, + "content": "to handle sequence changes.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 539, + 506, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 596, + 504, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "Synthetic polynomial auto-regression We created a simple auto-regression task in which data", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 607, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 619 + ], + "score": 1.0, + "content": "points are sampled from polynomial function chunks that change over time. The Main Network is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "a strong RNN–Gated Recurrent Unit (GRU [5]). We found that GRU integrated with a single-step", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "or multi-step Neurocoder converged much faster than all other baselines. The other conditional", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "computing counterparts (HyperNet [13], FiLM [28]) adapt by re-scaling weights or activation of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 650, + 426, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 426, + 663 + ], + "score": 1.0, + "content": "GRU, which were shown inferior to our modular approach (Appendix’s Fig. 6).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 595, + 506, + 663 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 667, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 86, + 666, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 86, + 668, + 100, + 679 + ], + "score": 1.0, + "content": "225", + "type": "text" + }, + { + "bbox": [ + 105, + 666, + 334, + 680 + ], + "score": 1.0, + "content": "Visualising the first singular vector attention weights in", + "type": "text" + }, + { + "bbox": [ + 334, + 667, + 353, + 678 + ], + "score": 0.89, + "content": "\\mathbf { M } _ { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 666, + 505, + 680 + ], + "score": 1.0, + "content": ", we find that the multi-step attention", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 86, + 680, + 100, + 690 + ], + "score": 1.0, + "content": "226", + "type": "text" + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "Neurocoder changes its attention following polynomial changes - it attends to the same singular pro-", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 86, + 691, + 99, + 700 + ], + "score": 1.0, + "content": "227", + "type": "text" + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "gram when processing data from the same polynomial and alters attention for data from a different", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 86, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "228", + "type": "text" + }, + { + "bbox": [ + 106, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "polynomial (Fig. 2(d)). 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Although single-step Neurocoder", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 86, + 206, + 99, + 215 + ], + "score": 1.0, + "content": "230", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "converges well, it did not discover the underlying structure of the data, and thus underperformed", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 86, + 216, + 99, + 227 + ], + "score": 1.0, + "content": "231", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "the multi-step Neurocoder. We hypothesise that when recurrence is employed, usage-based atten-", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 226, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 86, + 227, + 99, + 236 + ], + "score": 1.0, + "content": "232", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 226, + 505, + 237 + ], + "score": 1.0, + "content": "tion takes effect, stipulating better memory utilisation and diverse attentions over timesteps. We ran", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 236, + 504, + 248 + ], + "spans": [ + { + "bbox": [ + 86, + 239, + 99, + 248 + ], + "score": 1.0, + "content": "233", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 236, + 504, + 248 + ], + "score": 1.0, + "content": "multi-step Neurocoder without usage-based attention. The results were worse than the full multi-", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 248, + 385, + 259 + ], + "spans": [ + { + "bbox": [ + 86, + 249, + 99, + 259 + ], + "score": 1.0, + "content": "234", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 248, + 385, + 259 + ], + "score": 1.0, + "content": "step Neurocoder, which confirms our hypothesis (Appendix’s Fig. 6).", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + } + ], + "index": 40, + "bbox_fs": [ + 86, + 666, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 145, + 70, + 466, + 150 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 145, + 70, + 466, + 150 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 70, + 466, + 150 + ], + "spans": [ + { + "bbox": [ + 145, + 70, + 466, + 150 + ], + "score": 0.978, + "html": "
MethodMN (MLP[17])MN (MLP ours)NSMNeurocoder
Adam55.16±1.3853.55±1.2754.85±2.8158.46±0.46
Adagrad58.08±1.0657.83±2.7458.42±1.8762.28±4.03
L266.00±3.7364.37±2.4062.83±7.2169.89±1.72
SI64.76±3.0964.41±3.3664.36±2.9967.96±3.22
EWC58.85±2.5958.41±2.3758.12±3.2465.66±1.25
O-EWC57.33±1.4457.78±1.8458.55±3.4073.97±1.50
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Final test accuracy (mean and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 163, + 181, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 181, + 175 + ], + "score": 1.0, + "content": "std.) over 10 runs.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 87, + 203, + 504, + 259 + ], + "lines": [ + { + "bbox": [ + 86, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 86, + 206, + 99, + 215 + ], + "score": 1.0, + "content": "230", + "type": "text" + }, + { + "bbox": [ + 105, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "converges well, it did not discover the underlying structure of the data, and thus underperformed", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 86, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 86, + 216, + 99, + 227 + ], + "score": 1.0, + "content": "231", + "type": "text" + }, + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "the multi-step Neurocoder. We hypothesise that when recurrence is employed, usage-based atten-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 86, + 226, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 86, + 227, + 99, + 236 + ], + "score": 1.0, + "content": "232", + "type": "text" + }, + { + "bbox": [ + 106, + 226, + 505, + 237 + ], + "score": 1.0, + "content": "tion takes effect, stipulating better memory utilisation and diverse attentions over timesteps. We ran", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 86, + 236, + 504, + 248 + ], + "spans": [ + { + "bbox": [ + 86, + 239, + 99, + 248 + ], + "score": 1.0, + "content": "233", + "type": "text" + }, + { + "bbox": [ + 106, + 236, + 504, + 248 + ], + "score": 1.0, + "content": "multi-step Neurocoder without usage-based attention. The results were worse than the full multi-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 86, + 248, + 385, + 259 + ], + "spans": [ + { + "bbox": [ + 86, + 249, + 99, + 259 + ], + "score": 1.0, + "content": "234", + "type": "text" + }, + { + "bbox": [ + 105, + 248, + 385, + 259 + ], + "score": 1.0, + "content": "step Neurocoder, which confirms our hypothesis (Appendix’s Fig. 6).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 104, + 272, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "Atari game reinforcement learning We used reinforcement learning as a further testbed to show", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 282, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 296 + ], + "score": 1.0, + "content": "the ability to adapt to environmental changes. We performed experiments on several Atari 2600", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "games [3] wherein the agent was implemented as the Asynchronous Advantage Actor-Critic (A3C", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 306, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 107, + 306, + 505, + 317 + ], + "score": 1.0, + "content": "[26]). In the Atari platform, agents are allowed to observe the screen snapshot of the games and act", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "to earn the highest score. We augmented the A3C by employing Neurocoder’s working programs", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "for feed-forward layers of the actor and critic networks, aiming to decompose the policy and value", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 338, + 428, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 428, + 351 + ], + "score": 1.0, + "content": "function into singular programs that were selected depending on the game state.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 87, + 354, + 505, + 432 + ], + "lines": [ + { + "bbox": [ + 86, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 86, + 356, + 100, + 366 + ], + "score": 1.0, + "content": "242", + "type": "text" + }, + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "Frostbite and Montezuma’s Revenge. These games are known to be challenging for A3C and other", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 86, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 86, + 367, + 100, + 376 + ], + "score": 1.0, + "content": "243", + "type": "text" + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "algorithms [26]. We trained A3C and HyperNet-based A3C for over 300 million steps, yet these", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 86, + 378, + 100, + 388 + ], + "score": 1.0, + "content": "244", + "type": "text" + }, + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "models did not show any sign of learning, performing equivalently to random agents. For such com-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 86, + 389, + 100, + 398 + ], + "score": 1.0, + "content": "245", + "type": "text" + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "plicated environments with sparse rewards, both the monolithic neural networks and the HyperNet’s", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 86, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 86, + 400, + 100, + 410 + ], + "score": 1.0, + "content": "246", + "type": "text" + }, + { + "bbox": [ + 105, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "unstored fast-weights fail to learn (almost zero scores). In contrast, Neurocoder enabled A3C to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 409, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 86, + 411, + 99, + 419 + ], + "score": 1.0, + "content": "247", + "type": "text" + }, + { + "bbox": [ + 105, + 409, + 506, + 421 + ], + "score": 1.0, + "content": "achieve from 1, 500 to 3, 000 scores on these environments (Fig. 3), confirming the importance of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 86, + 419, + 383, + 433 + ], + "spans": [ + { + "bbox": [ + 86, + 421, + 100, + 432 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 105, + 419, + 383, + 433 + ], + "score": 1.0, + "content": "decomposing a complex solution to smaller, simple stored programs.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 96, + 446, + 400, + 458 + ], + "lines": [ + { + "bbox": [ + 93, + 444, + 401, + 460 + ], + "spans": [ + { + "bbox": [ + 93, + 444, + 401, + 460 + ], + "score": 1.0, + "content": "49 3.3 Multi-task learning - Solving mutliple algorithms simultenously", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 105, + 467, + 504, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "Here we explore the modular learning capability of Neurocoder in multi-task setting. Inspired by al-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "score": 1.0, + "content": "gorithmic sequencing tasks [22], we created a challenging sequential multi-task benchmark wherein", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "the input sequence is a series of sub-sequences from 4 algorithms: Copy, Repeat Copy, Associative", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "Recall and Priority Sort [11]. Each sub-sequence, following a task identification vector, represents", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 300, + 523 + ], + "score": 1.0, + "content": "the input for each task. 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Appendix’s Fig. 7 demonstrates that Neurocoder was performant in both test settings,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 571, + 504, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 249, + 583 + ], + "score": 1.0, + "content": "not only achieving lowest error on", + "type": "text" + }, + { + "bbox": [ + 249, + 571, + 277, + 581 + ], + "score": 0.9, + "content": "n = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 571, + 475, + 583 + ], + "score": 1.0, + "content": ", but also being the only one generalised well to", + "type": "text" + }, + { + "bbox": [ + 476, + 571, + 504, + 581 + ], + "score": 0.88, + "content": "n = 8", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 580, + 284, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 284, + 595 + ], + "score": 1.0, + "content": "scenario, which was unseen during training.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 99, + 608, + 470, + 620 + ], + "lines": [ + { + "bbox": [ + 104, + 606, + 472, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 210, + 623 + ], + "score": 1.0, + "content": "3.4 Continual learning", + "type": "text" + }, + { + "bbox": [ + 211, + 611, + 217, + 617 + ], + "score": 0.3, + "content": "-", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 606, + 472, + 623 + ], + "score": 1.0, + "content": "Learning tasks sequentially without catastrophic forgetting", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 504, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "In continual learning, standard neural networks often suffer from “catastrophic forgetting” in which", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 640, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 506, + 652 + ], + "score": 1.0, + "content": "they cannot retain knowledge acquired from old tasks upon learning new ones [10]. Our Neurocoder", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "offers natural mitigation of such catastrophic forgetting in neural networks by attending to different", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 662, + 304, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 304, + 674 + ], + "score": 1.0, + "content": "singular programs whilst learning different tasks.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 104, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "In this case, in addition to the Main Network, we examine several continual learning algorithms", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "with and without Neurocoder. 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We demonstrate that Neurocoder", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 145, + 70, + 466, + 150 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 145, + 70, + 466, + 150 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 70, + 466, + 150 + ], + "spans": [ + { + "bbox": [ + 145, + 70, + 466, + 150 + ], + "score": 0.978, + "html": "
MethodMN (MLP[17])MN (MLP ours)NSMNeurocoder
Adam55.16±1.3853.55±1.2754.85±2.8158.46±0.46
Adagrad58.08±1.0657.83±2.7458.42±1.8762.28±4.03
L266.00±3.7364.37±2.4062.83±7.2169.89±1.72
SI64.76±3.0964.41±3.3664.36±2.9967.96±3.22
EWC58.85±2.5958.41±2.3758.12±3.2465.66±1.25
O-EWC57.33±1.4457.78±1.8458.55±3.4073.97±1.50
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We performed experiments on several Atari 2600", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "games [3] wherein the agent was implemented as the Asynchronous Advantage Actor-Critic (A3C", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 306, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 107, + 306, + 505, + 317 + ], + "score": 1.0, + "content": "[26]). In the Atari platform, agents are allowed to observe the screen snapshot of the games and act", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "to earn the highest score. 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These games are known to be challenging for A3C and other", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 86, + 367, + 100, + 376 + ], + "score": 1.0, + "content": "243", + "type": "text" + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "algorithms [26]. We trained A3C and HyperNet-based A3C for over 300 million steps, yet these", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 86, + 378, + 100, + 388 + ], + "score": 1.0, + "content": "244", + "type": "text" + }, + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "models did not show any sign of learning, performing equivalently to random agents. For such com-", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 86, + 389, + 100, + 398 + ], + "score": 1.0, + "content": "245", + "type": "text" + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "plicated environments with sparse rewards, both the monolithic neural networks and the HyperNet’s", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 86, + 400, + 100, + 410 + ], + "score": 1.0, + "content": "246", + "type": "text" + }, + { + "bbox": [ + 105, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "unstored fast-weights fail to learn (almost zero scores). 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Inspired by al-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 490 + ], + "score": 1.0, + "content": "gorithmic sequencing tasks [22], we created a challenging sequential multi-task benchmark wherein", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "the input sequence is a series of sub-sequences from 4 algorithms: Copy, Repeat Copy, Associative", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "Recall and Priority Sort [11]. 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In each input sequence,", + "type": "text" + }, + { + "bbox": [ + 300, + 513, + 308, + 520 + ], + "score": 0.62, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "tasks were sampled from the set of 4 algorithms", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 521, + 443, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 443, + 534 + ], + "score": 1.0, + "content": "randomly with replacement and the output sequences were created correspondingly.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 466, + 506, + 534 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 538, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "We trained a MANN–Neural Turing Machine (NTM [11]) Main Network with FiLM, HyperNet and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 294, + 561 + ], + "score": 1.0, + "content": "our Neurocoder augmentation on sequences of", + "type": "text" + }, + { + "bbox": [ + 294, + 550, + 320, + 559 + ], + "score": 0.89, + "content": "n = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 548, + 461, + 561 + ], + "score": 1.0, + "content": "tasks, and tested with sequences of", + "type": "text" + }, + { + "bbox": [ + 462, + 549, + 487, + 559 + ], + "score": 0.9, + "content": "n = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 133, + 570 + ], + "score": 0.9, + "content": "n = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "tasks. Appendix’s Fig. 7 demonstrates that Neurocoder was performant in both test settings,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 571, + 504, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 249, + 583 + ], + "score": 1.0, + "content": "not only achieving lowest error on", + "type": "text" + }, + { + "bbox": [ + 249, + 571, + 277, + 581 + ], + "score": 0.9, + "content": "n = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 571, + 475, + 583 + ], + "score": 1.0, + "content": ", but also being the only one generalised well to", + "type": "text" + }, + { + "bbox": [ + 476, + 571, + 504, + 581 + ], + "score": 0.88, + "content": "n = 8", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 580, + 284, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 284, + 595 + ], + "score": 1.0, + "content": "scenario, which was unseen during training.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 537, + 505, + 595 + ] + }, + { + "type": "title", + "bbox": [ + 99, + 608, + 470, + 620 + ], + "lines": [ + { + "bbox": [ + 104, + 606, + 472, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 210, + 623 + ], + "score": 1.0, + "content": "3.4 Continual learning", + "type": "text" + }, + { + "bbox": [ + 211, + 611, + 217, + 617 + ], + "score": 0.3, + "content": "-", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 606, + 472, + 623 + ], + "score": 1.0, + "content": "Learning tasks sequentially without catastrophic forgetting", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 629, + 504, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "In continual learning, standard neural networks often suffer from “catastrophic forgetting” in which", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 640, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 506, + 652 + ], + "score": 1.0, + "content": "they cannot retain knowledge acquired from old tasks upon learning new ones [10]. Our Neurocoder", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "offers natural mitigation of such catastrophic forgetting in neural networks by attending to different", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 662, + 304, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 304, + 674 + ], + "score": 1.0, + "content": "singular programs whilst learning different tasks.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 629, + 506, + 674 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 678, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "In this case, in addition to the Main Network, we examine several continual learning algorithms", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "with and without Neurocoder. These algorithms, including Elastic Weight Consolidation (EWC", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 713 + ], + "score": 1.0, + "content": "[41]) and Synaptic Intelligence (SI [41]), work by regularising the loss function and thus can be", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 711, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 339, + 724 + ], + "score": 1.0, + "content": "easily combined with Neurocoder by modifying the loss", + "type": "text" + }, + { + "bbox": [ + 339, + 711, + 362, + 722 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { t a s k }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 711, + 506, + 724 + ], + "score": 1.0, + "content": ". We demonstrate that Neurocoder", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 677, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 94, + 72, + 505, + 95 + ], + "lines": [ + { + "bbox": [ + 93, + 73, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 93, + 73, + 505, + 86 + ], + "score": 1.0, + "content": "70 can improve these continual learning algorithms without requiring additional assumptions as in other", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 94, + 82, + 413, + 98 + ], + "spans": [ + { + "bbox": [ + 94, + 82, + 413, + 98 + ], + "score": 1.0, + "content": "1 approaches [25, 36, 34] that either utilise task embedding or replay memory.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 100, + 108, + 504, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 108, + 505, + 121 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 505, + 121 + ], + "score": 1.0, + "content": "Split MNIST We first considered the split MNIST dataset–a standard continual learning bench-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 119, + 505, + 132 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 505, + 132 + ], + "score": 1.0, + "content": "mark wherein the original MNIST was split into a 5 2-way classification tasks, consecutively pre-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 130, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 505, + 143 + ], + "score": 1.0, + "content": "sented to a Multi-layer Perceptron Main Network (MLP). We followed the benchmarking as in [17]", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 141, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 505, + 153 + ], + "score": 1.0, + "content": "in which various optimisers and state-of-the-art continual learning methods were examined under in-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 151, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 506, + 165 + ], + "score": 1.0, + "content": "cremental task and domain scenarios. We measured the performance of the MLP versus Neurocoder", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 505, + 175 + ], + "score": 1.0, + "content": "and NSM under each continual learning method. In both scenarios, Neurocoder was compatible", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "with all continual leaning methods, demonstrating superior performance over MLP and NSM with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 184, + 382, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 232, + 198 + ], + "score": 1.0, + "content": "performance gain between 1 to", + "type": "text" + }, + { + "bbox": [ + 232, + 185, + 252, + 195 + ], + "score": 0.88, + "content": "1 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 184, + 382, + 198 + ], + "score": 1.0, + "content": "(see Appendix’s Table 5 and 1).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 88, + 209, + 504, + 243 + ], + "lines": [ + { + "bbox": [ + 86, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 86, + 211, + 100, + 221 + ], + "score": 1.0, + "content": "280", + "type": "text" + }, + { + "bbox": [ + 104, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "Split CIFAR We verified the scalability of Neurocoder to more challenging datasets. We split", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 219, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 86, + 222, + 99, + 232 + ], + "score": 1.0, + "content": "281", + "type": "text" + }, + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "score": 1.0, + "content": "CIFAR datasets as in the split MNIST, resulting in 5-task 2-way split CIFAR10 and a 20-task 5-way", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 231, + 449, + 244 + ], + "spans": [ + { + "bbox": [ + 86, + 233, + 100, + 243 + ], + "score": 1.0, + "content": "282", + "type": "text" + }, + { + "bbox": [ + 104, + 231, + 449, + 244 + ], + "score": 1.0, + "content": "split CIFAR100. We used Main Network ResNet [15]–a very deep CNN architecture.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 87, + 247, + 504, + 303 + ], + "lines": [ + { + "bbox": [ + 86, + 247, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 86, + 250, + 99, + 258 + ], + "score": 1.0, + "content": "283", + "type": "text" + }, + { + "bbox": [ + 105, + 247, + 262, + 261 + ], + "score": 1.0, + "content": "When we stressed the orthogonal loss", + "type": "text" + }, + { + "bbox": [ + 263, + 248, + 293, + 258 + ], + "score": 0.86, + "content": "a = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 247, + 505, + 261 + ], + "score": 1.0, + "content": ") and used bigger program memory (100 slots), Neu-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 86, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 86, + 261, + 99, + 270 + ], + "score": 1.0, + "content": "284", + "type": "text" + }, + { + "bbox": [ + 105, + 258, + 277, + 271 + ], + "score": 1.0, + "content": "rocoder improved ResNet classification by", + "type": "text" + }, + { + "bbox": [ + 277, + 258, + 297, + 269 + ], + "score": 0.88, + "content": "1 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 258, + 314, + 271 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 315, + 258, + 334, + 269 + ], + "score": 0.89, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "on CIFAR10 and CIFAR100, respectively.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 86, + 271, + 99, + 280 + ], + "score": 1.0, + "content": "285", + "type": "text" + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "When we integrated Neurocoder with Synaptic Intelligence (SI [41]), the performance was further", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 85, + 278, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 85, + 282, + 100, + 293 + ], + "score": 1.0, + "content": "286", + "type": "text" + }, + { + "bbox": [ + 105, + 278, + 312, + 294 + ], + "score": 1.0, + "content": "improved, maintaining a stable performance above", + "type": "text" + }, + { + "bbox": [ + 313, + 280, + 333, + 291 + ], + "score": 0.89, + "content": "8 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 278, + 506, + 294 + ], + "score": 1.0, + "content": "accuracy for CIFAR10 and outperforming", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 85, + 291, + 358, + 304 + ], + "spans": [ + { + "bbox": [ + 85, + 293, + 100, + 303 + ], + "score": 1.0, + "content": "287", + "type": "text" + }, + { + "bbox": [ + 105, + 291, + 178, + 304 + ], + "score": 1.0, + "content": "using SI alone by", + "type": "text" + }, + { + "bbox": [ + 179, + 291, + 198, + 302 + ], + "score": 0.88, + "content": "1 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 291, + 358, + 304 + ], + "score": 1.0, + "content": "for CIFAR100 (see Appendix’s Fig. 8).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 88, + 319, + 180, + 333 + ], + "lines": [ + { + "bbox": [ + 84, + 318, + 181, + 335 + ], + "spans": [ + { + "bbox": [ + 84, + 318, + 181, + 335 + ], + "score": 1.0, + "content": "288 4 Discussion", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 105, + 345, + 504, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "Our experiments demonstrate that Neurocoder is capable of re-coding Neural Programs in distinctive", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "neural networks, amplifying their capabilities in diverse learning scenarios: instance-based, sequen-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 505, + 380 + ], + "score": 1.0, + "content": "tial, multi-task and continual learning. This consistently results in significant performance increase,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "and further creates novel robustness to pattern shift and catastrophic forgetting. This ability for each", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "architecture to re-code itself is made possible without changing the way it is trained, or majorly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 399, + 418, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 418, + 412 + ], + "score": 1.0, + "content": "increasing the number of parameters it needs to learn (see Appendix Table 7).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 101, + 416, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "The MNIST problem illustrates the reasoning process of Neurocoder when classifying digit images", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "score": 1.0, + "content": "wherein its singular program assignment resembles a binary tree decision-making process - it shows", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "how some singular programs are shared, others are not. The polynomial auto-regression problem", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "highlights the importance of efficient memory utilisation in re-constructing the working program", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "score": 1.0, + "content": "enabling discovery of hidden structures in sequential data. Training our framework with reinforce-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "ment learning, we enable neural agents to solve complex games wherein traditional methods fail", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "or learn slowly. Neurocoder also works well with multi-task setting, as shown in the challenging", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "score": 1.0, + "content": "multi-algorithm benchmark. Finally, continual learning problems show that Neurocoder mitigates", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 503, + 419, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 419, + 516 + ], + "score": 1.0, + "content": "catastrophic forgetting efficiently under different learning settings/algorithms.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 86, + 520, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 86, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 86, + 522, + 99, + 531 + ], + "score": 1.0, + "content": "304", + "type": "text" + }, + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "Our solution offers a single framework that is scalable and adaptable to various problems and learn-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 85, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 85, + 532, + 100, + 543 + ], + "score": 1.0, + "content": "305", + "type": "text" + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "ing paradigms. Unlike previous attempts to employ a bank of separate big programs [20, 35, 22],", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 85, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 85, + 543, + 100, + 554 + ], + "score": 1.0, + "content": "306", + "type": "text" + }, + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "Neurocoder maintains only shareable, smaller components that can reconstruct the whole program", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 85, + 553, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 85, + 554, + 100, + 565 + ], + "score": 1.0, + "content": "307", + "type": "text" + }, + { + "bbox": [ + 105, + 553, + 506, + 565 + ], + "score": 1.0, + "content": "space, thereby heavily utilising the parameters and preventing the model from proliferating. We", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 86, + 566, + 99, + 575 + ], + "score": 1.0, + "content": "308", + "type": "text" + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "note that Neurocoder is orthogonal to approaches employing tensor decomposition to reduce the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 86, + 573, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 86, + 577, + 99, + 586 + ], + "score": 1.0, + "content": "309", + "type": "text" + }, + { + "bbox": [ + 105, + 573, + 505, + 588 + ], + "score": 1.0, + "content": "number of parameters or hasten the computation [27, 23]. Neurocoder composes rather than decom-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 85, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 85, + 587, + 100, + 597 + ], + "score": 1.0, + "content": "310", + "type": "text" + }, + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "score": 1.0, + "content": "pose the neural weights. Our aim is not only to enable efficient parameter usage, but also achieve", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 85, + 596, + 496, + 609 + ], + "spans": [ + { + "bbox": [ + 85, + 598, + 100, + 608 + ], + "score": 1.0, + "content": "311", + "type": "text" + }, + { + "bbox": [ + 105, + 596, + 496, + 609 + ], + "score": 1.0, + "content": "general-purpose computing power, outperforming other methods in numerous learning problems.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 86, + 612, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 86, + 611, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 86, + 614, + 100, + 624 + ], + "score": 1.0, + "content": "312", + "type": "text" + }, + { + "bbox": [ + 105, + 611, + 506, + 625 + ], + "score": 1.0, + "content": "One limitation of this work is the number of additional hyperparameters, which prevents us from", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 85, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 85, + 625, + 100, + 636 + ], + "score": 1.0, + "content": "313", + "type": "text" + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "fully tuning Neurocoder. Our research aims to add new capabilities to current neural networks to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 85, + 635, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 85, + 636, + 100, + 646 + ], + "score": 1.0, + "content": "314", + "type": "text" + }, + { + "bbox": [ + 105, + 635, + 506, + 646 + ], + "score": 1.0, + "content": "improve their performance and make them robust in different learning scenario. Hence, we do", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 85, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 85, + 647, + 100, + 657 + ], + "score": 1.0, + "content": "315", + "type": "text" + }, + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "not see any intermediate negative societal impact. 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This consistently results in significant performance increase,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "and further creates novel robustness to pattern shift and catastrophic forgetting. This ability for each", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "architecture to re-code itself is made possible without changing the way it is trained, or majorly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 399, + 418, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 418, + 412 + ], + "score": 1.0, + "content": "increasing the number of parameters it needs to learn (see Appendix Table 7).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 346, + 505, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 101, + 416, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "The MNIST problem illustrates the reasoning process of Neurocoder when classifying digit images", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "score": 1.0, + "content": "wherein its singular program assignment resembles a binary tree decision-making process - it shows", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "how some singular programs are shared, others are not. The polynomial auto-regression problem", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "highlights the importance of efficient memory utilisation in re-constructing the working program", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "score": 1.0, + "content": "enabling discovery of hidden structures in sequential data. Training our framework with reinforce-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "ment learning, we enable neural agents to solve complex games wherein traditional methods fail", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "or learn slowly. Neurocoder also works well with multi-task setting, as shown in the challenging", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 505, + 506 + ], + "score": 1.0, + "content": "multi-algorithm benchmark. Finally, continual learning problems show that Neurocoder mitigates", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 503, + 419, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 419, + 516 + ], + "score": 1.0, + "content": "catastrophic forgetting efficiently under different learning settings/algorithms.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 415, + 506, + 516 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 520, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 86, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 86, + 522, + 99, + 531 + ], + "score": 1.0, + "content": "304", + "type": "text" + }, + { + "bbox": [ + 105, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "Our solution offers a single framework that is scalable and adaptable to various problems and learn-", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 85, + 532, + 100, + 543 + ], + "score": 1.0, + "content": "305", + "type": "text" + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "ing paradigms. 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We", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 563, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 86, + 566, + 99, + 575 + ], + "score": 1.0, + "content": "308", + "type": "text" + }, + { + "bbox": [ + 105, + 563, + 506, + 577 + ], + "score": 1.0, + "content": "note that Neurocoder is orthogonal to approaches employing tensor decomposition to reduce the", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 573, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 86, + 577, + 99, + 586 + ], + "score": 1.0, + "content": "309", + "type": "text" + }, + { + "bbox": [ + 105, + 573, + 505, + 588 + ], + "score": 1.0, + "content": "number of parameters or hasten the computation [27, 23]. Neurocoder composes rather than decom-", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 85, + 587, + 100, + 597 + ], + "score": 1.0, + "content": "310", + "type": "text" + }, + { + "bbox": [ + 105, + 584, + 506, + 599 + ], + "score": 1.0, + "content": "pose the neural weights. Our aim is not only to enable efficient parameter usage, but also achieve", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 596, + 496, + 609 + ], + "spans": [ + { + "bbox": [ + 85, + 598, + 100, + 608 + ], + "score": 1.0, + "content": "311", + "type": "text" + }, + { + "bbox": [ + 105, + 596, + 496, + 609 + ], + "score": 1.0, + "content": "general-purpose computing power, outperforming other methods in numerous learning problems.", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 611, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 86, + 614, + 100, + 624 + ], + "score": 1.0, + "content": "312", + "type": "text" + }, + { + "bbox": [ + 105, + 611, + 506, + 625 + ], + "score": 1.0, + "content": "One limitation of this work is the number of additional hyperparameters, which prevents us from", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 623, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 85, + 625, + 100, + 636 + ], + "score": 1.0, + "content": "313", + "type": "text" + }, + { + "bbox": [ + 105, + 623, + 506, + 636 + ], + "score": 1.0, + "content": "fully tuning Neurocoder. Our research aims to add new capabilities to current neural networks to", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 635, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 85, + 636, + 100, + 646 + ], + "score": 1.0, + "content": "314", + "type": "text" + }, + { + "bbox": [ + 105, + 635, + 506, + 646 + ], + "score": 1.0, + "content": "improve their performance and make them robust in different learning scenario. Hence, we do", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 645, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 85, + 647, + 100, + 657 + ], + "score": 1.0, + "content": "315", + "type": "text" + }, + { + "bbox": [ + 105, + 645, + 505, + 657 + ], + "score": 1.0, + "content": "not see any intermediate negative societal impact. 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We can also further extend Neurocoder’s", + "type": "text" + } + ], + "index": 47, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 85, + 680, + 100, + 690 + ], + "score": 1.0, + "content": "318", + "type": "text" + }, + { + "bbox": [ + 106, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "ability by allowing a growing Program Memory, in which the model decides to add or erase memory", + "type": "text" + } + ], + "index": 48, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 85, + 691, + 100, + 700 + ], + "score": 1.0, + "content": "319", + "type": "text" + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "slots as the number of data patterns grows or shrinks beyond the current program space’s capacity.", + "type": "text" + } + ], + "index": 49, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 86, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "320", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 505, + 713 + ], + "score": 1.0, + "content": "Such a system represents a more flexible general-purpose computer that can dynamically allocate", + "type": "text" + } + ], + "index": 50, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 711, + 361, + 723 + ], + "spans": [ + { + "bbox": [ + 85, + 713, + 99, + 723 + ], + "score": 1.0, + "content": "321", + "type": "text" + }, + { + "bbox": [ + 104, + 711, + 361, + 723 + ], + "score": 1.0, + "content": "computing resources by itself without human pre-specification.", + "type": "text" + } + ], + "index": 51, + "is_list_start_line": true + } + ], + "index": 37.5, + "bbox_fs": [ + 85, + 520, + 506, + 609 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 612, + 505, + 722 + ], + "lines": [], + "index": 46.5, + "bbox_fs": [ + 85, + 611, + 506, + 723 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 93, + 70, + 507, + 725 + ], + "lines": [ + { + "bbox": [ + 105, + 70, + 165, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 70, + 165, + 86 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 109, + 90, + 507, + 105 + ], + "spans": [ + { + "bbox": [ + 109, + 90, + 507, + 105 + ], + "score": 1.0, + "content": "[1] Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Dan Klein. 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MethodMN (MLP[17])MN (MLP ours)NSMNeurocoder
Adam55.16±1.3853.55±1.2754.85±2.8158.46±0.46
Adagrad58.08±1.0657.83±2.7458.42±1.8762.28±4.03
L266.00±3.7364.37±2.4062.83±7.2169.89±1.72
SI64.76±3.0964.41±3.3664.36±2.9967.96±3.22
EWC58.85±2.5958.41±2.3758.12±3.2465.66±1.25
O-EWC57.33±1.4457.78±1.8458.55±3.4073.97±1.50
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diff --git a/parse/train/ZS394D3djsg/ZS394D3djsg.md b/parse/train/ZS394D3djsg/ZS394D3djsg.md new file mode 100644 index 0000000000000000000000000000000000000000..2bcfc633a79a7ac7dda9ccd2c1974c6e0b5329f5 --- /dev/null +++ b/parse/train/ZS394D3djsg/ZS394D3djsg.md @@ -0,0 +1,457 @@ +# Fitting large mixture models using stochastic component selection + +Anonymous Author(s) +Affiliation +Address +email + +# Abstract + +1 Traditional methods for unsupervised learning of finite mixture models require to +2 evaluate the likelihood of all components of the mixture. This becomes computa +3 tionally prohibitive when the number of components is large, as it is, for example, +4 in the sum-product (transform) networks. As a remedy, we propose an approach +5 combining the expectation maximization and the Metropolis-Hastings algorithm +6 to evaluate only a small number of, stochastically sampled, components, thus +7 substantially reducing the computational cost. We put emphasis on generality of +8 our method, equipping it with the ability to train both shallow and deep mixture +9 models which involve complex, and possibly nonlinear, transformations. The +10 performance of our method is illustrated in a variety of synthetic and real-data +11 contexts, considering deep models, such as mixtures of normalizing flows and +12 sum-product (transform) networks. + +# 13 1 Introduction + +14 Finite mixture models [40] constitute a fundamental class of density estimation models. They +15 have been successfully applied in diverse fields, including bioinformatics [49], econometrics [10], +16 engineering [33], etc. A mixture model relies on a weighted sum of probability distributions—here +17 referred to as components—to cluster $N$ unlabelled datapoints into $K$ categories. The traditional +18 maximum likelihood techniques train the model by optimizing either (i) the marginal likelihood via +19 gradient-descent [50] or (ii) the evidence lower bound via variational methods [4], including the +20 expectation-maximization (EM) [13]. The dependence structure among approximate, variational, +21 distributions then ranges from the fully independent (mean-field) [25] to fully dependent [30]. The +22 sampling-based techniques target the posterior distribution using sequential Monte Carlo [9] or +23 Markov chain Monte Carlo [52], e.g. via the Gibbs [34] or Metropolis-Hastings sampling [38]. The +24 computational cost of these methods typically scales with $\mathcal { O } ( T K N D )$ operations, where $N$ and $K$ +25 are defined above, $T$ is the number of iterations and $D$ is the dimension of data. +26 Various methods to decrease the computational cost via any factor in $\mathcal { O } ( T K N D )$ have been proposed. +27 $T$ can be lowered by proper initialization, e.g. the optimal seeding [5]; an efficient step-size schedule, +28 e.g. the line-search [58]; or increased estimation precision, e.g. the variance reduction [8]. $N$ is often +29 reduced using the coreset methods, which approximate the original dataset by a weighted dataset +30 such that the exact and approximate marginal likelihoods are close. The weighted variants of the +31 variational [17, 59, 6] and sampling-based [39] methods then process the coresets. Reducing $D$ relies +32 on the compression of data into smaller representations via random projections [53, 2], which is +33 achieved in two ways: (i) each data item is projected into an individual representation [11]; (ii) all +34 data items are projected into an overall representation, commonly referred to as sketch [28, 22]. +35 Nevertheless, all the aforementioned techniques—including those with reduced computational cost— +36 evaluate all $K$ components. This is very demanding for large models, and the problem is even more +37 severe for mixtures involving intricate models, such as neural networks [21, 42], Gaussian processes +38 [57], normalizing flows [48]; or deep mixtures, including sum-product (transform) networks [45, 47], +39 deep Gaussian mixture models [55], etc. In spite of this, a little attention has been paid to the design +40 of algorithms which does not evaluate all $K$ components. The notable exceptions are the sparse EM +41 algorithm [24] and the truncated variational EM algorithm [18], see Table 1 and Section 5 for details. +42 Moreover, the methods are mostly tailored for a specific class of mixture models, e.g. the Gaussian +43 mixture models. + +Table 1: The computational features of various EM algorithms. We compare whether the methods (i) perform the computations with a reduced number of data (minibatching), (ii) update a lower number of statistics, (iii) make less evaluations of the conditional likelihood, and (iv) are suitable for training of deep models. Here, EM, SA, S, T, MC and MH stand for expectation-maximization, stochastic approximation, sparse, truncated, Monte Carlo and Metropolis-Hastings, respectively. + +
Feature/AlgorithmEM [13]SAEM [44]SSAEM [24]TSAEM [18]MCSAEM [1]MHSAEM (ours)
B<Ndatapoints×√
M<K statisticsxx厂
M<Klikelihoodsxx×X
deep models×××X×
+ +44 In this paper, we make the following contributions: + +45 • We propose an EM-based algorithm which relies on the MH sampler to stochastically evaluate less +46 components in mixture models, substantially reducing the computational cost. +7 • We design our method to enable optimization of fairly generic EM objective functions, making it +48 suitable for training of both shallow and deep mixture models. +49 • We apply our approach to Gaussian mixture mdoels (GMMs) and their generalizations: sum +50 product-transform networks (SPTNs) and mixtures of real-valued non-volume preserving (real +51 NVP) flows [15], reaching approximately $1 0 0 \times$ speed-up compared to state-of-the-art methods. + +# 52 2 Problem formulation + +A finite mixture model characterizes the relation between an observed (known) variable, 53 $\boldsymbol { x } \in \times \subseteq \mathbb { R } ^ { D }$ , 54 and a latent (unknown) variable, $z \in Z : = \{ 1 , \dots , K \}$ , via the marginal (incomplete-data) likelihood 55 in the following form: + +$$ +p _ { \theta } ( x ) = \sum _ { k = 1 } ^ { K } p _ { \eta _ { k } } ( x | z = k ) p _ { \pi _ { k } } ( z = k ) , +$$ + +56 where $\theta : = ( \pi _ { 1 } , \eta _ { 1 } , \dots , \pi _ { K } , \eta _ { K } ) \in \Theta$ are unknown parameters. Here, $\eta _ { z }$ are the parameters of the 57 conditional likelihood, $p _ { \eta _ { z } } ( x | z )$ , and $\pi _ { z }$ is the weight which parameterizes the prior, $p _ { \pi _ { z } } ( z ) = \pi _ { z }$ , and satisfies 58 $0 \leq \pi _ { k } \leq 1$ for each $k \in { \mathord { \mathbb { Z } } }$ and $\textstyle \sum _ { k = 1 } ^ { K } \pi _ { k } = 1$ . + +59 Given a set of independent and identically distributed data, $\mathbf { x } : = ( x _ { i } ) _ { i = 1 } ^ { N }$ , our goal is to learn the +60 unknown parameters of the marginal log-likelihood, + +$$ +\mathcal { L } ( \theta ) : = \log p _ { \theta } ( \mathbf { x } ) = \sum _ { i = 1 } ^ { N } \log \sum _ { k = 1 } ^ { K } p _ { \eta _ { k } } ( x _ { i } | z _ { i } = k ) p _ { \pi _ { k } } ( z _ { i } = k ) . +$$ + +61 The marginalization in (2) is tractable for almost all forms of $p _ { \eta _ { z } } ( x | z )$ . Indeed, we consider $p _ { \eta _ { z } } ( x | z )$ +62 to belong to an arbitrary family of $\eta _ { z }$ -differentiable probability distributions. However, we assume that +63 $K$ is high, making the marginalization in (2) computationally costly, thus rendering the optimization +64 objective presumably intractable. Therefore, we want to design a computationally efficient algorithm, +65 requiring only $M < K$ evaluations of $p _ { \eta _ { z } } ( x | z )$ at each iteration. +67 The maximum likelihood estimation seeks the parameters maximizing the marginal log-likelihood, +68 $\theta ^ { M L } : = \arg \operatorname* { m a x } _ { \theta \in \Theta } \mathcal { L } ( \theta )$ . The traditional EM algorithm [13] addresses this task indirectly, i.e. by +69 optimizing the evidence lower bound (ELBO), + +$$ +\mathcal { L } ( \theta ) \geq \mathcal { Q } ( \theta ) + \mathcal { H } ( \hat { \theta } ) : = \mathrm { E L B O } ( \hat { \theta } ) , +$$ + +where 70 $\mathcal { H } ( \hat { \theta } ) : = - \mathsf E _ { p _ { \hat { \theta } } ( \mathbf { z } | \mathbf { x } ) } [ \log p _ { \hat { \theta } } ( \mathbf { z } | \mathbf { x } ) ]$ is the differential entropy at an estimate, $\hat { \theta } \in \Theta$ , and + +$$ +\mathcal { Q } ( \theta ) : = \mathsf { E } _ { p _ { \hat { \theta } } ( \mathbf { z } | \mathbf { x } ) } [ \log p _ { \theta } ( \mathbf { z } , \mathbf { x } ) ] = \sum _ { i = 1 } ^ { N } \sum _ { k = 1 } ^ { K } p _ { \theta } ( z _ { i } = k | x _ { i } ) \log p _ { \theta } ( z _ { i } = k , x _ { i } ) +$$ + +71 is the EM objective function. Here, $p _ { \theta } ( \mathbf { z } , \mathbf { x } )$ is the joint (complete-data) likelihood, and $p _ { \theta } ( \mathbf { z } | \mathbf { x } )$ is +72 the posterior distribution over the latent variables $\dot { \mathbf { z } } : = ( z _ { i } ) _ { i = 1 } ^ { N }$ . Given an initial value, $\theta _ { 0 }$ , the EM +73 algorithm produces a sequence of estimates, $( \theta _ { t } ) _ { t = 1 } ^ { T }$ , by alternating between the expectation (E) and +74 maximization (M) steps, + +$$ +\begin{array} { r l } & { \mathrm { E \mathrm { - } s t e p } ; ~ \mathcal { Q } _ { t - 1 } ( \theta ) , } \\ & { \mathrm { M \mathrm { - } s t e p } ; ~ \theta _ { t } : = \arg \operatorname* { m a x } _ { \theta \in \Theta } \mathcal { Q } _ { t - 1 } ( \theta ) . } \end{array} +$$ + +75 This sequence is guaranteed to monotonically tighten the ELBO, arriving at a local optimum of (2) +76 under mild regularity assumptions [56]. +77 The EM algorithm is computationally expensive, since (4) evaluates $p _ { \theta } ( z _ { i } , x _ { i } )$ for each $z _ { i } \in \mathbb { Z }$ and +78 $i \in ( 1 , \ldots , N )$ . This has to be performed for all $t \in ( 1 , \ldots , T )$ in (5). Albeit the marginal factor, +79 $p _ { \pi _ { z } } ( z )$ , is just the cheap categorical distribution, the conditional factor, $p _ { \eta _ { z } } ( x | z )$ , typically involves +80 high-dimensional operations (e.g., the inversion of the full $D \times D$ -dimensional covariance matrices +81 in the GMMs). Moreover, the M-step (6) is also expensive for large $K$ . This holds despite that (6) +82 can be reduced to closed-form updates of expected sufficient statistics for $p _ { \eta _ { z } } ( x | z )$ belonging to the +83 exponential family [44] (again, due to high $D$ ). All in all, the computational complexity of the EM +84 algorithm scales with $\mathcal { O } ( T D N K )$ . + +If (6) cannot be computed under a closed-form solution, one can resort to direct gradient-descent optimization of $\mathcal { Q } ( \boldsymbol { \theta } )$ , where arg max is replaced by one (or more) step(s) of a gradient descent technique. The EM algorithm is then referred to as the generalized EM algorithm [56]. + +# 88 4 The generalized MHSAEM algorithm + +89 We design a version of the generalized EM algorithm suitable for scenarios where (4) can represent +90 deep, discrete, latent variable models, thus being parameterized by possibly complex nonlinear +91 transformations. We particularly focus on decreasing the the number of operations in the generalized +92 EM algorithm from $\mathcal { O } ( T D N K )$ to $\mathcal { O } ( T D B M )$ , where $B \ll N$ and $M \ll K$ . + +# 4.1 E-step + +94 We reduce the cost of evaluating the EM objective function (4) by combining the minibatching (as +95 used many times before) and the Monte Carlo sampling. Namely, the specific application of the latter +96 to generic mixture models is the key contribution of this paper. +97 Minibatching. At each iteration, $t$ , we compute the conditional expectation in (4) only for a subset— +98 here referred to as a minibatch—of the original full dataset, i.e. $( x _ { i } ) _ { i \in I }$ . Here, $I$ is a set of $B \ll N$ +99 indices, $i$ , sampled uniformly without replacement from $( 1 , \ldots , \dot { N } )$ . This substantially decreases the +100 necessary computations compared to the full sweep over all $N$ datapoints [23]. +101 Monte Carlo sampling. For each $i \in I$ , we want to draw $M \ll K$ random samples from $p _ { \theta } ( z _ { i } | x _ { i } )$ in +102 order to obtain a Monte Carlo estimate of (4). The straightforward way to do this would be to draw +103 the samples directly from $p _ { \theta } ( z _ { i } | x _ { i } )$ . However, direct sampling from $p _ { \theta } ( z _ { i } | x _ { i } )$ does not lead to any +104 substantial decrease in the number of operations. This is caused by the fact that even for a single +105 sample of $z _ { i }$ , we have to first compute the normalizing factor, $p _ { \theta } ( x _ { i } )$ , to obtaining the posterior, +106 $p _ { \theta } ( z _ { i } | x _ { i } )$ . This requires $K$ expensive evaluations of $p _ { \theta } ( z _ { i } , x _ { i } )$ , which is precisely what we want to +107 avoid. Our approach is to resort to the Markov chain Monte Carlo (MCMC), which allows us to +108 sample from $p _ { \theta } ( z _ { i } | x _ { i } )$ , with the computational complexity decreasing to only a single evaluation of +109 $p _ { \theta } ( z _ { i } , x _ { i } )$ per a single sample of $z _ { i }$ . +110 MCMC methods obviate the computation of the normalizing factor in $p _ { \theta } ( z _ { i } | x _ { i } )$ by simulating a +111 Markov chain, $( z _ { i , t } ) _ { t = 1 } ^ { T }$ , from a transition kernel, $z _ { i , t } \sim P ( z _ { i , t - 1 } , \cdot )$ , which leaves $p _ { \theta } ( z _ { i } | x _ { i } )$ as its +112 unique stationary (invariant) distribution, starting from an initial value $z _ { i , 0 }$ . The specific form of $P$ +113 determines the structure of an MCMC method. We chose the Metropolis-Hastings (MH) sampler, +114 which represents $P ( z _ { i , t - 1 } , z _ { i , t } )$ as follows: given $\bar { z } _ { i } : = z _ { i , t - 1 }$ , draw a sample from the proposal +115 distribution $z _ { i } \sim q ( \cdot | \bar { z } _ { i } )$ , compute the acceptance ratio, + +$$ +\alpha ( \bar { z } _ { i } , z _ { i } ) : = \operatorname* { m i n } \biggr \{ 1 , \frac { p _ { \eta _ { z _ { i } , t - 1 } } ( x _ { i } | z _ { i } ) \pi _ { z _ { i } , t - 1 } q ( \bar { z } _ { i } | z _ { i } ) } { p _ { \eta _ { \bar { z } _ { i } , t - 1 } } ( x _ { i } | \bar { z } _ { i } ) \pi _ { \bar { z } _ { i } , t - 1 } q ( z _ { i } | \bar { z } _ { i } ) } \biggr \} , +$$ + +116 and, if $u < \alpha \big ( \bar { z } _ { i } , z _ { i } \big )$ —where $u$ is drawn from a uniform distribution, Uniform $( 0 , 1 )$ —accept the +117 sample and set $z _ { i , t } = z _ { i }$ ; otherwise, set $z _ { i , t } = \bar { z } _ { i }$ . For each $i \in I$ and $t \in ( 1 , \ldots , T )$ , we repeat +118 this process $M$ times, construing a set $\mathbf { z } _ { i , t } = ( z _ { i , t } ^ { 1 } , \dots , z _ { i , t } ^ { M } )$ . Therefore, at every current iteration, +119 120 $t$ , we caking $\bar { z } _ { i } = z _ { i , t - 1 } ^ { M }$ extend the chain from the point where we left at the previous iteration, . Under mild regularity assumptions [52], the chain passes the transiti $t - 1$ , byriod +121 (the burn-in phase), and the samples can then be used to approximate the conditional expectation in +122 (4) as follows: + +$$ +\widehat { \mathcal { Q } } _ { t - 1 } ( \theta ) = \frac { 1 } { M } \sum _ { i \in I } \sum _ { z \in \mathbf { z } _ { i , t } } \log p _ { \eta _ { z } } ( x _ { i } | z ) \pi _ { z } . +$$ + +123 Note that, to ensure this approach is truly efficient, we have to draw only $M \ll K$ samples at each +124 iteration, $t$ ; otherwise, for $M \approx K$ , we may rather compute the exact marginalization in (4), since it +125 is tractable (but computationally costly). + +# 4.2 M-step + +127 Assume for a moment that (6) with $\mathcal { Q } _ { t - 1 } ( \theta )$ given by (8) has a closed-form solution, yielding an +128 estimate of $\theta$ . Such an estimate would have a high variance, converging only for $M \to \infty$ and +129 $T \to \infty$ [19]. The main reason is that the samples would not be reused over the iterations, $t$ , +130 thus wasting computational resources. We consider that there is no closed-form solution of (6), +131 and—to ensure that the samples (and thus computations) are recycled over the iterations—we use +132 the stochastic approximation (SA) [51] to optimize (8). This is analogous to applying a stochastic +133 gradient-descent method, $\theta _ { t } = \theta _ { t - 1 } + \gamma _ { t } \nabla _ { \theta } \tilde { \mathcal { Q } } _ { t - 1 } ( \theta )$ , where $\gamma _ { t }$ is the step-size, satisfying the Robbins +134 Monro constraints, $\begin{array} { r } { \gamma _ { t } \in [ 0 , 1 ] , \sum _ { t \geq 1 } \gamma _ { t } = \infty , \sum _ { t \geq 1 } \gamma _ { t } ^ { 2 } < \infty , } \end{array}$ and $\nabla _ { \theta }$ is the gradient w.r.t. $\theta$ . In this +135 way, the computations made in $\nabla _ { \boldsymbol { \theta } } \widehat { \mathcal { Q } }$ are accumulated via $\theta _ { t }$ and reused over the iterations. +136 The parameters $\eta _ { z }$ have a different form based on a specific case of $p _ { \eta _ { z } } ( x | z )$ , whereas $\pi _ { z }$ is a +137 permanent structure in (1). Therefore, without loss of generality, we split (6) into a generic part and a +138 fixed part as follows: + +$$ +\begin{array} { r l } & { \eta _ { k , t } = \eta _ { k , t - 1 } + \gamma _ { t } \nabla _ { \eta _ { k } } \widehat { \mathcal { Q } } _ { t - 1 } ( \theta ) , } \\ & { \nu _ { k , t } = \nu _ { k , t - 1 } + \gamma _ { t } \nabla _ { \nu _ { k } } \widehat { \mathcal { Q } } _ { t - 1 } ( \theta ) , } \end{array} +$$ + +139 where—to ensure that the probabilities, $( \pi _ { k , t } ) _ { k = 1 } ^ { K }$ , satisfy the constraints (Section 2)—we transform +140 $\nabla _ { \pi _ { k } } \widehat { \mathcal { Q } }$ via $\nu _ { k } = \log \pi _ { k }$ and optimize w.r.t. $\nu _ { k }$ . Then, to obtain $( \pi _ { k , t } ) _ { k = 1 } ^ { K }$ from $\nu _ { t } : = ( \nu _ { k , t } ) _ { k = 1 } ^ { K }$ , we +141 k b use the softmax function, i.e. $\pi _ { k , t } : = \mathrm { s o f t m a x } ( \pmb { \nu } _ { t } ) _ { k } : = \exp ( \nu _ { k , t } ) / \sum _ { l = 1 } ^ { K } \exp ( \nu _ { l , t } )$ . + +Computing the gradients for all pairs of $( \nu _ { k } , \eta _ { k } ) _ { k = 1 } ^ { K }$ would be inefficient, especially since ${ \bf z } _ { i , t }$ contains only a small number of unique values of Z for $M \ll K$ . Consequently, we compute $\dot { \nabla } _ { \eta _ { k } } \widehat { \mathcal { Q } }$ and $\nabla _ { \nu _ { k } } \widehat { \mathcal { Q } }$ only for $k \in { \mathrm { u n i q u e } } ( \mathbf { z } _ { i , t } )$ . We summarize the proposed approach in Algorithm 1. + +# 4.3 Proposal distribution + +146 The choice of the proposal distribution has a significant impact on the speed of convergence and the computational cost of the proposed algorithm. Here, we discuss various possible choices of 147 $q \big ( z _ { i } | \bar { z } _ { i } \big )$ . + +Input: $\theta _ { 0 }$ , $( \mathbf { z } _ { i , 0 } ) _ { i = 1 } ^ { N }$ , $( \mathbf { x } _ { i } ) _ { i = 1 } ^ { N }$ +Output: $( \theta _ { t } ) _ { t = 1 } ^ { T }$ for $t \in ( 1 , \ldots , T )$ or until convergence do form the set $\dot { I } = ( i _ { j } ) _ { j = 1 } ^ { B }$ by sampling (without replacement) $B$ indices $i \sim ( 1 , \dots , N )$ for $i \in I$ do set $\bar { z } _ { i }$ as the last element of $\mathbf { z } _ { i , t - 1 }$ for $j \in ( 1 , \ldots , M )$ do sample $z _ { i } \sim q ( z _ { i } | \bar { z } _ { i } )$ sample $u \sim \mathrm { U n i f o r m } ( 0 , 1 )$ compute $\alpha ( \bar { z } _ { i } , z _ { i } )$ in (7) if $u < \alpha \big ( \bar { z } _ { i } , z _ { i } \big )$ then set $z _ { i , t } ^ { j } = z _ { i }$ and $\bar { z } _ { i } = z _ { i }$ else set $z _ { i , t } ^ { j } = \bar { z } _ { i }$ end if end for set $\mathbf { z } _ { i , t } = ( z _ { i , t } ^ { 1 } , \dots , z _ { i , t } ^ { M } )$ end for compute (8) compute (9) for $k \in { \mathrm { u n i q u e } } ( \mathbf { z } _ { i , t } )$ compute $\pi _ { k , t } : = \mathrm { s o f t m a x } ( \pmb { \nu } _ { t } ) _ { k }$ for $k \in { \mathord { \mathbb { Z } } }$ end for + +148 Optimal proposal $( O )$ . The optimal proposal distribution is $q ( z _ { i } | \bar { z } _ { i } ) : = q ( z _ { i } ) : = p _ { \theta } ( z _ { i } | x _ { i } )$ . This +149 ensures that the acceptance rate (7) is always $\alpha ( \bar { z } _ { i } , z _ { i } ) = 1$ . However, the need to perform $K$ +150 expensive evaluations of $p _ { \theta } ( z _ { i } , x _ { i } )$ before sampling from $p _ { \theta } ( z _ { i } | x _ { i } )$ is the reason we resorted to +151 the MH sampler in the first place. We consider this case only to set the upper limit on admissible +152 computational cost and to study the impact of sub-optimal proposal distribtions. +153 Uniform proposal $( U )$ . The uniform distribution on the discrete interval from 1 to $K$ , i.e. $q ( z _ { i } | \bar { z } _ { i } ) : =$ +154 $q ( z _ { i } ) : = \mathrm { U n i f o r m } ( 1 , K )$ , is the simplest and computationally cheapest variant of the proposal +155 distribution. However, due to poor mixing properties, the algorithm may converge slowly for high $K$ . +156 Tabular proposal with forgetting $( T F )$ . The key requirement to design a proposal distribution is to +157 restrict its computational complexity somewhere between that of the $\mathrm { U }$ and $\mathrm { o }$ proposals. One way to +158 satisfy this constraint is to use the Markov chain, $( z _ { i , t } ) _ { t = 1 } ^ { T }$ , to learn a transition kernel, $p ( z _ { i } | \bar { z } _ { i } )$ , see, +159 e.g. [3]. Unfortunately, this would require us to store a table with $K ^ { 2 }$ entries for each $i \in ( 1 , \ldots , N )$ , +160 161 which is very demanding evethe Markov chain and define: $q ( z _ { i } | \bar { z } _ { i } ) : = q _ { \alpha _ { i } } ( z _ { i } ) : = \mathcal { C } ( \alpha _ { i } )$ $K$ $N$ herefore, where $\mathcal { C } ( \pmb { \alpha } _ { i } ) \propto \Pi _ { k = 1 } ^ { K } \alpha _ { k , i } ^ { \bar { 1 ( } z _ { i } = k ) }$ nce inis the +162 categorical distribution with the weights $\pmb { \alpha } _ { i } : = ( \alpha _ { 1 , i } , \dots , \alpha _ { K , i } )$ . For $\mathcal { L } ( \pmb { \alpha } _ { i } ) : = \Sigma _ { \tau = 1 } ^ { t } \log q _ { \pmb { \alpha } _ { i } } ( z _ { i , \tau } )$ +163 we obtain an estimate of $\alpha _ { i }$ at iteration $t$ as follows: $\begin{array} { r } { \alpha _ { i , t } : = \mathrm { \ a r g m a x } _ { \alpha _ { i } } \mathcal L ( \alpha _ { i } ) \ = \ \frac { n _ { i , t } } { t } } \end{array}$ t , with +164 $n _ { i , t } = \Sigma _ { \tau = 1 } ^ { t } \mathbf { e } _ { z _ { i , t } }$ , where $\mathbf { e } _ { k }$ is the standard basis vector (a one-hot vector) with one at $k$ th position +165 and zeros otherwise. This can be further rewritten into a recursive form: $n _ { i , t } = n _ { i , t - 1 } + \mathbf { e } _ { z _ { i , t } }$ or, +166 using the Robbins-Monro step-size, $n _ { i , t } = ( { \bf 1 } - { \bf e } _ { z _ { i , t } } \gamma _ { t } ) \odot n _ { i , t - 1 } + \gamma _ { t } { \bf e } _ { z _ { i , t } }$ , where 1 is the vector of +167 ones, and $\odot$ is the Hadamard product. We refer to this case simply as “table with forgetting” (TF) +168 due to that it represents $N \times K$ table in the memory and $\gamma _ { t }$ is a forgetting factor. + +# 169 5 Related work + +Stochastic approximation expectation-maximization. The application of SA to prevent the evaluation +1 of all $K$ components in mixture models has been overlooked for a long time. The reason is that the +72 original motivation to combine the EM algorithm with SA is to address the analytical intractability +73 of the expected value under $p _ { \theta } ( z | x )$ in (4), which is, however, almost always tractable for mixture +74 models. The intractability issue is addressed by either the Monte Carlo SAEM (MCSAEM) [12] +75 or the Markov chain Monte Carlo SAEM (MCMCSAEM) [31]. Applying the former approach to +76 mixture models would be inefficient, since it evaluates $K$ joint distributions, $p _ { \theta } ( z , x )$ , before drawing +177 $M$ samples from $p _ { \theta } ( z | x )$ . Therefore, this method reduces only the computational cost of updating the +178 sufficient statistics. This is addressed by the latter approach, where $M < K$ samples from a proposal +179 distribution, $q ( z | x )$ , is used to calculate $p _ { \theta } ( z , x )$ and also the sufficient statistics. However, all these +180 methods process all data at every iteration, providing only a limited advantage over the conventional +181 EM algorithm. Minibatch versions of these techniques have recently been proposed [27, 32, 1]. +182 All the above methods commonly assume $p _ { \theta } ( z , x )$ belonging to the exponential family. This provides +183 a convenient, but limiting, property which allows (6) to be computed under a closed-form solution. +184 The main contribution of our work is to release this restrictive assumption by admitting that $p _ { \theta } ( z , x )$ +185 (and thus $\mathcal { Q }$ ) is given by possibly complex and intractable transformations. +186 Sparse and truncated variational techniques. There is only a small body of methods explicitly +187 reducing the number of evaluated components. Their common aspect is that they follow from the +188 variational framework, where the exact posterior, $p _ { \theta } ( z | x )$ , is approximated by a variational posterior, +189 $q ( z | x )$ . This sparse, approximate, posterior is defined over a lower number of components, $M \ll K$ , +190 such that only the important components are selected, relying on relaxation of the hard EM algorithm +191 from taking a single $M = 1$ assignment [26] to taking multiple $M \ll K$ assignments. The sparse +192 SAEM (SSAEM) algorithm [24] selects the components by a quick partial sorting of the posterior +193 probabilities, $p _ { \theta } ( z | x )$ . Again, this requires $K$ evaluations of $p _ { \theta } ( z , x )$ before the sorting, thus only +194 reducing the amount of updated statistics. Similarly, the truncated SAEM (TSAEM) algorithm [18] +195 selects $M < K$ cluster-to-cluster and $\bar { M } < K$ cluster-to-datapoint minimal Euclidean distances, +196 preventing the problem in the SSAEM algorithm. However, all these distances are evaluated for all +197 components in a pairwise manner, leading to $K ^ { 2 }$ -computational complexity, which makes the saving +198 dubious. Similarly as before, these methods assume $p _ { \theta } ( z , x )$ to belong to the exponential family. + +![](images/0c516376ee7ecf27c7edaec6674fb19224d829bc10798ac37d421adee064aa0e.jpg) +Figure 1: The training log-likelihood, $\mathcal { L } ( \boldsymbol { \theta } _ { t } )$ , versus the computational time (in seconds). Here, on the $\mathbf { X }$ -axis, the computational time at a current iteration, $t$ , is obtained by accumulating the time from the previous iterations. corresponds to $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ , where $t _ { 9 5 }$ is the iteration of reaching $9 5 \%$ of max $\mathcal { L } ( \boldsymbol { \theta } _ { t } )$ . The projection of $^ { \circ }$ on the $\mathbf { X }$ -axis gives the time to reach $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ . This experiment was performed with the following settings: $( D , \bar { K } , N , \omega , B , M , T ) = ( 1 0 , 1 0 0 , 1 0 k , 0 . 1 , \bar { 2 } 0 0 , 2 , 2 0 k )$ , see Section 6.1 for details. The results are averaged over five repetitions. + +199 We summarize the distinguishing features of the above discussed methods in Table 1. + +# 6 Experiments + +To demonstrate the key features of our algorithm—its low computational complexity, competitive learning performance, and generality—we use it below to train: (i) GMMs on synthetic datasets, and (ii) SPTNs [47] and (iii) mixtures of real NVP flows [48] on real datasets. All experiments have been performed on a Slurm cluster equipped with Intel Xeon Scalable Gold 6146 with 384GB of RAM. + +# 6.1 Gaussian mixture models + +Consider the special case of a data-generating distribution given by (1), with the components taking the form of the multivariate Gaussian distribution, $p _ { \eta _ { z } } ( x | z ) = \mathcal { N } ( x ; \mu _ { z } , \Sigma _ { z } )$ , where $\mu _ { z }$ is the mean value and $\Sigma _ { z }$ is the covariance matrix. The difficulty of learning GMMs heavily depends on the degree of interaction among all mixture components, hence having the ability to generate synthetic datasets with arbitrary overlap characteristics between all pairs of components is crucial for systematic + +![](images/0ced572977ae404a4e785379dc4987ce758b6f6930ca4d73adbe35955ed678b7.jpg) +Figure 2: The absolute error, $\mathrm { A E } = | \mathcal { L } ( \theta _ { t _ { 9 5 } } ) - \mathcal { L } ( \theta ) |$ , versus the computational time (in seconds). All experiments use the following settings: $( D , K , \dot { N } , \omega , B , M , T ) = \bar { ( } 1 0 , 1 0 0 , 1 0 k , 0 . 1 , 2 0 0 , 2 , 2 0 k )$ , where the number of components, $K$ , (left), the batchsize, $B$ , (middle) and the number of samples, $M$ , (right) change for different values denoted by $( + , \sqsupset , \circ , \pmb { \triangle } )$ . At each of these points (marks), we perform an experiment as illustrated in Figure 1, find $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ to compute the AE, and record the time corresponding to $t _ { 9 5 }$ . The results are averaged over five repetitions. + +211 evaluation of performance of learning algorithms [43]. Traditional techniques usually define overlap +212 (or separation) of components only in terms of their mean vectors and maximum eigenvalues of the +213 covariance matrices, not accounting for their rotation and mixing weights (see [36] for a detailed +214 treatment of the problem). We therefore use a more objective measure of the clustering complexity +215 defined by the total probability of misclassification [41], which allows to generate data with a +216 user-defined degree of maximum pairwise overlap, $\omega$ . +217 Experiment settings: We generate the parameters of (1), and the corresponding dataset, uniquely for a +218 given quadruple $( D , K , N , \omega )$ . Therefore, the parameters of the generative model are known and we +219 can measure and display the convergence of the training log-likelihood, $\mathcal { L } ( \boldsymbol { \theta } _ { t } )$ , compared to the exact +220 log-likelihood, $\mathcal { L } ( \boldsymbol { \theta } ) \dot { }$ , for $t = ( 1 , \ldots , T )$ . We are further interested in the absolute error between the +221 training log-likelihood at the iteration of reaching $9 5 \%$ of its maximum value, $t _ { 9 5 }$ , and the exact +222 log-likelihood, i.e. $\mathrm { A E } = | \mathcal { L } ( \theta _ { t _ { 9 5 } } ) - \mathcal { L } ( \theta ) |$ . +23 We also measure the computational time until reaching $t _ { 9 5 }$ . We have used $9 5 \%$ of the maximum +24 value instead of the maximum value to prevent cases, where the model oscillate around target value, +25 making the estimate of convergence time very noisy (for example MCSAEM in Figure 1). +226 Algorithms: The GMMs belong to the exponential family of probability distributions. This allows us +227 to find a closed-form, recursive, solution of (6), relying on a Robbins-Monro type of the step-size +228 sequence, $( \gamma _ { t } ) _ { t = 1 } ^ { T }$ , [7, 44]. In this setting, we compare our MHSAEM algorithm with a number of +229 related methods in Table 1. Note we use the acronyms U and TF to specify the proposal distribution of +230 the MHSAEM algorithm (Section 4.3). However, we do not use the O-proposal, since the MHSAEM +231 O algorithm is equivalent to the MCSAEM algorithm. All the SA-variants in Table 1 use a minibatch +232 of size $B$ . The key quantity to reduce the number of evaluated components and/or sufficient statistics +233 in the SSAEM, TSAEM, MCSAEM and MHSAEM algorithms is collectively denoted by $M$ (Section +234 5). Note that we always keep $M = \bar { M }$ in the TSAEM algorithm (see Figure 1 and 2 for concrete +235 numbers). We use the step-size given by $\gamma _ { t } = 1$ for $t = 1 , \ldots , 5 0$ and $\gamma _ { t } = 0 . 0 5$ otherwise. In +236 this section, to counteract the issue of attaining poor local optima, we equip all algorithms with the +237 anti-annealing schedule $( \beta _ { t } ) _ { t = 1 } ^ { T }$ , starting with $\beta _ { 1 } = 0 . 1$ , reaching $\beta _ { 2 / 3 T } = 1 . 2$ , and decreasing back +238 to $\beta _ { T } = 1 . 0$ , see [43] for details. The initial estimates of: (i) $\mu _ { k }$ are uniformly drawn from the unit +239 hyper-cube, (ii) $\Sigma _ { k }$ are fixed to unit diagonal matrix, and (iii) $\pi _ { k }$ are uniformly drawn from the unit +240 interval (followed by normalization). +241 Results: Figure 1 shows that the EM [13] and SAEM [44] algorithms take the longest time to +242 converge, attaining a poor local optima. On the other hand, the MCSAEM [1] and MHSAEM (U +243 and TF) algorithms achieve $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ closest to the likelihood $\mathcal { L } ( \boldsymbol { \theta } )$ of the true model. Moreover, both +244 MHSAEM algorithms reach this value in the shortest time compared to all the other methods. The +245 SSAEM [24] and TSAEM [18] algorithms are comparable in terms of the computational time, but +246 they both provide the lowest $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ . In Figure 2, we investigate sensitivity of fitting the model to +47 increasing values of $K$ , $B$ and $M$ by measuring the time and the likelihood again. In all the cases, +48 the proposed MHSAEM algorithms achieve the lowest AE in the shortest time. + +SSAEM and TSAEM algorithms failed to converge for $M > 2$ and for $K > 5 0$ respectively. We believe this is caused by selecting only $M$ maximal probabilities in the SSAEM (or distances in the TSAEM) algorithm (Section 5), which prevents certain, but not a negligible number of, components from being updated, thus providing only a crude approximation of $\bar { p } _ { \theta } \bar { ( } z | x )$ . The results then suffer from substantial variational gap to the exact log-likelihood (Figure 1). On the contrary, MH sampler provides samples which consistently approximate $p _ { \theta } ( z | x )$ despite evaluating much lower number of components in each step. + +# 6.2 Sum-product transform networks + +The sum product networks (SPNs) are a deep learning extension of finite mixture models. They can be interpreted as a mixture of trees [60], where each tree corresponds to a component. Therefore, they can be cast into the form of (1), but the number of components grows exponentially with their depth. In this section, we use recently proposed SPTNs which introduce additional transformation nodes to provide better expressiveness than the SPNs (SPTNs effectively generalize SPNs and flow models into one large family of models). + +Experimental settings: We use 19 real datasets from the UCI database [16, 37, 35, 54], preprocessed in the same way as in [46]. For each experiment, we randomly split the data into $64 \%$ , $16 \%$ and $20 \%$ for training, validation and testing, respectively. We calculate the average log-likelihood on the test set and measure again the time to reach $9 5 \%$ of the maximal training log-likelihood, $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ . + +To evaluate various (possibly shallow and/or deep) architectures of SPTNs, we fit each dataset with all the following combinations of hyper-parameters1: $s \in ( 8 , 3 2 , 1 2 8 )$ , $b \in ( 2 , 4 , 6 , 8 )$ , $l \in ( 2 , 3 , 4 )$ , where $s$ is the number of children of each sum node, $b$ is the number of partitions of each product node, and $l$ is the number of layers (one layer contains sum and product nodes). The number of components of the SPTN, after its conversion into (1), is given as follows: $K = s ^ { l }$ . Note that the maximum number of components for the investigated parameters of the SPTN is 268,435,456. To reduce the space of possible architectures, we restrict ourselves only to (i) the leaf nodes given by $\mathcal { N } ( 0 , \bf { I } )$ ; (ii) affine transformations fixed to the singular value decomposition, choosing the the Givens parameterization for the unitary matrices [47]; and (iii) no sharing of any type of nodes [47]. + +276 Algorithms: We evaluate only on the MHSAEM-U algorithm—due to its favourable computational +277 complexity and simplicity—and compare it with the stochastic gradient-descent (SGD) algorithm, +278 which is routinely used to train SP(T)Ns [45, 47]. In this case, SGD in each iteration performs +279 computations over all subtrees of the network, whereas the MHSAEM-U algorithm computes with +280 only $M = 1$ subtrees, thus we should observe speed-up of the computations. In our implementation, +281 both these methods perform optimization of their respective objective functions—the log-likelihood +282 (2) for SGD and the EM objective (8) for MHSAEM-U—via the use of the automatic differentiation +283 and the ADAM optimizer [29], using $B = 1 0 0$ and $T = 2 0 0 0 0$ . + +Results: Since each dataset might benefit from a different architecture, Table 6.2 shows the test log-likelihood of the architectures selected according to the best likelihood measured on the validation set and the corresponding speed-up. The test log-likelihoods reveal that the MHSAEM-U algorithm outperforms the SGD algorithm on 10 out of 19 datasets, which was not originally the goal, but the added stochasticity helps to escape poor local minima. The speed-up demonstrates lower computational complexity of the MHSAEM-U algorithm on 17 out of 19 datasets, which was the main goal. The magic-telescope and wine datasets show approximately $1 0 2 \times$ and $7 5 \times$ speed-up, respectively, while on very small datasets (pima-indians and iris), the SGD is faster due to effective implementation. In the supplementary material, we present Table 3, exhibiting the same trends on a fixed architecture. + +# 6.3 Mixtures of real NVP flows + +We consider another class of mixture models (1), where each component $p _ { \eta _ { z } } ( x | z )$ is transformed by the flow model—real NVP [15]. These transformations are parameterized via deep neural networks, allowing for flexible adjustment of the learning capacity of each component. + +Table 2: The speed-up and test log-likelihood, $\mathcal { L } ^ { \mathrm { t e s t } }$ , for the SGD and MHSAEM-U algorithms. The test log-likelihood (higher is better) is computed for the best model, with the corresponding $K$ , which is selected based on the validation log-likelihood. The speed-up is computed as the ratio of MHSAEM-U to SGD, i.e. their time to reach $9 5 \%$ of the training log-likelihood. The results are averaged over five repetitions. Then, the higher test log-likelihood is highlighted with bold blue, and and no speed-up is highlighted with red. The average rank is computed as the standard competition (“1224”) ranking [14] on each dataset (lower is better). + +
Sum-product transformnetworksMixtures of real NVP flows SGD
SGDMHSAEM-UMHSAEM-U
speed-upLtestKLtestKspeed-upLtestK 32LtestK
breast-cancer-wisconsin4.66-4.66641.4310240.63-99.85-39.31128
cardiotocography10.5559.5251231.0410249.8554.343256.08128
magic-telescope102.53-3.65512-5.0310243.74-3.978-4.228
pendigits4.890.881024-4.86163844.171.4680.488
pima-indians0.37-8.5464-7.62641.35-20.09128-16.33128
wall-following-robot3.431.841024-11.31638422.21-14.26128-17.56128
waveform-14.35-26.1464-23.9110243.72-34.128-33.428
waveform-24.82-26.2164-23.9110244.12-34.158-33.648
yeast20.5710.265125.18102414.496.611289.59128
ecoli1.86-5.564-0.2210242.15-11.37128-10.64128
ionosphere1.88-20.2764-5.935122.74-87.01128-42.75128
iris0.23-10.6564-1.49163843.28-16.34128-9.2132
page-blocks12.1812.215126.84102444.9517.1312817.9432
parkinsons1.46-21.85640.55123.09-566.58128-33.3132
sonar2.96-95.39512-69.29642.52-622.2128-88.81128
statlog-segment1.4447.3551226.531638438.4935.8412842.0432
statlog-vehicle2.97-4.2564-5.4510246.78-31.3432-26.43128
wine rank75.42-25.991024-13.2710242.05-171.58128-25.57128
1.561.441.831.17
+ +298 Experimental settings: We use the same experimental settings and evaluation metrics as in Section +299 6.2. We apply the mixture model on all datasets, changing the number of components as follows: +300 $K \in ( 8 , 3 2 , 1 2 8 )$ . Each real NVP-based component in the mixture model has (i) the translation +301 function parameterized via multi-layer perceptron with a single hidden layer of dimension 10, using +302 the rectified linear activation function; and (ii) the scale function parameterized via the same network +303 except with the hyperbolic tangent activation function. We do not use the batch normalization [15] and +304 we stack two layers of the translation-scale transformation (we have used implementation from [20]). + +Algorithms: The algorithms and their settings are the same as those in Section 6.2. + +306 Results: The experimental results are presented in right part of Table 6.2. They are similar to those +307 obtained in the previous section. In terms of the test log-likelihood, the MHSAEM-U algorithm +308 outperforms the SGD algorithm on all but three datasets, and it provides a substantial speed-up on all +309 datasets except one. The test likelihood of models with the real NVP flows is most of the time worse +310 than that of SPTNs with the affine transformations. As explained in the supplementary, this is due to +311 the overfitting, which has been observed in [47]. + +# 312 7 Conclusion + +313 This paper has presented a method to decrease computational complexity of fitting mixture models, +314 including their generalizations, such as sum-product-(transform) networks and mixtures of flow +315 models. The speed-up is achieved by evaluating and updating only a single component (per iteration), +316 where the Metropolis-Hasting algorithm ensures sampling of components from a proper posterior. An +317 experimental comparison on all three classes of models mentioned above confirmed the theoretical +318 expectations. The method significantly speeds-up the fitting time and, importantly, without sacrificing +319 the quality of the fit. In fact, the likelihood was better than that of the models fitted by the EM +320 algorithm or the SGD algorithm in more than $50 \%$ of cases. We attribute this to higher stochasticity, +321 which helps to escape from poor local minima. +322 In the experiments, the proposed method has used a uniform proposal distribution in the MH sampler. +323 Despite outperforming the alternative methods, we conjecture that this limits the speed of convergence. +324 Therefore, we believe that there is still a room for improvement in the implementation. We plan to +325 address these issues in future work. + +The presented method decreases the computational complexity of fitting large (and deep) mixture models, which leads to five to hundred time speed-up depending on a size of the problem (although negative exceptions occurs). We believe this line of research, which we want to continue, to have important benefits. First, it is directly related to decrease in energy consumption and in production of CO2 (we expect similar rates as the speedup). Second, it has a positive effect on financial aspects of deploying (and experimenting with) mixture models. Third, it decreases the hardware requirements, as in all experiments presented above the model was fitted on a single-core. + +# References + +[1] S. Allassonnière and J. Chevallier. A new class of stochastic EM algorithms: Escaping local maxima and handling intractable sampling. Computational Statistics & Data Analysis, 159:107159, 2021. +[2] S. Ayesha, M. K. Hanif, and R. Talib. Overview and comparative study of dimensionality reduction techniques for high dimensional data. Information Fusion, 59:44–58, 2020. +[3] D. S. Bai. Efficient estimation of transition probabilities in a Markov chain. The Annals of Statistics, pages 1305–1317, 1975. +[4] D. M. Blei, A. 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Cases where it was not achieved are highlighted in the experimental section. +(c) Did you discuss any potential negative societal impacts of your work? [No] We do not foresee any potential negative impact. +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] + +3. If you ran experiments... + +(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The code is available in a github repository. All dataset are public from the UCI database. +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 6. +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We report only average of Monte Carlo repetitions, the error bars were too small to have any visual impact in the reported logarithmic scale. +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] We use 20 datasets from UCI, we cite the required papers for each dataset, mostly the UCI database and few additional publications. +(b) Did you mention the license of the assets? [No] The data are publically available, we comply with the requirement on citing appropriate publications. +(c) Did you include any new assets either in the supplemental material or as a URL? [No] +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] + +5. If you used crowdsourcing or conducted research with human subjects... + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] +(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/train/ZS394D3djsg/ZS394D3djsg_content_list.json b/parse/train/ZS394D3djsg/ZS394D3djsg_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..ed6277929ded73a04b7c5ed4fe67adf376102f54 --- /dev/null +++ b/parse/train/ZS394D3djsg/ZS394D3djsg_content_list.json @@ -0,0 +1,1195 @@ +[ + { + "type": "text", + "text": "Fitting large mixture models using stochastic component selection ", + "text_level": 1, + "bbox": [ + 228, + 122, + 769, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ", + "bbox": [ + 423, + 226, + 580, + 281 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 318, + 535, + 334 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Traditional methods for unsupervised learning of finite mixture models require to \n2 evaluate the likelihood of all components of the mixture. This becomes computa \n3 tionally prohibitive when the number of components is large, as it is, for example, \n4 in the sum-product (transform) networks. As a remedy, we propose an approach \n5 combining the expectation maximization and the Metropolis-Hastings algorithm \n6 to evaluate only a small number of, stochastically sampled, components, thus \n7 substantially reducing the computational cost. We put emphasis on generality of \n8 our method, equipping it with the ability to train both shallow and deep mixture \n9 models which involve complex, and possibly nonlinear, transformations. The \n10 performance of our method is illustrated in a variety of synthetic and real-data \n11 contexts, considering deep models, such as mixtures of normalizing flows and \n12 sum-product (transform) networks. ", + "bbox": [ + 148, + 348, + 766, + 515 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "13 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 539, + 312, + 556 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "14 Finite mixture models [40] constitute a fundamental class of density estimation models. They \n15 have been successfully applied in diverse fields, including bioinformatics [49], econometrics [10], \n16 engineering [33], etc. A mixture model relies on a weighted sum of probability distributions—here \n17 referred to as components—to cluster $N$ unlabelled datapoints into $K$ categories. The traditional \n18 maximum likelihood techniques train the model by optimizing either (i) the marginal likelihood via \n19 gradient-descent [50] or (ii) the evidence lower bound via variational methods [4], including the \n20 expectation-maximization (EM) [13]. The dependence structure among approximate, variational, \n21 distributions then ranges from the fully independent (mean-field) [25] to fully dependent [30]. The \n22 sampling-based techniques target the posterior distribution using sequential Monte Carlo [9] or \n23 Markov chain Monte Carlo [52], e.g. via the Gibbs [34] or Metropolis-Hastings sampling [38]. The \n24 computational cost of these methods typically scales with $\\mathcal { O } ( T K N D )$ operations, where $N$ and $K$ \n25 are defined above, $T$ is the number of iterations and $D$ is the dimension of data. \n26 Various methods to decrease the computational cost via any factor in $\\mathcal { O } ( T K N D )$ have been proposed. \n27 $T$ can be lowered by proper initialization, e.g. the optimal seeding [5]; an efficient step-size schedule, \n28 e.g. the line-search [58]; or increased estimation precision, e.g. the variance reduction [8]. $N$ is often \n29 reduced using the coreset methods, which approximate the original dataset by a weighted dataset \n30 such that the exact and approximate marginal likelihoods are close. The weighted variants of the \n31 variational [17, 59, 6] and sampling-based [39] methods then process the coresets. Reducing $D$ relies \n32 on the compression of data into smaller representations via random projections [53, 2], which is \n33 achieved in two ways: (i) each data item is projected into an individual representation [11]; (ii) all \n34 data items are projected into an overall representation, commonly referred to as sketch [28, 22]. \n35 Nevertheless, all the aforementioned techniques—including those with reduced computational cost— \n36 evaluate all $K$ components. This is very demanding for large models, and the problem is even more \n37 severe for mixtures involving intricate models, such as neural networks [21, 42], Gaussian processes \n38 [57], normalizing flows [48]; or deep mixtures, including sum-product (transform) networks [45, 47], \n39 deep Gaussian mixture models [55], etc. In spite of this, a little attention has been paid to the design \n40 of algorithms which does not evaluate all $K$ components. The notable exceptions are the sparse EM \n41 algorithm [24] and the truncated variational EM algorithm [18], see Table 1 and Section 5 for details. \n42 Moreover, the methods are mostly tailored for a specific class of mixture models, e.g. the Gaussian \n43 mixture models. ", + "bbox": [ + 147, + 570, + 825, + 736 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 742, + 825, + 867 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 873, + 828, + 901 + ], + "page_idx": 0 + }, + { + "type": "table", + "img_path": "images/5e346a33d4bf96097d66d042d2edb3a796cd7475f2111dc5b8813d4424db6d24.jpg", + "table_caption": [ + "Table 1: The computational features of various EM algorithms. We compare whether the methods (i) perform the computations with a reduced number of data (minibatching), (ii) update a lower number of statistics, (iii) make less evaluations of the conditional likelihood, and (iv) are suitable for training of deep models. Here, EM, SA, S, T, MC and MH stand for expectation-maximization, stochastic approximation, sparse, truncated, Monte Carlo and Metropolis-Hastings, respectively. " + ], + "table_footnote": [], + "table_body": "
Feature/AlgorithmEM [13]SAEM [44]SSAEM [24]TSAEM [18]MCSAEM [1]MHSAEM (ours)
B<Ndatapoints×√
M<K statisticsxx厂
M<Klikelihoodsxx×X
deep models×××X×
", + "bbox": [ + 222, + 165, + 774, + 244 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 273, + 826, + 372 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "44 In this paper, we make the following contributions: ", + "bbox": [ + 147, + 377, + 509, + 392 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "45 • We propose an EM-based algorithm which relies on the MH sampler to stochastically evaluate less \n46 components in mixture models, substantially reducing the computational cost. \n7 • We design our method to enable optimization of fairly generic EM objective functions, making it \n48 suitable for training of both shallow and deep mixture models. \n49 • We apply our approach to Gaussian mixture mdoels (GMMs) and their generalizations: sum \n50 product-transform networks (SPTNs) and mixtures of real-valued non-volume preserving (real \n51 NVP) flows [15], reaching approximately $1 0 0 \\times$ speed-up compared to state-of-the-art methods. ", + "bbox": [ + 151, + 405, + 825, + 434 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 155, + 441, + 823, + 472 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 479, + 825, + 522 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "52 2 Problem formulation ", + "text_level": 1, + "bbox": [ + 148, + 553, + 382, + 570 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A finite mixture model characterizes the relation between an observed (known) variable, 53 $\\boldsymbol { x } \\in \\times \\subseteq \\mathbb { R } ^ { D }$ , 54 and a latent (unknown) variable, $z \\in Z : = \\{ 1 , \\dots , K \\}$ , via the marginal (incomplete-data) likelihood 55 in the following form: ", + "bbox": [ + 147, + 587, + 825, + 630 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/1fd8c64bc931f74af8171cacb46094a19b2d53d8b9408dc1581f5a44dfea91a9.jpg", + "text": "$$\np _ { \\theta } ( x ) = \\sum _ { k = 1 } ^ { K } p _ { \\eta _ { k } } ( x | z = k ) p _ { \\pi _ { k } } ( z = k ) ,\n$$", + "text_format": "latex", + "bbox": [ + 369, + 633, + 627, + 678 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "56 where $\\theta : = ( \\pi _ { 1 } , \\eta _ { 1 } , \\dots , \\pi _ { K } , \\eta _ { K } ) \\in \\Theta$ are unknown parameters. Here, $\\eta _ { z }$ are the parameters of the 57 conditional likelihood, $p _ { \\eta _ { z } } ( x | z )$ , and $\\pi _ { z }$ is the weight which parameterizes the prior, $p _ { \\pi _ { z } } ( z ) = \\pi _ { z }$ , and satisfies 58 $0 \\leq \\pi _ { k } \\leq 1$ for each $k \\in { \\mathord { \\mathbb { Z } } }$ and $\\textstyle \\sum _ { k = 1 } ^ { K } \\pi _ { k } = 1$ . ", + "bbox": [ + 145, + 685, + 826, + 734 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "59 Given a set of independent and identically distributed data, $\\mathbf { x } : = ( x _ { i } ) _ { i = 1 } ^ { N }$ , our goal is to learn the \n60 unknown parameters of the marginal log-likelihood, ", + "bbox": [ + 148, + 739, + 825, + 770 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/3a533a8160c37e7b9fa1209ffe45b29c0eb14f6404e4bc9e51f2458ababc808c.jpg", + "text": "$$\n\\mathcal { L } ( \\theta ) : = \\log p _ { \\theta } ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ { N } \\log \\sum _ { k = 1 } ^ { K } p _ { \\eta _ { k } } ( x _ { i } | z _ { i } = k ) p _ { \\pi _ { k } } ( z _ { i } = k ) .\n$$", + "text_format": "latex", + "bbox": [ + 295, + 780, + 700, + 824 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "61 The marginalization in (2) is tractable for almost all forms of $p _ { \\eta _ { z } } ( x | z )$ . Indeed, we consider $p _ { \\eta _ { z } } ( x | z )$ \n62 to belong to an arbitrary family of $\\eta _ { z }$ -differentiable probability distributions. However, we assume that \n63 $K$ is high, making the marginalization in (2) computationally costly, thus rendering the optimization \n64 objective presumably intractable. Therefore, we want to design a computationally efficient algorithm, \n65 requiring only $M < K$ evaluations of $p _ { \\eta _ { z } } ( x | z )$ at each iteration. \n67 The maximum likelihood estimation seeks the parameters maximizing the marginal log-likelihood, \n68 $\\theta ^ { M L } : = \\arg \\operatorname* { m a x } _ { \\theta \\in \\Theta } \\mathcal { L } ( \\theta )$ . The traditional EM algorithm [13] addresses this task indirectly, i.e. by \n69 optimizing the evidence lower bound (ELBO), ", + "bbox": [ + 147, + 840, + 826, + 912 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 121, + 825, + 162 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/df55efbf1dd9a5a5ef40f11a29cb92a7c818d04135e3c849687eb5724cd486b5.jpg", + "text": "$$\n\\mathcal { L } ( \\theta ) \\geq \\mathcal { Q } ( \\theta ) + \\mathcal { H } ( \\hat { \\theta } ) : = \\mathrm { E L B O } ( \\hat { \\theta } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 379, + 170, + 619, + 189 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where 70 $\\mathcal { H } ( \\hat { \\theta } ) : = - \\mathsf E _ { p _ { \\hat { \\theta } } ( \\mathbf { z } | \\mathbf { x } ) } [ \\log p _ { \\hat { \\theta } } ( \\mathbf { z } | \\mathbf { x } ) ]$ is the differential entropy at an estimate, $\\hat { \\theta } \\in \\Theta$ , and ", + "bbox": [ + 140, + 198, + 771, + 215 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8dc49814f958acdf8ce0e11b963a6e6101e3e5d1439603310c411ba01588b3e9.jpg", + "text": "$$\n\\mathcal { Q } ( \\theta ) : = \\mathsf { E } _ { p _ { \\hat { \\theta } } ( \\mathbf { z } | \\mathbf { x } ) } [ \\log p _ { \\theta } ( \\mathbf { z } , \\mathbf { x } ) ] = \\sum _ { i = 1 } ^ { N } \\sum _ { k = 1 } ^ { K } p _ { \\theta } ( z _ { i } = k | x _ { i } ) \\log p _ { \\theta } ( z _ { i } = k , x _ { i } )\n$$", + "text_format": "latex", + "bbox": [ + 254, + 222, + 743, + 267 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "71 is the EM objective function. Here, $p _ { \\theta } ( \\mathbf { z } , \\mathbf { x } )$ is the joint (complete-data) likelihood, and $p _ { \\theta } ( \\mathbf { z } | \\mathbf { x } )$ is \n72 the posterior distribution over the latent variables $\\dot { \\mathbf { z } } : = ( z _ { i } ) _ { i = 1 } ^ { N }$ . Given an initial value, $\\theta _ { 0 }$ , the EM \n73 algorithm produces a sequence of estimates, $( \\theta _ { t } ) _ { t = 1 } ^ { T }$ , by alternating between the expectation (E) and \n74 maximization (M) steps, ", + "bbox": [ + 145, + 273, + 825, + 332 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b0f69e94b819f6dec2b2f46df7d892633bd8e60b66ed31dbb1985f882bcb2f60.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { E \\mathrm { - } s t e p } ; ~ \\mathcal { Q } _ { t - 1 } ( \\theta ) , } \\\\ & { \\mathrm { M \\mathrm { - } s t e p } ; ~ \\theta _ { t } : = \\arg \\operatorname* { m a x } _ { \\theta \\in \\Theta } \\mathcal { Q } _ { t - 1 } ( \\theta ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 387, + 337, + 607, + 382 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "75 This sequence is guaranteed to monotonically tighten the ELBO, arriving at a local optimum of (2) \n76 under mild regularity assumptions [56]. \n77 The EM algorithm is computationally expensive, since (4) evaluates $p _ { \\theta } ( z _ { i } , x _ { i } )$ for each $z _ { i } \\in \\mathbb { Z }$ and \n78 $i \\in ( 1 , \\ldots , N )$ . This has to be performed for all $t \\in ( 1 , \\ldots , T )$ in (5). Albeit the marginal factor, \n79 $p _ { \\pi _ { z } } ( z )$ , is just the cheap categorical distribution, the conditional factor, $p _ { \\eta _ { z } } ( x | z )$ , typically involves \n80 high-dimensional operations (e.g., the inversion of the full $D \\times D$ -dimensional covariance matrices \n81 in the GMMs). Moreover, the M-step (6) is also expensive for large $K$ . This holds despite that (6) \n82 can be reduced to closed-form updates of expected sufficient statistics for $p _ { \\eta _ { z } } ( x | z )$ belonging to the \n83 exponential family [44] (again, due to high $D$ ). All in all, the computational complexity of the EM \n84 algorithm scales with $\\mathcal { O } ( T D N K )$ . ", + "bbox": [ + 143, + 388, + 825, + 417 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 422, + 825, + 535 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "If (6) cannot be computed under a closed-form solution, one can resort to direct gradient-descent optimization of $\\mathcal { Q } ( \\boldsymbol { \\theta } )$ , where arg max is replaced by one (or more) step(s) of a gradient descent technique. The EM algorithm is then referred to as the generalized EM algorithm [56]. ", + "bbox": [ + 160, + 540, + 823, + 583 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "88 4 The generalized MHSAEM algorithm ", + "text_level": 1, + "bbox": [ + 148, + 602, + 521, + 619 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "89 We design a version of the generalized EM algorithm suitable for scenarios where (4) can represent \n90 deep, discrete, latent variable models, thus being parameterized by possibly complex nonlinear \n91 transformations. We particularly focus on decreasing the the number of operations in the generalized \n92 EM algorithm from $\\mathcal { O } ( T D N K )$ to $\\mathcal { O } ( T D B M )$ , where $B \\ll N$ and $M \\ll K$ . ", + "bbox": [ + 147, + 633, + 825, + 690 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4.1 E-step ", + "text_level": 1, + "bbox": [ + 161, + 705, + 256, + 720 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "94 We reduce the cost of evaluating the EM objective function (4) by combining the minibatching (as \n95 used many times before) and the Monte Carlo sampling. Namely, the specific application of the latter \n96 to generic mixture models is the key contribution of this paper. \n97 Minibatching. At each iteration, $t$ , we compute the conditional expectation in (4) only for a subset— \n98 here referred to as a minibatch—of the original full dataset, i.e. $( x _ { i } ) _ { i \\in I }$ . Here, $I$ is a set of $B \\ll N$ \n99 indices, $i$ , sampled uniformly without replacement from $( 1 , \\ldots , \\dot { N } )$ . This substantially decreases the \n100 necessary computations compared to the full sweep over all $N$ datapoints [23]. \n101 Monte Carlo sampling. For each $i \\in I$ , we want to draw $M \\ll K$ random samples from $p _ { \\theta } ( z _ { i } | x _ { i } )$ in \n102 order to obtain a Monte Carlo estimate of (4). The straightforward way to do this would be to draw \n103 the samples directly from $p _ { \\theta } ( z _ { i } | x _ { i } )$ . However, direct sampling from $p _ { \\theta } ( z _ { i } | x _ { i } )$ does not lead to any \n104 substantial decrease in the number of operations. This is caused by the fact that even for a single \n105 sample of $z _ { i }$ , we have to first compute the normalizing factor, $p _ { \\theta } ( x _ { i } )$ , to obtaining the posterior, \n106 $p _ { \\theta } ( z _ { i } | x _ { i } )$ . This requires $K$ expensive evaluations of $p _ { \\theta } ( z _ { i } , x _ { i } )$ , which is precisely what we want to \n107 avoid. Our approach is to resort to the Markov chain Monte Carlo (MCMC), which allows us to \n108 sample from $p _ { \\theta } ( z _ { i } | x _ { i } )$ , with the computational complexity decreasing to only a single evaluation of \n109 $p _ { \\theta } ( z _ { i } , x _ { i } )$ per a single sample of $z _ { i }$ . \n110 MCMC methods obviate the computation of the normalizing factor in $p _ { \\theta } ( z _ { i } | x _ { i } )$ by simulating a \n111 Markov chain, $( z _ { i , t } ) _ { t = 1 } ^ { T }$ , from a transition kernel, $z _ { i , t } \\sim P ( z _ { i , t - 1 } , \\cdot )$ , which leaves $p _ { \\theta } ( z _ { i } | x _ { i } )$ as its \n112 unique stationary (invariant) distribution, starting from an initial value $z _ { i , 0 }$ . The specific form of $P$ \n113 determines the structure of an MCMC method. We chose the Metropolis-Hastings (MH) sampler, \n114 which represents $P ( z _ { i , t - 1 } , z _ { i , t } )$ as follows: given $\\bar { z } _ { i } : = z _ { i , t - 1 }$ , draw a sample from the proposal \n115 distribution $z _ { i } \\sim q ( \\cdot | \\bar { z } _ { i } )$ , compute the acceptance ratio, ", + "bbox": [ + 147, + 731, + 825, + 773 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 779, + 825, + 835 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 842, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 90, + 825, + 147 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 152, + 825, + 238 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4e2865ff49fb5c3c34577e0a148cbc695663514dff1302f39bca47fd492f0e3e.jpg", + "text": "$$\n\\alpha ( \\bar { z } _ { i } , z _ { i } ) : = \\operatorname* { m i n } \\biggr \\{ 1 , \\frac { p _ { \\eta _ { z _ { i } , t - 1 } } ( x _ { i } | z _ { i } ) \\pi _ { z _ { i } , t - 1 } q ( \\bar { z } _ { i } | z _ { i } ) } { p _ { \\eta _ { \\bar { z } _ { i } , t - 1 } } ( x _ { i } | \\bar { z } _ { i } ) \\pi _ { \\bar { z } _ { i } , t - 1 } q ( z _ { i } | \\bar { z } _ { i } ) } \\biggr \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 321, + 241, + 676, + 279 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "116 and, if $u < \\alpha \\big ( \\bar { z } _ { i } , z _ { i } \\big )$ —where $u$ is drawn from a uniform distribution, Uniform $( 0 , 1 )$ —accept the \n117 sample and set $z _ { i , t } = z _ { i }$ ; otherwise, set $z _ { i , t } = \\bar { z } _ { i }$ . For each $i \\in I$ and $t \\in ( 1 , \\ldots , T )$ , we repeat \n118 this process $M$ times, construing a set $\\mathbf { z } _ { i , t } = ( z _ { i , t } ^ { 1 } , \\dots , z _ { i , t } ^ { M } )$ . Therefore, at every current iteration, \n119 120 $t$ , we caking $\\bar { z } _ { i } = z _ { i , t - 1 } ^ { M }$ extend the chain from the point where we left at the previous iteration, . Under mild regularity assumptions [52], the chain passes the transiti $t - 1$ , byriod \n121 (the burn-in phase), and the samples can then be used to approximate the conditional expectation in \n122 (4) as follows: ", + "bbox": [ + 140, + 282, + 825, + 382 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e43005c66ee8eb883ede3aeefb994947aa4f14879ab1b3689fc3176b2f635909.jpg", + "text": "$$\n\\widehat { \\mathcal { Q } } _ { t - 1 } ( \\theta ) = \\frac { 1 } { M } \\sum _ { i \\in I } \\sum _ { z \\in \\mathbf { z } _ { i , t } } \\log p _ { \\eta _ { z } } ( x _ { i } | z ) \\pi _ { z } .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 378, + 637, + 419 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "123 Note that, to ensure this approach is truly efficient, we have to draw only $M \\ll K$ samples at each \n124 iteration, $t$ ; otherwise, for $M \\approx K$ , we may rather compute the exact marginalization in (4), since it \n125 is tractable (but computationally costly). ", + "bbox": [ + 142, + 420, + 825, + 463 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.2 M-step ", + "text_level": 1, + "bbox": [ + 166, + 478, + 263, + 493 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "127 Assume for a moment that (6) with $\\mathcal { Q } _ { t - 1 } ( \\theta )$ given by (8) has a closed-form solution, yielding an \n128 estimate of $\\theta$ . Such an estimate would have a high variance, converging only for $M \\to \\infty$ and \n129 $T \\to \\infty$ [19]. The main reason is that the samples would not be reused over the iterations, $t$ , \n130 thus wasting computational resources. We consider that there is no closed-form solution of (6), \n131 and—to ensure that the samples (and thus computations) are recycled over the iterations—we use \n132 the stochastic approximation (SA) [51] to optimize (8). This is analogous to applying a stochastic \n133 gradient-descent method, $\\theta _ { t } = \\theta _ { t - 1 } + \\gamma _ { t } \\nabla _ { \\theta } \\tilde { \\mathcal { Q } } _ { t - 1 } ( \\theta )$ , where $\\gamma _ { t }$ is the step-size, satisfying the Robbins \n134 Monro constraints, $\\begin{array} { r } { \\gamma _ { t } \\in [ 0 , 1 ] , \\sum _ { t \\geq 1 } \\gamma _ { t } = \\infty , \\sum _ { t \\geq 1 } \\gamma _ { t } ^ { 2 } < \\infty , } \\end{array}$ and $\\nabla _ { \\theta }$ is the gradient w.r.t. $\\theta$ . In this \n135 way, the computations made in $\\nabla _ { \\boldsymbol { \\theta } } \\widehat { \\mathcal { Q } }$ are accumulated via $\\theta _ { t }$ and reused over the iterations. \n136 The parameters $\\eta _ { z }$ have a different form based on a specific case of $p _ { \\eta _ { z } } ( x | z )$ , whereas $\\pi _ { z }$ is a \n137 permanent structure in (1). Therefore, without loss of generality, we split (6) into a generic part and a \n138 fixed part as follows: ", + "bbox": [ + 140, + 502, + 826, + 636 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 641, + 828, + 684 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e1b6543f038daf2078a131f2fa387f7478babb5133579989565df178123b0c85.jpg", + "text": "$$\n\\begin{array} { r l } & { \\eta _ { k , t } = \\eta _ { k , t - 1 } + \\gamma _ { t } \\nabla _ { \\eta _ { k } } \\widehat { \\mathcal { Q } } _ { t - 1 } ( \\theta ) , } \\\\ & { \\nu _ { k , t } = \\nu _ { k , t - 1 } + \\gamma _ { t } \\nabla _ { \\nu _ { k } } \\widehat { \\mathcal { Q } } _ { t - 1 } ( \\theta ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 390, + 688, + 606, + 732 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "139 where—to ensure that the probabilities, $( \\pi _ { k , t } ) _ { k = 1 } ^ { K }$ , satisfy the constraints (Section 2)—we transform \n140 $\\nabla _ { \\pi _ { k } } \\widehat { \\mathcal { Q } }$ via $\\nu _ { k } = \\log \\pi _ { k }$ and optimize w.r.t. $\\nu _ { k }$ . Then, to obtain $( \\pi _ { k , t } ) _ { k = 1 } ^ { K }$ from $\\nu _ { t } : = ( \\nu _ { k , t } ) _ { k = 1 } ^ { K }$ , we \n141 k b use the softmax function, i.e. $\\pi _ { k , t } : = \\mathrm { s o f t m a x } ( \\pmb { \\nu } _ { t } ) _ { k } : = \\exp ( \\nu _ { k , t } ) / \\sum _ { l = 1 } ^ { K } \\exp ( \\nu _ { l , t } )$ . ", + "bbox": [ + 142, + 734, + 825, + 787 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Computing the gradients for all pairs of $( \\nu _ { k } , \\eta _ { k } ) _ { k = 1 } ^ { K }$ would be inefficient, especially since ${ \\bf z } _ { i , t }$ contains only a small number of unique values of Z for $M \\ll K$ . Consequently, we compute $\\dot { \\nabla } _ { \\eta _ { k } } \\widehat { \\mathcal { Q } }$ and $\\nabla _ { \\nu _ { k } } \\widehat { \\mathcal { Q } }$ only for $k \\in { \\mathrm { u n i q u e } } ( \\mathbf { z } _ { i , t } )$ . We summarize the proposed approach in Algorithm 1. ", + "bbox": [ + 160, + 792, + 825, + 843 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.3 Proposal distribution ", + "text_level": 1, + "bbox": [ + 171, + 857, + 361, + 872 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "146 The choice of the proposal distribution has a significant impact on the speed of convergence and the computational cost of the proposed algorithm. Here, we discuss various possible choices of 147 $q \\big ( z _ { i } | \\bar { z } _ { i } \\big )$ . ", + "bbox": [ + 145, + 882, + 828, + 912 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Input: $\\theta _ { 0 }$ , $( \\mathbf { z } _ { i , 0 } ) _ { i = 1 } ^ { N }$ , $( \\mathbf { x } _ { i } ) _ { i = 1 } ^ { N }$ \nOutput: $( \\theta _ { t } ) _ { t = 1 } ^ { T }$ for $t \\in ( 1 , \\ldots , T )$ or until convergence do form the set $\\dot { I } = ( i _ { j } ) _ { j = 1 } ^ { B }$ by sampling (without replacement) $B$ indices $i \\sim ( 1 , \\dots , N )$ for $i \\in I$ do set $\\bar { z } _ { i }$ as the last element of $\\mathbf { z } _ { i , t - 1 }$ for $j \\in ( 1 , \\ldots , M )$ do sample $z _ { i } \\sim q ( z _ { i } | \\bar { z } _ { i } )$ sample $u \\sim \\mathrm { U n i f o r m } ( 0 , 1 )$ compute $\\alpha ( \\bar { z } _ { i } , z _ { i } )$ in (7) if $u < \\alpha \\big ( \\bar { z } _ { i } , z _ { i } \\big )$ then set $z _ { i , t } ^ { j } = z _ { i }$ and $\\bar { z } _ { i } = z _ { i }$ else set $z _ { i , t } ^ { j } = \\bar { z } _ { i }$ end if end for set $\\mathbf { z } _ { i , t } = ( z _ { i , t } ^ { 1 } , \\dots , z _ { i , t } ^ { M } )$ end for compute (8) compute (9) for $k \\in { \\mathrm { u n i q u e } } ( \\mathbf { z } _ { i , t } )$ compute $\\pi _ { k , t } : = \\mathrm { s o f t m a x } ( \\pmb { \\nu } _ { t } ) _ { k }$ for $k \\in { \\mathord { \\mathbb { Z } } }$ end for ", + "bbox": [ + 173, + 108, + 787, + 421 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "148 Optimal proposal $( O )$ . The optimal proposal distribution is $q ( z _ { i } | \\bar { z } _ { i } ) : = q ( z _ { i } ) : = p _ { \\theta } ( z _ { i } | x _ { i } )$ . This \n149 ensures that the acceptance rate (7) is always $\\alpha ( \\bar { z } _ { i } , z _ { i } ) = 1$ . However, the need to perform $K$ \n150 expensive evaluations of $p _ { \\theta } ( z _ { i } , x _ { i } )$ before sampling from $p _ { \\theta } ( z _ { i } | x _ { i } )$ is the reason we resorted to \n151 the MH sampler in the first place. We consider this case only to set the upper limit on admissible \n152 computational cost and to study the impact of sub-optimal proposal distribtions. \n153 Uniform proposal $( U )$ . The uniform distribution on the discrete interval from 1 to $K$ , i.e. $q ( z _ { i } | \\bar { z } _ { i } ) : =$ \n154 $q ( z _ { i } ) : = \\mathrm { U n i f o r m } ( 1 , K )$ , is the simplest and computationally cheapest variant of the proposal \n155 distribution. However, due to poor mixing properties, the algorithm may converge slowly for high $K$ . \n156 Tabular proposal with forgetting $( T F )$ . The key requirement to design a proposal distribution is to \n157 restrict its computational complexity somewhere between that of the $\\mathrm { U }$ and $\\mathrm { o }$ proposals. One way to \n158 satisfy this constraint is to use the Markov chain, $( z _ { i , t } ) _ { t = 1 } ^ { T }$ , to learn a transition kernel, $p ( z _ { i } | \\bar { z } _ { i } )$ , see, \n159 e.g. [3]. Unfortunately, this would require us to store a table with $K ^ { 2 }$ entries for each $i \\in ( 1 , \\ldots , N )$ , \n160 161 which is very demanding evethe Markov chain and define: $q ( z _ { i } | \\bar { z } _ { i } ) : = q _ { \\alpha _ { i } } ( z _ { i } ) : = \\mathcal { C } ( \\alpha _ { i } )$ $K$ $N$ herefore, where $\\mathcal { C } ( \\pmb { \\alpha } _ { i } ) \\propto \\Pi _ { k = 1 } ^ { K } \\alpha _ { k , i } ^ { \\bar { 1 ( } z _ { i } = k ) }$ nce inis the \n162 categorical distribution with the weights $\\pmb { \\alpha } _ { i } : = ( \\alpha _ { 1 , i } , \\dots , \\alpha _ { K , i } )$ . For $\\mathcal { L } ( \\pmb { \\alpha } _ { i } ) : = \\Sigma _ { \\tau = 1 } ^ { t } \\log q _ { \\pmb { \\alpha } _ { i } } ( z _ { i , \\tau } )$ \n163 we obtain an estimate of $\\alpha _ { i }$ at iteration $t$ as follows: $\\begin{array} { r } { \\alpha _ { i , t } : = \\mathrm { \\ a r g m a x } _ { \\alpha _ { i } } \\mathcal L ( \\alpha _ { i } ) \\ = \\ \\frac { n _ { i , t } } { t } } \\end{array}$ t , with \n164 $n _ { i , t } = \\Sigma _ { \\tau = 1 } ^ { t } \\mathbf { e } _ { z _ { i , t } }$ , where $\\mathbf { e } _ { k }$ is the standard basis vector (a one-hot vector) with one at $k$ th position \n165 and zeros otherwise. This can be further rewritten into a recursive form: $n _ { i , t } = n _ { i , t - 1 } + \\mathbf { e } _ { z _ { i , t } }$ or, \n166 using the Robbins-Monro step-size, $n _ { i , t } = ( { \\bf 1 } - { \\bf e } _ { z _ { i , t } } \\gamma _ { t } ) \\odot n _ { i , t - 1 } + \\gamma _ { t } { \\bf e } _ { z _ { i , t } }$ , where 1 is the vector of \n167 ones, and $\\odot$ is the Hadamard product. We refer to this case simply as “table with forgetting” (TF) \n168 due to that it represents $N \\times K$ table in the memory and $\\gamma _ { t }$ is a forgetting factor. ", + "bbox": [ + 140, + 449, + 825, + 518 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 525, + 826, + 568 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 573, + 826, + 763 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "169 5 Related work ", + "text_level": 1, + "bbox": [ + 145, + 782, + 316, + 799 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Stochastic approximation expectation-maximization. The application of SA to prevent the evaluation \n1 of all $K$ components in mixture models has been overlooked for a long time. The reason is that the \n72 original motivation to combine the EM algorithm with SA is to address the analytical intractability \n73 of the expected value under $p _ { \\theta } ( z | x )$ in (4), which is, however, almost always tractable for mixture \n74 models. The intractability issue is addressed by either the Monte Carlo SAEM (MCSAEM) [12] \n75 or the Markov chain Monte Carlo SAEM (MCMCSAEM) [31]. Applying the former approach to \n76 mixture models would be inefficient, since it evaluates $K$ joint distributions, $p _ { \\theta } ( z , x )$ , before drawing \n177 $M$ samples from $p _ { \\theta } ( z | x )$ . Therefore, this method reduces only the computational cost of updating the \n178 sufficient statistics. This is addressed by the latter approach, where $M < K$ samples from a proposal \n179 distribution, $q ( z | x )$ , is used to calculate $p _ { \\theta } ( z , x )$ and also the sufficient statistics. However, all these \n180 methods process all data at every iteration, providing only a limited advantage over the conventional \n181 EM algorithm. Minibatch versions of these techniques have recently been proposed [27, 32, 1]. \n182 All the above methods commonly assume $p _ { \\theta } ( z , x )$ belonging to the exponential family. This provides \n183 a convenient, but limiting, property which allows (6) to be computed under a closed-form solution. \n184 The main contribution of our work is to release this restrictive assumption by admitting that $p _ { \\theta } ( z , x )$ \n185 (and thus $\\mathcal { Q }$ ) is given by possibly complex and intractable transformations. \n186 Sparse and truncated variational techniques. There is only a small body of methods explicitly \n187 reducing the number of evaluated components. Their common aspect is that they follow from the \n188 variational framework, where the exact posterior, $p _ { \\theta } ( z | x )$ , is approximated by a variational posterior, \n189 $q ( z | x )$ . This sparse, approximate, posterior is defined over a lower number of components, $M \\ll K$ , \n190 such that only the important components are selected, relying on relaxation of the hard EM algorithm \n191 from taking a single $M = 1$ assignment [26] to taking multiple $M \\ll K$ assignments. The sparse \n192 SAEM (SSAEM) algorithm [24] selects the components by a quick partial sorting of the posterior \n193 probabilities, $p _ { \\theta } ( z | x )$ . Again, this requires $K$ evaluations of $p _ { \\theta } ( z , x )$ before the sorting, thus only \n194 reducing the amount of updated statistics. Similarly, the truncated SAEM (TSAEM) algorithm [18] \n195 selects $M < K$ cluster-to-cluster and $\\bar { M } < K$ cluster-to-datapoint minimal Euclidean distances, \n196 preventing the problem in the SSAEM algorithm. However, all these distances are evaluated for all \n197 components in a pairwise manner, leading to $K ^ { 2 }$ -computational complexity, which makes the saving \n198 dubious. Similarly as before, these methods assume $p _ { \\theta } ( z , x )$ to belong to the exponential family. ", + "bbox": [ + 155, + 814, + 823, + 911 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/0c516376ee7ecf27c7edaec6674fb19224d829bc10798ac37d421adee064aa0e.jpg", + "image_caption": [ + "Figure 1: The training log-likelihood, $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t } )$ , versus the computational time (in seconds). Here, on the $\\mathbf { X }$ -axis, the computational time at a current iteration, $t$ , is obtained by accumulating the time from the previous iterations. corresponds to $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ , where $t _ { 9 5 }$ is the iteration of reaching $9 5 \\%$ of max $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t } )$ . The projection of $^ { \\circ }$ on the $\\mathbf { X }$ -axis gives the time to reach $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ . This experiment was performed with the following settings: $( D , \\bar { K } , N , \\omega , B , M , T ) = ( 1 0 , 1 0 0 , 1 0 k , 0 . 1 , \\bar { 2 } 0 0 , 2 , 2 0 k )$ , see Section 6.1 for details. The results are averaged over five repetitions. " + ], + "image_footnote": [], + "bbox": [ + 285, + 92, + 699, + 236 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 356, + 825, + 426 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 433, + 825, + 488 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 494, + 826, + 674 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "199 We summarize the distinguishing features of the above discussed methods in Table 1. ", + "bbox": [ + 151, + 680, + 728, + 695 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6 Experiments ", + "text_level": 1, + "bbox": [ + 166, + 714, + 312, + 731 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To demonstrate the key features of our algorithm—its low computational complexity, competitive learning performance, and generality—we use it below to train: (i) GMMs on synthetic datasets, and (ii) SPTNs [47] and (iii) mixtures of real NVP flows [48] on real datasets. All experiments have been performed on a Slurm cluster equipped with Intel Xeon Scalable Gold 6146 with 384GB of RAM. ", + "bbox": [ + 174, + 744, + 825, + 800 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6.1 Gaussian mixture models ", + "text_level": 1, + "bbox": [ + 171, + 816, + 388, + 830 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Consider the special case of a data-generating distribution given by (1), with the components taking the form of the multivariate Gaussian distribution, $p _ { \\eta _ { z } } ( x | z ) = \\mathcal { N } ( x ; \\mu _ { z } , \\Sigma _ { z } )$ , where $\\mu _ { z }$ is the mean value and $\\Sigma _ { z }$ is the covariance matrix. The difficulty of learning GMMs heavily depends on the degree of interaction among all mixture components, hence having the ability to generate synthetic datasets with arbitrary overlap characteristics between all pairs of components is crucial for systematic ", + "bbox": [ + 173, + 842, + 823, + 911 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/0ced572977ae404a4e785379dc4987ce758b6f6930ca4d73adbe35955ed678b7.jpg", + "image_caption": [ + "Figure 2: The absolute error, $\\mathrm { A E } = | \\mathcal { L } ( \\theta _ { t _ { 9 5 } } ) - \\mathcal { L } ( \\theta ) |$ , versus the computational time (in seconds). All experiments use the following settings: $( D , K , \\dot { N } , \\omega , B , M , T ) = \\bar { ( } 1 0 , 1 0 0 , 1 0 k , 0 . 1 , 2 0 0 , 2 , 2 0 k )$ , where the number of components, $K$ , (left), the batchsize, $B$ , (middle) and the number of samples, $M$ , (right) change for different values denoted by $( + , \\sqsupset , \\circ , \\pmb { \\triangle } )$ . At each of these points (marks), we perform an experiment as illustrated in Figure 1, find $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ to compute the AE, and record the time corresponding to $t _ { 9 5 }$ . The results are averaged over five repetitions. " + ], + "image_footnote": [], + "bbox": [ + 196, + 89, + 805, + 250 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "211 evaluation of performance of learning algorithms [43]. Traditional techniques usually define overlap \n212 (or separation) of components only in terms of their mean vectors and maximum eigenvalues of the \n213 covariance matrices, not accounting for their rotation and mixing weights (see [36] for a detailed \n214 treatment of the problem). We therefore use a more objective measure of the clustering complexity \n215 defined by the total probability of misclassification [41], which allows to generate data with a \n216 user-defined degree of maximum pairwise overlap, $\\omega$ . \n217 Experiment settings: We generate the parameters of (1), and the corresponding dataset, uniquely for a \n218 given quadruple $( D , K , N , \\omega )$ . Therefore, the parameters of the generative model are known and we \n219 can measure and display the convergence of the training log-likelihood, $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t } )$ , compared to the exact \n220 log-likelihood, $\\mathcal { L } ( \\boldsymbol { \\theta } ) \\dot { }$ , for $t = ( 1 , \\ldots , T )$ . We are further interested in the absolute error between the \n221 training log-likelihood at the iteration of reaching $9 5 \\%$ of its maximum value, $t _ { 9 5 }$ , and the exact \n222 log-likelihood, i.e. $\\mathrm { A E } = | \\mathcal { L } ( \\theta _ { t _ { 9 5 } } ) - \\mathcal { L } ( \\theta ) |$ . \n23 We also measure the computational time until reaching $t _ { 9 5 }$ . We have used $9 5 \\%$ of the maximum \n24 value instead of the maximum value to prevent cases, where the model oscillate around target value, \n25 making the estimate of convergence time very noisy (for example MCSAEM in Figure 1). \n226 Algorithms: The GMMs belong to the exponential family of probability distributions. This allows us \n227 to find a closed-form, recursive, solution of (6), relying on a Robbins-Monro type of the step-size \n228 sequence, $( \\gamma _ { t } ) _ { t = 1 } ^ { T }$ , [7, 44]. In this setting, we compare our MHSAEM algorithm with a number of \n229 related methods in Table 1. Note we use the acronyms U and TF to specify the proposal distribution of \n230 the MHSAEM algorithm (Section 4.3). However, we do not use the O-proposal, since the MHSAEM \n231 O algorithm is equivalent to the MCSAEM algorithm. All the SA-variants in Table 1 use a minibatch \n232 of size $B$ . The key quantity to reduce the number of evaluated components and/or sufficient statistics \n233 in the SSAEM, TSAEM, MCSAEM and MHSAEM algorithms is collectively denoted by $M$ (Section \n234 5). Note that we always keep $M = \\bar { M }$ in the TSAEM algorithm (see Figure 1 and 2 for concrete \n235 numbers). We use the step-size given by $\\gamma _ { t } = 1$ for $t = 1 , \\ldots , 5 0$ and $\\gamma _ { t } = 0 . 0 5$ otherwise. In \n236 this section, to counteract the issue of attaining poor local optima, we equip all algorithms with the \n237 anti-annealing schedule $( \\beta _ { t } ) _ { t = 1 } ^ { T }$ , starting with $\\beta _ { 1 } = 0 . 1$ , reaching $\\beta _ { 2 / 3 T } = 1 . 2$ , and decreasing back \n238 to $\\beta _ { T } = 1 . 0$ , see [43] for details. The initial estimates of: (i) $\\mu _ { k }$ are uniformly drawn from the unit \n239 hyper-cube, (ii) $\\Sigma _ { k }$ are fixed to unit diagonal matrix, and (iii) $\\pi _ { k }$ are uniformly drawn from the unit \n240 interval (followed by normalization). \n241 Results: Figure 1 shows that the EM [13] and SAEM [44] algorithms take the longest time to \n242 converge, attaining a poor local optima. On the other hand, the MCSAEM [1] and MHSAEM (U \n243 and TF) algorithms achieve $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ closest to the likelihood $\\mathcal { L } ( \\boldsymbol { \\theta } )$ of the true model. Moreover, both \n244 MHSAEM algorithms reach this value in the shortest time compared to all the other methods. The \n245 SSAEM [24] and TSAEM [18] algorithms are comparable in terms of the computational time, but \n246 they both provide the lowest $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ . In Figure 2, we investigate sensitivity of fitting the model to \n47 increasing values of $K$ , $B$ and $M$ by measuring the time and the likelihood again. In all the cases, \n48 the proposed MHSAEM algorithms achieve the lowest AE in the shortest time. ", + "bbox": [ + 140, + 387, + 825, + 470 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 477, + 825, + 561 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 156, + 566, + 826, + 608 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 614, + 826, + 821 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 828, + 825, + 912 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 151, + 90, + 825, + 119 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "SSAEM and TSAEM algorithms failed to converge for $M > 2$ and for $K > 5 0$ respectively. We believe this is caused by selecting only $M$ maximal probabilities in the SSAEM (or distances in the TSAEM) algorithm (Section 5), which prevents certain, but not a negligible number of, components from being updated, thus providing only a crude approximation of $\\bar { p } _ { \\theta } \\bar { ( } z | x )$ . The results then suffer from substantial variational gap to the exact log-likelihood (Figure 1). On the contrary, MH sampler provides samples which consistently approximate $p _ { \\theta } ( z | x )$ despite evaluating much lower number of components in each step. ", + "bbox": [ + 173, + 126, + 825, + 223 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.2 Sum-product transform networks ", + "text_level": 1, + "bbox": [ + 166, + 239, + 446, + 253 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The sum product networks (SPNs) are a deep learning extension of finite mixture models. They can be interpreted as a mixture of trees [60], where each tree corresponds to a component. Therefore, they can be cast into the form of (1), but the number of components grows exponentially with their depth. In this section, we use recently proposed SPTNs which introduce additional transformation nodes to provide better expressiveness than the SPNs (SPTNs effectively generalize SPNs and flow models into one large family of models). ", + "bbox": [ + 173, + 265, + 825, + 348 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Experimental settings: We use 19 real datasets from the UCI database [16, 37, 35, 54], preprocessed in the same way as in [46]. For each experiment, we randomly split the data into $64 \\%$ , $16 \\%$ and $20 \\%$ for training, validation and testing, respectively. We calculate the average log-likelihood on the test set and measure again the time to reach $9 5 \\%$ of the maximal training log-likelihood, $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ . ", + "bbox": [ + 171, + 354, + 825, + 410 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To evaluate various (possibly shallow and/or deep) architectures of SPTNs, we fit each dataset with all the following combinations of hyper-parameters1: $s \\in ( 8 , 3 2 , 1 2 8 )$ , $b \\in ( 2 , 4 , 6 , 8 )$ , $l \\in ( 2 , 3 , 4 )$ , where $s$ is the number of children of each sum node, $b$ is the number of partitions of each product node, and $l$ is the number of layers (one layer contains sum and product nodes). The number of components of the SPTN, after its conversion into (1), is given as follows: $K = s ^ { l }$ . Note that the maximum number of components for the investigated parameters of the SPTN is 268,435,456. To reduce the space of possible architectures, we restrict ourselves only to (i) the leaf nodes given by $\\mathcal { N } ( 0 , \\bf { I } )$ ; (ii) affine transformations fixed to the singular value decomposition, choosing the the Givens parameterization for the unitary matrices [47]; and (iii) no sharing of any type of nodes [47]. ", + "bbox": [ + 161, + 416, + 825, + 541 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "276 Algorithms: We evaluate only on the MHSAEM-U algorithm—due to its favourable computational \n277 complexity and simplicity—and compare it with the stochastic gradient-descent (SGD) algorithm, \n278 which is routinely used to train SP(T)Ns [45, 47]. In this case, SGD in each iteration performs \n279 computations over all subtrees of the network, whereas the MHSAEM-U algorithm computes with \n280 only $M = 1$ subtrees, thus we should observe speed-up of the computations. In our implementation, \n281 both these methods perform optimization of their respective objective functions—the log-likelihood \n282 (2) for SGD and the EM objective (8) for MHSAEM-U—via the use of the automatic differentiation \n283 and the ADAM optimizer [29], using $B = 1 0 0$ and $T = 2 0 0 0 0$ . ", + "bbox": [ + 142, + 547, + 825, + 659 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Results: Since each dataset might benefit from a different architecture, Table 6.2 shows the test log-likelihood of the architectures selected according to the best likelihood measured on the validation set and the corresponding speed-up. The test log-likelihoods reveal that the MHSAEM-U algorithm outperforms the SGD algorithm on 10 out of 19 datasets, which was not originally the goal, but the added stochasticity helps to escape poor local minima. The speed-up demonstrates lower computational complexity of the MHSAEM-U algorithm on 17 out of 19 datasets, which was the main goal. The magic-telescope and wine datasets show approximately $1 0 2 \\times$ and $7 5 \\times$ speed-up, respectively, while on very small datasets (pima-indians and iris), the SGD is faster due to effective implementation. In the supplementary material, we present Table 3, exhibiting the same trends on a fixed architecture. ", + "bbox": [ + 171, + 665, + 825, + 803 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.3 Mixtures of real NVP flows ", + "text_level": 1, + "bbox": [ + 173, + 819, + 400, + 834 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We consider another class of mixture models (1), where each component $p _ { \\eta _ { z } } ( x | z )$ is transformed by the flow model—real NVP [15]. These transformations are parameterized via deep neural networks, allowing for flexible adjustment of the learning capacity of each component. ", + "bbox": [ + 176, + 845, + 825, + 887 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/82dc83ba06aea51dfe211f3df1e7f4eb07c99ebac951998df4d56e6c0d99003a.jpg", + "table_caption": [ + "Table 2: The speed-up and test log-likelihood, $\\mathcal { L } ^ { \\mathrm { t e s t } }$ , for the SGD and MHSAEM-U algorithms. The test log-likelihood (higher is better) is computed for the best model, with the corresponding $K$ , which is selected based on the validation log-likelihood. The speed-up is computed as the ratio of MHSAEM-U to SGD, i.e. their time to reach $9 5 \\%$ of the training log-likelihood. The results are averaged over five repetitions. Then, the higher test log-likelihood is highlighted with bold blue, and and no speed-up is highlighted with red. The average rank is computed as the standard competition (“1224”) ranking [14] on each dataset (lower is better). " + ], + "table_footnote": [], + "table_body": "
Sum-product transformnetworksMixtures of real NVP flows SGD
SGDMHSAEM-UMHSAEM-U
speed-upLtestKLtestKspeed-upLtestK 32LtestK
breast-cancer-wisconsin4.66-4.66641.4310240.63-99.85-39.31128
cardiotocography10.5559.5251231.0410249.8554.343256.08128
magic-telescope102.53-3.65512-5.0310243.74-3.978-4.228
pendigits4.890.881024-4.86163844.171.4680.488
pima-indians0.37-8.5464-7.62641.35-20.09128-16.33128
wall-following-robot3.431.841024-11.31638422.21-14.26128-17.56128
waveform-14.35-26.1464-23.9110243.72-34.128-33.428
waveform-24.82-26.2164-23.9110244.12-34.158-33.648
yeast20.5710.265125.18102414.496.611289.59128
ecoli1.86-5.564-0.2210242.15-11.37128-10.64128
ionosphere1.88-20.2764-5.935122.74-87.01128-42.75128
iris0.23-10.6564-1.49163843.28-16.34128-9.2132
page-blocks12.1812.215126.84102444.9517.1312817.9432
parkinsons1.46-21.85640.55123.09-566.58128-33.3132
sonar2.96-95.39512-69.29642.52-622.2128-88.81128
statlog-segment1.4447.3551226.531638438.4935.8412842.0432
statlog-vehicle2.97-4.2564-5.4510246.78-31.3432-26.43128
wine rank75.42-25.991024-13.2710242.05-171.58128-25.57128
1.561.441.831.17
", + "bbox": [ + 186, + 194, + 813, + 422 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "298 Experimental settings: We use the same experimental settings and evaluation metrics as in Section \n299 6.2. We apply the mixture model on all datasets, changing the number of components as follows: \n300 $K \\in ( 8 , 3 2 , 1 2 8 )$ . Each real NVP-based component in the mixture model has (i) the translation \n301 function parameterized via multi-layer perceptron with a single hidden layer of dimension 10, using \n302 the rectified linear activation function; and (ii) the scale function parameterized via the same network \n303 except with the hyperbolic tangent activation function. We do not use the batch normalization [15] and \n304 we stack two layers of the translation-scale transformation (we have used implementation from [20]). ", + "bbox": [ + 147, + 455, + 825, + 553 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Algorithms: The algorithms and their settings are the same as those in Section 6.2. ", + "bbox": [ + 161, + 559, + 710, + 574 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "306 Results: The experimental results are presented in right part of Table 6.2. They are similar to those \n307 obtained in the previous section. In terms of the test log-likelihood, the MHSAEM-U algorithm \n308 outperforms the SGD algorithm on all but three datasets, and it provides a substantial speed-up on all \n309 datasets except one. The test likelihood of models with the real NVP flows is most of the time worse \n310 than that of SPTNs with the affine transformations. As explained in the supplementary, this is due to \n311 the overfitting, which has been observed in [47]. ", + "bbox": [ + 140, + 580, + 825, + 664 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "312 7 Conclusion ", + "text_level": 1, + "bbox": [ + 142, + 689, + 299, + 707 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "313 This paper has presented a method to decrease computational complexity of fitting mixture models, \n314 including their generalizations, such as sum-product-(transform) networks and mixtures of flow \n315 models. The speed-up is achieved by evaluating and updating only a single component (per iteration), \n316 where the Metropolis-Hasting algorithm ensures sampling of components from a proper posterior. An \n317 experimental comparison on all three classes of models mentioned above confirmed the theoretical \n318 expectations. The method significantly speeds-up the fitting time and, importantly, without sacrificing \n319 the quality of the fit. In fact, the likelihood was better than that of the models fitted by the EM \n320 algorithm or the SGD algorithm in more than $50 \\%$ of cases. We attribute this to higher stochasticity, \n321 which helps to escape from poor local minima. \n322 In the experiments, the proposed method has used a uniform proposal distribution in the MH sampler. \n323 Despite outperforming the alternative methods, we conjecture that this limits the speed of convergence. \n324 Therefore, we believe that there is still a room for improvement in the implementation. We plan to \n325 address these issues in future work. ", + "bbox": [ + 140, + 724, + 825, + 849 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 856, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The presented method decreases the computational complexity of fitting large (and deep) mixture models, which leads to five to hundred time speed-up depending on a size of the problem (although negative exceptions occurs). We believe this line of research, which we want to continue, to have important benefits. First, it is directly related to decrease in energy consumption and in production of CO2 (we expect similar rates as the speedup). Second, it has a positive effect on financial aspects of deploying (and experimenting with) mixture models. Third, it decreases the hardware requirements, as in all experiments presented above the model was fitted on a single-core. 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", + "bbox": [ + 155, + 85, + 830, + 920 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "60 [58] W. Xiang, A. Karfoul, C. Yang, H. Shu, and R. L. B. Jeannès. An exact line search scheme to \n61 accelerate the EM algorithm: Application to Gaussian mixture models identification. Journal of \n62 computational science, 41:101073, 2020. \n63 [59] M. Zhang, Y. Fu, K. M. Bennett, and T. Wu. Computational efficient variational Bayesian Gaus \n64 sian mixture models via coreset. In 2016 International Conference on Computer, Information \n65 and Telecommunication Systems (CITS), pages 1–5. IEEE, 2016. \n66 [60] H. Zhao, P. Poupart, and G. Gordon. A unified approach for learning the parameters of sum \n67 product networks. In Proceedings of the 30th International Conference on Neural Information \n68 Processing Systems, pages 433–441, 2016. ", + "bbox": [ + 150, + 90, + 826, + 237 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 116, + 339, + 131 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] Our main contribution is computational speedup. Cases where it was not achieved are highlighted in the experimental section. \n(c) Did you discuss any potential negative societal impacts of your work? [No] We do not foresee any potential negative impact. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 238, + 135, + 825, + 267 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 215, + 272, + 493, + 286 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ", + "bbox": [ + 238, + 290, + 736, + 321 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 212, + 325, + 393, + 340 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The code is available in a github repository. All dataset are public from the UCI database. \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 6. \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We report only average of Monte Carlo repetitions, the error bars were too small to have any visual impact in the reported logarithmic scale. \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] ", + "bbox": [ + 238, + 344, + 825, + 489 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 215, + 494, + 823, + 510 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] We use 20 datasets from UCI, we cite the required papers for each dataset, mostly the UCI database and few additional publications. \n(b) Did you mention the license of the assets? [No] The data are publically available, we comply with the requirement on citing appropriate publications. \n(c) Did you include any new assets either in the supplemental material or as a URL? [No] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 238, + 513, + 825, + 662 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 215, + 666, + 707, + 681 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? 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They", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 90, + 465, + 100, + 474 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "have been successfully applied in diverse fields, including bioinformatics [49], econometrics [10],", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 89, + 477, + 100, + 486 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "engineering [33], etc. A mixture model relies on a weighted sum of probability distributions—here", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 89, + 487, + 100, + 497 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 105, + 484, + 262, + 498 + ], + "score": 1.0, + "content": "referred to as components—to cluster", + "type": "text" + }, + { + "bbox": [ + 263, + 486, + 273, + 495 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 484, + 383, + 498 + ], + "score": 1.0, + "content": "unlabelled datapoints into", + "type": "text" + }, + { + "bbox": [ + 383, + 486, + 393, + 495 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "categories. The traditional", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 495, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 89, + 498, + 100, + 508 + ], + "score": 1.0, + "content": "18", + "type": "text" + }, + { + "bbox": [ + 105, + 495, + 506, + 509 + ], + "score": 1.0, + "content": "maximum likelihood techniques train the model by optimizing either (i) the marginal likelihood via", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 89, + 509, + 100, + 519 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "gradient-descent [50] or (ii) the evidence lower bound via variational methods [4], including the", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 89, + 520, + 100, + 530 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "expectation-maximization (EM) [13]. 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The EM algorithm is then referred to as the generalized EM algorithm [56].", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 91, + 477, + 319, + 491 + ], + "lines": [ + { + "bbox": [ + 86, + 476, + 320, + 495 + ], + "spans": [ + { + "bbox": [ + 86, + 476, + 320, + 495 + ], + "score": 1.0, + "content": "88 4 The generalized MHSAEM algorithm", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 90, + 502, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 89, + 501, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 89, + 504, + 100, + 514 + ], + "score": 1.0, + "content": "89", + "type": "text" + }, + { + "bbox": [ + 105, + 501, + 506, + 516 + ], + "score": 1.0, + "content": "We design a version of the generalized EM algorithm suitable for scenarios where (4) can represent", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 89, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 89, + 515, + 100, + 525 + ], + "score": 1.0, + "content": "90", + "type": "text" + }, + { + "bbox": [ + 106, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "deep, discrete, latent variable models, thus being parameterized by possibly complex nonlinear", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 89, + 524, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 89, + 526, + 100, + 535 + ], + "score": 1.0, + "content": "91", + "type": "text" + }, + { + "bbox": [ + 106, + 524, + 505, + 536 + ], + "score": 1.0, + "content": "transformations. 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At each iteration,", + "type": "text" + }, + { + "bbox": [ + 236, + 619, + 240, + 628 + ], + "score": 0.65, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 617, + 507, + 631 + ], + "score": 1.0, + "content": ", we compute the conditional expectation in (4) only for a subset—", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 628, + 504, + 642 + ], + "spans": [ + { + "bbox": [ + 88, + 631, + 100, + 641 + ], + "score": 1.0, + "content": "98", + "type": "text" + }, + { + "bbox": [ + 105, + 628, + 362, + 642 + ], + "score": 1.0, + "content": "here referred to as a minibatch—of the original full dataset, i.e.", + "type": "text" + }, + { + "bbox": [ + 362, + 629, + 392, + 641 + ], + "score": 0.91, + "content": "( x _ { i } ) _ { i \\in I }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 628, + 421, + 642 + ], + "score": 1.0, + "content": ". 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Under mild regularity assumptions [52], the chain passes the transiti", + "type": "text" + }, + { + "bbox": [ + 467, + 260, + 488, + 270 + ], + "score": 0.87, + "content": "t - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 259, + 510, + 288 + ], + "score": 1.0, + "content": ", byriod", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 86, + 282, + 100, + 293 + ], + "score": 1.0, + "content": "121", + "type": "text" + }, + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "(the burn-in phase), and the samples can then be used to approximate the conditional expectation in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 292, + 166, + 304 + ], + "spans": [ + { + "bbox": [ + 86, + 293, + 100, + 304 + ], + "score": 1.0, + "content": "122", + "type": "text" + }, + { + "bbox": [ + 104, + 292, + 166, + 304 + ], + "score": 1.0, + "content": "(4) as follows:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 300, + 390, + 332 + ], + "lines": [ + { + "bbox": [ + 220, + 300, + 390, + 332 + ], + "spans": [ + { + "bbox": [ + 220, + 300, + 390, + 332 + ], + "score": 0.95, + "content": "\\widehat { \\mathcal { Q } } _ { t - 1 } ( \\theta ) = \\frac { 1 } { M } \\sum _ { i \\in I } \\sum _ { z \\in \\mathbf { z } _ { i , t } } \\log p _ { \\eta _ { z } } ( x _ { i } | z ) \\pi _ { z } .", + "type": "interline_equation", + "image_path": "e43005c66ee8eb883ede3aeefb994947aa4f14879ab1b3689fc3176b2f635909.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 220, + 300, + 390, + 316.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 220, + 316.0, + 390, + 332.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 333, + 505, + 367 + ], + "lines": [ + { + "bbox": [ + 87, + 332, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 87, + 335, + 100, + 344 + ], + "score": 1.0, + "content": "123", + "type": "text" + }, + { + "bbox": [ + 105, + 332, + 401, + 346 + ], + "score": 1.0, + "content": "Note that, to ensure this approach is truly efficient, we have to draw only", + "type": "text" + }, + { + "bbox": [ + 402, + 334, + 438, + 344 + ], + "score": 0.9, + "content": "M \\ll K", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 332, + 505, + 346 + ], + "score": 1.0, + "content": "samples at each", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 87, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 87, + 346, + 100, + 354 + ], + "score": 1.0, + "content": "124", + "type": "text" + }, + { + "bbox": [ + 105, + 344, + 144, + 356 + ], + "score": 1.0, + "content": "iteration,", + "type": "text" + }, + { + "bbox": [ + 145, + 345, + 149, + 354 + ], + "score": 0.71, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 344, + 211, + 356 + ], + "score": 1.0, + "content": "; otherwise, for", + "type": "text" + }, + { + "bbox": [ + 211, + 344, + 245, + 354 + ], + "score": 0.9, + "content": "M \\approx K", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 344, + 506, + 356 + ], + "score": 1.0, + "content": ", we may rather compute the exact marginalization in (4), since it", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 87, + 354, + 271, + 369 + ], + "spans": [ + { + "bbox": [ + 87, + 357, + 100, + 367 + ], + "score": 1.0, + "content": "125", + "type": "text" + }, + { + "bbox": [ + 103, + 354, + 271, + 369 + ], + "score": 1.0, + "content": "is tractable (but computationally costly).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 102, + 379, + 161, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 163, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 163, + 394 + ], + "score": 1.0, + "content": "4.2 M-step", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 86, + 398, + 506, + 504 + ], + "lines": [ + { + "bbox": [ + 86, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 86, + 401, + 99, + 410 + ], + "score": 1.0, + "content": "127", + "type": "text" + }, + { + "bbox": [ + 105, + 399, + 253, + 411 + ], + "score": 1.0, + "content": "Assume for a moment that (6) with", + "type": "text" + }, + { + "bbox": [ + 254, + 399, + 289, + 411 + ], + "score": 0.93, + "content": "\\mathcal { Q } _ { t - 1 } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "given by (8) has a closed-form solution, yielding an", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 410, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 86, + 412, + 99, + 421 + ], + "score": 1.0, + "content": "128", + "type": "text" + }, + { + "bbox": [ + 106, + 410, + 155, + 422 + ], + "score": 1.0, + "content": "estimate of", + "type": "text" + }, + { + "bbox": [ + 155, + 410, + 161, + 420 + ], + "score": 0.77, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 410, + 446, + 422 + ], + "score": 1.0, + "content": ". Such an estimate would have a high variance, converging only for", + "type": "text" + }, + { + "bbox": [ + 446, + 410, + 486, + 420 + ], + "score": 0.9, + "content": "M \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 420, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 86, + 422, + 100, + 433 + ], + "score": 1.0, + "content": "129", + "type": "text" + }, + { + "bbox": [ + 106, + 421, + 145, + 431 + ], + "score": 0.88, + "content": "T \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 420, + 497, + 433 + ], + "score": 1.0, + "content": "[19]. 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We consider that there is no closed-form solution of (6),", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 86, + 444, + 99, + 455 + ], + "score": 1.0, + "content": "131", + "type": "text" + }, + { + "bbox": [ + 106, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "and—to ensure that the samples (and thus computations) are recycled over the iterations—we use", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 86, + 456, + 99, + 465 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 105, + 453, + 506, + 467 + ], + "score": 1.0, + "content": "the stochastic approximation (SA) [51] to optimize (8). This is analogous to applying a stochastic", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 464, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 86, + 469, + 99, + 478 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 105, + 465, + 207, + 480 + ], + "score": 1.0, + "content": "gradient-descent method,", + "type": "text" + }, + { + "bbox": [ + 207, + 464, + 313, + 478 + ], + "score": 0.88, + "content": "\\theta _ { t } = \\theta _ { t - 1 } + \\gamma _ { t } \\nabla _ { \\theta } \\tilde { \\mathcal { Q } } _ { t - 1 } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 465, + 342, + 480 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 342, + 468, + 352, + 478 + ], + "score": 0.83, + "content": "\\gamma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 465, + 506, + 480 + ], + "score": 1.0, + "content": "is the step-size, satisfying the Robbins-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 476, + 507, + 493 + ], + "spans": [ + { + "bbox": [ + 86, + 479, + 99, + 489 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 104, + 476, + 182, + 493 + ], + "score": 1.0, + "content": "Monro constraints,", + "type": "text" + }, + { + "bbox": [ + 183, + 478, + 354, + 491 + ], + "score": 0.3, + "content": "\\begin{array} { r } { \\gamma _ { t } \\in [ 0 , 1 ] , \\sum _ { t \\geq 1 } \\gamma _ { t } = \\infty , \\sum _ { t \\geq 1 } \\gamma _ { t } ^ { 2 } < \\infty , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 476, + 372, + 493 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 372, + 478, + 386, + 489 + ], + "score": 0.89, + "content": "\\nabla _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 476, + 467, + 493 + ], + "score": 1.0, + "content": "is the gradient w.r.t.", + "type": "text" + }, + { + "bbox": [ + 468, + 478, + 473, + 487 + ], + "score": 0.73, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 476, + 507, + 493 + ], + "score": 1.0, + "content": ". In this", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 87, + 491, + 470, + 505 + ], + "spans": [ + { + "bbox": [ + 87, + 494, + 99, + 504 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 105, + 491, + 232, + 505 + ], + "score": 1.0, + "content": "way, the computations made in", + "type": "text" + }, + { + "bbox": [ + 232, + 491, + 255, + 504 + ], + "score": 0.91, + "content": "\\nabla _ { \\boldsymbol { \\theta } } \\widehat { \\mathcal { Q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 491, + 337, + 505 + ], + "score": 1.0, + "content": "are accumulated via", + "type": "text" + }, + { + "bbox": [ + 338, + 493, + 347, + 504 + ], + "score": 0.87, + "content": "\\theta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 491, + 470, + 505 + ], + "score": 1.0, + "content": "and reused over the iterations.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 88, + 508, + 507, + 542 + ], + "lines": [ + { + "bbox": [ + 87, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 87, + 511, + 100, + 519 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 104, + 508, + 173, + 522 + ], + "score": 1.0, + "content": "The parameters", + "type": "text" + }, + { + "bbox": [ + 174, + 511, + 184, + 520 + ], + "score": 0.84, + "content": "\\eta _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 508, + 395, + 522 + ], + "score": 1.0, + "content": "have a different form based on a specific case of", + "type": "text" + }, + { + "bbox": [ + 395, + 509, + 432, + 521 + ], + "score": 0.92, + "content": "p _ { \\eta _ { z } } ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 508, + 473, + 522 + ], + "score": 1.0, + "content": ", whereas", + "type": "text" + }, + { + "bbox": [ + 474, + 511, + 485, + 520 + ], + "score": 0.86, + "content": "\\pi _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 87, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 87, + 522, + 100, + 531 + ], + "score": 1.0, + "content": "137", + "type": "text" + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "permanent structure in (1). 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Such an estimate would have a high variance, converging only for", + "type": "text" + }, + { + "bbox": [ + 446, + 410, + 486, + 420 + ], + "score": 0.9, + "content": "M \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 410, + 505, + 422 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 420, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 86, + 422, + 100, + 433 + ], + "score": 1.0, + "content": "129", + "type": "text" + }, + { + "bbox": [ + 106, + 421, + 145, + 431 + ], + "score": 0.88, + "content": "T \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 420, + 497, + 433 + ], + "score": 1.0, + "content": "[19]. The main reason is that the samples would not be reused over the iterations,", + "type": "text" + }, + { + "bbox": [ + 497, + 422, + 502, + 431 + ], + "score": 0.69, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 420, + 507, + 433 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 86, + 434, + 99, + 443 + ], + "score": 1.0, + "content": "130", + "type": "text" + }, + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "thus wasting computational resources. We consider that there is no closed-form solution of (6),", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 86, + 444, + 99, + 455 + ], + "score": 1.0, + "content": "131", + "type": "text" + }, + { + "bbox": [ + 106, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "and—to ensure that the samples (and thus computations) are recycled over the iterations—we use", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 453, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 86, + 456, + 99, + 465 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 105, + 453, + 506, + 467 + ], + "score": 1.0, + "content": "the stochastic approximation (SA) [51] to optimize (8). This is analogous to applying a stochastic", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 464, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 86, + 469, + 99, + 478 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 105, + 465, + 207, + 480 + ], + "score": 1.0, + "content": "gradient-descent method,", + "type": "text" + }, + { + "bbox": [ + 207, + 464, + 313, + 478 + ], + "score": 0.88, + "content": "\\theta _ { t } = \\theta _ { t - 1 } + \\gamma _ { t } \\nabla _ { \\theta } \\tilde { \\mathcal { Q } } _ { t - 1 } ( \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 465, + 342, + 480 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 342, + 468, + 352, + 478 + ], + "score": 0.83, + "content": "\\gamma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 465, + 506, + 480 + ], + "score": 1.0, + "content": "is the step-size, satisfying the Robbins-", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 476, + 507, + 493 + ], + "spans": [ + { + "bbox": [ + 86, + 479, + 99, + 489 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 104, + 476, + 182, + 493 + ], + "score": 1.0, + "content": "Monro constraints,", + "type": "text" + }, + { + "bbox": [ + 183, + 478, + 354, + 491 + ], + "score": 0.3, + "content": "\\begin{array} { r } { \\gamma _ { t } \\in [ 0 , 1 ] , \\sum _ { t \\geq 1 } \\gamma _ { t } = \\infty , \\sum _ { t \\geq 1 } \\gamma _ { t } ^ { 2 } < \\infty , } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 476, + 372, + 493 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 372, + 478, + 386, + 489 + ], + "score": 0.89, + "content": "\\nabla _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 476, + 467, + 493 + ], + "score": 1.0, + "content": "is the gradient w.r.t.", + "type": "text" + }, + { + "bbox": [ + 468, + 478, + 473, + 487 + ], + "score": 0.73, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 476, + 507, + 493 + ], + "score": 1.0, + "content": ". In this", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 491, + 470, + 505 + ], + "spans": [ + { + "bbox": [ + 87, + 494, + 99, + 504 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 105, + 491, + 232, + 505 + ], + "score": 1.0, + "content": "way, the computations made in", + "type": "text" + }, + { + "bbox": [ + 232, + 491, + 255, + 504 + ], + "score": 0.91, + "content": "\\nabla _ { \\boldsymbol { \\theta } } \\widehat { \\mathcal { Q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 491, + 337, + 505 + ], + "score": 1.0, + "content": "are accumulated via", + "type": "text" + }, + { + "bbox": [ + 338, + 493, + 347, + 504 + ], + "score": 0.87, + "content": "\\theta _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 491, + 470, + 505 + ], + "score": 1.0, + "content": "and reused over the iterations.", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 508, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 87, + 511, + 100, + 519 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 104, + 508, + 173, + 522 + ], + "score": 1.0, + "content": "The parameters", + "type": "text" + }, + { + "bbox": [ + 174, + 511, + 184, + 520 + ], + "score": 0.84, + "content": "\\eta _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 508, + 395, + 522 + ], + "score": 1.0, + "content": "have a different form based on a specific case of", + "type": "text" + }, + { + "bbox": [ + 395, + 509, + 432, + 521 + ], + "score": 0.92, + "content": "p _ { \\eta _ { z } } ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 508, + 473, + 522 + ], + "score": 1.0, + "content": ", whereas", + "type": "text" + }, + { + "bbox": [ + 474, + 511, + 485, + 520 + ], + "score": 0.86, + "content": "\\pi _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 508, + 506, + 522 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 87, + 522, + 100, + 531 + ], + "score": 1.0, + "content": "137", + "type": "text" + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "permanent structure in (1). Therefore, without loss of generality, we split (6) into a generic part and a", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 531, + 192, + 542 + ], + "spans": [ + { + "bbox": [ + 86, + 533, + 100, + 542 + ], + "score": 1.0, + "content": "138", + "type": "text" + }, + { + "bbox": [ + 105, + 531, + 192, + 542 + ], + "score": 1.0, + "content": "fixed part as follows:", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + } + ], + "index": 28, + "bbox_fs": [ + 86, + 399, + 507, + 505 + ] + }, + { + "type": "index", + "bbox": [ + 88, + 508, + 507, + 542 + ], + "lines": [], + "index": 34, + "bbox_fs": [ + 86, + 508, + 506, + 542 + ], + "lines_deleted": true + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 545, + 371, + 580 + ], + "lines": [ + { + "bbox": [ + 239, + 545, + 371, + 580 + ], + "spans": [ + { + "bbox": [ + 239, + 545, + 371, + 580 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\eta _ { k , t } = \\eta _ { k , t - 1 } + \\gamma _ { t } \\nabla _ { \\eta _ { k } } \\widehat { \\mathcal { Q } } _ { t - 1 } ( \\theta ) , } \\\\ & { \\nu _ { k , t } = \\nu _ { k , t - 1 } + \\gamma _ { t } \\nabla _ { \\nu _ { k } } \\widehat { \\mathcal { Q } } _ { t - 1 } ( \\theta ) , } \\end{array}", + "type": "interline_equation", + "image_path": "e1b6543f038daf2078a131f2fa387f7478babb5133579989565df178123b0c85.jpg" + } + ] + } + ], + "index": 36.5, + "virtual_lines": [ + { + "bbox": [ + 239, + 545, + 371, + 562.5 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 239, + 562.5, + 371, + 580.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "index", + "bbox": [ + 87, + 582, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 86, + 580, + 507, + 598 + ], + "spans": [ + { + "bbox": [ + 86, + 585, + 100, + 595 + ], + "score": 1.0, + "content": "139", + "type": "text" + }, + { + "bbox": [ + 104, + 580, + 266, + 598 + ], + "score": 1.0, + "content": "where—to ensure that the probabilities,", + "type": "text" + }, + { + "bbox": [ + 266, + 583, + 305, + 596 + ], + "score": 0.94, + "content": "( \\pi _ { k , t } ) _ { k = 1 } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 580, + 507, + 598 + ], + "score": 1.0, + "content": ", satisfy the constraints (Section 2)—we transform", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 591, + 508, + 614 + ], + "spans": [ + { + "bbox": [ + 86, + 598, + 100, + 608 + ], + "score": 1.0, + "content": "140", + "type": "text" + }, + { + "bbox": [ + 106, + 595, + 134, + 609 + ], + "score": 0.95, + "content": "\\nabla _ { \\pi _ { k } } \\widehat { \\mathcal { Q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 591, + 150, + 614 + ], + "score": 1.0, + "content": "via", + "type": "text" + }, + { + "bbox": [ + 150, + 596, + 200, + 609 + ], + "score": 0.9, + "content": "\\nu _ { k } = \\log \\pi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 591, + 279, + 614 + ], + "score": 1.0, + "content": "and optimize w.r.t.", + "type": "text" + }, + { + "bbox": [ + 279, + 599, + 290, + 608 + ], + "score": 0.86, + "content": "\\nu _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 591, + 359, + 614 + ], + "score": 1.0, + "content": ". 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However, due to poor mixing properties, the algorithm may converge slowly for high", + "type": "text" + }, + { + "bbox": [ + 493, + 438, + 503, + 448 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 438, + 507, + 451 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 86, + 454, + 506, + 605 + ], + "lines": [ + { + "bbox": [ + 86, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 86, + 456, + 100, + 466 + ], + "score": 1.0, + "content": "156", + "type": "text" + }, + { + "bbox": [ + 105, + 454, + 241, + 468 + ], + "score": 1.0, + "content": "Tabular proposal with forgetting", + "type": "text" + }, + { + "bbox": [ + 241, + 455, + 260, + 466 + ], + "score": 0.66, + "content": "( T F )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 454, + 506, + 468 + ], + "score": 1.0, + "content": ". The key requirement to design a proposal distribution is to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 86, + 466, + 100, + 477 + ], + "score": 1.0, + "content": "157", + "type": "text" + }, + { + "bbox": [ + 104, + 464, + 378, + 478 + ], + "score": 1.0, + "content": "restrict its computational complexity somewhere between that of the", + "type": "text" + }, + { + "bbox": [ + 378, + 466, + 387, + 475 + ], + "score": 0.41, + "content": "\\mathrm { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 464, + 404, + 478 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 404, + 466, + 413, + 475 + ], + "score": 0.38, + "content": "\\mathrm { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "proposals. One way to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 475, + 507, + 491 + ], + "spans": [ + { + "bbox": [ + 86, + 478, + 100, + 489 + ], + "score": 1.0, + "content": "158", + "type": "text" + }, + { + "bbox": [ + 105, + 475, + 303, + 491 + ], + "score": 1.0, + "content": "satisfy this constraint is to use the Markov chain,", + "type": "text" + }, + { + "bbox": [ + 304, + 475, + 339, + 489 + ], + "score": 0.94, + "content": "( z _ { i , t } ) _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 475, + 452, + 491 + ], + "score": 1.0, + "content": ", to learn a transition kernel,", + "type": "text" + }, + { + "bbox": [ + 452, + 476, + 484, + 488 + ], + "score": 0.92, + "content": "p ( z _ { i } | \\bar { z } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 475, + 507, + 491 + ], + "score": 1.0, + "content": ", see,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 487, + 507, + 501 + ], + "spans": [ + { + "bbox": [ + 86, + 489, + 100, + 500 + ], + "score": 1.0, + "content": "159", + "type": "text" + }, + { + "bbox": [ + 104, + 487, + 363, + 501 + ], + "score": 1.0, + "content": "e.g. [3]. Unfortunately, this would require us to store a table with", + "type": "text" + }, + { + "bbox": [ + 363, + 488, + 378, + 498 + ], + "score": 0.89, + "content": "K ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 487, + 442, + 501 + ], + "score": 1.0, + "content": "entries for each", + "type": "text" + }, + { + "bbox": [ + 442, + 489, + 502, + 500 + ], + "score": 0.9, + "content": "i \\in ( 1 , \\ldots , N )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 487, + 507, + 501 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 498, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 86, + 500, + 100, + 525 + ], + "score": 1.0, + "content": "160 161", + "type": "text" + }, + { + "bbox": [ + 101, + 498, + 230, + 533 + ], + "score": 1.0, + "content": "which is very demanding evethe Markov chain and define:", + "type": "text" + }, + { + "bbox": [ + 230, + 512, + 349, + 525 + ], + "score": 0.92, + "content": "q ( z _ { i } | \\bar { z } _ { i } ) : = q _ { \\alpha _ { i } } ( z _ { i } ) : = \\mathcal { C } ( \\alpha _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 500, + 303, + 509 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 500, + 333, + 509 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 498, + 381, + 533 + ], + "score": 1.0, + "content": "herefore, where", + "type": "text" + }, + { + "bbox": [ + 381, + 510, + 478, + 525 + ], + "score": 0.92, + "content": "\\mathcal { C } ( \\pmb { \\alpha } _ { i } ) \\propto \\Pi _ { k = 1 } ^ { K } \\alpha _ { k , i } ^ { \\bar { 1 ( } z _ { i } = k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 498, + 506, + 533 + ], + "score": 1.0, + "content": "nce inis the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 86, + 522, + 502, + 542 + ], + "spans": [ + { + "bbox": [ + 86, + 528, + 100, + 537 + ], + "score": 1.0, + "content": "162", + "type": "text" + }, + { + "bbox": [ + 103, + 522, + 272, + 542 + ], + "score": 1.0, + "content": "categorical distribution with the weights", + "type": "text" + }, + { + "bbox": [ + 273, + 526, + 365, + 538 + ], + "score": 0.9, + "content": "\\pmb { \\alpha } _ { i } : = ( \\alpha _ { 1 , i } , \\dots , \\alpha _ { K , i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 522, + 387, + 542 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 387, + 526, + 502, + 538 + ], + "score": 0.85, + "content": "\\mathcal { L } ( \\pmb { \\alpha } _ { i } ) : = \\Sigma _ { \\tau = 1 } ^ { t } \\log q _ { \\pmb { \\alpha } _ { i } } ( z _ { i , \\tau } )", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 534, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 86, + 538, + 100, + 549 + ], + "score": 1.0, + "content": "163", + "type": "text" + }, + { + "bbox": [ + 104, + 534, + 214, + 552 + ], + "score": 1.0, + "content": "we obtain an estimate of", + "type": "text" + }, + { + "bbox": [ + 215, + 538, + 227, + 548 + ], + "score": 0.86, + "content": "\\alpha _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 534, + 278, + 552 + ], + "score": 1.0, + "content": "at iteration", + "type": "text" + }, + { + "bbox": [ + 278, + 538, + 283, + 547 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 534, + 337, + 552 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + }, + { + "bbox": [ + 337, + 538, + 479, + 550 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\alpha _ { i , t } : = \\mathrm { \\ a r g m a x } _ { \\alpha _ { i } } \\mathcal L ( \\alpha _ { i } ) \\ = \\ \\frac { n _ { i , t } } { t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "t , with", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 86, + 546, + 507, + 565 + ], + "spans": [ + { + "bbox": [ + 86, + 550, + 100, + 561 + ], + "score": 1.0, + "content": "164", + "type": "text" + }, + { + "bbox": [ + 107, + 549, + 176, + 562 + ], + "score": 0.92, + "content": "n _ { i , t } = \\Sigma _ { \\tau = 1 } ^ { t } \\mathbf { e } _ { z _ { i , t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 546, + 206, + 565 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 207, + 550, + 218, + 560 + ], + "score": 0.87, + "content": "\\mathbf { e } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 546, + 454, + 565 + ], + "score": 1.0, + "content": "is the standard basis vector (a one-hot vector) with one at", + "type": "text" + }, + { + "bbox": [ + 454, + 550, + 460, + 559 + ], + "score": 0.44, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 546, + 507, + 565 + ], + "score": 1.0, + "content": "th position", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 559, + 507, + 574 + ], + "spans": [ + { + "bbox": [ + 86, + 561, + 100, + 572 + ], + "score": 1.0, + "content": "165", + "type": "text" + }, + { + "bbox": [ + 105, + 559, + 406, + 574 + ], + "score": 1.0, + "content": "and zeros otherwise. This can be further rewritten into a recursive form:", + "type": "text" + }, + { + "bbox": [ + 406, + 561, + 491, + 573 + ], + "score": 0.91, + "content": "n _ { i , t } = n _ { i , t - 1 } + \\mathbf { e } _ { z _ { i , t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 559, + 507, + 574 + ], + "score": 1.0, + "content": "or,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 571, + 507, + 586 + ], + "spans": [ + { + "bbox": [ + 86, + 574, + 100, + 583 + ], + "score": 1.0, + "content": "166", + "type": "text" + }, + { + "bbox": [ + 105, + 571, + 249, + 586 + ], + "score": 1.0, + "content": "using the Robbins-Monro step-size,", + "type": "text" + }, + { + "bbox": [ + 249, + 572, + 406, + 585 + ], + "score": 0.9, + "content": "n _ { i , t } = ( { \\bf 1 } - { \\bf e } _ { z _ { i , t } } \\gamma _ { t } ) \\odot n _ { i , t - 1 } + \\gamma _ { t } { \\bf e } _ { z _ { i , t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 571, + 507, + 586 + ], + "score": 1.0, + "content": ", where 1 is the vector of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 86, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 86, + 584, + 100, + 595 + ], + "score": 1.0, + "content": "167", + "type": "text" + }, + { + "bbox": [ + 105, + 582, + 147, + 596 + ], + "score": 1.0, + "content": "ones, and", + "type": "text" + }, + { + "bbox": [ + 147, + 584, + 157, + 593 + ], + "score": 0.83, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "is the Hadamard product. We refer to this case simply as “table with forgetting” (TF)", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 86, + 593, + 431, + 607 + ], + "spans": [ + { + "bbox": [ + 86, + 596, + 100, + 605 + ], + "score": 1.0, + "content": "168", + "type": "text" + }, + { + "bbox": [ + 104, + 593, + 202, + 607 + ], + "score": 1.0, + "content": "due to that it represents", + "type": "text" + }, + { + "bbox": [ + 202, + 594, + 234, + 604 + ], + "score": 0.91, + "content": "N \\times K", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 593, + 334, + 607 + ], + "score": 1.0, + "content": "table in the memory and", + "type": "text" + }, + { + "bbox": [ + 334, + 595, + 344, + 605 + ], + "score": 0.86, + "content": "\\gamma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 593, + 431, + 607 + ], + "score": 1.0, + "content": "is a forgetting factor.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 89, + 620, + 194, + 633 + ], + "lines": [ + { + "bbox": [ + 84, + 619, + 196, + 635 + ], + "spans": [ + { + "bbox": [ + 84, + 619, + 196, + 635 + ], + "score": 1.0, + "content": "169 5 Related work", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 95, + 645, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 102, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 102, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "Stochastic approximation expectation-maximization. The application of SA to prevent the evaluation", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 95, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 95, + 661, + 99, + 665 + ], + "score": 1.0, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 105, + 657, + 130, + 668 + ], + "score": 1.0, + "content": "of all", + "type": "text" + }, + { + "bbox": [ + 130, + 657, + 140, + 666 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "components in mixture models has been overlooked for a long time. The reason is that the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 93, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 93, + 671, + 99, + 677 + ], + "score": 1.0, + "content": "72", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "original motivation to combine the EM algorithm with SA is to address the analytical intractability", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 93, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 93, + 681, + 99, + 688 + ], + "score": 1.0, + "content": "73", + "type": "text" + }, + { + "bbox": [ + 105, + 678, + 221, + 690 + ], + "score": 1.0, + "content": "of the expected value under", + "type": "text" + }, + { + "bbox": [ + 221, + 680, + 252, + 690 + ], + "score": 0.95, + "content": "p _ { \\theta } ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "in (4), which is, however, almost always tractable for mixture", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 93, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 93, + 692, + 99, + 698 + ], + "score": 1.0, + "content": "74", + "type": "text" + }, + { + "bbox": [ + 104, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "models. The intractability issue is addressed by either the Monte Carlo SAEM (MCSAEM) [12]", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 93, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 93, + 704, + 99, + 709 + ], + "score": 1.0, + "content": "75", + "type": "text" + }, + { + "bbox": [ + 104, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "or the Markov chain Monte Carlo SAEM (MCMCSAEM) [31]. 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Unfortunately, this would require us to store a table with", + "type": "text" + }, + { + "bbox": [ + 363, + 488, + 378, + 498 + ], + "score": 0.89, + "content": "K ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 487, + 442, + 501 + ], + "score": 1.0, + "content": "entries for each", + "type": "text" + }, + { + "bbox": [ + 442, + 489, + 502, + 500 + ], + "score": 0.9, + "content": "i \\in ( 1 , \\ldots , N )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 487, + 507, + 501 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 498, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 86, + 500, + 100, + 525 + ], + "score": 1.0, + "content": "160 161", + "type": "text" + }, + { + "bbox": [ + 101, + 498, + 230, + 533 + ], + "score": 1.0, + "content": "which is very demanding evethe Markov chain and define:", + "type": "text" + }, + { + "bbox": [ + 230, + 512, + 349, + 525 + ], + "score": 0.92, + "content": "q ( z _ { i } | \\bar { z } _ { i } ) : = q _ { \\alpha _ { i } } ( z _ { i } ) : = \\mathcal { C } ( \\alpha _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 500, + 303, + 509 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 500, + 333, + 509 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 498, + 381, + 533 + ], + "score": 1.0, + "content": "herefore, where", + "type": "text" + }, + { + "bbox": [ + 381, + 510, + 478, + 525 + ], + "score": 0.92, + "content": "\\mathcal { C } ( \\pmb { \\alpha } _ { i } ) \\propto \\Pi _ { k = 1 } ^ { K } \\alpha _ { k , i } ^ { \\bar { 1 ( } z _ { i } = k ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 498, + 506, + 533 + ], + "score": 1.0, + "content": "nce inis the", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 522, + 502, + 542 + ], + "spans": [ + { + "bbox": [ + 86, + 528, + 100, + 537 + ], + "score": 1.0, + "content": "162", + "type": "text" + }, + { + "bbox": [ + 103, + 522, + 272, + 542 + ], + "score": 1.0, + "content": "categorical distribution with the weights", + "type": "text" + }, + { + "bbox": [ + 273, + 526, + 365, + 538 + ], + "score": 0.9, + "content": "\\pmb { \\alpha } _ { i } : = ( \\alpha _ { 1 , i } , \\dots , \\alpha _ { K , i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 522, + 387, + 542 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 387, + 526, + 502, + 538 + ], + "score": 0.85, + "content": "\\mathcal { L } ( \\pmb { \\alpha } _ { i } ) : = \\Sigma _ { \\tau = 1 } ^ { t } \\log q _ { \\pmb { \\alpha } _ { i } } ( z _ { i , \\tau } )", + "type": "inline_equation" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 534, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 86, + 538, + 100, + 549 + ], + "score": 1.0, + "content": "163", + "type": "text" + }, + { + "bbox": [ + 104, + 534, + 214, + 552 + ], + "score": 1.0, + "content": "we obtain an estimate of", + "type": "text" + }, + { + "bbox": [ + 215, + 538, + 227, + 548 + ], + "score": 0.86, + "content": "\\alpha _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 534, + 278, + 552 + ], + "score": 1.0, + "content": "at iteration", + "type": "text" + }, + { + "bbox": [ + 278, + 538, + 283, + 547 + ], + "score": 0.74, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 534, + 337, + 552 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + }, + { + "bbox": [ + 337, + 538, + 479, + 550 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\alpha _ { i , t } : = \\mathrm { \\ a r g m a x } _ { \\alpha _ { i } } \\mathcal L ( \\alpha _ { i } ) \\ = \\ \\frac { n _ { i , t } } { t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 537, + 506, + 549 + ], + "score": 1.0, + "content": "t , with", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 546, + 507, + 565 + ], + "spans": [ + { + "bbox": [ + 86, + 550, + 100, + 561 + ], + "score": 1.0, + "content": "164", + "type": "text" + }, + { + "bbox": [ + 107, + 549, + 176, + 562 + ], + "score": 0.92, + "content": "n _ { i , t } = \\Sigma _ { \\tau = 1 } ^ { t } \\mathbf { e } _ { z _ { i , t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 546, + 206, + 565 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 207, + 550, + 218, + 560 + ], + "score": 0.87, + "content": "\\mathbf { e } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 546, + 454, + 565 + ], + "score": 1.0, + "content": "is the standard basis vector (a one-hot vector) with one at", + "type": "text" + }, + { + "bbox": [ + 454, + 550, + 460, + 559 + ], + "score": 0.44, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 546, + 507, + 565 + ], + "score": 1.0, + "content": "th position", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 559, + 507, + 574 + ], + "spans": [ + { + "bbox": [ + 86, + 561, + 100, + 572 + ], + "score": 1.0, + "content": "165", + "type": "text" + }, + { + "bbox": [ + 105, + 559, + 406, + 574 + ], + "score": 1.0, + "content": "and zeros otherwise. This can be further rewritten into a recursive form:", + "type": "text" + }, + { + "bbox": [ + 406, + 561, + 491, + 573 + ], + "score": 0.91, + "content": "n _ { i , t } = n _ { i , t - 1 } + \\mathbf { e } _ { z _ { i , t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 559, + 507, + 574 + ], + "score": 1.0, + "content": "or,", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 571, + 507, + 586 + ], + "spans": [ + { + "bbox": [ + 86, + 574, + 100, + 583 + ], + "score": 1.0, + "content": "166", + "type": "text" + }, + { + "bbox": [ + 105, + 571, + 249, + 586 + ], + "score": 1.0, + "content": "using the Robbins-Monro step-size,", + "type": "text" + }, + { + "bbox": [ + 249, + 572, + 406, + 585 + ], + "score": 0.9, + "content": "n _ { i , t } = ( { \\bf 1 } - { \\bf e } _ { z _ { i , t } } \\gamma _ { t } ) \\odot n _ { i , t - 1 } + \\gamma _ { t } { \\bf e } _ { z _ { i , t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 571, + 507, + 586 + ], + "score": 1.0, + "content": ", where 1 is the vector of", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 86, + 584, + 100, + 595 + ], + "score": 1.0, + "content": "167", + "type": "text" + }, + { + "bbox": [ + 105, + 582, + 147, + 596 + ], + "score": 1.0, + "content": "ones, and", + "type": "text" + }, + { + "bbox": [ + 147, + 584, + 157, + 593 + ], + "score": 0.83, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "is the Hadamard product. We refer to this case simply as “table with forgetting” (TF)", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 593, + 431, + 607 + ], + "spans": [ + { + "bbox": [ + 86, + 596, + 100, + 605 + ], + "score": 1.0, + "content": "168", + "type": "text" + }, + { + "bbox": [ + 104, + 593, + 202, + 607 + ], + "score": 1.0, + "content": "due to that it represents", + "type": "text" + }, + { + "bbox": [ + 202, + 594, + 234, + 604 + ], + "score": 0.91, + "content": "N \\times K", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 593, + 334, + 607 + ], + "score": 1.0, + "content": "table in the memory and", + "type": "text" + }, + { + "bbox": [ + 334, + 595, + 344, + 605 + ], + "score": 0.86, + "content": "\\gamma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 593, + 431, + 607 + ], + "score": 1.0, + "content": "is a forgetting factor.", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + } + ], + "index": 23, + "bbox_fs": [ + 85, + 355, + 506, + 413 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 416, + 506, + 450 + ], + "lines": [], + "index": 27, + "bbox_fs": [ + 86, + 415, + 507, + 451 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 86, + 454, + 506, + 605 + ], + "lines": [], + "index": 34.5, + "bbox_fs": [ + 86, + 454, + 507, + 607 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 89, + 620, + 194, + 633 + ], + "lines": [ + { + "bbox": [ + 84, + 619, + 196, + 635 + ], + "spans": [ + { + "bbox": [ + 84, + 619, + 196, + 635 + ], + "score": 1.0, + "content": "169 5 Related work", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "index", + "bbox": [ + 95, + 645, + 504, + 722 + ], + "lines": [ + { + "bbox": [ + 102, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 102, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "Stochastic approximation expectation-maximization. The application of SA to prevent the evaluation", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 95, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 95, + 661, + 99, + 665 + ], + "score": 1.0, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 105, + 657, + 130, + 668 + ], + "score": 1.0, + "content": "of all", + "type": "text" + }, + { + "bbox": [ + 130, + 657, + 140, + 666 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "components in mixture models has been overlooked for a long time. The reason is that the", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 93, + 671, + 99, + 677 + ], + "score": 1.0, + "content": "72", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "original motivation to combine the EM algorithm with SA is to address the analytical intractability", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 93, + 681, + 99, + 688 + ], + "score": 1.0, + "content": "73", + "type": "text" + }, + { + "bbox": [ + 105, + 678, + 221, + 690 + ], + "score": 1.0, + "content": "of the expected value under", + "type": "text" + }, + { + "bbox": [ + 221, + 680, + 252, + 690 + ], + "score": 0.95, + "content": "p _ { \\theta } ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "in (4), which is, however, almost always tractable for mixture", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 93, + 692, + 99, + 698 + ], + "score": 1.0, + "content": "74", + "type": "text" + }, + { + "bbox": [ + 104, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "models. The intractability issue is addressed by either the Monte Carlo SAEM (MCSAEM) [12]", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 93, + 704, + 99, + 709 + ], + "score": 1.0, + "content": "75", + "type": "text" + }, + { + "bbox": [ + 104, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "or the Markov chain Monte Carlo SAEM (MCMCSAEM) [31]. Applying the former approach to", + "type": "text" + } + ], + "index": 47, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 93, + 714, + 99, + 721 + ], + "score": 1.0, + "content": "76", + "type": "text" + }, + { + "bbox": [ + 104, + 710, + 321, + 723 + ], + "score": 1.0, + "content": "mixture models would be inefficient, since it evaluates", + "type": "text" + }, + { + "bbox": [ + 321, + 713, + 331, + 721 + ], + "score": 0.86, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 710, + 406, + 723 + ], + "score": 1.0, + "content": "joint distributions,", + "type": "text" + }, + { + "bbox": [ + 406, + 711, + 439, + 723 + ], + "score": 0.93, + "content": "p _ { \\theta } ( z , x )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 710, + 505, + 723 + ], + "score": 1.0, + "content": ", before drawing", + "type": "text" + } + ], + "index": 48, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 282, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 86, + 284, + 100, + 294 + ], + "score": 1.0, + "content": "177", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 107, + 284, + 118, + 293 + ], + "score": 0.69, + "content": "M", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 119, + 282, + 174, + 296 + ], + "score": 1.0, + "content": "samples from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 174, + 283, + 206, + 295 + ], + "score": 0.92, + "content": "p _ { \\theta } ( z | x )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 207, + 282, + 505, + 296 + ], + "score": 1.0, + "content": ". Therefore, this method reduces only the computational cost of updating the", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 86, + 296, + 99, + 305 + ], + "score": 1.0, + "content": "178", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 294, + 370, + 306 + ], + "score": 1.0, + "content": "sufficient statistics. This is addressed by the latter approach, where", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 371, + 294, + 405, + 304 + ], + "score": 0.91, + "content": "M < K", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 406, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "samples from a proposal", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 304, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 87, + 307, + 99, + 316 + ], + "score": 1.0, + "content": "179", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 304, + 157, + 318 + ], + "score": 1.0, + "content": "distribution,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 157, + 305, + 184, + 317 + ], + "score": 0.91, + "content": "q ( z | x )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 185, + 304, + 265, + 318 + ], + "score": 1.0, + "content": ", is used to calculate", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 266, + 305, + 299, + 317 + ], + "score": 0.93, + "content": "p _ { \\theta } ( z , x )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 300, + 304, + 505, + 318 + ], + "score": 1.0, + "content": "and also the sufficient statistics. However, all these", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 86, + 317, + 100, + 328 + ], + "score": 1.0, + "content": "180", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "methods process all data at every iteration, providing only a limited advantage over the conventional", + "type": "text", + "cross_page": true + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 326, + 488, + 340 + ], + "spans": [ + { + "bbox": [ + 86, + 328, + 99, + 339 + ], + "score": 1.0, + "content": "181", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 326, + 488, + 340 + ], + "score": 1.0, + "content": "EM algorithm. Minibatch versions of these techniques have recently been proposed [27, 32, 1].", + "type": "text", + "cross_page": true + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 86, + 345, + 99, + 354 + ], + "score": 1.0, + "content": "182", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 343, + 271, + 357 + ], + "score": 1.0, + "content": "All the above methods commonly assume", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 271, + 343, + 304, + 355 + ], + "score": 0.92, + "content": "p _ { \\theta } ( z , x )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 305, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "belonging to the exponential family. 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This provides", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 354, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 86, + 356, + 100, + 365 + ], + "score": 1.0, + "content": "183", + "type": "text" + }, + { + "bbox": [ + 105, + 354, + 506, + 366 + ], + "score": 1.0, + "content": "a convenient, but limiting, property which allows (6) to be computed under a closed-form solution.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 363, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 86, + 366, + 100, + 376 + ], + "score": 1.0, + "content": "184", + "type": "text" + }, + { + "bbox": [ + 105, + 363, + 471, + 379 + ], + "score": 1.0, + "content": "The main contribution of our work is to release this restrictive assumption by admitting that", + "type": "text" + }, + { + "bbox": [ + 471, + 365, + 505, + 377 + ], + "score": 0.93, + "content": "p _ { \\theta } ( z , x )", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 376, + 407, + 389 + ], + "spans": [ + { + "bbox": [ + 86, + 378, + 99, + 387 + ], + "score": 1.0, + "content": "185", + "type": "text" + }, + { + "bbox": [ + 105, + 376, + 145, + 389 + ], + "score": 1.0, + "content": "(and thus", + "type": "text" + }, + { + "bbox": [ + 146, + 376, + 155, + 387 + ], + "score": 0.76, + "content": "\\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 376, + 407, + 389 + ], + "score": 1.0, + "content": ") is given by possibly complex and intractable transformations.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 86, + 392, + 506, + 534 + ], + "lines": [ + { + "bbox": [ + 86, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 86, + 394, + 100, + 403 + ], + "score": 1.0, + "content": "186", + "type": "text" + }, + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "Sparse and truncated variational techniques. 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We therefore use a more objective measure of the clustering complexity", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 85, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 85, + 353, + 100, + 362 + ], + "score": 1.0, + "content": "215", + "type": "text" + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "defined by the total probability of misclassification [41], which allows to generate data with a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 85, + 362, + 323, + 374 + ], + "spans": [ + { + "bbox": [ + 85, + 363, + 100, + 373 + ], + "score": 1.0, + "content": "216", + "type": "text" + }, + { + "bbox": [ + 105, + 362, + 311, + 374 + ], + "score": 1.0, + "content": "user-defined degree of maximum pairwise overlap,", + "type": "text" + }, + { + "bbox": [ + 312, + 364, + 319, + 371 + ], + "score": 0.7, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 362, + 323, + 374 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 86, + 378, + 505, + 445 + ], + "lines": [ + { + "bbox": [ + 86, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 86, + 380, + 100, + 390 + ], + "score": 1.0, + "content": "217", + "type": "text" + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "Experiment settings: We generate the parameters of (1), and the corresponding dataset, uniquely for a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 86, + 391, + 99, + 400 + ], + "score": 1.0, + "content": "218", + "type": "text" + }, + { + "bbox": [ + 105, + 388, + 173, + 402 + ], + "score": 1.0, + "content": "given quadruple", + "type": "text" + }, + { + "bbox": [ + 173, + 389, + 227, + 401 + ], + "score": 0.92, + "content": "( D , K , N , \\omega )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 388, + 506, + 402 + ], + "score": 1.0, + "content": ". Therefore, the parameters of the generative model are known and we", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 85, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 85, + 401, + 100, + 412 + ], + "score": 1.0, + "content": "219", + "type": "text" + }, + { + "bbox": [ + 106, + 400, + 388, + 412 + ], + "score": 1.0, + "content": "can measure and display the convergence of the training log-likelihood,", + "type": "text" + }, + { + "bbox": [ + 389, + 399, + 412, + 411 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 400, + 505, + 412 + ], + "score": 1.0, + "content": ", compared to the exact", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 86, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 86, + 412, + 100, + 422 + ], + "score": 1.0, + "content": "220", + "type": "text" + }, + { + "bbox": [ + 106, + 410, + 168, + 423 + ], + "score": 1.0, + "content": "log-likelihood,", + "type": "text" + }, + { + "bbox": [ + 168, + 410, + 188, + 423 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } ) \\dot { }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 410, + 207, + 423 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 207, + 411, + 266, + 423 + ], + "score": 0.93, + "content": "t = ( 1 , \\ldots , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 410, + 505, + 423 + ], + "score": 1.0, + "content": ". We are further interested in the absolute error between the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 86, + 424, + 99, + 433 + ], + "score": 1.0, + "content": "221", + "type": "text" + }, + { + "bbox": [ + 105, + 421, + 313, + 434 + ], + "score": 1.0, + "content": "training log-likelihood at the iteration of reaching", + "type": "text" + }, + { + "bbox": [ + 313, + 422, + 333, + 432 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 421, + 430, + 434 + ], + "score": 1.0, + "content": "of its maximum value,", + "type": "text" + }, + { + "bbox": [ + 430, + 423, + 444, + 433 + ], + "score": 0.87, + "content": "t _ { 9 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 421, + 506, + 434 + ], + "score": 1.0, + "content": ", and the exact", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 86, + 432, + 282, + 446 + ], + "spans": [ + { + "bbox": [ + 86, + 434, + 99, + 444 + ], + "score": 1.0, + "content": "222", + "type": "text" + }, + { + "bbox": [ + 105, + 432, + 182, + 446 + ], + "score": 1.0, + "content": "log-likelihood, i.e.", + "type": "text" + }, + { + "bbox": [ + 183, + 432, + 278, + 445 + ], + "score": 0.91, + "content": "\\mathrm { A E } = | \\mathcal { L } ( \\theta _ { t _ { 9 5 } } ) - \\mathcal { L } ( \\theta ) |", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 432, + 282, + 446 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 96, + 449, + 506, + 482 + ], + "lines": [ + { + "bbox": [ + 93, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 93, + 449, + 334, + 461 + ], + "score": 1.0, + "content": "23 We also measure the computational time until reaching", + "type": "text" + }, + { + "bbox": [ + 335, + 450, + 348, + 460 + ], + "score": 0.87, + "content": "t _ { 9 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 449, + 413, + 461 + ], + "score": 1.0, + "content": ". We have used", + "type": "text" + }, + { + "bbox": [ + 413, + 449, + 433, + 460 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "of the maximum", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 92, + 459, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 92, + 459, + 507, + 473 + ], + "score": 1.0, + "content": "24 value instead of the maximum value to prevent cases, where the model oscillate around target value,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 92, + 470, + 468, + 483 + ], + "spans": [ + { + "bbox": [ + 92, + 470, + 468, + 483 + ], + "score": 1.0, + "content": "25 making the estimate of convergence time very noisy (for example MCSAEM in Figure 1).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 86, + 487, + 506, + 651 + ], + "lines": [ + { + "bbox": [ + 86, + 487, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 86, + 489, + 99, + 498 + ], + "score": 1.0, + "content": "226", + "type": "text" + }, + { + "bbox": [ + 104, + 487, + 506, + 499 + ], + "score": 1.0, + "content": "Algorithms: The GMMs belong to the exponential family of probability distributions. This allows us", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 86, + 500, + 99, + 509 + ], + "score": 1.0, + "content": "227", + "type": "text" + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "to find a closed-form, recursive, solution of (6), relying on a Robbins-Monro type of the step-size", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 507, + 507, + 524 + ], + "spans": [ + { + "bbox": [ + 86, + 511, + 99, + 520 + ], + "score": 1.0, + "content": "228", + "type": "text" + }, + { + "bbox": [ + 104, + 507, + 149, + 524 + ], + "score": 1.0, + "content": "sequence,", + "type": "text" + }, + { + "bbox": [ + 149, + 508, + 180, + 521 + ], + "score": 0.92, + "content": "( \\gamma _ { t } ) _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 507, + 507, + 524 + ], + "score": 1.0, + "content": ", [7, 44]. In this setting, we compare our MHSAEM algorithm with a number of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 85, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 85, + 521, + 100, + 532 + ], + "score": 1.0, + "content": "229", + "type": "text" + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "related methods in Table 1. Note we use the acronyms U and TF to specify the proposal distribution of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 86, + 533, + 99, + 542 + ], + "score": 1.0, + "content": "230", + "type": "text" + }, + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "the MHSAEM algorithm (Section 4.3). However, we do not use the O-proposal, since the MHSAEM-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 86, + 544, + 99, + 553 + ], + "score": 1.0, + "content": "231", + "type": "text" + }, + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "O algorithm is equivalent to the MCSAEM algorithm. All the SA-variants in Table 1 use a minibatch", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 86, + 555, + 99, + 564 + ], + "score": 1.0, + "content": "232", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 135, + 565 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 135, + 553, + 144, + 563 + ], + "score": 0.71, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 552, + 506, + 565 + ], + "score": 1.0, + "content": ". The key quantity to reduce the number of evaluated components and/or sufficient statistics", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 85, + 564, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 85, + 565, + 100, + 575 + ], + "score": 1.0, + "content": "233", + "type": "text" + }, + { + "bbox": [ + 105, + 564, + 457, + 576 + ], + "score": 1.0, + "content": "in the SSAEM, TSAEM, MCSAEM and MHSAEM algorithms is collectively denoted by", + "type": "text" + }, + { + "bbox": [ + 457, + 564, + 469, + 574 + ], + "score": 0.69, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 564, + 506, + 576 + ], + "score": 1.0, + "content": "(Section", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 85, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 85, + 576, + 100, + 586 + ], + "score": 1.0, + "content": "234", + "type": "text" + }, + { + "bbox": [ + 105, + 574, + 228, + 587 + ], + "score": 1.0, + "content": "5). Note that we always keep", + "type": "text" + }, + { + "bbox": [ + 229, + 574, + 266, + 584 + ], + "score": 0.91, + "content": "M = \\bar { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "in the TSAEM algorithm (see Figure 1 and 2 for concrete", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 86, + 587, + 100, + 596 + ], + "score": 1.0, + "content": "235", + "type": "text" + }, + { + "bbox": [ + 104, + 584, + 277, + 599 + ], + "score": 1.0, + "content": "numbers). We use the step-size given by", + "type": "text" + }, + { + "bbox": [ + 278, + 586, + 308, + 597 + ], + "score": 0.91, + "content": "\\gamma _ { t } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 584, + 325, + 599 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 326, + 586, + 383, + 596 + ], + "score": 0.9, + "content": "t = 1 , \\ldots , 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 584, + 402, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 402, + 585, + 446, + 596 + ], + "score": 0.9, + "content": "\\gamma _ { t } = 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 584, + 506, + 599 + ], + "score": 1.0, + "content": "otherwise. In", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 85, + 595, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 85, + 598, + 100, + 608 + ], + "score": 1.0, + "content": "236", + "type": "text" + }, + { + "bbox": [ + 105, + 595, + 506, + 610 + ], + "score": 1.0, + "content": "this section, to counteract the issue of attaining poor local optima, we equip all algorithms with the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 605, + 507, + 622 + ], + "spans": [ + { + "bbox": [ + 86, + 609, + 99, + 618 + ], + "score": 1.0, + "content": "237", + "type": "text" + }, + { + "bbox": [ + 104, + 605, + 203, + 622 + ], + "score": 1.0, + "content": "anti-annealing schedule", + "type": "text" + }, + { + "bbox": [ + 203, + 606, + 234, + 619 + ], + "score": 0.93, + "content": "( \\beta _ { t } ) _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 605, + 290, + 622 + ], + "score": 1.0, + "content": ", starting with", + "type": "text" + }, + { + "bbox": [ + 290, + 608, + 327, + 618 + ], + "score": 0.91, + "content": "\\beta _ { 1 } = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 605, + 367, + 622 + ], + "score": 1.0, + "content": ", reaching", + "type": "text" + }, + { + "bbox": [ + 368, + 608, + 419, + 620 + ], + "score": 0.92, + "content": "\\beta _ { 2 / 3 T } = 1 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 605, + 507, + 622 + ], + "score": 1.0, + "content": ", and decreasing back", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 86, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 86, + 620, + 99, + 629 + ], + "score": 1.0, + "content": "238", + "type": "text" + }, + { + "bbox": [ + 105, + 617, + 117, + 631 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 618, + 156, + 630 + ], + "score": 0.91, + "content": "\\beta _ { T } = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 617, + 351, + 631 + ], + "score": 1.0, + "content": ", see [43] for details. The initial estimates of: (i)", + "type": "text" + }, + { + "bbox": [ + 351, + 620, + 363, + 629 + ], + "score": 0.85, + "content": "\\mu _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "are uniformly drawn from the unit", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 85, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 85, + 630, + 100, + 641 + ], + "score": 1.0, + "content": "239", + "type": "text" + }, + { + "bbox": [ + 105, + 628, + 171, + 642 + ], + "score": 1.0, + "content": "hyper-cube, (ii)", + "type": "text" + }, + { + "bbox": [ + 171, + 630, + 184, + 640 + ], + "score": 0.89, + "content": "\\Sigma _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 628, + 353, + 642 + ], + "score": 1.0, + "content": "are fixed to unit diagonal matrix, and (iii)", + "type": "text" + }, + { + "bbox": [ + 353, + 631, + 365, + 640 + ], + "score": 0.87, + "content": "\\pi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "are uniformly drawn from the unit", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 640, + 255, + 652 + ], + "spans": [ + { + "bbox": [ + 86, + 642, + 100, + 651 + ], + "score": 1.0, + "content": "240", + "type": "text" + }, + { + "bbox": [ + 105, + 640, + 255, + 652 + ], + "score": 1.0, + "content": "interval (followed by normalization).", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 86, + 656, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 86, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 86, + 658, + 99, + 668 + ], + "score": 1.0, + "content": "241", + "type": "text" + }, + { + "bbox": [ + 104, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "Results: Figure 1 shows that the EM [13] and SAEM [44] algorithms take the longest time to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 86, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 86, + 669, + 99, + 678 + ], + "score": 1.0, + "content": "242", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "converge, attaining a poor local optima. On the other hand, the MCSAEM [1] and MHSAEM (U", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 86, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 86, + 680, + 100, + 690 + ], + "score": 1.0, + "content": "243", + "type": "text" + }, + { + "bbox": [ + 105, + 678, + 217, + 691 + ], + "score": 1.0, + "content": "and TF) algorithms achieve", + "type": "text" + }, + { + "bbox": [ + 218, + 678, + 248, + 690 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 678, + 347, + 691 + ], + "score": 1.0, + "content": "closest to the likelihood", + "type": "text" + }, + { + "bbox": [ + 347, + 678, + 367, + 690 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "of the true model. Moreover, both", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 86, + 688, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 86, + 691, + 100, + 700 + ], + "score": 1.0, + "content": "244", + "type": "text" + }, + { + "bbox": [ + 105, + 688, + 506, + 701 + ], + "score": 1.0, + "content": "MHSAEM algorithms reach this value in the shortest time compared to all the other methods. The", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 86, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 86, + 702, + 100, + 711 + ], + "score": 1.0, + "content": "245", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "SSAEM [24] and TSAEM [18] algorithms are comparable in terms of the computational time, but", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 86, + 711, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 99, + 722 + ], + "score": 1.0, + "content": "246", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 224, + 724 + ], + "score": 1.0, + "content": "they both provide the lowest", + "type": "text" + }, + { + "bbox": [ + 224, + 711, + 255, + 723 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 711, + 506, + 724 + ], + "score": 1.0, + "content": ". 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All", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 219, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 270, + 232 + ], + "score": 1.0, + "content": "experiments use the following settings:", + "type": "text" + }, + { + "bbox": [ + 270, + 219, + 502, + 231 + ], + "score": 0.87, + "content": "( D , K , \\dot { N } , \\omega , B , M , T ) = \\bar { ( } 1 0 , 1 0 0 , 1 0 k , 0 . 1 , 2 0 0 , 2 , 2 0 k )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 219, + 507, + 232 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 229, + 507, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 245, + 244 + ], + "score": 1.0, + "content": "where the number of components,", + "type": "text" + }, + { + "bbox": [ + 246, + 231, + 256, + 241 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 229, + 342, + 244 + ], + "score": 1.0, + "content": ", (left), the batchsize,", + "type": "text" + }, + { + "bbox": [ + 343, + 231, + 352, + 241 + ], + "score": 0.76, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 229, + 507, + 244 + ], + "score": 1.0, + "content": ", (middle) and the number of samples,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 240, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 107, + 241, + 118, + 252 + ], + "score": 0.68, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 240, + 312, + 254 + ], + "score": 1.0, + "content": ", (right) change for different values denoted by", + "type": "text" + }, + { + "bbox": [ + 312, + 242, + 351, + 252 + ], + "score": 0.86, + "content": "( + , \\sqsupset , \\circ , \\pmb { \\triangle } )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 240, + 506, + 254 + ], + "score": 1.0, + "content": ". At each of these points (marks), we", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 252, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 316, + 265 + ], + "score": 1.0, + "content": "perform an experiment as illustrated in Figure 1, find", + "type": "text" + }, + { + "bbox": [ + 316, + 253, + 347, + 264 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 252, + 506, + 265 + ], + "score": 1.0, + "content": "to compute the AE, and record the time", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 263, + 377, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 176, + 276 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + }, + { + "bbox": [ + 176, + 264, + 189, + 275 + ], + "score": 0.85, + "content": "t _ { 9 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 263, + 377, + 276 + ], + "score": 1.0, + "content": ". The results are averaged over five repetitions.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "index", + "bbox": [ + 86, + 307, + 505, + 373 + ], + "lines": [ + { + "bbox": [ + 85, + 306, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 85, + 309, + 99, + 319 + ], + "score": 1.0, + "content": "211", + "type": "text" + }, + { + "bbox": [ + 104, + 306, + 505, + 320 + ], + "score": 1.0, + "content": "evaluation of performance of learning algorithms [43]. Traditional techniques usually define overlap", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 85, + 320, + 100, + 330 + ], + "score": 1.0, + "content": "212", + "type": "text" + }, + { + "bbox": [ + 105, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "(or separation) of components only in terms of their mean vectors and maximum eigenvalues of the", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 85, + 331, + 100, + 341 + ], + "score": 1.0, + "content": "213", + "type": "text" + }, + { + "bbox": [ + 104, + 329, + 505, + 341 + ], + "score": 1.0, + "content": "covariance matrices, not accounting for their rotation and mixing weights (see [36] for a detailed", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 85, + 342, + 100, + 351 + ], + "score": 1.0, + "content": "214", + "type": "text" + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "treatment of the problem). We therefore use a more objective measure of the clustering complexity", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 85, + 353, + 100, + 362 + ], + "score": 1.0, + "content": "215", + "type": "text" + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "defined by the total probability of misclassification [41], which allows to generate data with a", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 362, + 323, + 374 + ], + "spans": [ + { + "bbox": [ + 85, + 363, + 100, + 373 + ], + "score": 1.0, + "content": "216", + "type": "text" + }, + { + "bbox": [ + 105, + 362, + 311, + 374 + ], + "score": 1.0, + "content": "user-defined degree of maximum pairwise overlap,", + "type": "text" + }, + { + "bbox": [ + 312, + 364, + 319, + 371 + ], + "score": 0.7, + "content": "\\omega", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 362, + 323, + 374 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 86, + 380, + 100, + 390 + ], + "score": 1.0, + "content": "217", + "type": "text" + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "Experiment settings: We generate the parameters of (1), and the corresponding dataset, uniquely for a", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 86, + 391, + 99, + 400 + ], + "score": 1.0, + "content": "218", + "type": "text" + }, + { + "bbox": [ + 105, + 388, + 173, + 402 + ], + "score": 1.0, + "content": "given quadruple", + "type": "text" + }, + { + "bbox": [ + 173, + 389, + 227, + 401 + ], + "score": 0.92, + "content": "( D , K , N , \\omega )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 388, + 506, + 402 + ], + "score": 1.0, + "content": ". Therefore, the parameters of the generative model are known and we", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 85, + 401, + 100, + 412 + ], + "score": 1.0, + "content": "219", + "type": "text" + }, + { + "bbox": [ + 106, + 400, + 388, + 412 + ], + "score": 1.0, + "content": "can measure and display the convergence of the training log-likelihood,", + "type": "text" + }, + { + "bbox": [ + 389, + 399, + 412, + 411 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 400, + 505, + 412 + ], + "score": 1.0, + "content": ", compared to the exact", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 86, + 412, + 100, + 422 + ], + "score": 1.0, + "content": "220", + "type": "text" + }, + { + "bbox": [ + 106, + 410, + 168, + 423 + ], + "score": 1.0, + "content": "log-likelihood,", + "type": "text" + }, + { + "bbox": [ + 168, + 410, + 188, + 423 + ], + "score": 0.91, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } ) \\dot { }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 410, + 207, + 423 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 207, + 411, + 266, + 423 + ], + "score": 0.93, + "content": "t = ( 1 , \\ldots , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 410, + 505, + 423 + ], + "score": 1.0, + "content": ". We are further interested in the absolute error between the", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 86, + 424, + 99, + 433 + ], + "score": 1.0, + "content": "221", + "type": "text" + }, + { + "bbox": [ + 105, + 421, + 313, + 434 + ], + "score": 1.0, + "content": "training log-likelihood at the iteration of reaching", + "type": "text" + }, + { + "bbox": [ + 313, + 422, + 333, + 432 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 421, + 430, + 434 + ], + "score": 1.0, + "content": "of its maximum value,", + "type": "text" + }, + { + "bbox": [ + 430, + 423, + 444, + 433 + ], + "score": 0.87, + "content": "t _ { 9 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 421, + 506, + 434 + ], + "score": 1.0, + "content": ", and the exact", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 432, + 282, + 446 + ], + "spans": [ + { + "bbox": [ + 86, + 434, + 99, + 444 + ], + "score": 1.0, + "content": "222", + "type": "text" + }, + { + "bbox": [ + 105, + 432, + 182, + 446 + ], + "score": 1.0, + "content": "log-likelihood, i.e.", + "type": "text" + }, + { + "bbox": [ + 183, + 432, + 278, + 445 + ], + "score": 0.91, + "content": "\\mathrm { A E } = | \\mathcal { L } ( \\theta _ { t _ { 9 5 } } ) - \\mathcal { L } ( \\theta ) |", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 432, + 282, + 446 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 93, + 449, + 334, + 461 + ], + "score": 1.0, + "content": "23 We also measure the computational time until reaching", + "type": "text" + }, + { + "bbox": [ + 335, + 450, + 348, + 460 + ], + "score": 0.87, + "content": "t _ { 9 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 449, + 413, + 461 + ], + "score": 1.0, + "content": ". We have used", + "type": "text" + }, + { + "bbox": [ + 413, + 449, + 433, + 460 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 449, + 505, + 461 + ], + "score": 1.0, + "content": "of the maximum", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 459, + 507, + 473 + ], + "spans": [ + { + "bbox": [ + 92, + 459, + 507, + 473 + ], + "score": 1.0, + "content": "24 value instead of the maximum value to prevent cases, where the model oscillate around target value,", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 470, + 468, + 483 + ], + "spans": [ + { + "bbox": [ + 92, + 470, + 468, + 483 + ], + "score": 1.0, + "content": "25 making the estimate of convergence time very noisy (for example MCSAEM in Figure 1).", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 487, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 86, + 489, + 99, + 498 + ], + "score": 1.0, + "content": "226", + "type": "text" + }, + { + "bbox": [ + 104, + 487, + 506, + 499 + ], + "score": 1.0, + "content": "Algorithms: The GMMs belong to the exponential family of probability distributions. This allows us", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 86, + 500, + 99, + 509 + ], + "score": 1.0, + "content": "227", + "type": "text" + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "to find a closed-form, recursive, solution of (6), relying on a Robbins-Monro type of the step-size", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 507, + 507, + 524 + ], + "spans": [ + { + "bbox": [ + 86, + 511, + 99, + 520 + ], + "score": 1.0, + "content": "228", + "type": "text" + }, + { + "bbox": [ + 104, + 507, + 149, + 524 + ], + "score": 1.0, + "content": "sequence,", + "type": "text" + }, + { + "bbox": [ + 149, + 508, + 180, + 521 + ], + "score": 0.92, + "content": "( \\gamma _ { t } ) _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 507, + 507, + 524 + ], + "score": 1.0, + "content": ", [7, 44]. In this setting, we compare our MHSAEM algorithm with a number of", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 519, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 85, + 521, + 100, + 532 + ], + "score": 1.0, + "content": "229", + "type": "text" + }, + { + "bbox": [ + 105, + 519, + 506, + 533 + ], + "score": 1.0, + "content": "related methods in Table 1. Note we use the acronyms U and TF to specify the proposal distribution of", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 86, + 533, + 99, + 542 + ], + "score": 1.0, + "content": "230", + "type": "text" + }, + { + "bbox": [ + 106, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "the MHSAEM algorithm (Section 4.3). However, we do not use the O-proposal, since the MHSAEM-", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 86, + 544, + 99, + 553 + ], + "score": 1.0, + "content": "231", + "type": "text" + }, + { + "bbox": [ + 106, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "O algorithm is equivalent to the MCSAEM algorithm. All the SA-variants in Table 1 use a minibatch", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 86, + 555, + 99, + 564 + ], + "score": 1.0, + "content": "232", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 135, + 565 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 135, + 553, + 144, + 563 + ], + "score": 0.71, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 552, + 506, + 565 + ], + "score": 1.0, + "content": ". The key quantity to reduce the number of evaluated components and/or sufficient statistics", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 564, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 85, + 565, + 100, + 575 + ], + "score": 1.0, + "content": "233", + "type": "text" + }, + { + "bbox": [ + 105, + 564, + 457, + 576 + ], + "score": 1.0, + "content": "in the SSAEM, TSAEM, MCSAEM and MHSAEM algorithms is collectively denoted by", + "type": "text" + }, + { + "bbox": [ + 457, + 564, + 469, + 574 + ], + "score": 0.69, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 564, + 506, + 576 + ], + "score": 1.0, + "content": "(Section", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 85, + 576, + 100, + 586 + ], + "score": 1.0, + "content": "234", + "type": "text" + }, + { + "bbox": [ + 105, + 574, + 228, + 587 + ], + "score": 1.0, + "content": "5). Note that we always keep", + "type": "text" + }, + { + "bbox": [ + 229, + 574, + 266, + 584 + ], + "score": 0.91, + "content": "M = \\bar { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "in the TSAEM algorithm (see Figure 1 and 2 for concrete", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 584, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 86, + 587, + 100, + 596 + ], + "score": 1.0, + "content": "235", + "type": "text" + }, + { + "bbox": [ + 104, + 584, + 277, + 599 + ], + "score": 1.0, + "content": "numbers). We use the step-size given by", + "type": "text" + }, + { + "bbox": [ + 278, + 586, + 308, + 597 + ], + "score": 0.91, + "content": "\\gamma _ { t } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 584, + 325, + 599 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 326, + 586, + 383, + 596 + ], + "score": 0.9, + "content": "t = 1 , \\ldots , 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 584, + 402, + 599 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 402, + 585, + 446, + 596 + ], + "score": 0.9, + "content": "\\gamma _ { t } = 0 . 0 5", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 584, + 506, + 599 + ], + "score": 1.0, + "content": "otherwise. In", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 595, + 506, + 610 + ], + "spans": [ + { + "bbox": [ + 85, + 598, + 100, + 608 + ], + "score": 1.0, + "content": "236", + "type": "text" + }, + { + "bbox": [ + 105, + 595, + 506, + 610 + ], + "score": 1.0, + "content": "this section, to counteract the issue of attaining poor local optima, we equip all algorithms with the", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 605, + 507, + 622 + ], + "spans": [ + { + "bbox": [ + 86, + 609, + 99, + 618 + ], + "score": 1.0, + "content": "237", + "type": "text" + }, + { + "bbox": [ + 104, + 605, + 203, + 622 + ], + "score": 1.0, + "content": "anti-annealing schedule", + "type": "text" + }, + { + "bbox": [ + 203, + 606, + 234, + 619 + ], + "score": 0.93, + "content": "( \\beta _ { t } ) _ { t = 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 605, + 290, + 622 + ], + "score": 1.0, + "content": ", starting with", + "type": "text" + }, + { + "bbox": [ + 290, + 608, + 327, + 618 + ], + "score": 0.91, + "content": "\\beta _ { 1 } = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 605, + 367, + 622 + ], + "score": 1.0, + "content": ", reaching", + "type": "text" + }, + { + "bbox": [ + 368, + 608, + 419, + 620 + ], + "score": 0.92, + "content": "\\beta _ { 2 / 3 T } = 1 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 605, + 507, + 622 + ], + "score": 1.0, + "content": ", and decreasing back", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 86, + 620, + 99, + 629 + ], + "score": 1.0, + "content": "238", + "type": "text" + }, + { + "bbox": [ + 105, + 617, + 117, + 631 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 618, + 156, + 630 + ], + "score": 0.91, + "content": "\\beta _ { T } = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 617, + 351, + 631 + ], + "score": 1.0, + "content": ", see [43] for details. The initial estimates of: (i)", + "type": "text" + }, + { + "bbox": [ + 351, + 620, + 363, + 629 + ], + "score": 0.85, + "content": "\\mu _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "are uniformly drawn from the unit", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 85, + 630, + 100, + 641 + ], + "score": 1.0, + "content": "239", + "type": "text" + }, + { + "bbox": [ + 105, + 628, + 171, + 642 + ], + "score": 1.0, + "content": "hyper-cube, (ii)", + "type": "text" + }, + { + "bbox": [ + 171, + 630, + 184, + 640 + ], + "score": 0.89, + "content": "\\Sigma _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 628, + 353, + 642 + ], + "score": 1.0, + "content": "are fixed to unit diagonal matrix, and (iii)", + "type": "text" + }, + { + "bbox": [ + 353, + 631, + 365, + 640 + ], + "score": 0.87, + "content": "\\pi _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "are uniformly drawn from the unit", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 640, + 255, + 652 + ], + "spans": [ + { + "bbox": [ + 86, + 642, + 100, + 651 + ], + "score": 1.0, + "content": "240", + "type": "text" + }, + { + "bbox": [ + 105, + 640, + 255, + 652 + ], + "score": 1.0, + "content": "interval (followed by normalization).", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 86, + 658, + 99, + 668 + ], + "score": 1.0, + "content": "241", + "type": "text" + }, + { + "bbox": [ + 104, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "Results: Figure 1 shows that the EM [13] and SAEM [44] algorithms take the longest time to", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 667, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 86, + 669, + 99, + 678 + ], + "score": 1.0, + "content": "242", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 505, + 679 + ], + "score": 1.0, + "content": "converge, attaining a poor local optima. 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They can", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 220, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 234 + ], + "score": 1.0, + "content": "be interpreted as a mixture of trees [60], where each tree corresponds to a component. Therefore,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 244 + ], + "score": 1.0, + "content": "they can be cast into the form of (1), but the number of components grows exponentially with their", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "depth. In this section, we use recently proposed SPTNs which introduce additional transformation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "nodes to provide better expressiveness than the SPNs (SPTNs effectively generalize SPNs and flow", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 271, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 271, + 278 + ], + "score": 1.0, + "content": "models into one large family of models).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 105, + 281, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "Experimental settings: We use 19 real datasets from the UCI database [16, 37, 35, 54], preprocessed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 425, + 305 + ], + "score": 1.0, + "content": "in the same way as in [46]. For each experiment, we randomly split the data into", + "type": "text" + }, + { + "bbox": [ + 425, + 293, + 444, + 303 + ], + "score": 0.85, + "content": "64 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 292, + 448, + 305 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 448, + 293, + 468, + 303 + ], + "score": 0.82, + "content": "16 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 292, + 485, + 305 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 485, + 292, + 505, + 303 + ], + "score": 0.87, + "content": "20 \\%", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "for training, validation and testing, respectively. We calculate the average log-likelihood on the test", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 313, + 479, + 328 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 265, + 328 + ], + "score": 1.0, + "content": "set and measure again the time to reach", + "type": "text" + }, + { + "bbox": [ + 266, + 314, + 285, + 325 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 313, + 444, + 328 + ], + "score": 1.0, + "content": "of the maximal training log-likelihood,", + "type": "text" + }, + { + "bbox": [ + 444, + 314, + 475, + 326 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 313, + 479, + 328 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 99, + 330, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "To evaluate various (possibly shallow and/or deep) architectures of SPTNs, we fit each dataset with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 324, + 354 + ], + "score": 1.0, + "content": "all the following combinations of hyper-parameters1:", + "type": "text" + }, + { + "bbox": [ + 324, + 341, + 387, + 353 + ], + "score": 0.75, + "content": "s \\in ( 8 , 3 2 , 1 2 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 340, + 392, + 354 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 392, + 341, + 450, + 353 + ], + "score": 0.63, + "content": "b \\in ( 2 , 4 , 6 , 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 340, + 454, + 354 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 454, + 341, + 502, + 353 + ], + "score": 0.59, + "content": "l \\in ( 2 , 3 , 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 340, + 506, + 354 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 133, + 364 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 354, + 140, + 362 + ], + "score": 0.64, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 352, + 322, + 364 + ], + "score": 1.0, + "content": "is the number of children of each sum node,", + "type": "text" + }, + { + "bbox": [ + 322, + 353, + 328, + 362 + ], + "score": 0.62, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "is the number of partitions of each product", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 149, + 376 + ], + "score": 1.0, + "content": "node, and", + "type": "text" + }, + { + "bbox": [ + 150, + 363, + 154, + 373 + ], + "score": 0.55, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "is the number of layers (one layer contains sum and product nodes). The number of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 412, + 386 + ], + "score": 1.0, + "content": "components of the SPTN, after its conversion into (1), is given as follows:", + "type": "text" + }, + { + "bbox": [ + 412, + 373, + 444, + 384 + ], + "score": 0.9, + "content": "K = s ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 374, + 506, + 386 + ], + "score": 1.0, + "content": ". Note that the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "maximum number of components for the investigated parameters of the SPTN is 268,435,456. To", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "reduce the space of possible architectures, we restrict ourselves only to (i) the leaf nodes given by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 405, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 139, + 418 + ], + "score": 0.91, + "content": "\\mathcal { N } ( 0 , \\bf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 405, + 506, + 420 + ], + "score": 1.0, + "content": "; (ii) affine transformations fixed to the singular value decomposition, choosing the the Givens", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 417, + 477, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 477, + 430 + ], + "score": 1.0, + "content": "parameterization for the unitary matrices [47]; and (iii) no sharing of any type of nodes [47].", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 87, + 434, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 86, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 86, + 436, + 99, + 445 + ], + "score": 1.0, + "content": "276", + "type": "text" + }, + { + "bbox": [ + 104, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "Algorithms: We evaluate only on the MHSAEM-U algorithm—due to its favourable computational", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 86, + 446, + 100, + 457 + ], + "score": 1.0, + "content": "277", + "type": "text" + }, + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "complexity and simplicity—and compare it with the stochastic gradient-descent (SGD) algorithm,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 85, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 85, + 456, + 100, + 468 + ], + "score": 1.0, + "content": "278", + "type": "text" + }, + { + "bbox": [ + 105, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "which is routinely used to train SP(T)Ns [45, 47]. In this case, SGD in each iteration performs", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 85, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 85, + 468, + 100, + 479 + ], + "score": 1.0, + "content": "279", + "type": "text" + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "computations over all subtrees of the network, whereas the MHSAEM-U algorithm computes with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 85, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 85, + 479, + 100, + 490 + ], + "score": 1.0, + "content": "280", + "type": "text" + }, + { + "bbox": [ + 105, + 477, + 127, + 491 + ], + "score": 1.0, + "content": "only", + "type": "text" + }, + { + "bbox": [ + 127, + 478, + 157, + 488 + ], + "score": 0.9, + "content": "M = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "subtrees, thus we should observe speed-up of the computations. In our implementation,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 85, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 85, + 490, + 99, + 501 + ], + "score": 1.0, + "content": "281", + "type": "text" + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "both these methods perform optimization of their respective objective functions—the log-likelihood", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 498, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 86, + 502, + 99, + 511 + ], + "score": 1.0, + "content": "282", + "type": "text" + }, + { + "bbox": [ + 104, + 498, + 506, + 513 + ], + "score": 1.0, + "content": "(2) for SGD and the EM objective (8) for MHSAEM-U—via the use of the automatic differentiation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 86, + 510, + 363, + 523 + ], + "spans": [ + { + "bbox": [ + 86, + 513, + 100, + 522 + ], + "score": 1.0, + "content": "283", + "type": "text" + }, + { + "bbox": [ + 104, + 510, + 257, + 523 + ], + "score": 1.0, + "content": "and the ADAM optimizer [29], using", + "type": "text" + }, + { + "bbox": [ + 257, + 510, + 295, + 520 + ], + "score": 0.9, + "content": "B = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 510, + 312, + 523 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 510, + 359, + 520 + ], + "score": 0.9, + "content": "T = 2 0 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 510, + 363, + 523 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 105, + 527, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "score": 1.0, + "content": "Results: Since each dataset might benefit from a different architecture, Table 6.2 shows the test", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 538, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 549 + ], + "score": 1.0, + "content": "log-likelihood of the architectures selected according to the best likelihood measured on the validation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "set and the corresponding speed-up. The test log-likelihoods reveal that the MHSAEM-U algorithm", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "outperforms the SGD algorithm on 10 out of 19 datasets, which was not originally the goal, but", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "the added stochasticity helps to escape poor local minima. The speed-up demonstrates lower", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 581, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 593 + ], + "score": 1.0, + "content": "computational complexity of the MHSAEM-U algorithm on 17 out of 19 datasets, which was the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 591, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 405, + 606 + ], + "score": 1.0, + "content": "main goal. The magic-telescope and wine datasets show approximately", + "type": "text" + }, + { + "bbox": [ + 405, + 592, + 429, + 603 + ], + "score": 0.89, + "content": "1 0 2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 591, + 446, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 446, + 592, + 465, + 603 + ], + "score": 0.88, + "content": "7 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 591, + 507, + 606 + ], + "score": 1.0, + "content": "speed-up,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 603, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 615 + ], + "score": 1.0, + "content": "respectively, while on very small datasets (pima-indians and iris), the SGD is faster due to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 614, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 506, + 626 + ], + "score": 1.0, + "content": "effective implementation. In the supplementary material, we present Table 3, exhibiting the same", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 625, + 226, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 226, + 636 + ], + "score": 1.0, + "content": "trends on a fixed architecture.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 106, + 649, + 245, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 247, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 247, + 663 + ], + "score": 1.0, + "content": "6.3 Mixtures of real NVP flows", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 108, + 670, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 396, + 683 + ], + "score": 1.0, + "content": "We consider another class of mixture models (1), where each component", + "type": "text" + }, + { + "bbox": [ + 396, + 669, + 432, + 682 + ], + "score": 0.93, + "content": "p _ { \\eta _ { z } } ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 668, + 505, + 683 + ], + "score": 1.0, + "content": "is transformed by", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "the flow model—real NVP [15]. These transformations are parameterized via deep neural networks,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 691, + 413, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 413, + 704 + ], + "score": 1.0, + "content": "allowing for flexible adjustment of the learning capacity of each component.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 117, + 712, + 456, + 722 + ], + "lines": [ + { + "bbox": [ + 119, + 709, + 458, + 723 + ], + "spans": [ + { + "bbox": [ + 119, + 709, + 458, + 723 + ], + "score": 1.0, + "content": "1We have set a hard limit to train a single model to 24h, which is default on our Slurm cluster.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 93, + 72, + 505, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 92, + 71, + 507, + 96 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 99, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 334, + 113 + ], + "score": 1.0, + "content": "SSAEM and TSAEM algorithms failed to converge for", + "type": "text" + }, + { + "bbox": [ + 334, + 100, + 366, + 110 + ], + "score": 0.9, + "content": "M > 2", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 99, + 398, + 113 + ], + "score": 1.0, + "content": "and for", + "type": "text" + }, + { + "bbox": [ + 399, + 100, + 434, + 111 + ], + "score": 0.89, + "content": "K > 5 0", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 99, + 506, + 113 + ], + "score": 1.0, + "content": "respectively. We", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 110, + 506, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 263, + 124 + ], + "score": 1.0, + "content": "believe this is caused by selecting only", + "type": "text" + }, + { + "bbox": [ + 264, + 111, + 276, + 121 + ], + "score": 0.8, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 110, + 506, + 124 + ], + "score": 1.0, + "content": "maximal probabilities in the SSAEM (or distances in the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 505, + 134 + ], + "score": 1.0, + "content": "TSAEM) algorithm (Section 5), which prevents certain, but not a negligible number of, components", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 376, + 146 + ], + "score": 1.0, + "content": "from being updated, thus providing only a crude approximation of", + "type": "text" + }, + { + "bbox": [ + 376, + 132, + 408, + 145 + ], + "score": 0.92, + "content": "\\bar { p } _ { \\theta } \\bar { ( } z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 132, + 506, + 146 + ], + "score": 1.0, + "content": ". The results then suffer", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 143, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 506, + 157 + ], + "score": 1.0, + "content": "from substantial variational gap to the exact log-likelihood (Figure 1). On the contrary, MH sampler", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 305, + 168 + ], + "score": 1.0, + "content": "provides samples which consistently approximate", + "type": "text" + }, + { + "bbox": [ + 306, + 154, + 337, + 167 + ], + "score": 0.93, + "content": "p _ { \\theta } ( z | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "despite evaluating much lower number of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 165, + 209, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 209, + 179 + ], + "score": 1.0, + "content": "components in each step.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 99, + 506, + 179 + ] + }, + { + "type": "title", + "bbox": [ + 102, + 190, + 273, + 201 + ], + "lines": [ + { + "bbox": [ + 106, + 190, + 275, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 275, + 203 + ], + "score": 1.0, + "content": "6.2 Sum-product transform networks", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 210, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 223 + ], + "score": 1.0, + "content": "The sum product networks (SPNs) are a deep learning extension of finite mixture models. They can", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 220, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 234 + ], + "score": 1.0, + "content": "be interpreted as a mixture of trees [60], where each tree corresponds to a component. Therefore,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 244 + ], + "score": 1.0, + "content": "they can be cast into the form of (1), but the number of components grows exponentially with their", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "depth. In this section, we use recently proposed SPTNs which introduce additional transformation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "nodes to provide better expressiveness than the SPNs (SPTNs effectively generalize SPNs and flow", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 265, + 271, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 271, + 278 + ], + "score": 1.0, + "content": "models into one large family of models).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 209, + 506, + 278 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 281, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "Experimental settings: We use 19 real datasets from the UCI database [16, 37, 35, 54], preprocessed", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 292, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 425, + 305 + ], + "score": 1.0, + "content": "in the same way as in [46]. For each experiment, we randomly split the data into", + "type": "text" + }, + { + "bbox": [ + 425, + 293, + 444, + 303 + ], + "score": 0.85, + "content": "64 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 292, + 448, + 305 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 448, + 293, + 468, + 303 + ], + "score": 0.82, + "content": "16 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 292, + 485, + 305 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 485, + 292, + 505, + 303 + ], + "score": 0.87, + "content": "20 \\%", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "for training, validation and testing, respectively. We calculate the average log-likelihood on the test", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 313, + 479, + 328 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 265, + 328 + ], + "score": 1.0, + "content": "set and measure again the time to reach", + "type": "text" + }, + { + "bbox": [ + 266, + 314, + 285, + 325 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 313, + 444, + 328 + ], + "score": 1.0, + "content": "of the maximal training log-likelihood,", + "type": "text" + }, + { + "bbox": [ + 444, + 314, + 475, + 326 + ], + "score": 0.92, + "content": "\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 313, + 479, + 328 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 281, + 506, + 328 + ] + }, + { + "type": "text", + "bbox": [ + 99, + 330, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "To evaluate various (possibly shallow and/or deep) architectures of SPTNs, we fit each dataset with", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 324, + 354 + ], + "score": 1.0, + "content": "all the following combinations of hyper-parameters1:", + "type": "text" + }, + { + "bbox": [ + 324, + 341, + 387, + 353 + ], + "score": 0.75, + "content": "s \\in ( 8 , 3 2 , 1 2 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 340, + 392, + 354 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 392, + 341, + 450, + 353 + ], + "score": 0.63, + "content": "b \\in ( 2 , 4 , 6 , 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 340, + 454, + 354 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 454, + 341, + 502, + 353 + ], + "score": 0.59, + "content": "l \\in ( 2 , 3 , 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 340, + 506, + 354 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 133, + 364 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 354, + 140, + 362 + ], + "score": 0.64, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 352, + 322, + 364 + ], + "score": 1.0, + "content": "is the number of children of each sum node,", + "type": "text" + }, + { + "bbox": [ + 322, + 353, + 328, + 362 + ], + "score": 0.62, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "is the number of partitions of each product", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 149, + 376 + ], + "score": 1.0, + "content": "node, and", + "type": "text" + }, + { + "bbox": [ + 150, + 363, + 154, + 373 + ], + "score": 0.55, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "is the number of layers (one layer contains sum and product nodes). The number of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 412, + 386 + ], + "score": 1.0, + "content": "components of the SPTN, after its conversion into (1), is given as follows:", + "type": "text" + }, + { + "bbox": [ + 412, + 373, + 444, + 384 + ], + "score": 0.9, + "content": "K = s ^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 374, + 506, + 386 + ], + "score": 1.0, + "content": ". Note that the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "maximum number of components for the investigated parameters of the SPTN is 268,435,456. To", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "reduce the space of possible architectures, we restrict ourselves only to (i) the leaf nodes given by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 405, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 139, + 418 + ], + "score": 0.91, + "content": "\\mathcal { N } ( 0 , \\bf { I } )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 405, + 506, + 420 + ], + "score": 1.0, + "content": "; (ii) affine transformations fixed to the singular value decomposition, choosing the the Givens", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 417, + 477, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 477, + 430 + ], + "score": 1.0, + "content": "parameterization for the unitary matrices [47]; and (iii) no sharing of any type of nodes [47].", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 330, + 506, + 430 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 434, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 86, + 434, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 86, + 436, + 99, + 445 + ], + "score": 1.0, + "content": "276", + "type": "text" + }, + { + "bbox": [ + 104, + 434, + 505, + 446 + ], + "score": 1.0, + "content": "Algorithms: We evaluate only on the MHSAEM-U algorithm—due to its favourable computational", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 86, + 446, + 100, + 457 + ], + "score": 1.0, + "content": "277", + "type": "text" + }, + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "complexity and simplicity—and compare it with the stochastic gradient-descent (SGD) algorithm,", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 85, + 456, + 100, + 468 + ], + "score": 1.0, + "content": "278", + "type": "text" + }, + { + "bbox": [ + 105, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "which is routinely used to train SP(T)Ns [45, 47]. In this case, SGD in each iteration performs", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 85, + 468, + 100, + 479 + ], + "score": 1.0, + "content": "279", + "type": "text" + }, + { + "bbox": [ + 105, + 466, + 506, + 479 + ], + "score": 1.0, + "content": "computations over all subtrees of the network, whereas the MHSAEM-U algorithm computes with", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 477, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 85, + 479, + 100, + 490 + ], + "score": 1.0, + "content": "280", + "type": "text" + }, + { + "bbox": [ + 105, + 477, + 127, + 491 + ], + "score": 1.0, + "content": "only", + "type": "text" + }, + { + "bbox": [ + 127, + 478, + 157, + 488 + ], + "score": 0.9, + "content": "M = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 477, + 506, + 491 + ], + "score": 1.0, + "content": "subtrees, thus we should observe speed-up of the computations. In our implementation,", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 488, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 85, + 490, + 99, + 501 + ], + "score": 1.0, + "content": "281", + "type": "text" + }, + { + "bbox": [ + 105, + 488, + 506, + 502 + ], + "score": 1.0, + "content": "both these methods perform optimization of their respective objective functions—the log-likelihood", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 498, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 86, + 502, + 99, + 511 + ], + "score": 1.0, + "content": "282", + "type": "text" + }, + { + "bbox": [ + 104, + 498, + 506, + 513 + ], + "score": 1.0, + "content": "(2) for SGD and the EM objective (8) for MHSAEM-U—via the use of the automatic differentiation", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 510, + 363, + 523 + ], + "spans": [ + { + "bbox": [ + 86, + 513, + 100, + 522 + ], + "score": 1.0, + "content": "283", + "type": "text" + }, + { + "bbox": [ + 104, + 510, + 257, + 523 + ], + "score": 1.0, + "content": "and the ADAM optimizer [29], using", + "type": "text" + }, + { + "bbox": [ + 257, + 510, + 295, + 520 + ], + "score": 0.9, + "content": "B = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 510, + 312, + 523 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 510, + 359, + 520 + ], + "score": 0.9, + "content": "T = 2 0 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 510, + 363, + 523 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + } + ], + "index": 32.5, + "bbox_fs": [ + 85, + 434, + 506, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 527, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "score": 1.0, + "content": "Results: Since each dataset might benefit from a different architecture, Table 6.2 shows the test", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 538, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 549 + ], + "score": 1.0, + "content": "log-likelihood of the architectures selected according to the best likelihood measured on the validation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "set and the corresponding speed-up. The test log-likelihoods reveal that the MHSAEM-U algorithm", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 572 + ], + "score": 1.0, + "content": "outperforms the SGD algorithm on 10 out of 19 datasets, which was not originally the goal, but", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "the added stochasticity helps to escape poor local minima. The speed-up demonstrates lower", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 581, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 593 + ], + "score": 1.0, + "content": "computational complexity of the MHSAEM-U algorithm on 17 out of 19 datasets, which was the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 591, + 507, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 405, + 606 + ], + "score": 1.0, + "content": "main goal. The magic-telescope and wine datasets show approximately", + "type": "text" + }, + { + "bbox": [ + 405, + 592, + 429, + 603 + ], + "score": 0.89, + "content": "1 0 2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 591, + 446, + 606 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 446, + 592, + 465, + 603 + ], + "score": 0.88, + "content": "7 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 591, + 507, + 606 + ], + "score": 1.0, + "content": "speed-up,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 603, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 615 + ], + "score": 1.0, + "content": "respectively, while on very small datasets (pima-indians and iris), the SGD is faster due to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 614, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 506, + 626 + ], + "score": 1.0, + "content": "effective implementation. In the supplementary material, we present Table 3, exhibiting the same", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 625, + 226, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 226, + 636 + ], + "score": 1.0, + "content": "trends on a fixed architecture.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 526, + 507, + 636 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 649, + 245, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 247, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 247, + 663 + ], + "score": 1.0, + "content": "6.3 Mixtures of real NVP flows", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 108, + 670, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 668, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 396, + 683 + ], + "score": 1.0, + "content": "We consider another class of mixture models (1), where each component", + "type": "text" + }, + { + "bbox": [ + 396, + 669, + 432, + 682 + ], + "score": 0.93, + "content": "p _ { \\eta _ { z } } ( x | z )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 668, + 505, + 683 + ], + "score": 1.0, + "content": "is transformed by", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "the flow model—real NVP [15]. These transformations are parameterized via deep neural networks,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 691, + 413, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 413, + 704 + ], + "score": 1.0, + "content": "allowing for flexible adjustment of the learning capacity of each component.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 668, + 506, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 114, + 154, + 498, + 335 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 70, + 505, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 69, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 292, + 84 + ], + "score": 1.0, + "content": "Table 2: The speed-up and test log-likelihood,", + "type": "text" + }, + { + "bbox": [ + 292, + 70, + 310, + 81 + ], + "score": 0.87, + "content": "\\mathcal { L } ^ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 69, + 506, + 84 + ], + "score": 1.0, + "content": ", for the SGD and MHSAEM-U algorithms. The", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 81, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 492, + 93 + ], + "score": 1.0, + "content": "test log-likelihood (higher is better) is computed for the best model, with the corresponding", + "type": "text" + }, + { + "bbox": [ + 492, + 82, + 502, + 91 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 81, + 506, + 93 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 92, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 104 + ], + "score": 1.0, + "content": "which is selected based on the validation log-likelihood. The speed-up is computed as the ratio of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 103, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 294, + 115 + ], + "score": 1.0, + "content": "MHSAEM-U to SGD, i.e. their time to reach", + "type": "text" + }, + { + "bbox": [ + 295, + 103, + 315, + 114 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 103, + 506, + 115 + ], + "score": 1.0, + "content": "of the training log-likelihood. The results are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "averaged over five repetitions. 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Sum-product transformnetworksMixtures of real NVP flows SGD
SGDMHSAEM-UMHSAEM-U
speed-upLtestKLtestKspeed-upLtestK 32LtestK
breast-cancer-wisconsin4.66-4.66641.4310240.63-99.85-39.31128
cardiotocography10.5559.5251231.0410249.8554.343256.08128
magic-telescope102.53-3.65512-5.0310243.74-3.978-4.228
pendigits4.890.881024-4.86163844.171.4680.488
pima-indians0.37-8.5464-7.62641.35-20.09128-16.33128
wall-following-robot3.431.841024-11.31638422.21-14.26128-17.56128
waveform-14.35-26.1464-23.9110243.72-34.128-33.428
waveform-24.82-26.2164-23.9110244.12-34.158-33.648
yeast20.5710.265125.18102414.496.611289.59128
ecoli1.86-5.564-0.2210242.15-11.37128-10.64128
ionosphere1.88-20.2764-5.935122.74-87.01128-42.75128
iris0.23-10.6564-1.49163843.28-16.34128-9.2132
page-blocks12.1812.215126.84102444.9517.1312817.9432
parkinsons1.46-21.85640.55123.09-566.58128-33.3132
sonar2.96-95.39512-69.29642.52-622.2128-88.81128
statlog-segment1.4447.3551226.531638438.4935.8412842.0432
statlog-vehicle2.97-4.2564-5.4510246.78-31.3432-26.43128
wine rank75.42-25.991024-13.2710242.05-171.58128-25.57128
1.561.441.831.17
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Each real NVP-based component in the mixture model has (i) the translation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 88, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 88, + 397, + 99, + 405 + ], + "score": 1.0, + "content": "301", + "type": "text" + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "function parameterized via multi-layer perceptron with a single hidden layer of dimension 10, using", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 88, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 88, + 408, + 100, + 416 + ], + "score": 1.0, + "content": "302", + "type": "text" + }, + { + "bbox": [ + 105, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "the rectified linear activation function; and (ii) the scale function parameterized via the same network", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 88, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 88, + 418, + 100, + 428 + ], + "score": 1.0, + "content": "303", + "type": "text" + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "except with the hyperbolic tangent activation function. We do not use the batch normalization [15] and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 89, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 89, + 430, + 100, + 438 + ], + "score": 1.0, + "content": "304", + "type": "text" + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "we stack two layers of the translation-scale transformation (we have used implementation from [20]).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 99, + 443, + 435, + 455 + ], + "lines": [ + { + "bbox": [ + 104, + 442, + 437, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 437, + 457 + ], + "score": 1.0, + "content": "Algorithms: The algorithms and their settings are the same as those in Section 6.2.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 86, + 460, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 85, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 85, + 461, + 100, + 471 + ], + "score": 1.0, + "content": "306", + "type": "text" + }, + { + "bbox": [ + 104, + 460, + 506, + 472 + ], + "score": 1.0, + "content": "Results: The experimental results are presented in right part of Table 6.2. They are similar to those", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 85, + 471, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 85, + 472, + 100, + 482 + ], + "score": 1.0, + "content": "307", + "type": "text" + }, + { + "bbox": [ + 106, + 471, + 505, + 482 + ], + "score": 1.0, + "content": "obtained in the previous section. In terms of the test log-likelihood, the MHSAEM-U algorithm", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 85, + 482, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 85, + 483, + 100, + 493 + ], + "score": 1.0, + "content": "308", + "type": "text" + }, + { + "bbox": [ + 105, + 482, + 506, + 494 + ], + "score": 1.0, + "content": "outperforms the SGD algorithm on all but three datasets, and it provides a substantial speed-up on all", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 85, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 85, + 494, + 100, + 504 + ], + "score": 1.0, + "content": "309", + "type": "text" + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "datasets except one. The test likelihood of models with the real NVP flows is most of the time worse", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 85, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 85, + 505, + 100, + 515 + ], + "score": 1.0, + "content": "310", + "type": "text" + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "than that of SPTNs with the affine transformations. As explained in the supplementary, this is due to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 85, + 515, + 301, + 526 + ], + "spans": [ + { + "bbox": [ + 85, + 516, + 100, + 526 + ], + "score": 1.0, + "content": "311", + "type": "text" + }, + { + "bbox": [ + 106, + 515, + 301, + 526 + ], + "score": 1.0, + "content": "the overfitting, which has been observed in [47].", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 87, + 546, + 183, + 560 + ], + "lines": [ + { + "bbox": [ + 83, + 545, + 185, + 562 + ], + "spans": [ + { + "bbox": [ + 83, + 545, + 185, + 562 + ], + "score": 1.0, + "content": "312 7 Conclusion", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 86, + 574, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 86, + 575, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 86, + 577, + 99, + 586 + ], + "score": 1.0, + "content": "313", + "type": "text" + }, + { + "bbox": [ + 106, + 575, + 506, + 587 + ], + "score": 1.0, + "content": "This paper has presented a method to decrease computational complexity of fitting mixture models,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 86, + 587, + 99, + 596 + ], + "score": 1.0, + "content": "314", + "type": "text" + }, + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "including their generalizations, such as sum-product-(transform) networks and mixtures of flow", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 86, + 599, + 99, + 608 + ], + "score": 1.0, + "content": "315", + "type": "text" + }, + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "models. The speed-up is achieved by evaluating and updating only a single component (per iteration),", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 85, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 85, + 609, + 100, + 619 + ], + "score": 1.0, + "content": "316", + "type": "text" + }, + { + "bbox": [ + 106, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "where the Metropolis-Hasting algorithm ensures sampling of components from a proper posterior. An", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 618, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 86, + 620, + 99, + 629 + ], + "score": 1.0, + "content": "317", + "type": "text" + }, + { + "bbox": [ + 105, + 618, + 506, + 630 + ], + "score": 1.0, + "content": "experimental comparison on all three classes of models mentioned above confirmed the theoretical", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 85, + 628, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 85, + 631, + 100, + 641 + ], + "score": 1.0, + "content": "318", + "type": "text" + }, + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "score": 1.0, + "content": "expectations. The method significantly speeds-up the fitting time and, importantly, without sacrificing", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 85, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 85, + 641, + 100, + 652 + ], + "score": 1.0, + "content": "319", + "type": "text" + }, + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "the quality of the fit. In fact, the likelihood was better than that of the models fitted by the EM", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 85, + 649, + 507, + 664 + ], + "spans": [ + { + "bbox": [ + 85, + 653, + 100, + 663 + ], + "score": 1.0, + "content": "320", + "type": "text" + }, + { + "bbox": [ + 105, + 649, + 289, + 664 + ], + "score": 1.0, + "content": "algorithm or the SGD algorithm in more than", + "type": "text" + }, + { + "bbox": [ + 289, + 651, + 308, + 661 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 649, + 507, + 664 + ], + "score": 1.0, + "content": "of cases. We attribute this to higher stochasticity,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 661, + 295, + 675 + ], + "spans": [ + { + "bbox": [ + 86, + 663, + 99, + 674 + ], + "score": 1.0, + "content": "321", + "type": "text" + }, + { + "bbox": [ + 105, + 661, + 295, + 675 + ], + "score": 1.0, + "content": "which helps to escape from poor local minima.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 87, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 86, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 86, + 680, + 99, + 689 + ], + "score": 1.0, + "content": "322", + "type": "text" + }, + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "In the experiments, the proposed method has used a uniform proposal distribution in the MH sampler.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 688, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 86, + 691, + 99, + 700 + ], + "score": 1.0, + "content": "323", + "type": "text" + }, + { + "bbox": [ + 105, + 688, + 507, + 702 + ], + "score": 1.0, + "content": "Despite outperforming the alternative methods, we conjecture that this limits the speed of convergence.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 85, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 85, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "324", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Therefore, we believe that there is still a room for improvement in the implementation. We plan to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 711, + 249, + 722 + ], + "spans": [ + { + "bbox": [ + 86, + 712, + 99, + 722 + ], + "score": 1.0, + "content": "325", + "type": "text" + }, + { + "bbox": [ + 106, + 711, + 249, + 722 + ], + "score": 1.0, + "content": "address these issues in future work.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 114, + 154, + 498, + 335 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 70, + 505, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 69, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 104, + 69, + 292, + 84 + ], + "score": 1.0, + "content": "Table 2: The speed-up and test log-likelihood,", + "type": "text" + }, + { + "bbox": [ + 292, + 70, + 310, + 81 + ], + "score": 0.87, + "content": "\\mathcal { L } ^ { \\mathrm { t e s t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 69, + 506, + 84 + ], + "score": 1.0, + "content": ", for the SGD and MHSAEM-U algorithms. The", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 81, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 492, + 93 + ], + "score": 1.0, + "content": "test log-likelihood (higher is better) is computed for the best model, with the corresponding", + "type": "text" + }, + { + "bbox": [ + 492, + 82, + 502, + 91 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 81, + 506, + 93 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 92, + 506, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 104 + ], + "score": 1.0, + "content": "which is selected based on the validation log-likelihood. The speed-up is computed as the ratio of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 103, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 294, + 115 + ], + "score": 1.0, + "content": "MHSAEM-U to SGD, i.e. their time to reach", + "type": "text" + }, + { + "bbox": [ + 295, + 103, + 315, + 114 + ], + "score": 0.88, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 103, + 506, + 115 + ], + "score": 1.0, + "content": "of the training log-likelihood. The results are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "averaged over five repetitions. Then, the higher test log-likelihood is highlighted with bold blue, and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 506, + 138 + ], + "score": 1.0, + "content": "and no speed-up is highlighted with red. The average rank is computed as the standard competition", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 136, + 328, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 328, + 148 + ], + "score": 1.0, + "content": "(“1224”) ranking [14] on each dataset (lower is better).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "table_body", + "bbox": [ + 114, + 154, + 498, + 335 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 154, + 498, + 335 + ], + "spans": [ + { + "bbox": [ + 114, + 154, + 498, + 335 + ], + "score": 0.985, + "html": "
Sum-product transformnetworksMixtures of real NVP flows SGD
SGDMHSAEM-UMHSAEM-U
speed-upLtestKLtestKspeed-upLtestK 32LtestK
breast-cancer-wisconsin4.66-4.66641.4310240.63-99.85-39.31128
cardiotocography10.5559.5251231.0410249.8554.343256.08128
magic-telescope102.53-3.65512-5.0310243.74-3.978-4.228
pendigits4.890.881024-4.86163844.171.4680.488
pima-indians0.37-8.5464-7.62641.35-20.09128-16.33128
wall-following-robot3.431.841024-11.31638422.21-14.26128-17.56128
waveform-14.35-26.1464-23.9110243.72-34.128-33.428
waveform-24.82-26.2164-23.9110244.12-34.158-33.648
yeast20.5710.265125.18102414.496.611289.59128
ecoli1.86-5.564-0.2210242.15-11.37128-10.64128
ionosphere1.88-20.2764-5.935122.74-87.01128-42.75128
iris0.23-10.6564-1.49163843.28-16.34128-9.2132
page-blocks12.1812.215126.84102444.9517.1312817.9432
parkinsons1.46-21.85640.55123.09-566.58128-33.3132
sonar2.96-95.39512-69.29642.52-622.2128-88.81128
statlog-segment1.4447.3551226.531638438.4935.8412842.0432
statlog-vehicle2.97-4.2564-5.4510246.78-31.3432-26.43128
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We apply the mixture model on all datasets, changing the number of components as follows:", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 88, + 386, + 100, + 395 + ], + "score": 1.0, + "content": "300", + "type": "text" + }, + { + "bbox": [ + 107, + 384, + 177, + 395 + ], + "score": 0.78, + "content": "K \\in ( 8 , 3 2 , 1 2 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 383, + 506, + 396 + ], + "score": 1.0, + "content": ". Each real NVP-based component in the mixture model has (i) the translation", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 88, + 397, + 99, + 405 + ], + "score": 1.0, + "content": "301", + "type": "text" + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "function parameterized via multi-layer perceptron with a single hidden layer of dimension 10, using", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 405, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 88, + 408, + 100, + 416 + ], + "score": 1.0, + "content": "302", + "type": "text" + }, + { + "bbox": [ + 105, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "the rectified linear activation function; and (ii) the scale function parameterized via the same network", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 88, + 418, + 100, + 428 + ], + "score": 1.0, + "content": "303", + "type": "text" + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "except with the hyperbolic tangent activation function. We do not use the batch normalization [15] and", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 89, + 430, + 100, + 438 + ], + "score": 1.0, + "content": "304", + "type": "text" + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "we stack two layers of the translation-scale transformation (we have used implementation from [20]).", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + } + ], + "index": 13, + "bbox_fs": [ + 88, + 362, + 507, + 440 + ] + }, + { + "type": "text", + "bbox": [ + 99, + 443, + 435, + 455 + ], + "lines": [ + { + "bbox": [ + 104, + 442, + 437, + 457 + ], + "spans": [ + { + "bbox": [ + 104, + 442, + 437, + 457 + ], + "score": 1.0, + "content": "Algorithms: The algorithms and their settings are the same as those in Section 6.2.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 442, + 437, + 457 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 460, + 505, + 526 + ], + "lines": [ + { + "bbox": [ + 85, + 460, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 85, + 461, + 100, + 471 + ], + "score": 1.0, + "content": "306", + "type": "text" + }, + { + "bbox": [ + 104, + 460, + 506, + 472 + ], + "score": 1.0, + "content": "Results: The experimental results are presented in right part of Table 6.2. They are similar to those", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 471, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 85, + 472, + 100, + 482 + ], + "score": 1.0, + "content": "307", + "type": "text" + }, + { + "bbox": [ + 106, + 471, + 505, + 482 + ], + "score": 1.0, + "content": "obtained in the previous section. In terms of the test log-likelihood, the MHSAEM-U algorithm", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 482, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 85, + 483, + 100, + 493 + ], + "score": 1.0, + "content": "308", + "type": "text" + }, + { + "bbox": [ + 105, + 482, + 506, + 494 + ], + "score": 1.0, + "content": "outperforms the SGD algorithm on all but three datasets, and it provides a substantial speed-up on all", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 85, + 494, + 100, + 504 + ], + "score": 1.0, + "content": "309", + "type": "text" + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "datasets except one. The test likelihood of models with the real NVP flows is most of the time worse", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 503, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 85, + 505, + 100, + 515 + ], + "score": 1.0, + "content": "310", + "type": "text" + }, + { + "bbox": [ + 105, + 503, + 505, + 515 + ], + "score": 1.0, + "content": "than that of SPTNs with the affine transformations. As explained in the supplementary, this is due to", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 515, + 301, + 526 + ], + "spans": [ + { + "bbox": [ + 85, + 516, + 100, + 526 + ], + "score": 1.0, + "content": "311", + "type": "text" + }, + { + "bbox": [ + 106, + 515, + 301, + 526 + ], + "score": 1.0, + "content": "the overfitting, which has been observed in [47].", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + } + ], + "index": 20.5, + "bbox_fs": [ + 85, + 460, + 506, + 526 + ] + }, + { + "type": "title", + "bbox": [ + 87, + 546, + 183, + 560 + ], + "lines": [ + { + "bbox": [ + 83, + 545, + 185, + 562 + ], + "spans": [ + { + "bbox": [ + 83, + 545, + 185, + 562 + ], + "score": 1.0, + "content": "312 7 Conclusion", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "index", + "bbox": [ + 86, + 574, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 86, + 575, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 86, + 577, + 99, + 586 + ], + "score": 1.0, + "content": "313", + "type": "text" + }, + { + "bbox": [ + 106, + 575, + 506, + 587 + ], + "score": 1.0, + "content": "This paper has presented a method to decrease computational complexity of fitting mixture models,", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 86, + 587, + 99, + 596 + ], + "score": 1.0, + "content": "314", + "type": "text" + }, + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "including their generalizations, such as sum-product-(transform) networks and mixtures of flow", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 86, + 599, + 99, + 608 + ], + "score": 1.0, + "content": "315", + "type": "text" + }, + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "models. The speed-up is achieved by evaluating and updating only a single component (per iteration),", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 606, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 85, + 609, + 100, + 619 + ], + "score": 1.0, + "content": "316", + "type": "text" + }, + { + "bbox": [ + 106, + 606, + 506, + 620 + ], + "score": 1.0, + "content": "where the Metropolis-Hasting algorithm ensures sampling of components from a proper posterior. An", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 618, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 86, + 620, + 99, + 629 + ], + "score": 1.0, + "content": "317", + "type": "text" + }, + { + "bbox": [ + 105, + 618, + 506, + 630 + ], + "score": 1.0, + "content": "experimental comparison on all three classes of models mentioned above confirmed the theoretical", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 628, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 85, + 631, + 100, + 641 + ], + "score": 1.0, + "content": "318", + "type": "text" + }, + { + "bbox": [ + 105, + 628, + 506, + 643 + ], + "score": 1.0, + "content": "expectations. The method significantly speeds-up the fitting time and, importantly, without sacrificing", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 640, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 85, + 641, + 100, + 652 + ], + "score": 1.0, + "content": "319", + "type": "text" + }, + { + "bbox": [ + 105, + 640, + 505, + 652 + ], + "score": 1.0, + "content": "the quality of the fit. In fact, the likelihood was better than that of the models fitted by the EM", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 649, + 507, + 664 + ], + "spans": [ + { + "bbox": [ + 85, + 653, + 100, + 663 + ], + "score": 1.0, + "content": "320", + "type": "text" + }, + { + "bbox": [ + 105, + 649, + 289, + 664 + ], + "score": 1.0, + "content": "algorithm or the SGD algorithm in more than", + "type": "text" + }, + { + "bbox": [ + 289, + 651, + 308, + 661 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 649, + 507, + 664 + ], + "score": 1.0, + "content": "of cases. We attribute this to higher stochasticity,", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 661, + 295, + 675 + ], + "spans": [ + { + "bbox": [ + 86, + 663, + 99, + 674 + ], + "score": 1.0, + "content": "321", + "type": "text" + }, + { + "bbox": [ + 105, + 661, + 295, + 675 + ], + "score": 1.0, + "content": "which helps to escape from poor local minima.", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 678, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 86, + 680, + 99, + 689 + ], + "score": 1.0, + "content": "322", + "type": "text" + }, + { + "bbox": [ + 106, + 678, + 506, + 690 + ], + "score": 1.0, + "content": "In the experiments, the proposed method has used a uniform proposal distribution in the MH sampler.", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 688, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 86, + 691, + 99, + 700 + ], + "score": 1.0, + "content": "323", + "type": "text" + }, + { + "bbox": [ + 105, + 688, + 507, + 702 + ], + "score": 1.0, + "content": "Despite outperforming the alternative methods, we conjecture that this limits the speed of convergence.", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 85, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "324", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Therefore, we believe that there is still a room for improvement in the implementation. We plan to", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 711, + 249, + 722 + ], + "spans": [ + { + "bbox": [ + 86, + 712, + 99, + 722 + ], + "score": 1.0, + "content": "325", + "type": "text" + }, + { + "bbox": [ + 106, + 711, + 249, + 722 + ], + "score": 1.0, + "content": "address these issues in future work.", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + } + ], + "index": 29, + "bbox_fs": [ + 85, + 575, + 507, + 675 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 678, + 505, + 722 + ], + "lines": [], + "index": 35.5, + "bbox_fs": [ + 85, + 678, + 507, + 722 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 96, + 506, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 96, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 505, + 109 + ], + "score": 1.0, + "content": "The presented method decreases the computational complexity of fitting large (and deep) mixture", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 107, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 107, + 505, + 120 + ], + "score": 1.0, + "content": "models, which leads to five to hundred time speed-up depending on a size of the problem (although", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 118, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 506, + 130 + ], + "score": 1.0, + "content": "negative exceptions occurs). We believe this line of research, which we want to continue, to have", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 142 + ], + "score": 1.0, + "content": "important benefits. First, it is directly related to decrease in energy consumption and in production of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 506, + 153 + ], + "score": 1.0, + "content": "CO2 (we expect similar rates as the speedup). Second, it has a positive effect on financial aspects of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 507, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 507, + 164 + ], + "score": 1.0, + "content": "deploying (and experimenting with) mixture models. 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Computational efficient variational Bayesian Gaus-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 90, + 124, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 90, + 126, + 100, + 136 + ], + "score": 1.0, + "content": "64", + "type": "text" + }, + { + "bbox": [ + 127, + 124, + 505, + 138 + ], + "score": 1.0, + "content": "sian mixture models via coreset. In 2016 International Conference on Computer, Information", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 90, + 134, + 389, + 149 + ], + "spans": [ + { + "bbox": [ + 90, + 137, + 100, + 147 + ], + "score": 1.0, + "content": "65", + "type": "text" + }, + { + "bbox": [ + 127, + 134, + 389, + 149 + ], + "score": 1.0, + "content": "and Telecommunication Systems (CITS), pages 1–5. IEEE, 2016.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 90, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 90, + 156, + 100, + 165 + ], + "score": 1.0, + "content": "66", + "type": "text" + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "[60] H. Zhao, P. Poupart, and G. Gordon. A unified approach for learning the parameters of sum-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 90, + 165, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 90, + 167, + 100, + 177 + ], + "score": 1.0, + "content": "67", + "type": "text" + }, + { + "bbox": [ + 127, + 165, + 506, + 179 + ], + "score": 1.0, + "content": "product networks. 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Xiang, A. Karfoul, C. Yang, H. Shu, and R. L. B. Jeannès. An exact line search scheme to", + "type": "text" + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 84, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 90, + 86, + 99, + 95 + ], + "score": 1.0, + "content": "61", + "type": "text" + }, + { + "bbox": [ + 127, + 84, + 506, + 97 + ], + "score": 1.0, + "content": "accelerate the EM algorithm: Application to Gaussian mixture models identification. 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Computational efficient variational Bayesian Gaus-", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 124, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 90, + 126, + 100, + 136 + ], + "score": 1.0, + "content": "64", + "type": "text" + }, + { + "bbox": [ + 127, + 124, + 505, + 138 + ], + "score": 1.0, + "content": "sian mixture models via coreset. In 2016 International Conference on Computer, Information", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 134, + 389, + 149 + ], + "spans": [ + { + "bbox": [ + 90, + 137, + 100, + 147 + ], + "score": 1.0, + "content": "65", + "type": "text" + }, + { + "bbox": [ + 127, + 134, + 389, + 149 + ], + "score": 1.0, + "content": "and Telecommunication Systems (CITS), pages 1–5. IEEE, 2016.", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 90, + 156, + 100, + 165 + ], + "score": 1.0, + "content": "66", + "type": "text" + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "[60] H. Zhao, P. Poupart, and G. Gordon. A unified approach for learning the parameters of sum-", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 165, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 90, + 167, + 100, + 177 + ], + "score": 1.0, + "content": "67", + "type": "text" + }, + { + "bbox": [ + 127, + 165, + 506, + 179 + ], + "score": 1.0, + "content": "product networks. In Proceedings of the 30th International Conference on Neural Information", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 176, + 300, + 189 + ], + "spans": [ + { + "bbox": [ + 90, + 178, + 100, + 188 + ], + "score": 1.0, + "content": "68", + "type": "text" + }, + { + "bbox": [ + 127, + 176, + 300, + 189 + ], + "score": 1.0, + "content": "Processing Systems, pages 433–441, 2016.", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + } + ], + "index": 4, + "bbox_fs": [ + 90, + 73, + 506, + 189 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 131, + 92, + 208, + 104 + ], + "lines": [ + { + "bbox": [ + 129, + 91, + 210, + 105 + ], + "spans": [ + { + "bbox": [ + 129, + 91, + 210, + 105 + ], + "score": 1.0, + "content": "1. For all authors...", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 146, + 107, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 145, + 107, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 145, + 107, + 505, + 120 + ], + "score": 1.0, + "content": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 162, + 118, + 288, + 131 + ], + "spans": [ + { + "bbox": [ + 162, + 118, + 288, + 131 + ], + "score": 1.0, + "content": "contributions and scope? [Yes]", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 144, + 130, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 144, + 130, + 506, + 145 + ], + "score": 1.0, + "content": "(b) Did you describe the limitations of your work? [Yes] Our main contribution is compu-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 162, + 142, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 162, + 142, + 505, + 154 + ], + "score": 1.0, + "content": "tational speedup. Cases where it was not achieved are highlighted in the experimental", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 161, + 152, + 196, + 165 + ], + "spans": [ + { + "bbox": [ + 161, + 152, + 196, + 165 + ], + "score": 1.0, + "content": "section.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 146, + 165, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 146, + 165, + 506, + 179 + ], + "score": 1.0, + "content": "(c) Did you discuss any potential negative societal impacts of your work? [No] We do not", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 162, + 177, + 315, + 189 + ], + "spans": [ + { + "bbox": [ + 162, + 177, + 315, + 189 + ], + "score": 1.0, + "content": "foresee any potential negative impact.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 144, + 189, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 144, + 189, + 505, + 203 + ], + "score": 1.0, + "content": "(d) Have you read the ethics review guidelines and ensured that your paper conforms to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 161, + 200, + 214, + 213 + ], + "spans": [ + { + "bbox": [ + 161, + 200, + 214, + 213 + ], + "score": 1.0, + "content": "them? [Yes]", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 132, + 216, + 302, + 227 + ], + "lines": [ + { + "bbox": [ + 129, + 214, + 304, + 229 + ], + "spans": [ + { + "bbox": [ + 129, + 214, + 304, + 229 + ], + "score": 1.0, + "content": "2. If you are including theoretical results...", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 146, + 230, + 451, + 255 + ], + "lines": [ + { + "bbox": [ + 145, + 230, + 453, + 243 + ], + "spans": [ + { + "bbox": [ + 145, + 230, + 453, + 243 + ], + "score": 1.0, + "content": "(a) Did you state the full set of assumptions of all theoretical results? [N/A]", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 144, + 243, + 424, + 256 + ], + "spans": [ + { + "bbox": [ + 144, + 243, + 424, + 256 + ], + "score": 1.0, + "content": "(b) Did you include complete proofs of all theoretical results? [N/A]", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 130, + 258, + 241, + 270 + ], + "lines": [ + { + "bbox": [ + 128, + 256, + 243, + 272 + ], + "spans": [ + { + "bbox": [ + 128, + 256, + 243, + 272 + ], + "score": 1.0, + "content": "3. If you ran experiments...", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 146, + 273, + 505, + 388 + ], + "lines": [ + { + "bbox": [ + 146, + 273, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 146, + 273, + 506, + 285 + ], + "score": 1.0, + "content": "(a) Did you include the code, data, and instructions needed to reproduce the main exper-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 162, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 162, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "imental results (either in the supplemental material or as a URL)? [Yes] The code is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 162, + 295, + 473, + 307 + ], + "spans": [ + { + "bbox": [ + 162, + 295, + 473, + 307 + ], + "score": 1.0, + "content": "available in a github repository. All dataset are public from the UCI database.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 146, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 146, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 162, + 319, + 305, + 330 + ], + "spans": [ + { + "bbox": [ + 162, + 319, + 305, + 330 + ], + "score": 1.0, + "content": "were chosen)? [Yes] See Section 6.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 146, + 332, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 146, + 332, + 506, + 344 + ], + "score": 1.0, + "content": "(c) Did you report error bars (e.g., with respect to the random seed after running experi-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 162, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 162, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "ments multiple times)? [No] We report only average of Monte Carlo repetitions, the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 162, + 353, + 502, + 366 + ], + "spans": [ + { + "bbox": [ + 162, + 353, + 502, + 366 + ], + "score": 1.0, + "content": "error bars were too small to have any visual impact in the reported logarithmic scale.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 146, + 365, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 146, + 365, + 505, + 380 + ], + "score": 1.0, + "content": "(d) Did you include the total amount of compute and the type of resources used (e.g., type", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 162, + 377, + 369, + 389 + ], + "spans": [ + { + "bbox": [ + 162, + 377, + 369, + 389 + ], + "score": 1.0, + "content": "of GPUs, internal cluster, or cloud provider)? [Yes]", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 132, + 392, + 504, + 404 + ], + "lines": [ + { + "bbox": [ + 129, + 390, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 129, + 390, + 506, + 406 + ], + "score": 1.0, + "content": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 146, + 407, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 145, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 145, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "(a) If your work uses existing assets, did you cite the creators? 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[No]", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 146, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 146, + 479, + 160, + 490 + ], + "score": 1.0, + "content": "(d)", + "type": "text" + }, + { + "bbox": [ + 160, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "Did you discuss whether and how consent was obtained from people whose data you’re", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 162, + 490, + 253, + 501 + ], + "spans": [ + { + "bbox": [ + 162, + 490, + 253, + 501 + ], + "score": 1.0, + "content": "using/curating? [N/A]", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 147, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 147, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "(e) Did you discuss whether the data you are using/curating contains personally identifiable", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 161, + 513, + 325, + 525 + ], + "spans": [ + { + "bbox": [ + 161, + 513, + 325, + 525 + ], + "score": 1.0, + "content": "information or offensive content? [N/A]", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 132, + 528, + 433, + 540 + ], + "lines": [ + { + "bbox": [ + 128, + 526, + 433, + 542 + ], + "spans": [ + { + "bbox": [ + 128, + 526, + 433, + 542 + ], + "score": 1.0, + "content": "5. If you used crowdsourcing or conducted research with human subjects...", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 146, + 543, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 145, + 542, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 145, + 542, + 506, + 556 + ], + "score": 1.0, + "content": "(a) Did you include the full text of instructions given to participants and screenshots, if", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 161, + 554, + 237, + 566 + ], + "spans": [ + { + "bbox": [ + 161, + 554, + 237, + 566 + ], + "score": 1.0, + "content": "applicable? [N/A]", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 145, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 145, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "(b) Did you describe any potential participant risks, with links to Institutional Review", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 161, + 577, + 342, + 590 + ], + "spans": [ + { + "bbox": [ + 161, + 577, + 342, + 590 + ], + "score": 1.0, + "content": "Board (IRB) approvals, if applicable? [N/A]", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 147, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 147, + 590, + 505, + 603 + ], + "score": 1.0, + "content": "(c) Did you include the estimated hourly wage paid to participants and the total amount", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 162, + 601, + 333, + 614 + ], + "spans": [ + { + "bbox": [ + 162, + 601, + 333, + 614 + ], + "score": 1.0, + "content": "spent on participant compensation? 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[N/A]", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 144, + 243, + 424, + 256 + ], + "spans": [ + { + "bbox": [ + 144, + 243, + 424, + 256 + ], + "score": 1.0, + "content": "(b) Did you include complete proofs of all theoretical results? [N/A]", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 144, + 230, + 453, + 256 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 258, + 241, + 270 + ], + "lines": [ + { + "bbox": [ + 128, + 256, + 243, + 272 + ], + "spans": [ + { + "bbox": [ + 128, + 256, + 243, + 272 + ], + "score": 1.0, + "content": "3. 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All dataset are public from the UCI database.", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 146, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 319, + 305, + 330 + ], + "spans": [ + { + "bbox": [ + 162, + 319, + 305, + 330 + ], + "score": 1.0, + "content": "were chosen)? [Yes] See Section 6.", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 332, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 146, + 332, + 506, + 344 + ], + "score": 1.0, + "content": "(c) Did you report error bars (e.g., with respect to the random seed after running experi-", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 162, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "ments multiple times)? [No] We report only average of Monte Carlo repetitions, the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 162, + 353, + 502, + 366 + ], + "spans": [ + { + "bbox": [ + 162, + 353, + 502, + 366 + ], + "score": 1.0, + "content": "error bars were too small to have any visual impact in the reported logarithmic scale.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 146, + 365, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 146, + 365, + 505, + 380 + ], + "score": 1.0, + "content": "(d) Did you include the total amount of compute and the type of resources used (e.g., type", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 377, + 369, + 389 + ], + "spans": [ + { + "bbox": [ + 162, + 377, + 369, + 389 + ], + "score": 1.0, + "content": "of GPUs, internal cluster, or cloud provider)? 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[Yes] We use 20 datasets", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 161, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 161, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "from UCI, we cite the required papers for each dataset, mostly the UCI database and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 162, + 429, + 276, + 441 + ], + "spans": [ + { + "bbox": [ + 162, + 429, + 276, + 441 + ], + "score": 1.0, + "content": "few additional publications.", + "type": "text" + } + ], + "index": 27, + "is_list_end_line": true + }, + { + "bbox": [ + 145, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 145, + 442, + 505, + 454 + ], + "score": 1.0, + "content": "(b) Did you mention the license of the assets? [No] The data are publically available, we", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 453, + 419, + 465 + ], + "spans": [ + { + "bbox": [ + 162, + 453, + 419, + 465 + ], + "score": 1.0, + "content": "comply with the requirement on citing appropriate publications.", + "type": "text" + } + ], + "index": 29, + "is_list_end_line": true + }, + { + "bbox": [ + 146, + 465, + 503, + 478 + ], + "spans": [ + { + "bbox": [ + 146, + 465, + 503, + 478 + ], + "score": 1.0, + "content": "(c) Did you include any new assets either in the supplemental material or as a URL? [No]", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 146, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 146, + 479, + 160, + 490 + ], + "score": 1.0, + "content": "(d)", + "type": "text" + }, + { + "bbox": [ + 160, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "Did you discuss whether and how consent was obtained from people whose data you’re", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 162, + 490, + 253, + 501 + ], + "spans": [ + { + "bbox": [ + 162, + 490, + 253, + 501 + ], + "score": 1.0, + "content": "using/curating? 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Input:00,(Zi0)=1,(xi)=1 Output: (0t)t=1 for t ∈ (1,...,T) or until convergence do
form the set I = (ij)=1 by sampling (without replacement) B indices i~ (1,...,N) fori∈Ido
set zi as the last element of Zi,t-1
for j ∈(1,...,M) do
sample zi~q(zilzi) sample u~ Uniform(0,1)
compute α(zi, zi) in (7)
ifu<α(zi,zi） then
set 2,t = zi and zi = zi else
=2i
end if
end for set Zi,t = M
end for
compute (8)
compute (9) for k ∈ unique(zi,t)
compute πk,t := softmax(Vt)k for k ∈ Z end for
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Sum-product transformnetworksMixtures of real NVP flows SGD
SGDMHSAEM-UMHSAEM-U
speed-upLtestKLtestKspeed-upLtestK 32LtestK
breast-cancer-wisconsin4.66-4.66641.4310240.63-99.85-39.31128
cardiotocography10.5559.5251231.0410249.8554.343256.08128
magic-telescope102.53-3.65512-5.0310243.74-3.978-4.228
pendigits4.890.881024-4.86163844.171.4680.488
pima-indians0.37-8.5464-7.62641.35-20.09128-16.33128
wall-following-robot3.431.841024-11.31638422.21-14.26128-17.56128
waveform-14.35-26.1464-23.9110243.72-34.128-33.428
waveform-24.82-26.2164-23.9110244.12-34.158-33.648
yeast20.5710.265125.18102414.496.611289.59128
ecoli1.86-5.564-0.2210242.15-11.37128-10.64128
ionosphere1.88-20.2764-5.935122.74-87.01128-42.75128
iris0.23-10.6564-1.49163843.28-16.34128-9.2132
page-blocks12.1812.215126.84102444.9517.1312817.9432
parkinsons1.46-21.85640.55123.09-566.58128-33.3132
sonar2.96-95.39512-69.29642.52-622.2128-88.81128
statlog-segment1.4447.3551226.531638438.4935.8412842.0432
statlog-vehicle2.97-4.2564-5.4510246.78-31.3432-26.43128
wine rank75.42-25.991024-13.2710242.05-171.58128-25.57128
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EXPLANATIONS THAT ARE HARD TO VARY + +Giambattista Parascandolo1, 2, \* Alexander Neitz1, \* +Antonio Orvieto2 Luigi Gresele1, 3 Bernhard Schölkopf1, 2 +${ } ^ { 1 } \mathbf { M P I }$ for Intelligent Systems, Tübingen, 2ETH, Zürich, $^ { 3 } \mathbf { M P I }$ for Biological Cybernetics, Tübingen +˚equal contribution + +# ABSTRACT + +In this paper, we investigate the principle that good explanations are hard to vary in the context of deep learning. We show that averaging gradients across examples – akin to a logical OR $( \vee )$ of patterns – can favor memorization and ‘patchwork’ solutions that sew together different strategies, instead of identifying invariances. To inspect this, we first formalize a notion of consistency for minima of the loss surface, which measures to what extent a minimum appears only when examples are pooled. We then propose and experimentally validate a simple alternative algorithm based on a logical AND $( \wedge )$ , that focuses on invariances and prevents memorization in a set of real-world tasks. Finally, using a synthetic dataset with a clear distinction between invariant and spurious mechanisms, we dissect learning signals and compare this approach to well-established regularizers. + +# 1 INTRODUCTION + +Consider the top of Figure 1, which shows a view from above of the loss surface obtained as we vary a two dimensional parameter vector $\theta = \left( \theta _ { 1 } , \theta _ { 2 } \right)$ , for a fictional dataset containing two observations $x _ { A }$ and $x _ { B }$ . Note the two global minima on the top-right and bottom-left. Depending on the initial values of $\theta$ — marked as white circles — gradient descent converges to one of the two minima. Judging solely by the value of the loss function, which is zero in both cases, the two minima look equally good. + +However, looking at the loss surfaces for $x _ { A }$ and $x _ { B }$ separately, as shown below, a crucial difference between those two minima appears: Starting from the same initial parameter configurations and following the gradient of the loss, $\nabla _ { \boldsymbol { \theta } } \mathcal { L } ( \boldsymbol { \theta } , \boldsymbol { x } _ { i } )$ , the probability of finding the same minimum on the top-right in either case is zero. In contrast, the minimum in the lower-left corner has a significant overlap across the two loss surfaces, so gradient descent can converge to it even if training on $x _ { A }$ (or $x _ { B }$ ) only. Note that after averaging there is no way to tell what the two loss surfaces looked like: Are we destroying information that is potentially important? + +![](images/38033836d6c1b78bc80896191f206749b79f36ee8e69d1dff08f5e95a1853e34.jpg) +Figure 1: Loss landscapes of a two-parameter model. Averaging gradients forgoes information that can identify patterns shared across different environments. + +In this paper, we argue that the answer is yes. In particular, we hypothesize that if the goal is to find invariant mechanisms in the data, these can be identified by finding explanations (e.g. model parameters) that are hard to vary across examples. A notion of invariance implies something that stays the same, as something else changes. We assume that data comes from different environments: An invariant mechanism is shared across all, generalizes out of distribution (o.o.d.), but might be hard to model; each environment also has spurious explanations that are easy to spot (‘shortcuts’), but do not generalize o.o.d. From the point of view of causal modeling, such invariant mechanisms can be interpreted as conditional distributions of the targets given causal features of the inputs; invariance of such conditionals is expected if they represent causal mechanisms, that is — stable properties of the physical world (see e.g. Hoover (1990)). Generalizing o.o.d. means therefore that the predictor should perform equally well on data coming from different settings, as long as they share the causal mechanisms. + +We formalize a notion of consistency, which characterizes to what extent a minimum of the loss surface appears only when data from different environments are pooled. Minima with low consistency are ‘patchwork’ solutions, which (we hypothesize) sew together different strategies and should not be expected to generalize to new environments. An intuitive description of this principle was proposed by physicist David Deutsch: “good explanations are hard to vary” (Deutsch, 2011). + +Using the notion of consistency, we define Invariant Learning Consistency (ILC), a measure of the expected consistency of the solution found by a learning algorithm on a given hypothesis class. The ILC can be improved by changing the hypothesis class or the learning algorithm, and in the last part of the paper we focus on the latter. We then analyse why current practices in deep learning provide little incentive for networks to learn invariances, and show that standard training is instead set up with the explicit objective of greedily maximizing speed of learning, i.e., progress on the training loss. When learning “as fast as possible” is not the main objective, we show we can trade-off some “learning speed” for prioritizing learning the invariances. A practical instantiation of ILC leads to o.o.d. generalization on a challenging synthetic task where several established regularizers fail to generalize; moreover, following the memorization task from Zhang et al. (2017), ILC prevents convergence on CIFAR-10 with random labels, as no shared mechanism is present, and similarly when a portion of training labels is incorrect. Lastly, we set up a behavioural cloning task based on the game CoinRun (Cobbe et al., 2019b), and observe better generalization on new unseen levels. + +An example. Take these two second-hand books of chess puzzles. We can learn the two independent shortcuts (blue arrows for the left book OR handwritten solutions on the right), or actually learn to play chess (the invariant mechanism). While both strategies solve other problems from the same books (i.i.d.), only the latter generalises to new chess puzzle books (o.o.d.). How to distinguish the two? We would not have learned about the red arrows had we trained on the book on the right, and vice versa with the hand-written notes. + +![](images/4e958bc770fbe8a283a4b825875d313db41b01e0fd8c86aa895fd80d473209b2.jpg) + +# 2 EXPLANATIONS THAT ARE HARD TO VARY + +We consider datasets $\{ \mathcal { D } ^ { e } \} _ { e \in \mathcal { E } }$ , with $| \mathcal { E } | = d$ , and $\mathcal { D } ^ { e } = ( x _ { i } ^ { e } , y _ { i } ^ { e } )$ , $i _ { e } = 1 , \dots , n ^ { e }$ . Here $x _ { i } ^ { e } \in \mathcal { X } \subseteq \mathbb { R } ^ { m }$ is the vector containing the observed inputs, and $y _ { i } ^ { e } \in \mathcal { V } \subseteq \mathbb { R } ^ { p }$ the targets. The superscript $e \in { \mathcal { E } }$ indexes some aspect of the data collection process, and can be interpreted as an environment label. Our objective is to infer a function $f : \mathcal { X } \mathcal { Y }$ — which we call mechanism — assigning a target $y _ { i } ^ { e }$ to each input $\boldsymbol { x } _ { i } ^ { e }$ ; as explained in the introduction, we assume that such function is shared across all environments. For estimation purposes, $f$ may be parametrized by a neural network with continuous activations; for weights $\theta \in \Theta \subseteq \mathbb { R } ^ { n }$ , we denote the neural network output at $x \in \mathcal { X }$ as $f _ { \boldsymbol { \theta } } ( \boldsymbol { x } )$ . + +Gradient-based optimization. To find an appropriate model $f _ { \theta }$ , standard optimizers rely on gradients from a pooled loss function $\mathcal { L } : \mathbb { R } ^ { n } \mathbb { R }$ . This function measures the average performance ofř the neural network when predicting data labels, across all environments: ř $\begin{array} { r } { \mathcal { L } ( \boldsymbol { \theta } ) : = \dot { \frac { 1 } { | \mathcal { E } | } } \sum _ { e \in \mathcal { E } } \mathcal { L } _ { e } ( \boldsymbol { \theta } ) } \end{array}$ , with $\begin{array} { r } { \begin{array} { r } { \mathcal { L } _ { e } ( \theta ) : = \frac { 1 } { | \mathcal { D } ^ { e } | } \sum _ { ( x _ { i } ^ { e } , y _ { i } ^ { e } ) \in \mathcal { D } ^ { e } } \ell ( f ( x _ { i } ^ { e } ; \theta ) , y _ { i } ^ { e } ) } \end{array} } \end{array}$ ; where $\ell : \mathbb { R } ^ { p } \times \mathbb { R } ^ { p } \to [ 0 , + \infty )$ is usually chosen to be the $L 2$ loss or the cross-entropy loss. The parameter updates according to gradient descent (GD) are given by $\theta _ { \mathrm { G D } } ^ { k + 1 } = \theta _ { \mathrm { G D } } ^ { k } - \eta \bar { \nabla } \bar { \mathcal { L } } ( \theta _ { \mathrm { G D } } ^ { k } )$ , where $\eta > 0$ is the learning rate. Under some standard assumptions (Lee et al., 2016), $( \theta _ { \mathrm { G D } } ^ { k } ) _ { k \geqslant 0 }$ converges to a local minimizer of , with probability one. + +When do we not learn invariances? We start by describing what might prevent learning invariances in standard gradient-based optimization. + +(i) Training stops once the loss is low enough. If optimization learned spurious patterns by the time it converged, invariances will not be learned anymore. This depends on the rate at which different patterns are learned. The rates at which invariant patterns emerge (and vice-versa, the spurious patterns do not) can be improved by e.g.: (a) careful architecture design, e.g. as done by hardcoding spatial equivariance in convolutional networks; (b) fine-tuning models pre-trained on large amounts of data, where strong features already emerged and can be readily selected. + +(ii) Learning signals: everything looks relevant for a dataset of size 1. Due to the summation in the definition of the pooled loss $\mathcal { L }$ , gradients for each example are computed independently. Informally, each signal is identical to the one for an equivalent dataset of size 1, where every pattern appears relevant to the task. To find invariant patterns across examples, if we compute our training signals on each of them independently, we have to rely on the way these are aggregated.1 + +(iii) Aggregating gradients: averaging maximizes learning speed. The default method to pool gradients is the arithmetic mean. GD applied to $\mathcal { L }$ is designed to minimize the pooled loss by prioritizing descent speed.2 Indeed, a step of GD is equivalent to finding a tight3 quadratic upper bound $\hat { \mathcal { L } }$ to $\mathcal { L }$ , and then jumping to the minimizer of this approximation (Nocedal and Wright, 2006). While speed is often desirable, by construction GD ignores one potentially crucial piece of information: The gradient $\nabla \mathcal { L }$ is the result of averaging signals $\nabla { \mathcal { L } } _ { e }$ , which correspond to the patterns visible from each environment at this stage of optimization. In other words, GD with average gradients greedily maximizes for learning speed, but in some situations we would like to trade some convergence speed for invariance. + +For instance, instead of performing an arithmetic mean between gradients (logical OR), we might want to look towards a logical AND, which can be characterized as a geometric mean. Fig. 1 shows how a sum can be seen as a logical OR: the two orthogonal gradients from data $A$ and data $B$ at (0.5,0.5) point to different directions, yet both are kept in the combined gradient.4 In Sec. 2.3 we elaborate on this idea and on implementing a logical AND between gradients. Before presenting this discussion, we take some time to better motivate the need for invariant learning consistency and to construct a precise mathematical definition of consistency. + +![](images/835eb492d9c071bc01f3a6e37b6c8100e1cc95943f631aa6c9fbfe0b09d88e7c.jpg) +Figure 2: Inconsistency in gradient directions. + +# 2.1 FORMAL DEFINITION OF ILC + +Let $\Theta _ { \mathcal { A } } ^ { * }$ be the set of convergence points of algorithm $\mathcal { A }$ when trained using all environments (pooled data): that is, $\Theta _ { \mathcal { A } } ^ { \ast } = \{ \theta ^ { \ast } \in \Theta \mid \exists \theta ^ { 0 } \in \mathbb { R } ^ { n }$ s.t. $A _ { \infty } ( \theta ^ { 0 } , { \mathcal { E } } ) = \theta ^ { * } \}$ . For instance, if $\mathcal { A }$ is gradient A descent, the result of Lee et al. (2016) implies that $\Theta _ { \mathcal { A } } ^ { * }$ is the set of local minimizers of the pooled loss $\mathcal { L }$ . To each $\theta ^ { \ast } \in \Theta _ { A } ^ { \ast }$ A , we want to associate a consistency score, quantifying the concept “good $\theta ^ { * }$ Aare hard to vary”. In other words, we would like the score to capture the consistency of the loss landscape around $\theta ^ { * }$ across the different environments. For example, in Fig. 1 the loss landscape near the bottom-left minimizer is consistent across environments, while the top-right minimizer is not. + +![](images/eb9ca484544d5d5cf5c2980ba2b3999fd126d44ac36d9c5259ef498ccebf057c.jpg) + +Let us characterize the landscape around $\theta ^ { * }$ from the perspective of a fixed environment $e \in { \mathcal { E } }$ . We define the set $N _ { e , \theta ^ { * } } ^ { \epsilon }$ to be the largest path-connected region of space containing both $\theta ^ { * }$ and the set $\{ \theta \in \Theta \mathrm { s . t . } | \mathcal { L } _ { e } ( \theta ) - \mathcal { L } _ { e } ( \theta ^ { * } ) | \leqslant$ $\epsilon \}$ , with $\epsilon > 0$ . In other words, if $\theta \in N _ { e , \theta ^ { * } } ^ { \epsilon }$ then there exist a path-connected region in parameter space including $\theta ^ { * }$ and $\theta$ where each parameter also is in N e,θ˚ and its loss on environment $e$ is comparable. From the perspective of environment $e$ , all these points are equivalent to $\theta ^ { * }$ . We would like to evaluate the elements of this set with respect to a different environment ˇ $e ^ { \prime } \neq e$ . We will say that $e ^ { \prime }$ is consistent with $e$ in $\theta ^ { * }$ if $\operatorname* { m a x } _ { \theta \in N _ { e , \theta ^ { * } } ^ { \epsilon } } \mathbf { \bar { | } } \mathcal L _ { e ^ { \prime } } ( \theta ) - \mathcal L _ { e } ( \theta ) \vert$ is small. Repeating this reasoning for all environment pairs, we arrive at the following inconsistency score: + +![](images/33d19950daa9e64b7d688eda9692ea92abc78d7ced3f1ad6fae22b8df1765b16.jpg) +Loss surface for data B + +$$ +\mathcal { T } ^ { \epsilon } ( \theta ^ { * } ) : = \operatorname* { m a x } _ { ( e , e ^ { \prime } ) \in \mathcal { E } ^ { 2 } } \operatorname* { m a x } _ { \theta \in N _ { e , \theta ^ { * } } ^ { \epsilon } } | \mathcal { L } _ { e ^ { \prime } } ( \theta ) - \mathcal { L } _ { e } ( \theta ^ { * } ) | . +$$ + +This consistency is our formalization of the principle “good explanations are hard to vary”. Finally, we can write down an invariant learning consistency score for $\mathcal { A }$ : + +$$ +\operatorname { I L C } ( \mathcal { A } , p _ { \theta ^ { 0 } } ) : = - \mathbb { E } _ { \theta ^ { 0 } \sim p ( \theta ^ { 0 } ) } \left[ \mathcal { T } ^ { \epsilon } ( \mathcal { A } _ { \infty } ( \theta ^ { 0 } , \mathcal { E } ) \right] . +$$ + +That is, the learning consistency of an algorithm measures the expected consistency across environments of the minimizer it converges to on the pooled data. + +Example: low consistency of a classic patchwork solution. One-hidden-layer networks with sigmoid activations and enough neurons can approximate any function $f ^ { * } : [ 0 , \dot { 1 } ] \to \mathbb { R }$ (Cybenko, 1989). In appendix A.1 we show how the construction used to obtain the weights leads to a maximally inconsistent solution according to $\mathcal { T } ^ { \epsilon } ( \theta ^ { * } )$ , which would not be expected to generalize o.o.d. + +# 2.2 ILC AS A LOGICAL AND BETWEEN LANDSCAPES + +Here we draw a connection between our definition of inconsistency and the local geometric properties of the loss landscapes. For the sake of clarity, we consider two environments $A$ and $B$ ) and assume $\theta ^ { * }$ to be a local minimizer (with zero loss) for both environments. Using a Taylor approximation5, we get $\begin{array} { r } { \mathcal { L } ( \theta ) \approx \frac { 1 } { 2 } ( \theta - \theta ^ { * } ) ^ { \top } H _ { A + B } ( \theta - \theta ^ { * } ) } \end{array}$ for $\lVert \dot { \theta } - \theta ^ { * } \rVert \approx 0$ , where $H _ { A + B } = \left( H _ { A } + \mathbf { \bar { \mathit { H } } } _ { B } \right) / 2$ is the arithmetic mean of the Hessians $H _ { A } : = \nabla ^ { 2 } \mathcal { L } _ { A } ( \theta ^ { * } )$ and $\Dot { H _ { B } } : = \nabla ^ { 2 } \mathcal { L } _ { A } ( \theta ^ { * } )$ . $H _ { A + B }$ does not capture the possibly conflicting geometries of landscape $A$ or $B$ : It performs a “logical OR” on the dominant eigendirections. In contrast, the geometric mean, or Karcher mean, $H _ { A \land B }$ (Ando et al., 2004) is affected by the inconsistencies between landscapes: It performs a “logical AND”. In appendix A.2, we give a formal definition of $H _ { A \land B }$ and show that for diagonal Hessians, Ipθ˚q ď 2p detpHA\`BqdetpHA^Bq q + +![](images/638768a396b51f0d355670863efe7197718799d15f49a554a924db13e7580926.jpg) +.5 -1 -0.5 0 0.5 1 1.5Figure 3: Plotted are contour lines $\theta ^ { \top } H ^ { - 1 } \theta \quad = \quad 1$ for $H _ { A } \ = \ \mathrm { d i a g } ( 0 . 0 5 , 1 )$ and $H _ { B } = \mathrm { d i a g } ( 1 , 0 . 0 5 )$ . $H _ { A \land B }$ retains the original volumes, while for $H _ { A + B }$ it is $5 \times$ bigger. This magnification shows inconsistency of $A$ and $B$ . + +for the geometric mean of positive numbers, $0 \leqslant \operatorname* { d e t } ( H _ { A \wedge B } ) \leqslant \operatorname* { d e t } ( H _ { A + B } )$ ; thus, inconsistency is lowest when shapes of $A$ and $B$ are similar – exactly as in the bottom-left minimizer of Fig. 1. + +From Hessians to gradients. We just saw that the consistency of $\theta ^ { * }$ is linked to the geometric mean of the Hessians $\{ \bar { H _ { e } } ( \theta ^ { * } ) \} _ { e \in \mathcal { E } }$ . Under the simplifying assumption that each ś $H _ { e }$ is diagonal6 and allś eigenvalues $\lambda _ { i } ^ { e }$ are positive, their geometric mean is $\begin{array} { r } { H ^ { \wedge } : = \mathrm { d i a g } ( ( \prod _ { e \in \mathcal { E } } \lambda _ { 1 } ^ { e } ) ^ { 1 / | \varepsilon | } , \dots , ( \prod _ { e \in \mathcal { E } } \lambda _ { n } ^ { e } ) ^ { 1 / | \varepsilon | } ) } \end{array}$ . The curvature of the corresponding loss in the $i$ -th eigendirection depends on how consistent the curvatures of each environment are in that direction. Consider now optimizing from a point ř ř $\theta ^ { k }$ ; gradient descent reads $\theta ^ { k + 1 } = \theta ^ { k } - \eta H ^ { + } ( \theta ^ { k } - \theta ^ { * } )$ , where $\begin{array} { r } { H ^ { + } : = \mathrm { d i a g } \big ( \frac { 1 } { | \mathcal { E } | } \sum _ { e \in \mathcal { E } } \breve { \lambda _ { 1 } ^ { e } } , \dots , \frac { 1 } { | \mathcal { E } | } \sum _ { e \in \mathcal { E } } \lambda _ { n } ^ { e } \big ) } \end{array}$ . For $\eta$ small enough7, we have $\begin{array} { r } { \vert \theta _ { i } ^ { k + 1 } - \theta _ { i } ^ { * } \vert = ( 1 - \eta \frac { 1 } { \vert \mathcal { E } \vert } \sum _ { e \in \mathcal { E } } \lambda _ { i } ^ { e } ) \vert \theta _ { i } ^ { k } - \theta _ { i } ^ { * } \vert } \end{array}$ . As noted, this choice maximises the speed of convergence to $\theta ^ { * }$ , but does not take into account whether this minimizer is consistent. We can reduce the speed of convergence on directions where landscapes have different curvatures – which would lead to a high inconsistency – by following the gradients from the geometric mean of the landscapes, as opposed to the arithmetic mean. I.e, we substitute the full gradient $\nabla { \mathcal { L } } ( \theta ) = H ^ { + } ( \theta ^ { k } - \theta ^ { * } )$ with $\nabla \mathcal { L } ^ { \wedge } ( \theta ) = H ^ { \wedge } ( \theta ^ { k } - \theta ^ { * } )$ . Also, we have that8 $\begin{array} { r } { \nabla \mathcal { L } ^ { \wedge } ( \theta ) = \big ( \prod _ { e \in \mathcal { E } } \nabla \mathcal { L } _ { e } ( \theta ) \big ) ^ { 1 / | \varepsilon | } } \end{array}$ : to reduce the speed of convergence in directions with inconsistency, we can take the element-wise geometric mean of gradients from different environments (see also Fig. 11 in the appendix). + +# 2.3 MASKING GRADIENTS WITH A LOGICAL AND + +The element-wise geometric mean of gradients, instead of the arithmetic mean, increases consistency in the convex quadratic case. However, there are a few practical limitations: + +(i) The geometric mean is only defined when all the signs are consistent. It is still to be defined how sign inconsistencies, which can occur in non-convex settings, should be dealt with. + +(ii) It provides little flexibility for ‘partial’ agreement: Even a single zero gradient component in one environment stops optimization in that direction. + +(iii) For numerical stability, it needs to be computed in log domain (more computationally expensive). + +(iv) Adaptive step-size schemes (e.g. Adam (Kingma and Ba, 2015)) rescale the signal componentwise for local curvature adaptation. The exact magnitude of the geometric mean would be ignored and most of the difference from arithmetic averaging will come from the zero-ed components. + +(i) can be overcome by treating different signs as zeros, resulting in a geometric mean of 0 if there is any sign disagreement across environments for a gradient component. For (ii) we can allow for some disagreement (with a hyperparameter), by not masking out if there is a large percentage of environments with gradients in that direction. (iii) and (iv) can be addressed together: Since the final magnitude will be rescaled except for masked components, i.e. where the geometric mean is 0, we can use the average gradients (fast to compute) and mask out the components based on the sign agreement (computable avoiding the log domain). + +The AND-mask. We translate the reasoning we just presented to a practical algorithm that we will refer to as the AND-mask. In its most simple implementation, we zero out those gradient components with respect to weights that have inconsistent signs across environments. Formally, the masked gradients at iteration $k$ are $m _ { t } ( \theta ^ { k } ) \odot \nabla { \mathcal { L } } ( \theta ^ { k } )$ , where $m _ { t } ( \theta ^ { k } )$ vanishes for any component where there are less than $t \in \{ d / 2 , d / 2 + 1 , \ldots , d \}$ agreeing gradient signs across environments $d$ is the number of environments in the batch), and is equal to one otherwise. For convenience, our implementation of the AND-mask uses a threshold $\tau \in [ 0 , 1 ]$ as hyper-parameter instead of ř $t$ , such that $\begin{array} { r } { t = \frac { d } { 2 } ( \tau + 1 ) } \end{array}$ Mathematically, for every component $[ m _ { \tau } ] _ { j }$ of $m _ { \tau }$ , $\begin{array} { r } { [ m _ { \tau } ] _ { j } = \pmb { 1 } \left[ \tau d \leqslant | \sum _ { e } \mathrm { s i g n } ( [ \nabla \mathcal { L } _ { e } ] _ { j } ) | \right] } \end{array}$ . + +Computing the AND-mask has the same time and space complexity of standard gradient descent, i.e., linear in the number of examples that we average. Due to its simplicity and computational efficiency, this is the algorithm that we will use in the experiment section. As a first result, we show that following the AND-masked gradient leads to convergence in the directions made visible by the AND-mask. The proof is presented in appendix A.3. + +Proposition 1. Let $\mathcal { L }$ have $L$ -Lipschitz gradients and consider a learning rate $\eta \leqslant 1 / L$ . After $k$ iterations, AND-masked GD visits at least once a point $\theta$ where $\| m _ { t } ( \theta ) \odot \nabla \mathcal { L } ( \theta ) \| ^ { 2 } \leqslant \mathcal { O } ( 1 / k )$ . + +Behaviour in the face of randomness. Here we put the AND mask through a theoretical test: For gradients coming from different environments that are inconsistent (or even random), how fast does the AND mask reduce the magnitude of the step taken in parameter space, compared to standard GD? In case of inconsistency, the AND mask should quickly make the gradient steps more conservative. + +To assess this property, we consider a fixed set of $n$ parameters $\theta$ and gradients $\nabla { \mathcal { L } } _ { e }$ drawn independently from a multivariate Gaussian with zero mean and unit covariance. + +![](images/ad04a04ecd51008a3704226762cc27ef505decdf650968cedaac64cbe9c1e43f.jpg) +Figure 4: Magnitude of gradient (average or masked) on random data $( \left| \theta \right| =$ 3000, $t = 0 . 8 d$ ). + +Proposition 2. Consider the setting we just outlined, with $\begin{array} { r } { \mathcal { L } = ( 1 / d ) \sum _ { e = 1 } ^ { d } \mathcal { L } _ { e } } \end{array}$ . While $\mathbb { E } \| \nabla \mathcal { L } ( \theta ) \| ^ { 2 } =$ $\mathcal { O } ( n / d )$ , we have that $\forall t \in \{ d / 2 + 1 , \ldots , d \} , \exists c \in ( 1 , 2 ]$ such that $\mathbb { E } \| m _ { t } ( \theta ) \odot \nabla \mathcal { L } ( \theta ) \| ^ { 2 } \leqslant \mathcal { O } ( n / c ^ { d } )$ + +The proof is presented in Appendix A.4, and an illustration with numerical verification in Fig. 4 (the magnitudes of masked gradients (•) for more than 100 examples were always zero in the numerical verification). Intuitively, in the presence of purely random patterns, the AND-mask has a desirable property: it decreases the strength of these signals exponentially fast, as opposed to linearly. + +# 3 EXPERIMENTS + +Real-world datasets are generated by (causal) generative processes which share mechanisms (Pearl, 2009). However, mechanisms and spurious signals are often entangled, making it hard to assess what part of the learning signal is due to either. As the goal of this paper is to dissect these two components to understand how they ultimately contribute to the learning process, we create a simple synthetic dataset that allows us to control the complexity, intensity, and number of shortcuts in the data. After that, we evaluate whether spurious signals can be detected even in high-dimensional networks and datasets by testing the AND-mask on a memorization task similar to the one proposed in Zhang et al. (2017), and on a behavioral cloning task using the game CoinRun (Cobbe et al., 2019a). + +![](images/2c6666a00d98e65f93e0f8cfc43ecdc82f29522ed1a9935aadd3b95f63205d2e.jpg) +Figure 5: A 4-dimensional instantiation of the synthetic memorization dataset for visualization. Every example is a dot in both circles, and it can be classified by finding either of the “oracle” decision boundaries shown. + +# 3.1 THE SYNTHETIC MEMORIZATION DATASET + +We introduce a binary classification task. The input dimensionality is $d = d _ { M } + d _ { S }$ . While $p ( y | x _ { d _ { M } } )$ is the same across all environments (i.e. the mechanism), $p ( y | x _ { d _ { S } } , e )$ is not the same across all environments (the shortcuts). While the mechanism is shared, it needs a highly non-linear decision boundary to classify the data. The shortcuts are not shared across environments, but provide a simple way to classify the data, even when pooling all the environments together. See Figure 5 for a concrete example with $d _ { M }$ and $d _ { S }$ equal to 2, and two environments ( $A$ and $B$ ). The spirals (on $d _ { M }$ ) are invariant but hard to model. The shortcuts (on $d _ { S }$ ) are simple blobs but different in every environment: in $A$ , linearly separable through a vertical decision boundary, in $B$ with a horizontal one. If the two environments are pooled, a new diagonal decision boundary emerges on the shortcut dimensions as the most ‘natural’ one. While this perfectly classifies data in both environments $A$ and $B$ , critically it would have not been found by training on either partition $A$ or $B$ alone. The out-of-distribution (o.o.d.) test data has the same mechanism but random shortcuts. Therefore, any method relying exclusively on the shortcuts will have chance-level o.o.d. performance. Details about the dataset, baselines, and training curves are reported in appendix B. + +Despite the apparent simplicity of this dataset, note that it is challenging to find the invariant mechanism. In high dimensions, even with tens of pooled environments, the shortcuts allow for a simple classification rule under almost every classical definition of ‘simple’: the boundary is linear, it has a large margin, it can be expressed with small weights, it is fast to learn, robust to input noise, and has perfect accuracy and no i.i.d. generalization gap. Finding the complex decision boundary of the spirals, instead, is a fiddly process and arguably a much slower path towards small loss. + +Baselines. We evaluate several domain-agnostic baselines (all multilayer perceptrons) with some of the most common regularizers used in deep learning — Dropout, L1, L2, Batch normalization. We also consider methods that explicitly make use of the environment labels, namely: (i) Domain Adversarial Neural Networks (DANN) (Ganin et al., 2016), a method specifically designed to address domain adaptation by obfuscating domain information with an adversarial classifier; (ii) Invariant Risk Minimization (IRM) (Arjovsky et al., 2019), discussed in detail in appendix B. The AND-mask is trained with the same configurations in Table 1. + +Results. Fig. 6 shows training and test accuracy. DANN fails because it can align the representationlayer distributions from different environments using only shortcuts, such that they become indistinguishable to the domain-discriminating classifier. The AND-mask was the only method to achieve perfect test accuracy, by fitting the spirals instead of the shortcuts. In particular, the combination of the AND-mask with L1 or L2 regularization gave the most robust results overall, as they help suppress neurons that at initialization are tuned towards the shortcuts. + +![](images/1e0e4467ccc660e2ea7f279aa37ed5069560e29245353b3bd2c3a0696908dec2.jpg) +Figure 6: Results on the synthetic dataset. + +Correlations between average, memorization and generalization gradients. Due to the synthetic nature of the dataset, we can intervene on its data-generating process in order to examine the learning signals coming from the mechanisms and from the shortcuts. We isolate the two and measure their contribution to the average gradients, as we vary the agreement threshold of the mask. More precisely, we look at the gradients computed with respect to the weights of a randomly initialized network for different sets of data: (i) The original data, with mechanisms and shortcuts. (ii) Randomly permuting the dataset over the mechanisms dimensions, thus leaving the “memorization” signal of the shortcuts. (iii) Randomly permuting over the shortcuts dimensions, isolating the “generalization” signal of the mechanisms alone. Figure 7 shows the correlation between the components of the original average gradient (i) and the shortcut gradients ((ii), dashed line), and between the original average gradients and the mechanism gradients ((iii), solid line). + +While the signal from the mechanisms is present in the original average gradients (i.e. $\rho \approx 0 . 4$ for $\tau = 0$ ), its magnitude is smaller and it is ‘drowned’ by the memorization signal. Instead, increasing the threshold of the AND-mask (right side) suppresses memorization gradients due to the shortcuts, and for $\tau \approx 1$ most of the gradient components remaining contain signal from the mechanism. On the left side, we test the other side of our hypothesis: An XORmask zeroes out consistent gradients, preserves those with different signs, and results in a sharper decrease of the correlation with the mechanism gradients. + +![](images/f6c918dad9067762cd130aed6eaf781a1015531ba0a3e0f3f71834d439bdadd8.jpg) +Figure 7: Gradient correlations. + +# 3.2 EXPERIMENTS ON CIFAR-10 + +Memorization in a vision task. Zhang et al. (2017) showed that neural networks trained with standard regularizers — like L2 and Dropout — can still memorize large training datasets with shuffled labels, i.e. reaching $\approx 1 0 0 \%$ training accuracy. Their experiments raised significant questions about the generalization properties of neural networks and the role of regularizers in constraining the hypothesis class. Our hypothesis is that ILC — for example implemented as the AND-mask — should prevent memorization on a similar task with the shuffled labels, as gradients will tend to largely ‘disagree’ in the absence of a shared mechanism. However, when the labels are not shuffled, ILC should have a much weaker effect, as real shared mechanisms are still present in the data. + +To test our hypothesis, we ran an experiment that closely resembles the one in (Zhang et al., 2017) on CIFAR-10. We trained a ResNet on CIFAR-10 with random labels, with and without the AND-mask. In all experiments we used batch size 80, and treated each example as its own “environment”. Recall that standard gradient averaging is equivalent to an AND-mask with threshold 0. As shown in Figure 8, the ResNet with standard average gradients memorized the data, while slightly increasing the threshold for the AND-mask quickly prevented memorization (dark blue line). In contrast, training the same networks on the dataset with the original labels resulted in both of them converging and generalizing to the test set, confirming that the mask did not significantly affect the generalization error with a general underlying mechanism in the data. + +Note that there is no standard notion of environments in CIFAR10, which is why we treated every example as coming from its own environment. This assumption is not unreasonable, as every image in the dataset was literally collected in a different physical environment. If anything, it is the standard i.i.d. assumption that hides this variety behind a notion of a single distribution encompassing all environments. The results of this experiment further support this interpretation, and can serve as evidence that — in some cases — we might be able to identify invariances even without an explicit partition into environments, as this can be already identified at the level of individual examples. + +![](images/f6190599675b973b7ae2e4d3afc55b69d7b084c707dcc9dcb178991769022b01.jpg) +Figure 8: As the AND-mask threshold increases, memorization on CIFAR-10 with random labels is quickly hindered. + +Label noise. Following up on this experiment, we test how the AND-mask performs in the presence of label noise, i.e. when a portion of the labels in the training set are randomly shuffled $2 5 \%$ here). According to our hypothesis, gradients computed on examples with random labels should disagree and get masked out by the AND-mask, while signal from correctly labeled data should contribute to update the model. As shown in Figure 9, the performance on the incorrectly labeled portion of the dataset is well below chance for the AND-mask (as it predicts correctly despite the wrong labels), while the baseline again memorizes the incorrect labels. On the test set (with untouched labels), the baseline peaks early then decreases as the model overfits, while the AND-mask slowly but steadily improves. + +![](images/cd69c7bf4b0dcb56f492eb009b3fe37088a338122999723f978315d05295eae2.jpg) +Figure 9: The AND-mask prevents overfitting to the incorrectly labeled portion of the training set (left) without hurting the test accuracy (right). + +# 3.3 BEHAVIORAL CLONING ON COINRUN + +CoinRun (Cobbe et al., 2019b) is a game introduced to test how RL agents generalize to novel situations. The agent needs to collect coins, jumping on top of walls and boxes and avoiding enemies.9 Each level is procedurally generated — i.e. it has a different combination of sprites, background, and layout — but the physics and goals are invariant. Cobbe et al. (2019b) showed that state-of-the-art RL algorithms fail to model these invariant mechanisms, performing poorly on new levels unless trained on thousands of them. To test our hypothesis, we set up a behavioral cloning task using CoinRun.10 We start by pre-training a strong policy $\pi ^ { * }$ using standard PPO (Schulman et al., 2017) for 400M steps on the full distribution of levels. We then generate a dataset of pairs $( s , \pi ^ { * } ( a | s ) )$ from the on-policy distribution. The training data consists of 1000 states from each of 64 levels, while test data comes from 2000 levels. A ResNet- $1 8 ~ \hat { \pi } _ { \boldsymbol { \theta } }$ is then trained to minimize the loss $D _ { \mathrm { K L } } ( \pi ^ { * } | | \hat { \pi } _ { \boldsymbol { \theta } } )$ on the training set. We compare the generalization performance of regular Adam to a version that uses the AND-mask. For each method we ran an automatic hyperparameter optimization study using Tree-structured Parzen Estimation (Bergstra et al., 2013) of 1024 trials. + +Despite the theoretical computational efficiency of computing the ANDmask as presented in Section 2.3 (i.e., linear time and memory in the size of the mini-batch, just like classic SGD), current deep learning frameworks like PyTorch (Paszke et al., 2017) have optimized routines that sum gradients across examples in a mini-batch before it is possible to efficiently compute the AND-mask. We therefore test the AND-mask in a slightly different way. In training, in each iteration we sample a batch of data from a randomly chosen level out of the 64 available (and cycle through them all once per epoch). We then apply the AND-mask ‘temporally’, only allowing gradients that are consistent across time (and therefore across levels). See Algorithm 1 in appendix B.6 for a detailed description of this alternative formulation of the AND-mask. The figure shows the minimum test loss for the 10 best runs, supporting the hypothesis that the AND-mask helps identify invariant mechanisms across different levels. + +![](images/f98c901e8f6899d5e94e32da3effe7ac6f1e48a2f2b95afa4d08c00b4e8b8a48.jpg) + +# 4 RELATED WORK + +Generalization and covariate shift. The classic formulation of statistical learning theory (Vapnik) concerns learning from independent and identically distributed samples. The case where the distribution of the covariates at test time differs from the one observed during training is termed covariate shift (Sugiyama et al., 2007; Quionero-Candela et al., 2009; Sugiyama and Kawanabe, 2012). Standard solutions involve re-weighting of the training examples, but require the additional assumption of overlapping supports for train and test distributions. + +Causal models and invariances. As we mentioned in the Introduction, causality provides a strong motivation for our work, based on the notion that statistical dependencies are epiphenomena of an underlying causal model (Pearl, 2009; Peters et al., 2017). The causal description identifies stable elements – e.g. physical mechanisms – connecting causes and effects, which are expected to remain invariant under interventions or changing external conditions (Haavelmo, 1943; Schölkopf et al., 2012)). This motivates our notion of invariant mechanisms, and inspired related notions which have been proposed for robust regression (Rojas-Carulla et al., 2018; Heinze-Deml et al., 2018; Arjovsky et al., 2019; Hermann and Lampinen, 2020; Ahuja et al., 2020; Krueger et al., 2020). We discuss this in more detail in appendix C.1. + +Domain generalization. ILC can be used in a setting of domain generalization (Muandet et al., 2013), but it is not limited to it: as demonstrated in the experiments in Section 3.2, the AND-mask can be applied even if domain labels are not available. In contrast, by treating every example as a single domain, methods relying on domain classifiers (like DANN Ganin et al. (2016) or Balaji et al. (2018)) would require as many output units as there are training examples (i.e. $5 0 \mathrm { ^ { \circ } 0 0 0 }$ for CIFAR-10). + +Gradient agreement. Looking at gradient agreement to learn meaningful representations in neural networks has been explored in (Du et al., 2018; Eshratifar et al., 2018; Fort et al., 2019; Zhang et al., 2019b). These approaches mainly rely on a measure of cosine similarity between gradients, which we did not consider here for two main reasons: $( i )$ It is a ‘global’ property of the gradients, and it would not allow us to extract precise information about different patterns in the network; $( i i )$ It is unclear how to extend it beyond pairs of vectors, and for pairwise interactions its computational cost scales quadratic in the number of examples used. + +# 5 CONCLUSIONS + +Generalizing out of distribution is one of the most significant open challenges in machine learning, and relying on invariances across environments or examples may be key in certain contexts. In this paper we analyzed how neural networks trained by averaging gradients across examples might converge to solutions that ignore the invariances, especially if these are harder to learn than spurious patterns. We argued that if learning signals are collected on one example at the time — as it is the case for gradients, e.g., computed with backpropagation — the way these signals are aggregated can play a significant role in the patterns that will ultimately be expressed: Averaging gradients in particular can be too permissive, acting as a logical $O R$ of a collection of distinct patterns, and lead to a ‘patchwork’ solution. We introduced and formalized the concept of Invariant Learning Consistency, and showed how to learn invariances even in the face of alternative explanations that — although spurious — fulfill most characteristics of a good solution. The AND-mask is but one of multiple possible ways to improve consistency, and it is unlikely to be a practical algorithm for all applications. However, we believe this should not distract from the general idea which we are trying to put forward — namely, that it is worthwhile to study learning of explanations that are hard to vary, with the longer term goal of advancing our understanding of learning, memorization and generalization. + +# ACKNOWLEDGMENTS + +We wish to thank Sebastian Gomez, Luca Biggio, Julius von Kügelgen, Paolo Penna, Ioannis Anagno, Ricards Marcinkevics, Sidak Pal Singh, Damien Teney for feedback on the manuscript, and thank Nando de Freitas for fruitful discussions in the early stage of this project. We also thank the Max Planck ETH Center for Learning Systems for supporting Giambattista Parascandolo, and the International Max Planck Research School for Intelligent Systems for supporting Alexander Neitz. + +# REFERENCES + +L. Adolphs, J. Kohler, and A. Lucchi. 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URL http://proceedings.mlr.press/ v97/zhang19q.html. + +# A APPENDIX TO SECTION 2 + +A.1 A CLASSIC EXAMPLE OF A PATCHWORK SOLUTION + +Consider a neural network with one hidden layer consisting of two neurons and sigmoidal activations: + +$$ +f _ { \theta } ( x ) = \theta _ { 5 } \sigma ( \theta _ { 1 } x + \theta _ { 2 } ) + \theta _ { 6 } \sigma ( \theta _ { 3 } x + \theta _ { 4 } ) , \quad \sigma ( z ) : = 1 / ( 1 + e ^ { - z } ) . +$$ + +We want to learn the continuous function $f ^ { * } : [ 0 , 1 ] \to [ 0 , 2 ]$ defined as + +$$ +f ^ { * } ( x ) = \left\{ \begin{array} { l l } { 0 } & { x \in [ 0 , 0 . 4 ) ; } \\ { 1 0 ( x - 0 . 4 ) } & { x \in [ 0 . 4 , 0 . 5 ) ; } \\ { 1 } & { x \in [ 0 . 5 , 0 . 7 ) ; } \\ { 1 0 ( x - 0 . 7 ) + 1 } & { x \in [ 0 . 7 , 0 . 8 ) ; } \\ { 2 } & { x \in [ 0 . 8 , 1 ] . } \end{array} \right. +$$ + +To perform this task, we have access to (noiseless) data from two environments: + +$$ +A : \{ ( x , f ( x ) ) \mid x \in [ 0 , 0 . 5 ) \} , \quad B : \{ ( x , f ( x ) ) \mid x \in [ 0 . 5 , 1 ] \} . +$$ + +There is a simple constructive way, provided by the universal function approximation theorem Cybenko (1989) to fit this function11 using $f _ { \theta }$ up to an arbitrarily small mean squared error $\mathcal { L } _ { A + B } ( \theta ^ { * } )$ . Leaving out the details of such a construction (Cybenko (1989) for details), the reader can check on the left panel of Figure 10 that $\theta ^ { * } = ( 1 0 0 , - 5 0 , 1 0 0 , - 7 5 , 1 , 1 )$ provides a good fit for both environments A and B — both ${ \mathcal { L } } _ { A } ( \theta ^ { * } )$ and ${ \mathcal { L } } _ { B } ( \theta ^ { * } )$ are small. + +![](images/eb35dbc64d91d70df2fc12fa12cfa328de9a0881fab9199faceabaf62636ab02.jpg) +Figure 10: Performance of the neural network in Equation 3 for two different parameters. Any reasonable modification on $\theta _ { 6 }$ (say $\pm 1$ ) leaves the performance on environment A unchanged, while the performance on environment B quickly degrades. + +However, it is easy to realize that $\theta ^ { * }$ — while being a solution which can be returned by gradient descent using the pooled data $\mathbf { A } { + } \mathbf { B }$ — is not consistent (formal definition given in the main paper in Section 2). Indeed, it is possible to modify $\tilde { \theta } ^ { * }$ such that the loss in environment A remains almost unchanged, while the loss in environment B gets larger. In particular, on the right panel of Figure 10, we show that $\tilde { \theta } ^ { * } = ( 1 0 0 , - 5 0 , 1 0 0 , - 7 5 , 1 , - 0 . 5 )$ is such that $\mathcal { L } _ { A } ( \theta ^ { * } ) \leqslant \bar { \mathcal { L } _ { A } ( \tilde { \theta } ^ { * } ) } + \epsilon$ (with $\epsilon$ very small) but $\mathcal { L } _ { B } ( \theta ^ { * } ) \ll \mathcal { L } _ { B } ( \tilde { \theta } ^ { * } )$ . According to our definition in Equation 1 (see main paper), we have ${ \cal T } ^ { \epsilon } ( \theta ^ { * } ) \leq | { \mathcal L } _ { B } ( \theta ^ { * } ) - { \mathcal L } _ { B } ( \tilde { \theta } ^ { * } ) |$ — that is a large number (low consistency). + +Remark 1 (Connection to out of distribution generalization). The main point of this analysis was to show an example of where our measure of consistency behaves according to expectations: A typical implementation of the universal approximation theorem — which one would not expect to generalize out of distribution, due to its ‘patchwork’ behavior — leads indeed to a very low consistency score. + +Geometric mean of matrices. Given an $n$ -tuple of $d \times d$ positive definite matrices $( A _ { j } ) _ { j = 1 } ^ { n }$ , the ean Ando et al. (2004) is the unique positive definite solution, where log is the matrix logarithm. This matrix average he it relevant to signal processing and medical imaging. T $X$ to the equationmany desirable Karcher mean +${ \ k { \stackrel { \smile } { \sum } } _ { i = 1 } ^ { m } } \log ( \dot { A } _ { i } ^ { - 1 } X ) = 0$ +can also be written as $\begin{array} { r } { \operatorname * { a r g m i n } _ { X \in S ^ { + + } ( d ) } \bar { f } ( X ) = \frac { 1 } { 2 m } \bar { \sum _ { i = 1 } ^ { m } } d ( A _ { i } , X ) ^ { 2 } } \end{array}$ , where $d$ is the Riemannian +distance in the manifold of SPD matrices $S ^ { + + } ( d )$ . + +![](images/ad592e94eca81ce1c9da75f174ffeb9e7cd7d2510f2e4b3249bdbae3b184dfa4.jpg) +Figure 11: While the arithmetic mean of the two loss surfaces on the left is identical in all three cases (third column), the geometric mean has weaker and weaker gradients (black arrow) the more inconsistent the two loss surfaces become. + +Link between consistency and geometric means. Here we show how the consistency score introduced in Equation 1 can be linked (in a simplified setting) to a comparison between the arithmetic and geometric means of the Hessians approximating the landscapes of two separate environments $A$ and $B$ . + +At the local minimizer $\theta ^ { * } = 0$ , we assume that $\mathcal { L } _ { A } = \mathcal { L } _ { B } = 0$ and consider the local quadratic approximations $\begin{array} { r } { \ \mathcal { L } _ { A } ( \theta ) \ = \ \frac { 1 } { 2 } \theta ^ { \top } H _ { A } \theta } \end{array}$ and $\begin{array} { r } { \ \mathcal { L } _ { B } ( \theta ) \ = \ \frac { 1 } { 2 } \theta ^ { \top } H _ { B } \theta } \end{array}$ . Here, we make the additional simplifying assumption that $H _ { A }$ and $H _ { B }$ are diagonal (or, more broadly, co-diagonalizable): $H _ { A } \mathbf { \bar { \Sigma } } = \mathrm { d i a g } ( \lambda _ { 1 } ^ { A } , \cdot \cdot \cdot , \lambda _ { n } ^ { A } )$ , $H _ { B } = \mathrm { d i a g } ( \lambda _ { 1 } ^ { B } , \cdot \cdot \cdot , \lambda _ { n } ^ { B } )$ , with $\lambda _ { i } ^ { A } \geqslant 0$ and $\lambda _ { i } ^ { B } \geqslant 0$ for all $i = 1 , \ldots , n$ The arithmetic and geometric means (noted as $H _ { A + B }$ and $H _ { A \land B }$ ) of these matrices are defined in this simplified setting as follows: + +$$ +H _ { A + B } = \mathrm { d i a g } \left( \frac { 1 } { 2 } ( \lambda _ { 1 } ^ { A } + \lambda _ { 1 } ^ { B } ) , \cdots , \frac { 1 } { 2 } ( \lambda _ { n } ^ { A } + \lambda _ { n } ^ { B } ) \right) , \quad H _ { A \wedge B } = \mathrm { d i a g } \left( \sqrt { \lambda _ { 1 } ^ { A } \lambda _ { 1 } ^ { B } } , \cdots , \sqrt { \lambda _ { n } ^ { A } \lambda _ { n } ^ { B } } \right) . +$$ + +As motivated in the main paper and in Figure 12, one can link the consistency of two landscapes to a comparison between the geometric and arithmetic means of the corresponding Hessians. + +Proposition 3. In the setting we just described, the consistency score in Equation 1 can be estimated as follows: + +$$ +{ \mathcal { T } } ^ { \epsilon } ( \theta ^ { * } ) \leqslant 2 \epsilon \left( { \frac { \operatorname* { d e t } \bigl ( H _ { A + B } \bigr ) } { \operatorname* { d e t } \bigl ( H _ { A \wedge B } \bigr ) } } \right) ^ { 2 } . +$$ + +![](images/18902103c5165a84f4ea044ec9a748571660c465a1a120f77d810e8def45c95e.jpg) +Figure 12: Plotted are contour lines $\theta ^ { \top } H ^ { - 1 } \theta = 1$ for $H _ { A } = \mathrm { d i a g } ( 0 . 0 1 , 1 )$ and $H _ { B } = \mathrm { d i a g } ( 1 , 0 . 0 1 )$ . It is convenient to provide this visualization because it is linked to the matrix determinant: a $\mathrm { V o l } ( \{ \theta ^ { \top } H ^ { - 1 } \theta = 1 \} ) =$ $\pi { \sqrt { \operatorname* { d e t } ( H ) } }$ . The geometric average retains the volume of the original ellipses, while the volume of $H _ { A + B }$ is 25 times bigger. This magnification indicates that landscape $A$ is not consistent with landscape $B$ . + +Before showing the proof, we note that the proposition gives a lower bound on the consistency. That is, it provides a pessimistic estimate. Yet, as we motivated, this estimate has a nice geometric interpretation. However, as we outline in a remark after the proof, this estimate is tight in two important limit cases. + +Proof. In this setting, Equation 1 gives + +$$ +\mathcal { T } ^ { \epsilon } ( \theta ^ { * } ) : = \operatorname* { m a x } \left\{ \operatorname* { m a x } _ { \mathcal { L } _ { A } ( \theta ) \leqslant \epsilon } \mathcal { L } _ { B } ( \theta ) , \operatorname* { m a x } _ { \mathcal { L } _ { B } ( \theta ) \leqslant \epsilon } \mathcal { L } _ { A } ( \theta ) \right\} . +$$ + +Recall that + +$$ +\mathcal { L } _ { A } ( \theta ) = \frac { 1 } { 2 } \theta ^ { \top } H _ { A } \theta = \frac { 1 } { 2 } \sum _ { i } \lambda _ { i } ^ { A } \theta _ { i } ^ { 2 } . +$$ + +Hence, this is a simple quadratic program with quadratic constraints, and + +$$ +\operatorname* { m a x } _ { \mathcal { L } _ { A } ( \theta ) \leqslant \epsilon } \mathcal { L } _ { B } ( \theta ) = \operatorname* { m a x } _ { \frac { 1 } { 2 } \sum _ { i } \lambda _ { i } ^ { A } \theta _ { i } ^ { 2 } \leqslant \epsilon } \frac { 1 } { 2 } \sum _ { i } \lambda _ { i } ^ { B } \theta _ { i } ^ { 2 } . +$$ + +Further, we can change variables and introduce $\tilde { \theta } _ { i } = \theta _ { i } \sqrt { \lambda _ { i } ^ { A } / 2 }$ . The problem gets even simpler: + +$$ +\operatorname* { m a x } _ { \mathcal { L } _ { A } ( \theta ) \leqslant \epsilon } \mathcal { L } _ { B } ( \theta ) = \operatorname* { m a x } _ { \| \tilde { \theta } \| ^ { 2 } \leqslant \epsilon } \sum _ { i } \frac { \lambda _ { i } ^ { B } } { \lambda _ { i } ^ { A } } \tilde { \theta } _ { i } ^ { 2 } = \epsilon \cdot \operatorname* { m a x } _ { i } \frac { \lambda _ { i } ^ { B } } { \lambda _ { i } ^ { A } } . +$$ + +All in all, we get + +$$ +\begin{array} { r l } { T ^ { \epsilon } ( \theta ^ { * } ) = \epsilon \operatorname* { m a x } \Bigg \{ \underset { i } { \operatorname* { m a x } } \frac { \lambda _ { i } ^ { B } } { \lambda _ { i } ^ { A } } , \underset { i } { \operatorname* { m a x } } \frac { \lambda _ { i } ^ { A } } { \lambda _ { i } ^ { B } } \Bigg \} } & { } \\ { = \epsilon \cdot \underset { i } { \operatorname* { m a x } } \operatorname* { m a x } \left\{ \frac { \lambda _ { i } ^ { B } } { \lambda _ { i } ^ { A } } , \frac { \lambda _ { i } ^ { A } } { \lambda _ { B } ^ { B } } \right\} } & { } \\ { \leqslant \epsilon \cdot \underset { i } { \operatorname* { m a x } } \left( \frac { \lambda _ { i } ^ { B } } { \lambda _ { A } ^ { A } } + \frac { \lambda _ { i } ^ { A } } { \lambda _ { B } ^ { B } } \right) } & { } \\ { = \epsilon \cdot \underset { i } { \operatorname* { m a x } } \left\{ \frac { ( \lambda _ { i } ^ { B } ) ^ { 2 } + ( \lambda _ { i } ^ { A } ) ^ { 2 } } { \lambda _ { B } ^ { B } \lambda _ { i } ^ { A } } \right\} } & { } \\ { \leqslant \epsilon \cdot \underset { i } { \operatorname* { m a x } } \left\{ \frac { ( \lambda _ { i } ^ { B } + \lambda _ { i } ^ { A } ) ^ { 2 } } { \lambda _ { B } ^ { B } \lambda _ { i } ^ { A } } \right\} . } \end{array} +$$ + +This means + +$$ +\sqrt { { \cal T } ^ { \epsilon } ( \theta ^ { * } ) } \leqslant \epsilon \operatorname* { m a x } _ { i } \frac { \lambda _ { i } ^ { B } + \lambda _ { i } ^ { A } } { \sqrt { \lambda _ { i } ^ { B } \lambda _ { i } ^ { A } } } = 2 \epsilon \operatorname* { m a x } _ { i } \frac { ( \lambda _ { i } ^ { B } + \lambda _ { i } ^ { A } ) / 2 } { \sqrt { \lambda _ { i } ^ { B } \lambda _ { i } ^ { A } } } \leqslant 2 \epsilon \frac { \prod _ { i } ( \lambda _ { i } ^ { B } + \lambda _ { i } ^ { A } ) / 2 } { \prod _ { i } \sqrt { \lambda _ { i } ^ { B } \lambda _ { i } ^ { A } } } = 2 \epsilon \frac { \operatorname* { d e t } ( H _ { A + B } ) } { \operatorname* { d e t } ( H _ { A \wedge B } ) } , +$$ + +where the first inequality comes from the monotonicity of the square root function, and the second inequality comes from the fact that (i) the geometric mean is always smaller or equal than theś arithmetic mean and (ii) for any sequence of numbers $\alpha _ { i } > 1$ , $\begin{array} { r } { \operatorname* { m a x } _ { i } \dot { \alpha _ { i } } \leqslant \prod _ { i } \alpha _ { i } } \end{array}$ . □ + +Remark 2 (Sanity check). There are two important cases where we can test the bound above. First, if $H _ { A } = H _ { B }$ , then $T ^ { \epsilon } ( \theta ^ { * } ) = \epsilon$ , and the bound returns $T ^ { \epsilon } ( \theta ^ { * } ) \leqslant 2 \epsilon$ , since the geometric and arithmetic mean are the same. Next, say $\lambda _ { i } ^ { A } = 0$ but $\lambda _ { i } ^ { B } > 0$ ; then, both the bound and the inconsistency score are $\infty$ (highest possible inconsistency). + +# A.3 PROOF OF PROPOSITION 1 + +In this appendix section we consider the AND-masked GD algorithm, introduced at the end of Section 2. We recall that the masked gradients at iteration $k$ are $\mathsf { \bar { \rho } } m _ { t } ( \theta ^ { k } ) \odot \nabla \mathcal { L } ( \theta ^ { k } )$ , where $m _ { t } ( \theta ^ { k } )$ vanishes for any component where there are less than $t \in \{ d / 2 + 1 , \ldots , d \}$ agreeing gradient signs across environments, and is equal to one otherwise. In a full-batch setting, the algorithm is + +$$ +\theta ^ { k + 1 } = \theta ^ { k } - \eta m _ { t } ( \theta ^ { k } ) \odot \nabla { \mathcal { L } } ( \theta ^ { k } ) , +$$ + +(AND-masked GD) + +where $\eta > 0$ is the learning rate. + +Proposition 1. Let $\mathcal { L }$ have $L$ -Lipschitz gradients and consider a learning rate $\eta \leqslant 1 / L$ . After $k$ iterations, AND-masked GD visits at least once a point $\theta$ where $\| m _ { t } ( \theta ) \odot \nabla \mathcal { L } ( \theta ) \| ^ { 2 } \leqslant \mathcal { O } ( 1 / k )$ . + +Proof. Thanks to the component-wise $L$ -smoothness and using a Taylor expansion around $\theta ^ { i }$ we have + +$$ +\begin{array} { l } { \displaystyle \mathcal { L } ( \theta ^ { i + 1 } ) \leqslant \mathcal { L } ( \theta ^ { i } ) - \eta \langle \nabla \mathcal { L } ( \theta ^ { i } ) , m _ { t } ( \theta ^ { i } ) \odot \nabla \mathcal { L } ( \theta ^ { i } ) \rangle + \displaystyle \frac { L \eta ^ { 2 } } { 2 } \| m _ { t } ( \theta ^ { i } ) \odot \nabla \mathcal { L } ( \theta ^ { i } ) \| ^ { 2 } } \\ { \displaystyle \qquad = \mathcal { L } ( \theta ^ { i } ) - \left( \eta - \frac { L \eta ^ { 2 } } { 2 } \right) \| m _ { t } ( \theta ^ { i } ) \odot \nabla \mathcal { L } ( \theta ^ { i } ) \| ^ { 2 } . } \end{array} +$$ + +If we seek $\eta - L \eta ^ { 2 } / 2 \geqslant \eta / 2$ , then $\begin{array} { r } { \eta \leqslant \frac { 1 } { L } } \end{array}$ , as we assumed in the proposition statement. Therefore, $\mathcal { L } ( \theta ^ { i + 1 } ) \leqslant \mathcal { L } ( \theta ^ { i } ) - ( \eta / 2 ) \| m _ { t } ( \theta ^ { i } ) \odot \nabla \mathcal { L } ( \theta ^ { i } ) \| ^ { 2 }$ , for all $i \geqslant 0$ . Summing over $i$ from 0 to a desired iteration $k$ , we get + +$$ +\sum _ { i = 0 } ^ { k - 1 } ( \eta / 2 ) \| m _ { t } ( \theta ^ { i } ) \odot \nabla \mathcal { L } ( \theta ^ { i } ) \| ^ { 2 } \leqslant \mathcal { L } ( \theta ^ { 0 } ) - \mathcal { L } ( \theta ^ { k } ) \leqslant \mathcal { L } ( \theta ^ { 0 } ) . +$$ + +Therefore, + +$$ +\operatorname* { m i n } _ { i = 0 , \dots , k } \| m _ { t } ( \theta ^ { i } ) \odot \nabla { \mathcal { L } } ( \theta ^ { i } ) \| ^ { 2 } \leqslant \frac { 1 } { k } \sum _ { i = 0 } ^ { k - 1 } ( \eta / 2 ) \| m _ { t } ( \theta ^ { i } ) \odot \nabla { \mathcal { L } } ( \theta ^ { i } ) \| ^ { 2 } \leqslant \frac { 2 { \mathcal { L } } ( \theta ^ { 0 } ) } { \eta k } . +$$ + +Hence, there exist an iteration $i ^ { * } \in \{ 0 , \ldots , k \}$ such that $\| m _ { t } ( { \theta ^ { i } } ^ { * } ) \odot \nabla { \mathcal { L } } ( { \theta ^ { i } } ^ { * } ) \| ^ { 2 } \leqslant { \mathcal { O } } ( 1 / k )$ + +# A.4 PROOF OF PROPOSITION 2 + +Here we fix parameters $\boldsymbol \theta \in \mathbb { R } ^ { n }$ and assume gradients $\nabla \mathcal { L } _ { e } ( \theta ) \in \mathbb { R } ^ { n }$ coming from environments $e \in \mathcal { E }$ are drawn independently from a multivariate Gaussian with zero mean and $\sigma ^ { 2 } I$ covariance. We want to show that, in this random setting, the AND-mask introduced in Section 2.3 decreases the magnitude of the gradient step. + +Proposition 2. Consider the setting we just outlined, with $\begin{array} { r } { \mathcal { L } = ( 1 / d ) \sum _ { e = 1 } ^ { d } \mathcal { L } _ { e } } \end{array}$ . While $\mathbb { E } \| \nabla \mathcal { L } ( \theta ) \| ^ { 2 } =$ $\mathcal { O } ( n / d )$ , we have that $\forall t \in \{ d / 2 + 1 , \ldots , d \} , \exists c \in ( 1 , 2 ]$ such that $\mathbb { E } \| m _ { t } ( \theta ) \odot \nabla \mathcal { L } ( \theta ) \| ^ { 2 } \leqslant \mathcal { O } ( n / c ^ { d } )$ + +Proof. Let us drop the argument $\theta$ for ease of notation. First, let us consider $\nabla \mathcal { L }$ (no gradient AND-mask): + +$$ +\mathbb { E } \left\| \frac { 1 } { d } \sum _ { i = 1 } ^ { d } \nabla { \mathcal { L } } _ { e _ { i } } \right\| ^ { 2 } = \frac { 1 } { d ^ { 2 } } \sum _ { i = 1 } ^ { d } \mathbb { E } \| \nabla { \mathcal { L } } _ { e _ { i } } \| ^ { 2 } = \frac { n \sigma ^ { 2 } } { d } , +$$ + +where in the first equality we used the fact that the $\nabla { \mathcal { L } } _ { e _ { i } }$ are uncorrelated and in the second the fact that $\mathbb { E } [ \| \nabla { \mathcal { L } } _ { e _ { i } } \| ^ { 2 } ]$ is the trace of the covariance of $\nabla { \mathcal { L } } _ { e _ { i } }$ . + +Next, assume we apply the element-wise AND-mask $m _ { t }$ to the gradients, which puts to zero the components (dimensions) where there are less than $t \in \{ d / 2 , \ldots , d \}$ equal signs. Since Gaussians are symmetric around zero, the probability of having exactly \` ˘ \` ˘ $u$ positive $j$ -th gradient component among $d$ environments is $\begin{array} { r } { P r ( p _ { j } = u ) = \left( \frac { 1 } { 2 } \right) ^ { d } { \binom { d } { u } } } \end{array}$ . Hence, the probability to keep the $j$ -th gradient direction (considering also negative consistency) is + +$$ +\begin{array} { l } { \displaystyle \operatorname* { P r } [ [ m _ { t } ] _ { j } = 1 ] = \sum _ { u = t } ^ { d } \operatorname* { P r } ( p _ { j } = u ) + \sum _ { u = 0 } ^ { d - t } \operatorname* { P r } ( p _ { j } = u ) } \\ { \displaystyle \quad = \bigg ( \frac { 1 } { 2 } \bigg ) ^ { d } \sum _ { k = t } ^ { d } \binom { d } { k } + \bigg ( \frac { 1 } { 2 } \bigg ) ^ { d } \sum _ { k = 0 } ^ { d - t } \binom { d } { k } } \\ { \displaystyle \quad = 2 \bigg ( \frac { 1 } { 2 } \bigg ) ^ { d } \sum _ { k = t } ^ { d } \binom { d } { k } . } \end{array} +$$ + +We would now like to compute $\begin{array} { r } { \mathbb { E } \left\| m _ { t } \odot \left( \frac { 1 } { d } \sum _ { i = 1 } ^ { d } \nabla { \mathcal { L } } _ { e _ { i } } \right) \right\| ^ { 2 } } \end{array}$ . The difficulty lies in the fact that the event $m _ { t } = 1$ makes gradients conditionally dependent. Indeed, conditioning on both $m _ { t } = 1$ and $[ \nabla { \mathcal { L } } _ { e } ] _ { j } > 0$ changes the distribution of $[ \nabla { \mathcal { L } } _ { e ^ { \prime } } ] _ { j }$ : this gradient entry is going to be more likely to be positive or negative, depending on the value of $[ \nabla { \mathcal { L } } _ { e } ] _ { j }$ and on the details of the gradient mask. To solve the issue, we our strategy is to reduce the discussion (without loss in generality and with no additional assumption) to the case where gradient entries have all the same sign and hence conditional independence is restored. + +We consider the following writing for the quantity we are interested in: + +$$ +\begin{array} { r l } { \mathbb { E } \| m _ { t } \operatorname { \wp } ( \displaystyle \frac { 1 } { d } \sum _ { i = 1 } ^ { d } \nabla { \mathcal { L } } _ { \epsilon _ { \epsilon } } ) \| ^ { 2 } = \displaystyle \sum _ { j = 1 } ^ { n } \mathbb { E } [ [ m _ { \epsilon } ] _ { j } ( \displaystyle \frac { 1 } { d } \sum _ { i = 1 } ^ { d } [ \nabla { \mathcal { L } } _ { \epsilon _ { \epsilon } } ] _ { i } ) ^ { 2 } ] } & { } \\ { = \displaystyle \sum _ { j = 1 } ^ { n } \sum _ { \tilde { p } _ { \tilde { \imath } } = 0 } ^ { d } \mathbb { E } [ [ m _ { \epsilon } ] _ { j } ( \displaystyle \frac { 1 } { d } \sum _ { i = 1 } ^ { d } [ \nabla { \mathcal { L } } _ { \epsilon _ { \epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \hat { p } _ { j } ] \mathbb { P } \mathrm { r } [ p _ { j } = \hat { p } _ { j } ] } \\ { = \displaystyle \sum _ { j = 1 } ^ { n } \sum _ { \tilde { p } _ { \tilde { \jmath } } = 0 } ^ { d - 1 } \sum _ { \tilde { p } = 0 } ^ { d } \mathbb { E } [ ( \displaystyle \frac { 1 } { d } \sum _ { i = 1 } ^ { d } [ \nabla { \mathcal { L } } _ { \epsilon _ { \epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \hat { p } _ { j } ] \mathbb { P } \mathrm { r } [ p _ { j } = \hat { p } _ { j } ] } \\ { = \displaystyle 2 \sum _ { j = 1 } ^ { n } \sum _ { \tilde { p } _ { \tilde { \jmath } } = - 1 } ^ { d } \mathbb { E } [ ( \displaystyle \frac { 1 } { d } \sum _ { i = 1 } ^ { d } [ \nabla { \mathcal { L } } _ { \epsilon _ { \epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \hat { p } _ { j } ] ( \displaystyle \frac { 1 } { 2 } ) ^ { d } ( \displaystyle \frac { d } { \hat { p } _ { j } } ) , } \end{array} +$$ + +where we used the definition of 2-norm, the law of total expectation, and the symmetry of the problem with respect to positive and negative numbers. Finally, since the gradient components within the same environment are conditionally independent, for any $j \in \{ 1 , \ldots , n \}$ we can write + +$$ +\mathbb { E } \left. m _ { t } \odot \left( \frac { 1 } { d } \sum _ { i = 1 } ^ { d } \nabla \mathcal { L } _ { e _ { i } } \right) \right. ^ { 2 } = 2 n \sum _ { \hat { p } _ { j } = t } ^ { d } \mathbb { E } \left[ \left( \frac { 1 } { d } \sum _ { i = 1 } ^ { d } [ \nabla \mathcal { L } _ { e _ { i } } ] _ { j } \right) ^ { 2 } \bigg | p _ { j } = \hat { p } _ { j } \right] \left( \frac { 1 } { 2 } \right) ^ { d } { \binom { d } { \hat { p } _ { j } } } . +$$ + +Finally, we note that the following bound holds: + +$$ +\mathbb { E } \left[ \left( \frac { 1 } { d } \sum _ { i = 1 } ^ { d } [ \nabla \mathcal { L } _ { e _ { i } } ] _ { j } \right) ^ { 2 } \middle | p _ { j } = \hat { p } _ { j } \leqslant d \right] \leqslant \mathbb { E } \left[ \left( \frac { 1 } { d } \sum _ { i = 1 } ^ { d } [ \nabla \mathcal { L } _ { e _ { i } } ] _ { j } \right) ^ { 2 } \middle | p _ { j } = d \right] . +$$ + +Indeed, if all environments lead to positive (or, symmetrically, negative) and non-interacting gradients in the $j$ -th direction, the average will be the biggest in norm. Moreover — crucially — conditioned on the event $p _ { j } \ = \ d$ , gradients coming from different environments are distributed as a positive half-normal distributions. Moreover, they are conditionally independent; this because, since they are all positive, the value of a gradient in one environment cannot influence the value of the gradient in another one. We remark that conditional independence on the right-hand side is therefore not an assumption, but is intrinsic to the upper bound. + +Putting it all together, we have + +$$ +\begin{array} { r l r } { \mathbb { E } \left\| m _ { t } \odot \left( \displaystyle \frac { 1 } { d } \sum _ { i = 1 } ^ { d } \nabla \mathcal { L } _ { e _ { i } } \right) \right\| ^ { 2 } \leqslant 2 n \displaystyle \sum _ { \hat { p } _ { j } = t } ^ { d } \mathbb { E } \left[ \left( \displaystyle \frac { 1 } { d } \sum _ { i = 1 } ^ { d } [ \nabla \mathcal { L } _ { e _ { i } } ] _ { j } \right) ^ { 2 } \bigg | p _ { j } = d \right] \left( \displaystyle \frac { 1 } { 2 } \right) ^ { d } \left( \displaystyle \frac { d } { \hat { p } _ { j } } \right) } & \\ & { \leqslant 2 n \displaystyle \sum _ { \hat { p } _ { j } = t } ^ { d } \sigma ^ { 2 } \left( \displaystyle \frac { 1 } { 2 } \right) ^ { d } \left( \displaystyle \frac { d } { \hat { p } _ { j } } \right) } & \\ & { \leqslant \sigma ^ { 2 } n ( d - t ) \Big ( \displaystyle \frac { d } { t } \Big ) \left( \displaystyle \frac { 1 } { 2 } \right) ^ { d - 1 } , } & \end{array} +$$ + +where in the second line we bounded the squared average of a sum of half normal distributions: let $\{ X _ { i } \} _ { i = 1 } ^ { d }$ be a family of uncorrelated positive half-normal distributions derived from a Gaussians witha mean zero and variance $\sigma ^ { 2 }$ , we have12 that $\mathbb { E } [ X _ { i } ] = \sigma { \sqrt { 2 / \pi } }$ and $\mathbb { E } [ X _ { i } ^ { 2 } ] = \sigma ^ { 2 }$ . Also, $\mathbb { E } [ X _ { i } X _ { j } ] =$ $\mathbb { E } [ X _ { i } ] \mathbb { E } [ X _ { j } ] \leqslant \sigma ^ { 2 }$ . Therefore, + +$$ +{ \mathbb E } \left[ \left( \frac { 1 } { d } \sum _ { i = 1 } ^ { d } X _ { i } \right) ^ { 2 } \right] = \frac { 1 } { d ^ { 2 } } \sum _ { i , j = 1 } ^ { d } { \mathbb E } [ X _ { i } X _ { j } ] \leqslant \sigma ^ { 2 } . +$$ + +Finally, if we set $r = t / d \in ( 0 . 5 , 1 ]$ , we have13 + +$$ +{ \binom { d } { t } } \sim \left( { \frac { 1 } { r ^ { r } ( 1 - r ) ^ { 1 - r } } } \right) ^ { d } +$$ + +as $d \to \infty$ (discarding all polynomial terms). Hence $\binom { d } { t }$ is of the form $q ^ { d }$ , with $1 \leqslant q < 2$ . So, the quantity σ2npd ´ tq dt 12 d´1 w ill be exponentially decreasing at a rate $O ( n / ( 2 - q ) ^ { d } )$ . Notably, if $t = d / 2$ , then we lose the exponential rate and get back to $\mathcal { O } ( n / d )$ . □ + +# B APPENDIX TO SECTION 3 + +We used Pytorch Paszke et al. (2017) to implement all experiments in this paper. Our codebase is publicly available at https://github.com/gibipara92/ learning-explanations-hard-to-vary. + +# B.1 SECTION 3.1 + +Table 1: Hyperparameter ranges for synthetic data experiments. The regularizers L1 and L2 are never combined; instead, one weight regularization type out of L1, L2 and none is selected and we sample from the respective range afterwards. + +
HyperparameterRanges
No.hidden units{256,512}
No.hidden layers{3,5}
Batch-size{64,128,256}
Optimizer{Adamβ1 =0.9,β2=0.99, SGD + momentum0.9}
Learning rate{1e-3,1e-2,1e-1}
Batch-normalization{Yes,No}
Dropout{0.0, 0.5}
L2 regularization{1e-5,1e-4,1e-3}
L1 regularization{1e-6, 1e-5, 1e-4}
+ +# B.2 DATASET + +Here we report more technical details about the synthetic dataset described in Section 3. Each example is constructed as follows: we first choose the label randomly to be either $+ 1$ or $- 1$ , with equal probability. The example is a vector with $d _ { S } + d _ { M }$ entries, consisting of the shortcut and the mechanism. In our experiments, $d _ { M } = 2$ and $d _ { S } = 3 2$ . + +The Gaussian shortcuts are obtained by first sampling one random vector $\mathbf { x } _ { s } \in \mathbb { R } ^ { d _ { S } }$ per environment. Its components $x _ { s , i }$ are sampled independently from a Normal distribution: $x _ { s , i } \sim \mathcal { N } ( 0 , 0 . 1 )$ . We use $\mathbf { x } _ { s }$ for class 1, and $- \mathbf { x } _ { s }$ for class -1. In the test set, all shortcut components are sampled i.i.d. from the same Normal distribution. Effectively, each example of the test set belongs to a different domain. The mechanism is implemented as the two interconnected spirals shown in Figure 13 by sampling the radius $r \sim \mathrm { U n i f } ( 0 . 0 8 , 1 . 0 )$ and then computing the angle as $\alpha = 2 \pi n r$ where $n$ is the number of revolutions of the spiral. We add uniform noise in the range $\left[ - 0 . 0 2 , 0 . 0 2 \right]$ to the radii afterwards. + +![](images/2703df68fff546c00ca1b4b0275d73937f6cdf748f6bd7fabe5dde2e5d70d63d.jpg) +Figure 13: The spirals used as the mechanism in the synthetic memorization dataset. + +The training dataset consists of 1280 examples per environment and we use $D = 3 2$ environments unless otherwise mentioned. The training datasets consists of 2000 examples. + +# B.3 EXPERIMENT + +We train all networks for $\lfloor 3 0 0 0 / D \rfloor$ epochs, dropping the learning rate by a factor 10 halfway through, and again at three-quarters of training. For computational reason, we stop each trial before completion if the training accuracy exceeds $9 7 \%$ and the test accuracy is below $60 \%$ . All networks are MLPs with LeakyReLU activation functions and a cross-entropy loss on the output. We run a hyperparameter search over the ranges shown in Table 1. For IRM and the AND-mask, we select the best-performing run and re-run it 50 times with different random seeds. For DANN and the standard baselines nothing produced results significantly better than chance. + +# B.3.1 STANDARD REGULARIZERS AND AND-MASK + +The networks with the L1, L2, Dropout and Batch-normalization regularizers, have hyperparameters that were randomly selected from Table 1. For the AND-mask we used the very same ranges. The regularizers L1 and L2 are never combined; instead, one weight regularization type out of L1, L2 and none is selected and we sample from the respective range afterwards. The parameters found to work best from the grid search were: agreement threshold of 1, 256 hidden units, 3 hidden layers, batch size 128, Adam with learning rate 1e-2, no batch norm, no dropout, L2-regularization with a coefficient of 1e-4, no L1-regularization. In practice, we often found it helpful to rescale the gradients after masking to compensate for the decreasing overall magnitude. We add the option for gradient rescaling as an additional hyperparameter, as we found it to help in several experiments. It rescales gradient components layer-wise after masking, by multiplying the remaining gradient components by $c$ , where $c$ is the ratio of the number of components in that layer over the number of non-masked components in that layer (i.e. the sum of the binary elements in the mask).14. We speculate that for very large layers, a less extreme normalization scheme or the additional use of gradient clipping might be appropriate. + +# B.3.2 DOMAIN ADVERSARIAL NEURAL NETWORKS + +The experiments using DANN follow a similar pattern. The model consists of an embedding network, a classification network, and a “domain discrimination” network. All three modules are two-layer multi-layer perceptrons (MLP). The number of hidden units of all MLPs are sampled from the range specified in Table 1, and we trained 100 models. Both label classifier and domain discriminator are applied to the output of the embedding network. The label classifier is trained to minimize the cross-entropy-loss between the predicted and the true label. Similarly, the domain discriminator is trained to minimize the loss between predicted and true domain-label. The embedding network is trained to minimize the regular task classification loss and at the same time to maximize the the domain-loss achieved by the domain discriminator. + +# B.3.3 INVARIANT RISK MINIMIZATION + +For the experiments using IRM we used the authors’ PyTorch implementation from https: //github.com/facebookresearch/InvariantRiskMinimization. We perform a random hyperparameter search over with the ranges shown in Table 2 + +Table 2: Hyperparameter ranges for IRM. + +
HyperparameterRanges
No.hidden units{256,512}
No.hidden layers{3,5}
Batch-size{64,128,256}
Optimizer{Adamβ1=0.9,β2=0.999, SGD + momentum0.9}
Batch-normalization{Yes,No}
Penalty weight{10.0,100.0,1000.0}
Number of annealing iterations{0,1,2,4,8}
Learning rate{1e-3,1e-2, 1e-1, 1}
+ +# B.3.4 CURVES FOR ALL EXPERIMENTS + +In Figure 14 we show the learning curves of training and test accuracy for the different methods. + +# B.3.5 CORRELATION PLOTS + +For the correlation plots in Figure 7 we used a randomly initialized MLP with the following configuration: 3 hidden layers, 256 hidden units. The dataset was using 16 environments and batches of size 1024. The lines in Figure 7 are linear least-squares regressions to the gradient data shown as scatter plots. We repeat the experiment 10 times with different network weight seeds, resulting in the 10 regression lines. Zero gradients are excluded from the regression computation, as most gradients are masked out by the product mask in both cases. + +![](images/f1c54ea4ad51704480b75ad6c14459bd6db2dfafae7f409ff4810c75f7d4bc0c.jpg) +Figure 14: Learning curves for the evaluated methods. The top row shows the accuracy on the training set, the bottom row shows the accuracy on the test set. + +# B.4 FURTHER VISUALIZATIONS AND EXPERIMENTS + +In Figure 15 we show how many environments need to be present for the baseline without AND-mask to switch the decision boundary from the shortcuts to the mechanism. Under the same experimental condition as in the main paper, the baseline first succeeds at 1024 environments. + +![](images/4ec4fe862dad3a66e23c73556d1743d289326e46a0506be20524dc68015f4975.jpg) +Figure 15: Relationship between number of training environments and test accuracy for the AND-mask method compared to the baseline. We show the best performance out of five runs using the settings that were used for the experiment in the main text. + +# B.5 SECTION 3.2: CIFAR-10 MEMORIZATION AND LABEL NOISE EXPERIMENTS + +Memorization experiment In Figure 16, we report the test performance (dashed lines) corresponding to the curves presented in the main paper for the CIFAR-10 memorization experiment. The test performance with standard labels decreases slower than the training performance as the threshold increases, and they eventually reach the same value. This is consistent with the hypothesis that by training on the consistent directions, the AND-mask selects the invariant patterns and prunes out the signals that are not invariant. + +![](images/8ca71fe755e2a88b3312a7912c3fb6fe9a5bde54794c3813b8551085371d1d41.jpg) +Figure 16: Dashed lines show test acc, solid lines show training acc. + +# Network architecture and training details Each trial + +trains the ResNet “FastResNet” from the PyTorch-Ignite example15 for 80 epochs on the full CIFAR-10 training set. We use the Adam optimizer with a learning rate of 5e´4, and a 0.1 learning rate decay at epoch 40 and 60. We fix the batch size to 80. We set up 14 trials by evaluating each of the AND-mask-thresholds $\{ 0 , 0 . 0 5 , 0 . 1 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 \}$ for two datasets: (a) unchanged CIFAR-10, (b) CIFAR-10 with the training labels replaced by random labels. Note that a threshold of 0 corresponds to not using the AND-mask. Each trial is run twice with separate random seeds. + +Label noise experiment We trained the same ResNet as for the experiment above, once with and once without the AND-mask. We ran each experiment with three different starting learning rates $\{ 5 \mathrm { e } { - } 4 , 1 \mathrm { e } { - } 3 , 5 \mathrm { e } { - } 3 \}$ and a learning rate decay at epoch 60. The baseline worked best with a learning rate of $\mathrm { 1 e { - 3 } }$ , while the AND-mask with 5e´3, likely to compensate for the masked out gradients. The AND-mask threshold that worked best was 0.2, which is consistent with the results obtain in the experiment above. + +# B.6 SECTION 3.3: BEHAVIORAL CLONING ON COINRUN + +The target policy $\pi ^ { * }$ is obtained by training PPO (Schulman et al., 2017) for 400M time steps using the code16 for the paper Cobbe et al. (2020). This policy is trained on the full distribution of levels in order to maximize its generality. We use $\pi ^ { * }$ to generate a behavioral cloning (BC) dataset, consisting of pairs $( s , \pi ^ { * } ( a | s ) )$ , where $s$ are the input-images ( $6 4 \times 6 4$ RGB) and $\pi ^ { * } ( a | s )$ is the discrete probability distribution over actions output by $\pi ^ { * }$ . + +The states are sampled randomly from trajectories generated by $\pi ^ { * }$ . In order to test for generalization performance, the BC training dataset is restricted to 64 distinct levels. We generate 1000 examples per training level. The test set consists of 2000 examples, each from a different level which does not appear in the training set. + +![](images/31904abde0e46daf3cc3ab7cd2a873c086e55035d97a7d88b06b70d71bcc27ed.jpg) +Figure 17: Screenshots of 6 levels of CoinRun (from OpenAI). + +A ResNet- $1 8 ~ \hat { \pi } _ { \boldsymbol { \theta } }$ is trained to minimize the loss $D _ { \mathrm { K L } } ( \pi ^ { * } | | \hat { \pi } _ { \boldsymbol { \theta } } )$ . We ran two automatic hyperparameter optimization studies using Tree-structured Parzen Estimation (TPE) (Bergstra et al., 2013) of 1024 trials each, with and without the AND-mask. The learning rate was decayed by a factor of 10 half-way at at $3 / _ { 4 }$ of the training epochs. + +The “temporal” version of the AND-mask used for this experiment is reported in Algorithm 1. + +# Algorithm 1: Temporal AND-mask Adam + +1 $\mathbf { m } \gets \beta _ { 1 } \cdot \mathbf { m } + ( 1 - \beta _ { 1 } ) \cdot \mathbf { g }$ +2 $\mathbf { v } \beta _ { 2 } \cdot \mathbf { v } + ( 1 - \beta _ { 2 } ) \cdot ( \mathbf { g } \circ \mathbf { g } )$ +3 $\mathbf { a } \beta _ { 3 } \cdot \mathbf { a } + ( 1 - \beta _ { 3 } ) \cdot \mathsf { e l e m w i s e \_ s i g n } ( \mathbf { g } )$ +4 $\mathbf { b } \gets \mathbb { 1 } [ | \mathbf { a } | \geqslant \tau ]$ +5 $\theta \gets \theta - \alpha ( \mathbf { m } \circ \mathbf { b } ) \oslash \sqrt { \mathbf { v } + \epsilon }$ + +In blue we highlight the additional lines compared to traditional Adam. The threshold $\tau$ and $\beta _ { 3 }$ are hyperparameters that we included in the $1 ^ { \cdot } 0 2 4$ trials of the search using Tree-structured Parsen Estimators. For the top 10 runs, hyperparameter values that were selected via the TPE search for the AND-mask are the following. + +Table 3: Hyperparameters for the 5 best runs using the AND-mask, from the TPE search. + +
Test KL divlrβ1β3Tweight decay
1.652e-20.00780.210.790.360.057
1.656e-20.00720.260.860.400.041
1.662e-20.00800.230.840.410.045
1.665e-20.00680.330.720.470.077
1.672e-20.00630.670.650.470.080
+ +We found that applying weight decay as a second independent update after the AND-mask routine improved performance. To keep the comparison fair, we added this as a switch in the hyperparameter search for the Adam baseline as well, and it improved performance there as well. + +![](images/58e7efa834f50663cd23c9bf7b8b533a909ff815082f83abdc760db4f08069bd.jpg) +Figure 18: Learning curves for the behavioral cloning experiment on CoinRun. Training loss is shown on the left, test loss is shown on the right. We show the mean over the top-10 runs for each method. The shaded regions correspond to the $9 5 \%$ confidence interval of the mean based on bootstrapping. + +# C APPENDIX TO SECTION 4 + +# C.1 RELATED WORK IN CAUSAL INFERENCE + +Causal graphs and causal factorizations The formalization of causality through directed acyclic graphs (Pearl, 2009) is a key element informing our exposition. According to such formalization, a causal model gives rise to each observed distribution. It is thereby possible to exploit properties of the causal factorization of the joint probability distribution over the observed variables. Clearly, there are many ways to factorize a joint distribution into conditionals; a distinguishing feature of the causal factorization is that many of the conditionals, which we can think of as physical mechanisms underlying the statistical dependencies represented, are expected to remain invariant under interventions or changing external conditions. This postulate has appeared in various forms in the literature (Haavelmo, 1943; Simon, 1953; Hurwicz, 1962; Pearl, 2009; Schölkopf et al., 2012).17 + +Causal models and robust regression Based on this insight, it was proposed that regression based on causal features should presents desirable invariance and robustness properties (Mooij et al., 2009; Schölkopf et al., 2012; Peters et al., 2016; Rojas-Carulla et al., 2018; Heinze-Deml et al., 2018; von Kügelgen et al., 2019; Parascandolo et al., 2018). In this view, the mechanisms can be considered as features of the patterns such that they support stable conditional probabilities. Thus learning the mechanisms may help achieve a stable performance across a number of conditions. Other works connecting causality and learning through invariances are (Subbaswamy et al., 2019; Heinze-Deml and Meinshausen, 2017), and perhaps – most related to our work – (Arjovsky et al., 2019): we presented a comparison with this method in the following section. + +Causal regularization Recently (Janzing, 2019) showed that biasing learning towards models of lower complexity might in some cases be beneficial for a notion of generalization from observational to interventional regimes. Our proposed solution is however different, in that we only indirectly deal with penalizing model complexity, and rather focus on our proposed notion of consistency. + +# C.2 LEARNING INVARIANCES IN THE DATA + +Here we are going to compare ILC to other approaches for learning invariances in the data with neural networks, and in particular to Invariant Risk Minimization (IRM) Arjovsky et al. (2019). The authors of IRM analyze a set up where minimizing training error might lead to models which absorb all the correlations found within the training data, thus failing to recover the relevant causal explanation. They consider a multi-environment setting and focus on the objective of extracting data representations that lead to invariant prediction across environments. + +While the high level objective is close to the one we focused on, the differences become clear when considering the definition of invariant predictors presented in Arjovsky et al. (2019): + +Definition 1. A data representation $\Phi : \mathcal { X } \to \mathcal { H }$ elicits an invariant predictor $w \circ \Phi$ across environments $\mathcal { E }$ if there is a classifier $w : \mathcal { H } \to \mathcal { V }$ simultaneously optimal for all environments, i.e., $w \in \arg \operatorname* { m i n } _ { \bar { w } : \mathcal { H } y } R ^ { e } ( \bar { w } \circ \Phi ) \forall e \in \mathcal { E }$ . + +In particular, the objective minimized by IRM is: + +$$ +\operatorname* { m i n } _ { \Phi : \mathcal { X } \to \mathcal { V } } \sum _ { e \in \mathcal { E } _ { \mathrm { t r } } } R ^ { e } ( \Phi ) + \lambda \cdot \left\| \nabla _ { w | w = 1 . 0 } R ^ { e } ( w \cdot \Phi ) \right\| ^ { 2 } +$$ + +where $\Phi$ are the logits predicted by the neural network and $w$ is a dummy scaling variable (see› Arjovsky et al. (2019)). The relevant part is the penalty term $\lambda \cdot \left\| \nabla _ { w | w = 1 . 0 } R ^ { e } ( w \cdot \Phi ) \right\| ^ { 2 }$ : One way to interpret it, is that the penalty is large on every environment where the distribution outputted by $\Phi$ could be made ‘closer’ to the distribution of the labels by either sharpening $\mathit { w } > 1 $ ) or softening it (i.e., closer to uniform $w < 1$ ). + +Let us consider the example from IRM, where the authors describe two datasets of images that each contain either a cow or a camel: In one of the datasets, there is grass on $80 \%$ of the images with cows, while in the other dataset there is grass on $90 \%$ of them. IRM then makes the point that we can learn to ignore grass as a feature, because its correlation with the label cow is inconsistent ( $80 \%$ vs $90 \%$ ). The setting we consider in this paper is slightly different: take our example from the CIFAR-10 experiments. Under our concept of invariance, we expect that (depending on the data generating process) even a single dataset where we treat every image as coming from its own ‘environment’ should be sufficient to discover invariances. Drawing a connection to the setting from IRM, we would argue that the second dataset should not be necessary to learn that ‘grass’ is not ‘cow’. If one treats every example as coming from its own environment, there is already sufficient information in the first dataset to realize that cows are not grass: Grass is predictive of cows only in $80 \%$ of the data, so grass cannot be ‘cow’. The actual cow on the other hand, should be present in $100 \%$ of the images, and as such it is the invariance we are looking for. Note that this is of course a much more strict definition of invariance: If our dataset contains images labeled as ’cows’ but that have no cows within them, we might start to discard the features of cows as well. \ No newline at end of file diff --git a/parse/train/hb1sDDSLbV/hb1sDDSLbV_content_list.json b/parse/train/hb1sDDSLbV/hb1sDDSLbV_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..24b20ddaa196bf5382ddde1a02751b4f68dd336c --- /dev/null +++ b/parse/train/hb1sDDSLbV/hb1sDDSLbV_content_list.json @@ -0,0 +1,3167 @@ +[ + { + "type": "text", + "text": "LEARNING EXPLANATIONS THAT ARE HARD TO VARY ", + "text_level": 1, + "bbox": [ + 174, + 101, + 808, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Giambattista Parascandolo1, 2, \\* Alexander Neitz1, \\* \nAntonio Orvieto2 Luigi Gresele1, 3 Bernhard Schölkopf1, 2 \n${ } ^ { 1 } \\mathbf { M P I }$ for Intelligent Systems, Tübingen, 2ETH, Zürich, $^ { 3 } \\mathbf { M P I }$ for Biological Cybernetics, Tübingen \n˚equal contribution ", + "bbox": [ + 181, + 142, + 792, + 202 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 452, + 220, + 544, + 236 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we investigate the principle that good explanations are hard to vary in the context of deep learning. We show that averaging gradients across examples – akin to a logical OR $( \\vee )$ of patterns – can favor memorization and ‘patchwork’ solutions that sew together different strategies, instead of identifying invariances. To inspect this, we first formalize a notion of consistency for minima of the loss surface, which measures to what extent a minimum appears only when examples are pooled. We then propose and experimentally validate a simple alternative algorithm based on a logical AND $( \\wedge )$ , that focuses on invariances and prevents memorization in a set of real-world tasks. Finally, using a synthetic dataset with a clear distinction between invariant and spurious mechanisms, we dissect learning signals and compare this approach to well-established regularizers. ", + "bbox": [ + 232, + 248, + 766, + 401 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 424, + 336, + 439 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Consider the top of Figure 1, which shows a view from above of the loss surface obtained as we vary a two dimensional parameter vector $\\theta = \\left( \\theta _ { 1 } , \\theta _ { 2 } \\right)$ , for a fictional dataset containing two observations $x _ { A }$ and $x _ { B }$ . Note the two global minima on the top-right and bottom-left. Depending on the initial values of $\\theta$ — marked as white circles — gradient descent converges to one of the two minima. Judging solely by the value of the loss function, which is zero in both cases, the two minima look equally good. ", + "bbox": [ + 174, + 450, + 537, + 590 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "However, looking at the loss surfaces for $x _ { A }$ and $x _ { B }$ separately, as shown below, a crucial difference between those two minima appears: Starting from the same initial parameter configurations and following the gradient of the loss, $\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } ( \\boldsymbol { \\theta } , \\boldsymbol { x } _ { i } )$ , the probability of finding the same minimum on the top-right in either case is zero. In contrast, the minimum in the lower-left corner has a significant overlap across the two loss surfaces, so gradient descent can converge to it even if training on $x _ { A }$ (or $x _ { B }$ ) only. Note that after averaging there is no way to tell what the two loss surfaces looked like: Are we destroying information that is potentially important? ", + "bbox": [ + 174, + 597, + 535, + 736 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/38033836d6c1b78bc80896191f206749b79f36ee8e69d1dff08f5e95a1853e34.jpg", + "image_caption": [ + "Figure 1: Loss landscapes of a two-parameter model. Averaging gradients forgoes information that can identify patterns shared across different environments. " + ], + "image_footnote": [], + "bbox": [ + 560, + 445, + 820, + 665 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 178, + 736, + 820, + 750 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we argue that the answer is yes. In particular, we hypothesize that if the goal is to find invariant mechanisms in the data, these can be identified by finding explanations (e.g. model parameters) that are hard to vary across examples. A notion of invariance implies something that stays the same, as something else changes. We assume that data comes from different environments: An invariant mechanism is shared across all, generalizes out of distribution (o.o.d.), but might be hard to model; each environment also has spurious explanations that are easy to spot (‘shortcuts’), but do not generalize o.o.d. From the point of view of causal modeling, such invariant mechanisms can be interpreted as conditional distributions of the targets given causal features of the inputs; invariance of such conditionals is expected if they represent causal mechanisms, that is — stable properties of the physical world (see e.g. Hoover (1990)). Generalizing o.o.d. means therefore that the predictor should perform equally well on data coming from different settings, as long as they share the causal mechanisms. ", + "bbox": [ + 174, + 757, + 825, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We formalize a notion of consistency, which characterizes to what extent a minimum of the loss surface appears only when data from different environments are pooled. Minima with low consistency are ‘patchwork’ solutions, which (we hypothesize) sew together different strategies and should not be expected to generalize to new environments. An intuitive description of this principle was proposed by physicist David Deutsch: “good explanations are hard to vary” (Deutsch, 2011). ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Using the notion of consistency, we define Invariant Learning Consistency (ILC), a measure of the expected consistency of the solution found by a learning algorithm on a given hypothesis class. The ILC can be improved by changing the hypothesis class or the learning algorithm, and in the last part of the paper we focus on the latter. We then analyse why current practices in deep learning provide little incentive for networks to learn invariances, and show that standard training is instead set up with the explicit objective of greedily maximizing speed of learning, i.e., progress on the training loss. When learning “as fast as possible” is not the main objective, we show we can trade-off some “learning speed” for prioritizing learning the invariances. A practical instantiation of ILC leads to o.o.d. generalization on a challenging synthetic task where several established regularizers fail to generalize; moreover, following the memorization task from Zhang et al. (2017), ILC prevents convergence on CIFAR-10 with random labels, as no shared mechanism is present, and similarly when a portion of training labels is incorrect. Lastly, we set up a behavioural cloning task based on the game CoinRun (Cobbe et al., 2019b), and observe better generalization on new unseen levels. ", + "bbox": [ + 174, + 180, + 825, + 361 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "An example. Take these two second-hand books of chess puzzles. We can learn the two independent shortcuts (blue arrows for the left book OR handwritten solutions on the right), or actually learn to play chess (the invariant mechanism). While both strategies solve other problems from the same books (i.i.d.), only the latter generalises to new chess puzzle books (o.o.d.). How to distinguish the two? We would not have learned about the red arrows had we trained on the book on the right, and vice versa with the hand-written notes. ", + "bbox": [ + 181, + 372, + 509, + 501 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/4e958bc770fbe8a283a4b825875d313db41b01e0fd8c86aa895fd80d473209b2.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 521, + 372, + 818, + 492 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 179, + 497, + 774, + 512 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 EXPLANATIONS THAT ARE HARD TO VARY ", + "text_level": 1, + "bbox": [ + 174, + 542, + 555, + 558 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We consider datasets $\\{ \\mathcal { D } ^ { e } \\} _ { e \\in \\mathcal { E } }$ , with $| \\mathcal { E } | = d$ , and $\\mathcal { D } ^ { e } = ( x _ { i } ^ { e } , y _ { i } ^ { e } )$ , $i _ { e } = 1 , \\dots , n ^ { e }$ . Here $x _ { i } ^ { e } \\in \\mathcal { X } \\subseteq \\mathbb { R } ^ { m }$ is the vector containing the observed inputs, and $y _ { i } ^ { e } \\in \\mathcal { V } \\subseteq \\mathbb { R } ^ { p }$ the targets. The superscript $e \\in { \\mathcal { E } }$ indexes some aspect of the data collection process, and can be interpreted as an environment label. Our objective is to infer a function $f : \\mathcal { X } \\mathcal { Y }$ — which we call mechanism — assigning a target $y _ { i } ^ { e }$ to each input $\\boldsymbol { x } _ { i } ^ { e }$ ; as explained in the introduction, we assume that such function is shared across all environments. For estimation purposes, $f$ may be parametrized by a neural network with continuous activations; for weights $\\theta \\in \\Theta \\subseteq \\mathbb { R } ^ { n }$ , we denote the neural network output at $x \\in \\mathcal { X }$ as $f _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } )$ . ", + "bbox": [ + 173, + 570, + 825, + 670 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Gradient-based optimization. To find an appropriate model $f _ { \\theta }$ , standard optimizers rely on gradients from a pooled loss function $\\mathcal { L } : \\mathbb { R } ^ { n } \\mathbb { R }$ . This function measures the average performance ofř the neural network when predicting data labels, across all environments: ř $\\begin{array} { r } { \\mathcal { L } ( \\boldsymbol { \\theta } ) : = \\dot { \\frac { 1 } { | \\mathcal { E } | } } \\sum _ { e \\in \\mathcal { E } } \\mathcal { L } _ { e } ( \\boldsymbol { \\theta } ) } \\end{array}$ , with $\\begin{array} { r } { \\begin{array} { r } { \\mathcal { L } _ { e } ( \\theta ) : = \\frac { 1 } { | \\mathcal { D } ^ { e } | } \\sum _ { ( x _ { i } ^ { e } , y _ { i } ^ { e } ) \\in \\mathcal { D } ^ { e } } \\ell ( f ( x _ { i } ^ { e } ; \\theta ) , y _ { i } ^ { e } ) } \\end{array} } \\end{array}$ ; where $\\ell : \\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { p } \\to [ 0 , + \\infty )$ is usually chosen to be the $L 2$ loss or the cross-entropy loss. The parameter updates according to gradient descent (GD) are given by $\\theta _ { \\mathrm { G D } } ^ { k + 1 } = \\theta _ { \\mathrm { G D } } ^ { k } - \\eta \\bar { \\nabla } \\bar { \\mathcal { L } } ( \\theta _ { \\mathrm { G D } } ^ { k } )$ , where $\\eta > 0$ is the learning rate. Under some standard assumptions (Lee et al., 2016), $( \\theta _ { \\mathrm { G D } } ^ { k } ) _ { k \\geqslant 0 }$ converges to a local minimizer of , with probability one. ", + "bbox": [ + 173, + 681, + 825, + 791 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "When do we not learn invariances? We start by describing what might prevent learning invariances in standard gradient-based optimization. ", + "bbox": [ + 173, + 804, + 825, + 833 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(i) Training stops once the loss is low enough. If optimization learned spurious patterns by the time it converged, invariances will not be learned anymore. This depends on the rate at which different patterns are learned. The rates at which invariant patterns emerge (and vice-versa, the spurious patterns do not) can be improved by e.g.: (a) careful architecture design, e.g. as done by hardcoding spatial equivariance in convolutional networks; (b) fine-tuning models pre-trained on large amounts of data, where strong features already emerged and can be readily selected. ", + "bbox": [ + 173, + 840, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(ii) Learning signals: everything looks relevant for a dataset of size 1. Due to the summation in the definition of the pooled loss $\\mathcal { L }$ , gradients for each example are computed independently. Informally, each signal is identical to the one for an equivalent dataset of size 1, where every pattern appears relevant to the task. To find invariant patterns across examples, if we compute our training signals on each of them independently, we have to rely on the way these are aggregated.1 ", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "(iii) Aggregating gradients: averaging maximizes learning speed. The default method to pool gradients is the arithmetic mean. GD applied to $\\mathcal { L }$ is designed to minimize the pooled loss by prioritizing descent speed.2 Indeed, a step of GD is equivalent to finding a tight3 quadratic upper bound $\\hat { \\mathcal { L } }$ to $\\mathcal { L }$ , and then jumping to the minimizer of this approximation (Nocedal and Wright, 2006). While speed is often desirable, by construction GD ignores one potentially crucial piece of information: The gradient $\\nabla \\mathcal { L }$ is the result of averaging signals $\\nabla { \\mathcal { L } } _ { e }$ , which correspond to the patterns visible from each environment at this stage of optimization. In other words, GD with average gradients greedily maximizes for learning speed, but in some situations we would like to trade some convergence speed for invariance. ", + "bbox": [ + 173, + 180, + 826, + 294 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For instance, instead of performing an arithmetic mean between gradients (logical OR), we might want to look towards a logical AND, which can be characterized as a geometric mean. Fig. 1 shows how a sum can be seen as a logical OR: the two orthogonal gradients from data $A$ and data $B$ at (0.5,0.5) point to different directions, yet both are kept in the combined gradient.4 In Sec. 2.3 we elaborate on this idea and on implementing a logical AND between gradients. Before presenting this discussion, we take some time to better motivate the need for invariant learning consistency and to construct a precise mathematical definition of consistency. ", + "bbox": [ + 174, + 294, + 653, + 417 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/835eb492d9c071bc01f3a6e37b6c8100e1cc95943f631aa6c9fbfe0b09d88e7c.jpg", + "image_caption": [ + "Figure 2: Inconsistency in gradient directions. " + ], + "image_footnote": [], + "bbox": [ + 671, + 303, + 818, + 383 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 FORMAL DEFINITION OF ILC ", + "text_level": 1, + "bbox": [ + 176, + 444, + 415, + 458 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $\\Theta _ { \\mathcal { A } } ^ { * }$ be the set of convergence points of algorithm $\\mathcal { A }$ when trained using all environments (pooled data): that is, $\\Theta _ { \\mathcal { A } } ^ { \\ast } = \\{ \\theta ^ { \\ast } \\in \\Theta \\mid \\exists \\theta ^ { 0 } \\in \\mathbb { R } ^ { n }$ s.t. $A _ { \\infty } ( \\theta ^ { 0 } , { \\mathcal { E } } ) = \\theta ^ { * } \\}$ . For instance, if $\\mathcal { A }$ is gradient A descent, the result of Lee et al. (2016) implies that $\\Theta _ { \\mathcal { A } } ^ { * }$ is the set of local minimizers of the pooled loss $\\mathcal { L }$ . To each $\\theta ^ { \\ast } \\in \\Theta _ { A } ^ { \\ast }$ A , we want to associate a consistency score, quantifying the concept “good $\\theta ^ { * }$ Aare hard to vary”. In other words, we would like the score to capture the consistency of the loss landscape around $\\theta ^ { * }$ across the different environments. For example, in Fig. 1 the loss landscape near the bottom-left minimizer is consistent across environments, while the top-right minimizer is not. ", + "bbox": [ + 173, + 473, + 826, + 571 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/eb9ca484544d5d5cf5c2980ba2b3999fd126d44ac36d9c5259ef498ccebf057c.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 570, + 575, + 676, + 678 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let us characterize the landscape around $\\theta ^ { * }$ from the perspective of a fixed environment $e \\in { \\mathcal { E } }$ . We define the set $N _ { e , \\theta ^ { * } } ^ { \\epsilon }$ to be the largest path-connected region of space containing both $\\theta ^ { * }$ and the set $\\{ \\theta \\in \\Theta \\mathrm { s . t . } | \\mathcal { L } _ { e } ( \\theta ) - \\mathcal { L } _ { e } ( \\theta ^ { * } ) | \\leqslant$ $\\epsilon \\}$ , with $\\epsilon > 0$ . In other words, if $\\theta \\in N _ { e , \\theta ^ { * } } ^ { \\epsilon }$ then there exist a path-connected region in parameter space including $\\theta ^ { * }$ and $\\theta$ where each parameter also is in N \u000fe,θ˚ and its loss on environment $e$ is comparable. From the perspective of environment $e$ , all these points are equivalent to $\\theta ^ { * }$ . We would like to evaluate the elements of this set with respect to a different environment ˇ $e ^ { \\prime } \\neq e$ . We will say that $e ^ { \\prime }$ is consistent with $e$ in $\\theta ^ { * }$ if $\\operatorname* { m a x } _ { \\theta \\in N _ { e , \\theta ^ { * } } ^ { \\epsilon } } \\mathbf { \\bar { | } } \\mathcal L _ { e ^ { \\prime } } ( \\theta ) - \\mathcal L _ { e } ( \\theta ) \\vert$ is small. Repeating this reasoning for all environment pairs, we arrive at the following inconsistency score: ", + "bbox": [ + 174, + 571, + 550, + 689 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/33d19950daa9e64b7d688eda9692ea92abc78d7ced3f1ad6fae22b8df1765b16.jpg", + "image_caption": [ + "Loss surface for data B " + ], + "image_footnote": [], + "bbox": [ + 699, + 588, + 805, + 666 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 689, + 825, + 748 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/a69693a9a6f57a5b78c26af8fee718ab41a3bce63580312e3f479ef37f36a8d5.jpg", + "text": "$$\n\\mathcal { T } ^ { \\epsilon } ( \\theta ^ { * } ) : = \\operatorname* { m a x } _ { ( e , e ^ { \\prime } ) \\in \\mathcal { E } ^ { 2 } } \\operatorname* { m a x } _ { \\theta \\in N _ { e , \\theta ^ { * } } ^ { \\epsilon } } | \\mathcal { L } _ { e ^ { \\prime } } ( \\theta ) - \\mathcal { L } _ { e } ( \\theta ^ { * } ) | .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 762, + 651, + 790 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This consistency is our formalization of the principle “good explanations are hard to vary”. Finally, we can write down an invariant learning consistency score for $\\mathcal { A }$ : ", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/15d53d902ccedfade86b78eb659b805fa5c09679e4e9226207ff64ac2c9b60ad.jpg", + "text": "$$\n\\operatorname { I L C } ( \\mathcal { A } , p _ { \\theta ^ { 0 } } ) : = - \\mathbb { E } _ { \\theta ^ { 0 } \\sim p ( \\theta ^ { 0 } ) } \\left[ \\mathcal { T } ^ { \\epsilon } ( \\mathcal { A } _ { \\infty } ( \\theta ^ { 0 } , \\mathcal { E } ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 346, + 136, + 651, + 156 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "That is, the learning consistency of an algorithm measures the expected consistency across environments of the minimizer it converges to on the pooled data. ", + "bbox": [ + 173, + 161, + 826, + 189 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Example: low consistency of a classic patchwork solution. One-hidden-layer networks with sigmoid activations and enough neurons can approximate any function $f ^ { * } : [ 0 , \\dot { 1 } ] \\to \\mathbb { R }$ (Cybenko, 1989). In appendix A.1 we show how the construction used to obtain the weights leads to a maximally inconsistent solution according to $\\mathcal { T } ^ { \\epsilon } ( \\theta ^ { * } )$ , which would not be expected to generalize o.o.d. ", + "bbox": [ + 173, + 193, + 825, + 250 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 ILC AS A LOGICAL AND BETWEEN LANDSCAPES ", + "text_level": 1, + "bbox": [ + 173, + 265, + 558, + 280 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here we draw a connection between our definition of inconsistency and the local geometric properties of the loss landscapes. For the sake of clarity, we consider two environments $A$ and $B$ ) and assume $\\theta ^ { * }$ to be a local minimizer (with zero loss) for both environments. Using a Taylor approximation5, we get $\\begin{array} { r } { \\mathcal { L } ( \\theta ) \\approx \\frac { 1 } { 2 } ( \\theta - \\theta ^ { * } ) ^ { \\top } H _ { A + B } ( \\theta - \\theta ^ { * } ) } \\end{array}$ for $\\lVert \\dot { \\theta } - \\theta ^ { * } \\rVert \\approx 0$ , where $H _ { A + B } = \\left( H _ { A } + \\mathbf { \\bar { \\mathit { H } } } _ { B } \\right) / 2$ is the arithmetic mean of the Hessians $H _ { A } : = \\nabla ^ { 2 } \\mathcal { L } _ { A } ( \\theta ^ { * } )$ and $\\Dot { H _ { B } } : = \\nabla ^ { 2 } \\mathcal { L } _ { A } ( \\theta ^ { * } )$ . $H _ { A + B }$ does not capture the possibly conflicting geometries of landscape $A$ or $B$ : It performs a “logical OR” on the dominant eigendirections. In contrast, the geometric mean, or Karcher mean, $H _ { A \\land B }$ (Ando et al., 2004) is affected by the inconsistencies between landscapes: It performs a “logical AND”. In appendix A.2, we give a formal definition of $H _ { A \\land B }$ and show that for diagonal Hessians, I\u000fpθ˚q ď 2\u000fp detpHA\\`BqdetpHA^Bq q ", + "bbox": [ + 173, + 285, + 633, + 470 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/638768a396b51f0d355670863efe7197718799d15f49a554a924db13e7580926.jpg", + "image_caption": [ + ".5 -1 -0.5 0 0.5 1 1.5Figure 3: Plotted are contour lines $\\theta ^ { \\top } H ^ { - 1 } \\theta \\quad = \\quad 1$ for $H _ { A } \\ = \\ \\mathrm { d i a g } ( 0 . 0 5 , 1 )$ and $H _ { B } = \\mathrm { d i a g } ( 1 , 0 . 0 5 )$ . $H _ { A \\land B }$ retains the original volumes, while for $H _ { A + B }$ it is $5 \\times$ bigger. This magnification shows inconsistency of $A$ and $B$ . " + ], + "image_footnote": [], + "bbox": [ + 671, + 263, + 808, + 361 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "for the geometric mean of positive numbers, $0 \\leqslant \\operatorname* { d e t } ( H _ { A \\wedge B } ) \\leqslant \\operatorname* { d e t } ( H _ { A + B } )$ ; thus, inconsistency is lowest when shapes of $A$ and $B$ are similar – exactly as in the bottom-left minimizer of Fig. 1. ", + "bbox": [ + 174, + 472, + 825, + 500 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "From Hessians to gradients. We just saw that the consistency of $\\theta ^ { * }$ is linked to the geometric mean of the Hessians $\\{ \\bar { H _ { e } } ( \\theta ^ { * } ) \\} _ { e \\in \\mathcal { E } }$ . Under the simplifying assumption that each ś $H _ { e }$ is diagonal6 and allś eigenvalues $\\lambda _ { i } ^ { e }$ are positive, their geometric mean is $\\begin{array} { r } { H ^ { \\wedge } : = \\mathrm { d i a g } ( ( \\prod _ { e \\in \\mathcal { E } } \\lambda _ { 1 } ^ { e } ) ^ { 1 / | \\varepsilon | } , \\dots , ( \\prod _ { e \\in \\mathcal { E } } \\lambda _ { n } ^ { e } ) ^ { 1 / | \\varepsilon | } ) } \\end{array}$ . The curvature of the corresponding loss in the $i$ -th eigendirection depends on how consistent the curvatures of each environment are in that direction. Consider now optimizing from a point ř ř $\\theta ^ { k }$ ; gradient descent reads $\\theta ^ { k + 1 } = \\theta ^ { k } - \\eta H ^ { + } ( \\theta ^ { k } - \\theta ^ { * } )$ , where $\\begin{array} { r } { H ^ { + } : = \\mathrm { d i a g } \\big ( \\frac { 1 } { | \\mathcal { E } | } \\sum _ { e \\in \\mathcal { E } } \\breve { \\lambda _ { 1 } ^ { e } } , \\dots , \\frac { 1 } { | \\mathcal { E } | } \\sum _ { e \\in \\mathcal { E } } \\lambda _ { n } ^ { e } \\big ) } \\end{array}$ . For $\\eta$ small enough7, we have $\\begin{array} { r } { \\vert \\theta _ { i } ^ { k + 1 } - \\theta _ { i } ^ { * } \\vert = ( 1 - \\eta \\frac { 1 } { \\vert \\mathcal { E } \\vert } \\sum _ { e \\in \\mathcal { E } } \\lambda _ { i } ^ { e } ) \\vert \\theta _ { i } ^ { k } - \\theta _ { i } ^ { * } \\vert } \\end{array}$ . As noted, this choice maximises the speed of convergence to $\\theta ^ { * }$ , but does not take into account whether this minimizer is consistent. We can reduce the speed of convergence on directions where landscapes have different curvatures – which would lead to a high inconsistency – by following the gradients from the geometric mean of the landscapes, as opposed to the arithmetic mean. I.e, we substitute the full gradient $\\nabla { \\mathcal { L } } ( \\theta ) = H ^ { + } ( \\theta ^ { k } - \\theta ^ { * } )$ with $\\nabla \\mathcal { L } ^ { \\wedge } ( \\theta ) = H ^ { \\wedge } ( \\theta ^ { k } - \\theta ^ { * } )$ . Also, we have that8 $\\begin{array} { r } { \\nabla \\mathcal { L } ^ { \\wedge } ( \\theta ) = \\big ( \\prod _ { e \\in \\mathcal { E } } \\nabla \\mathcal { L } _ { e } ( \\theta ) \\big ) ^ { 1 / | \\varepsilon | } } \\end{array}$ : to reduce the speed of convergence in directions with inconsistency, we can take the element-wise geometric mean of gradients from different environments (see also Fig. 11 in the appendix). ", + "bbox": [ + 173, + 507, + 826, + 715 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 MASKING GRADIENTS WITH A LOGICAL AND ", + "text_level": 1, + "bbox": [ + 173, + 726, + 534, + 739 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The element-wise geometric mean of gradients, instead of the arithmetic mean, increases consistency in the convex quadratic case. However, there are a few practical limitations: ", + "bbox": [ + 174, + 744, + 823, + 772 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(i) The geometric mean is only defined when all the signs are consistent. It is still to be defined how sign inconsistencies, which can occur in non-convex settings, should be dealt with. ", + "bbox": [ + 176, + 773, + 823, + 801 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(ii) It provides little flexibility for ‘partial’ agreement: Even a single zero gradient component in one environment stops optimization in that direction. ", + "bbox": [ + 178, + 804, + 823, + 832 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(iii) For numerical stability, it needs to be computed in log domain (more computationally expensive). ", + "bbox": [ + 171, + 103, + 825, + 118 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(iv) Adaptive step-size schemes (e.g. Adam (Kingma and Ba, 2015)) rescale the signal componentwise for local curvature adaptation. The exact magnitude of the geometric mean would be ignored and most of the difference from arithmetic averaging will come from the zero-ed components. ", + "bbox": [ + 174, + 123, + 823, + 165 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "(i) can be overcome by treating different signs as zeros, resulting in a geometric mean of 0 if there is any sign disagreement across environments for a gradient component. For (ii) we can allow for some disagreement (with a hyperparameter), by not masking out if there is a large percentage of environments with gradients in that direction. (iii) and (iv) can be addressed together: Since the final magnitude will be rescaled except for masked components, i.e. where the geometric mean is 0, we can use the average gradients (fast to compute) and mask out the components based on the sign agreement (computable avoiding the log domain). ", + "bbox": [ + 173, + 169, + 825, + 267 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The AND-mask. We translate the reasoning we just presented to a practical algorithm that we will refer to as the AND-mask. In its most simple implementation, we zero out those gradient components with respect to weights that have inconsistent signs across environments. Formally, the masked gradients at iteration $k$ are $m _ { t } ( \\theta ^ { k } ) \\odot \\nabla { \\mathcal { L } } ( \\theta ^ { k } )$ , where $m _ { t } ( \\theta ^ { k } )$ vanishes for any component where there are less than $t \\in \\{ d / 2 , d / 2 + 1 , \\ldots , d \\}$ agreeing gradient signs across environments $d$ is the number of environments in the batch), and is equal to one otherwise. For convenience, our implementation of the AND-mask uses a threshold $\\tau \\in [ 0 , 1 ]$ as hyper-parameter instead of ř $t$ , such that $\\begin{array} { r } { t = \\frac { d } { 2 } ( \\tau + 1 ) } \\end{array}$ Mathematically, for every component $[ m _ { \\tau } ] _ { j }$ of $m _ { \\tau }$ , $\\begin{array} { r } { [ m _ { \\tau } ] _ { j } = \\pmb { 1 } \\left[ \\tau d \\leqslant | \\sum _ { e } \\mathrm { s i g n } ( [ \\nabla \\mathcal { L } _ { e } ] _ { j } ) | \\right] } \\end{array}$ . ", + "bbox": [ + 173, + 280, + 825, + 393 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Computing the AND-mask has the same time and space complexity of standard gradient descent, i.e., linear in the number of examples that we average. Due to its simplicity and computational efficiency, this is the algorithm that we will use in the experiment section. As a first result, we show that following the AND-masked gradient leads to convergence in the directions made visible by the AND-mask. The proof is presented in appendix A.3. ", + "bbox": [ + 173, + 398, + 825, + 468 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 1. Let $\\mathcal { L }$ have $L$ -Lipschitz gradients and consider a learning rate $\\eta \\leqslant 1 / L$ . After $k$ iterations, AND-masked GD visits at least once a point $\\theta$ where $\\| m _ { t } ( \\theta ) \\odot \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } \\leqslant \\mathcal { O } ( 1 / k )$ . ", + "bbox": [ + 174, + 474, + 821, + 503 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Behaviour in the face of randomness. Here we put the AND mask through a theoretical test: For gradients coming from different environments that are inconsistent (or even random), how fast does the AND mask reduce the magnitude of the step taken in parameter space, compared to standard GD? In case of inconsistency, the AND mask should quickly make the gradient steps more conservative. ", + "bbox": [ + 174, + 520, + 583, + 618 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To assess this property, we consider a fixed set of $n$ parameters $\\theta$ and gradients $\\nabla { \\mathcal { L } } _ { e }$ drawn independently from a multivariate Gaussian with zero mean and unit covariance. ", + "bbox": [ + 174, + 625, + 580, + 666 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/ad04a04ecd51008a3704226762cc27ef505decdf650968cedaac64cbe9c1e43f.jpg", + "image_caption": [ + "Figure 4: Magnitude of gradient (average or masked) on random data $( \\left| \\theta \\right| =$ 3000, $t = 0 . 8 d$ ). " + ], + "image_footnote": [], + "bbox": [ + 617, + 520, + 795, + 622 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 2. Consider the setting we just outlined, with $\\begin{array} { r } { \\mathcal { L } = ( 1 / d ) \\sum _ { e = 1 } ^ { d } \\mathcal { L } _ { e } } \\end{array}$ . While $\\mathbb { E } \\| \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } =$ $\\mathcal { O } ( n / d )$ , we have that $\\forall t \\in \\{ d / 2 + 1 , \\ldots , d \\} , \\exists c \\in ( 1 , 2 ]$ such that $\\mathbb { E } \\| m _ { t } ( \\theta ) \\odot \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } \\leqslant \\mathcal { O } ( n / c ^ { d } )$ ", + "bbox": [ + 174, + 672, + 821, + 705 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The proof is presented in Appendix A.4, and an illustration with numerical verification in Fig. 4 (the magnitudes of masked gradients (•) for more than 100 examples were always zero in the numerical verification). Intuitively, in the presence of purely random patterns, the AND-mask has a desirable property: it decreases the strength of these signals exponentially fast, as opposed to linearly. ", + "bbox": [ + 174, + 710, + 825, + 767 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 785, + 326, + 801 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Real-world datasets are generated by (causal) generative processes which share mechanisms (Pearl, 2009). However, mechanisms and spurious signals are often entangled, making it hard to assess what part of the learning signal is due to either. As the goal of this paper is to dissect these two components to understand how they ultimately contribute to the learning process, we create a simple synthetic dataset that allows us to control the complexity, intensity, and number of shortcuts in the data. After that, we evaluate whether spurious signals can be detected even in high-dimensional networks and datasets by testing the AND-mask on a memorization task similar to the one proposed in Zhang et al. (2017), and on a behavioral cloning task using the game CoinRun (Cobbe et al., 2019a). ", + "bbox": [ + 174, + 813, + 825, + 925 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/2c6666a00d98e65f93e0f8cfc43ecdc82f29522ed1a9935aadd3b95f63205d2e.jpg", + "image_caption": [ + "Figure 5: A 4-dimensional instantiation of the synthetic memorization dataset for visualization. Every example is a dot in both circles, and it can be classified by finding either of the “oracle” decision boundaries shown. " + ], + "image_footnote": [], + "bbox": [ + 215, + 102, + 782, + 161 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 THE SYNTHETIC MEMORIZATION DATASET ", + "text_level": 1, + "bbox": [ + 176, + 217, + 508, + 231 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We introduce a binary classification task. The input dimensionality is $d = d _ { M } + d _ { S }$ . While $p ( y | x _ { d _ { M } } )$ is the same across all environments (i.e. the mechanism), $p ( y | x _ { d _ { S } } , e )$ is not the same across all environments (the shortcuts). While the mechanism is shared, it needs a highly non-linear decision boundary to classify the data. The shortcuts are not shared across environments, but provide a simple way to classify the data, even when pooling all the environments together. See Figure 5 for a concrete example with $d _ { M }$ and $d _ { S }$ equal to 2, and two environments ( $A$ and $B$ ). The spirals (on $d _ { M }$ ) are invariant but hard to model. The shortcuts (on $d _ { S }$ ) are simple blobs but different in every environment: in $A$ , linearly separable through a vertical decision boundary, in $B$ with a horizontal one. If the two environments are pooled, a new diagonal decision boundary emerges on the shortcut dimensions as the most ‘natural’ one. While this perfectly classifies data in both environments $A$ and $B$ , critically it would have not been found by training on either partition $A$ or $B$ alone. The out-of-distribution (o.o.d.) test data has the same mechanism but random shortcuts. Therefore, any method relying exclusively on the shortcuts will have chance-level o.o.d. performance. Details about the dataset, baselines, and training curves are reported in appendix B. ", + "bbox": [ + 173, + 239, + 825, + 434 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Despite the apparent simplicity of this dataset, note that it is challenging to find the invariant mechanism. In high dimensions, even with tens of pooled environments, the shortcuts allow for a simple classification rule under almost every classical definition of ‘simple’: the boundary is linear, it has a large margin, it can be expressed with small weights, it is fast to learn, robust to input noise, and has perfect accuracy and no i.i.d. generalization gap. Finding the complex decision boundary of the spirals, instead, is a fiddly process and arguably a much slower path towards small loss. ", + "bbox": [ + 174, + 440, + 825, + 523 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baselines. We evaluate several domain-agnostic baselines (all multilayer perceptrons) with some of the most common regularizers used in deep learning — Dropout, L1, L2, Batch normalization. We also consider methods that explicitly make use of the environment labels, namely: (i) Domain Adversarial Neural Networks (DANN) (Ganin et al., 2016), a method specifically designed to address domain adaptation by obfuscating domain information with an adversarial classifier; (ii) Invariant Risk Minimization (IRM) (Arjovsky et al., 2019), discussed in detail in appendix B. The AND-mask is trained with the same configurations in Table 1. ", + "bbox": [ + 173, + 532, + 825, + 628 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Results. Fig. 6 shows training and test accuracy. DANN fails because it can align the representationlayer distributions from different environments using only shortcuts, such that they become indistinguishable to the domain-discriminating classifier. The AND-mask was the only method to achieve perfect test accuracy, by fitting the spirals instead of the shortcuts. In particular, the combination of the AND-mask with L1 or L2 regularization gave the most robust results overall, as they help suppress neurons that at initialization are tuned towards the shortcuts. ", + "bbox": [ + 174, + 638, + 531, + 790 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/1e0e4467ccc660e2ea7f279aa37ed5069560e29245353b3bd2c3a0696908dec2.jpg", + "image_caption": [ + "Figure 6: Results on the synthetic dataset. " + ], + "image_footnote": [], + "bbox": [ + 555, + 638, + 805, + 767 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Correlations between average, memorization and generalization gradients. Due to the synthetic nature of the dataset, we can intervene on its data-generating process in order to examine the learning signals coming from the mechanisms and from the shortcuts. We isolate the two and measure their contribution to the average gradients, as we vary the agreement threshold of the mask. More precisely, we look at the gradients computed with respect to the weights of a randomly initialized network for different sets of data: (i) The original data, with mechanisms and shortcuts. (ii) Randomly permuting the dataset over the mechanisms dimensions, thus leaving the “memorization” signal of the shortcuts. (iii) Randomly permuting over the shortcuts dimensions, isolating the “generalization” signal of the mechanisms alone. Figure 7 shows the correlation between the components of the original average gradient (i) and the shortcut gradients ((ii), dashed line), and between the original average gradients and the mechanism gradients ((iii), solid line). ", + "bbox": [ + 173, + 799, + 826, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "While the signal from the mechanisms is present in the original average gradients (i.e. $\\rho \\approx 0 . 4$ for $\\tau = 0$ ), its magnitude is smaller and it is ‘drowned’ by the memorization signal. Instead, increasing the threshold of the AND-mask (right side) suppresses memorization gradients due to the shortcuts, and for $\\tau \\approx 1$ most of the gradient components remaining contain signal from the mechanism. On the left side, we test the other side of our hypothesis: An XORmask zeroes out consistent gradients, preserves those with different signs, and results in a sharper decrease of the correlation with the mechanism gradients. ", + "bbox": [ + 174, + 132, + 550, + 284 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/f6c918dad9067762cd130aed6eaf781a1015531ba0a3e0f3f71834d439bdadd8.jpg", + "image_caption": [ + "Figure 7: Gradient correlations. " + ], + "image_footnote": [], + "bbox": [ + 568, + 143, + 815, + 253 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 EXPERIMENTS ON CIFAR-10 ", + "text_level": 1, + "bbox": [ + 176, + 295, + 419, + 309 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Memorization in a vision task. Zhang et al. (2017) showed that neural networks trained with standard regularizers — like L2 and Dropout — can still memorize large training datasets with shuffled labels, i.e. reaching $\\approx 1 0 0 \\%$ training accuracy. Their experiments raised significant questions about the generalization properties of neural networks and the role of regularizers in constraining the hypothesis class. Our hypothesis is that ILC — for example implemented as the AND-mask — should prevent memorization on a similar task with the shuffled labels, as gradients will tend to largely ‘disagree’ in the absence of a shared mechanism. However, when the labels are not shuffled, ILC should have a much weaker effect, as real shared mechanisms are still present in the data. ", + "bbox": [ + 173, + 316, + 825, + 429 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To test our hypothesis, we ran an experiment that closely resembles the one in (Zhang et al., 2017) on CIFAR-10. We trained a ResNet on CIFAR-10 with random labels, with and without the AND-mask. In all experiments we used batch size 80, and treated each example as its own “environment”. Recall that standard gradient averaging is equivalent to an AND-mask with threshold 0. As shown in Figure 8, the ResNet with standard average gradients memorized the data, while slightly increasing the threshold for the AND-mask quickly prevented memorization (dark blue line). In contrast, training the same networks on the dataset with the original labels resulted in both of them converging and generalizing to the test set, confirming that the mask did not significantly affect the generalization error with a general underlying mechanism in the data. ", + "bbox": [ + 173, + 435, + 825, + 560 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Note that there is no standard notion of environments in CIFAR10, which is why we treated every example as coming from its own environment. This assumption is not unreasonable, as every image in the dataset was literally collected in a different physical environment. If anything, it is the standard i.i.d. assumption that hides this variety behind a notion of a single distribution encompassing all environments. The results of this experiment further support this interpretation, and can serve as evidence that — in some cases — we might be able to identify invariances even without an explicit partition into environments, as this can be already identified at the level of individual examples. ", + "bbox": [ + 174, + 568, + 583, + 734 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/f6190599675b973b7ae2e4d3afc55b69d7b084c707dcc9dcb178991769022b01.jpg", + "image_caption": [ + "Figure 8: As the AND-mask threshold increases, memorization on CIFAR-10 with random labels is quickly hindered. " + ], + "image_footnote": [], + "bbox": [ + 596, + 568, + 818, + 659 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Label noise. Following up on this experiment, we test how the AND-mask performs in the presence of label noise, i.e. when a portion of the labels in the training set are randomly shuffled $2 5 \\%$ here). According to our hypothesis, gradients computed on examples with random labels should disagree and get masked out by the AND-mask, while signal from correctly labeled data should contribute to update the model. As shown in Figure 9, the performance on the incorrectly labeled portion of the dataset is well below chance for the AND-mask (as it predicts correctly despite the wrong labels), while the baseline again memorizes the incorrect labels. On the test set (with untouched labels), the baseline peaks early then decreases as the model overfits, while the AND-mask slowly but steadily improves. ", + "bbox": [ + 174, + 743, + 584, + 922 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/cd69c7bf4b0dcb56f492eb009b3fe37088a338122999723f978315d05295eae2.jpg", + "image_caption": [ + "Figure 9: The AND-mask prevents overfitting to the incorrectly labeled portion of the training set (left) without hurting the test accuracy (right). " + ], + "image_footnote": [], + "bbox": [ + 602, + 742, + 820, + 837 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.3 BEHAVIORAL CLONING ON COINRUN ", + "text_level": 1, + "bbox": [ + 178, + 103, + 475, + 117 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "CoinRun (Cobbe et al., 2019b) is a game introduced to test how RL agents generalize to novel situations. The agent needs to collect coins, jumping on top of walls and boxes and avoiding enemies.9 Each level is procedurally generated — i.e. it has a different combination of sprites, background, and layout — but the physics and goals are invariant. Cobbe et al. (2019b) showed that state-of-the-art RL algorithms fail to model these invariant mechanisms, performing poorly on new levels unless trained on thousands of them. To test our hypothesis, we set up a behavioral cloning task using CoinRun.10 We start by pre-training a strong policy $\\pi ^ { * }$ using standard PPO (Schulman et al., 2017) for 400M steps on the full distribution of levels. We then generate a dataset of pairs $( s , \\pi ^ { * } ( a | s ) )$ from the on-policy distribution. The training data consists of 1000 states from each of 64 levels, while test data comes from 2000 levels. A ResNet- $1 8 ~ \\hat { \\pi } _ { \\boldsymbol { \\theta } }$ is then trained to minimize the loss $D _ { \\mathrm { K L } } ( \\pi ^ { * } | | \\hat { \\pi } _ { \\boldsymbol { \\theta } } )$ on the training set. We compare the generalization performance of regular Adam to a version that uses the AND-mask. For each method we ran an automatic hyperparameter optimization study using Tree-structured Parzen Estimation (Bergstra et al., 2013) of 1024 trials. ", + "bbox": [ + 173, + 131, + 825, + 310 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Despite the theoretical computational efficiency of computing the ANDmask as presented in Section 2.3 (i.e., linear time and memory in the size of the mini-batch, just like classic SGD), current deep learning frameworks like PyTorch (Paszke et al., 2017) have optimized routines that sum gradients across examples in a mini-batch before it is possible to efficiently compute the AND-mask. We therefore test the AND-mask in a slightly different way. In training, in each iteration we sample a batch of data from a randomly chosen level out of the 64 available (and cycle through them all once per epoch). We then apply the AND-mask ‘temporally’, only allowing gradients that are consistent across time (and therefore across levels). See Algorithm 1 in appendix B.6 for a detailed description of this alternative formulation of the AND-mask. The figure shows the minimum test loss for the 10 best runs, supporting the hypothesis that the AND-mask helps identify invariant mechanisms across different levels. ", + "bbox": [ + 174, + 311, + 666, + 503 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/f98c901e8f6899d5e94e32da3effe7ac6f1e48a2f2b95afa4d08c00b4e8b8a48.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 686, + 323, + 813, + 502 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 523, + 344, + 540 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Generalization and covariate shift. The classic formulation of statistical learning theory (Vapnik) concerns learning from independent and identically distributed samples. The case where the distribution of the covariates at test time differs from the one observed during training is termed covariate shift (Sugiyama et al., 2007; Quionero-Candela et al., 2009; Sugiyama and Kawanabe, 2012). Standard solutions involve re-weighting of the training examples, but require the additional assumption of overlapping supports for train and test distributions. ", + "bbox": [ + 174, + 550, + 825, + 633 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Causal models and invariances. As we mentioned in the Introduction, causality provides a strong motivation for our work, based on the notion that statistical dependencies are epiphenomena of an underlying causal model (Pearl, 2009; Peters et al., 2017). The causal description identifies stable elements – e.g. physical mechanisms – connecting causes and effects, which are expected to remain invariant under interventions or changing external conditions (Haavelmo, 1943; Schölkopf et al., 2012)). This motivates our notion of invariant mechanisms, and inspired related notions which have been proposed for robust regression (Rojas-Carulla et al., 2018; Heinze-Deml et al., 2018; Arjovsky et al., 2019; Hermann and Lampinen, 2020; Ahuja et al., 2020; Krueger et al., 2020). We discuss this in more detail in appendix C.1. ", + "bbox": [ + 174, + 633, + 825, + 757 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Domain generalization. ILC can be used in a setting of domain generalization (Muandet et al., 2013), but it is not limited to it: as demonstrated in the experiments in Section 3.2, the AND-mask can be applied even if domain labels are not available. In contrast, by treating every example as a single domain, methods relying on domain classifiers (like DANN Ganin et al. (2016) or Balaji et al. (2018)) would require as many output units as there are training examples (i.e. $5 0 \\mathrm { ^ { \\circ } 0 0 0 }$ for CIFAR-10). ", + "bbox": [ + 174, + 758, + 825, + 827 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Gradient agreement. Looking at gradient agreement to learn meaningful representations in neural networks has been explored in (Du et al., 2018; Eshratifar et al., 2018; Fort et al., 2019; Zhang et al., 2019b). These approaches mainly rely on a measure of cosine similarity between gradients, which we did not consider here for two main reasons: $( i )$ It is a ‘global’ property of the gradients, and it would not allow us to extract precise information about different patterns in the network; $( i i )$ It is unclear how to extend it beyond pairs of vectors, and for pairwise interactions its computational cost scales quadratic in the number of examples used. ", + "bbox": [ + 176, + 828, + 823, + 869 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 174, + 328, + 190 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Generalizing out of distribution is one of the most significant open challenges in machine learning, and relying on invariances across environments or examples may be key in certain contexts. In this paper we analyzed how neural networks trained by averaging gradients across examples might converge to solutions that ignore the invariances, especially if these are harder to learn than spurious patterns. We argued that if learning signals are collected on one example at the time — as it is the case for gradients, e.g., computed with backpropagation — the way these signals are aggregated can play a significant role in the patterns that will ultimately be expressed: Averaging gradients in particular can be too permissive, acting as a logical $O R$ of a collection of distinct patterns, and lead to a ‘patchwork’ solution. We introduced and formalized the concept of Invariant Learning Consistency, and showed how to learn invariances even in the face of alternative explanations that — although spurious — fulfill most characteristics of a good solution. The AND-mask is but one of multiple possible ways to improve consistency, and it is unlikely to be a practical algorithm for all applications. However, we believe this should not distract from the general idea which we are trying to put forward — namely, that it is worthwhile to study learning of explanations that are hard to vary, with the longer term goal of advancing our understanding of learning, memorization and generalization. ", + "bbox": [ + 173, + 204, + 825, + 411 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 434, + 356, + 449 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We wish to thank Sebastian Gomez, Luca Biggio, Julius von Kügelgen, Paolo Penna, Ioannis Anagno, Ricards Marcinkevics, Sidak Pal Singh, Damien Teney for feedback on the manuscript, and thank Nando de Freitas for fruitful discussions in the early stage of this project. 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", + "bbox": [ + 176, + 839, + 826, + 910 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A APPENDIX TO SECTION 2 ", + "text_level": 1, + "bbox": [ + 176, + 102, + 421, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 A CLASSIC EXAMPLE OF A PATCHWORK SOLUTION ", + "bbox": [ + 174, + 133, + 570, + 148 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Consider a neural network with one hidden layer consisting of two neurons and sigmoidal activations: ", + "bbox": [ + 163, + 160, + 826, + 176 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/1c3e1f1349f383088f95799608bda92f13aa62d011e38583a0fbed793bbfff9f.jpg", + "text": "$$\nf _ { \\theta } ( x ) = \\theta _ { 5 } \\sigma ( \\theta _ { 1 } x + \\theta _ { 2 } ) + \\theta _ { 6 } \\sigma ( \\theta _ { 3 } x + \\theta _ { 4 } ) , \\quad \\sigma ( z ) : = 1 / ( 1 + e ^ { - z } ) .\n$$", + "text_format": "latex", + "bbox": [ + 277, + 189, + 720, + 205 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We want to learn the continuous function $f ^ { * } : [ 0 , 1 ] \\to [ 0 , 2 ]$ defined as ", + "bbox": [ + 174, + 209, + 647, + 226 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/6a13e80a45253052ef835a0015cb8ae32cb16dbbc7a7556dafc018015a29d766.jpg", + "text": "$$\nf ^ { * } ( x ) = \\left\\{ \\begin{array} { l l } { 0 } & { x \\in [ 0 , 0 . 4 ) ; } \\\\ { 1 0 ( x - 0 . 4 ) } & { x \\in [ 0 . 4 , 0 . 5 ) ; } \\\\ { 1 } & { x \\in [ 0 . 5 , 0 . 7 ) ; } \\\\ { 1 0 ( x - 0 . 7 ) + 1 } & { x \\in [ 0 . 7 , 0 . 8 ) ; } \\\\ { 2 } & { x \\in [ 0 . 8 , 1 ] . } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 348, + 233, + 647, + 320 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "To perform this task, we have access to (noiseless) data from two environments: ", + "bbox": [ + 173, + 327, + 697, + 342 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/a6d9d3abf3d9676bfad743aa44245173df4e26b6f581bd28df47a570ac8d588e.jpg", + "text": "$$\nA : \\{ ( x , f ( x ) ) \\mid x \\in [ 0 , 0 . 5 ) \\} , \\quad B : \\{ ( x , f ( x ) ) \\mid x \\in [ 0 . 5 , 1 ] \\} .\n$$", + "text_format": "latex", + "bbox": [ + 294, + 349, + 704, + 367 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "There is a simple constructive way, provided by the universal function approximation theorem Cybenko (1989) to fit this function11 using $f _ { \\theta }$ up to an arbitrarily small mean squared error $\\mathcal { L } _ { A + B } ( \\theta ^ { * } )$ . Leaving out the details of such a construction (Cybenko (1989) for details), the reader can check on the left panel of Figure 10 that $\\theta ^ { * } = ( 1 0 0 , - 5 0 , 1 0 0 , - 7 5 , 1 , 1 )$ provides a good fit for both environments A and B — both ${ \\mathcal { L } } _ { A } ( \\theta ^ { * } )$ and ${ \\mathcal { L } } _ { B } ( \\theta ^ { * } )$ are small. ", + "bbox": [ + 173, + 375, + 828, + 445 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/eb35dbc64d91d70df2fc12fa12cfa328de9a0881fab9199faceabaf62636ab02.jpg", + "image_caption": [ + "Figure 10: Performance of the neural network in Equation 3 for two different parameters. Any reasonable modification on $\\theta _ { 6 }$ (say $\\pm 1$ ) leaves the performance on environment A unchanged, while the performance on environment B quickly degrades. " + ], + "image_footnote": [], + "bbox": [ + 212, + 470, + 789, + 660 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "However, it is easy to realize that $\\theta ^ { * }$ — while being a solution which can be returned by gradient descent using the pooled data $\\mathbf { A } { + } \\mathbf { B }$ — is not consistent (formal definition given in the main paper in Section 2). Indeed, it is possible to modify $\\tilde { \\theta } ^ { * }$ such that the loss in environment A remains almost unchanged, while the loss in environment B gets larger. In particular, on the right panel of Figure 10, we show that $\\tilde { \\theta } ^ { * } = ( 1 0 0 , - 5 0 , 1 0 0 , - 7 5 , 1 , - 0 . 5 )$ is such that $\\mathcal { L } _ { A } ( \\theta ^ { * } ) \\leqslant \\bar { \\mathcal { L } _ { A } ( \\tilde { \\theta } ^ { * } ) } + \\epsilon$ (with $\\epsilon$ very small) but $\\mathcal { L } _ { B } ( \\theta ^ { * } ) \\ll \\mathcal { L } _ { B } ( \\tilde { \\theta } ^ { * } )$ . According to our definition in Equation 1 (see main paper), we have ${ \\cal T } ^ { \\epsilon } ( \\theta ^ { * } ) \\leq | { \\mathcal L } _ { B } ( \\theta ^ { * } ) - { \\mathcal L } _ { B } ( \\tilde { \\theta } ^ { * } ) |$ — that is a large number (low consistency). ", + "bbox": [ + 173, + 723, + 825, + 830 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Remark 1 (Connection to out of distribution generalization). The main point of this analysis was to show an example of where our measure of consistency behaves according to expectations: A typical implementation of the universal approximation theorem — which one would not expect to generalize out of distribution, due to its ‘patchwork’ behavior — leads indeed to a very low consistency score. ", + "bbox": [ + 174, + 832, + 825, + 888 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Geometric mean of matrices. Given an $n$ -tuple of $d \\times d$ positive definite matrices $( A _ { j } ) _ { j = 1 } ^ { n }$ , the ean Ando et al. (2004) is the unique positive definite solution, where log is the matrix logarithm. This matrix average he it relevant to signal processing and medical imaging. T $X$ to the equationmany desirable Karcher mean \n${ \\ k { \\stackrel { \\smile } { \\sum } } _ { i = 1 } ^ { m } } \\log ( \\dot { A } _ { i } ^ { - 1 } X ) = 0$ \ncan also be written as $\\begin{array} { r } { \\operatorname * { a r g m i n } _ { X \\in S ^ { + + } ( d ) } \\bar { f } ( X ) = \\frac { 1 } { 2 m } \\bar { \\sum _ { i = 1 } ^ { m } } d ( A _ { i } , X ) ^ { 2 } } \\end{array}$ , where $d$ is the Riemannian \ndistance in the manifold of SPD matrices $S ^ { + + } ( d )$ . ", + "bbox": [ + 173, + 128, + 825, + 217 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/ad592e94eca81ce1c9da75f174ffeb9e7cd7d2510f2e4b3249bdbae3b184dfa4.jpg", + "image_caption": [ + "Figure 11: While the arithmetic mean of the two loss surfaces on the left is identical in all three cases (third column), the geometric mean has weaker and weaker gradients (black arrow) the more inconsistent the two loss surfaces become. " + ], + "image_footnote": [], + "bbox": [ + 238, + 229, + 761, + 565 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Link between consistency and geometric means. Here we show how the consistency score introduced in Equation 1 can be linked (in a simplified setting) to a comparison between the arithmetic and geometric means of the Hessians approximating the landscapes of two separate environments $A$ and $B$ . ", + "bbox": [ + 173, + 640, + 825, + 695 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "At the local minimizer $\\theta ^ { * } = 0$ , we assume that $\\mathcal { L } _ { A } = \\mathcal { L } _ { B } = 0$ and consider the local quadratic approximations $\\begin{array} { r } { \\ \\mathcal { L } _ { A } ( \\theta ) \\ = \\ \\frac { 1 } { 2 } \\theta ^ { \\top } H _ { A } \\theta } \\end{array}$ and $\\begin{array} { r } { \\ \\mathcal { L } _ { B } ( \\theta ) \\ = \\ \\frac { 1 } { 2 } \\theta ^ { \\top } H _ { B } \\theta } \\end{array}$ . Here, we make the additional simplifying assumption that $H _ { A }$ and $H _ { B }$ are diagonal (or, more broadly, co-diagonalizable): $H _ { A } \\mathbf { \\bar { \\Sigma } } = \\mathrm { d i a g } ( \\lambda _ { 1 } ^ { A } , \\cdot \\cdot \\cdot , \\lambda _ { n } ^ { A } )$ , $H _ { B } = \\mathrm { d i a g } ( \\lambda _ { 1 } ^ { B } , \\cdot \\cdot \\cdot , \\lambda _ { n } ^ { B } )$ , with $\\lambda _ { i } ^ { A } \\geqslant 0$ and $\\lambda _ { i } ^ { B } \\geqslant 0$ for all $i = 1 , \\ldots , n$ The arithmetic and geometric means (noted as $H _ { A + B }$ and $H _ { A \\land B }$ ) of these matrices are defined in this simplified setting as follows: ", + "bbox": [ + 173, + 702, + 825, + 786 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/564e77bac0632db3bd160c7ba220fd9c3c9e2dfcac42b211a3dc1ea1a5771f5f.jpg", + "text": "$$\nH _ { A + B } = \\mathrm { d i a g } \\left( \\frac { 1 } { 2 } ( \\lambda _ { 1 } ^ { A } + \\lambda _ { 1 } ^ { B } ) , \\cdots , \\frac { 1 } { 2 } ( \\lambda _ { n } ^ { A } + \\lambda _ { n } ^ { B } ) \\right) , \\quad H _ { A \\wedge B } = \\mathrm { d i a g } \\left( \\sqrt { \\lambda _ { 1 } ^ { A } \\lambda _ { 1 } ^ { B } } , \\cdots , \\sqrt { \\lambda _ { n } ^ { A } \\lambda _ { n } ^ { B } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 792, + 820, + 828 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "As motivated in the main paper and in Figure 12, one can link the consistency of two landscapes to a comparison between the geometric and arithmetic means of the corresponding Hessians. ", + "bbox": [ + 174, + 833, + 821, + 863 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proposition 3. In the setting we just described, the consistency score in Equation 1 can be estimated as follows: ", + "bbox": [ + 176, + 866, + 823, + 893 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/b7adb9e04414281e3352f0794331547efa43be680e37fb2969ea051af4c4c98f.jpg", + "text": "$$\n{ \\mathcal { T } } ^ { \\epsilon } ( \\theta ^ { * } ) \\leqslant 2 \\epsilon \\left( { \\frac { \\operatorname* { d e t } \\bigl ( H _ { A + B } \\bigr ) } { \\operatorname* { d e t } \\bigl ( H _ { A \\wedge B } \\bigr ) } } \\right) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 393, + 891, + 602, + 929 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/18902103c5165a84f4ea044ec9a748571660c465a1a120f77d810e8def45c95e.jpg", + "image_caption": [ + "Figure 12: Plotted are contour lines $\\theta ^ { \\top } H ^ { - 1 } \\theta = 1$ for $H _ { A } = \\mathrm { d i a g } ( 0 . 0 1 , 1 )$ and $H _ { B } = \\mathrm { d i a g } ( 1 , 0 . 0 1 )$ . It is convenient to provide this visualization because it is linked to the matrix determinant: a $\\mathrm { V o l } ( \\{ \\theta ^ { \\top } H ^ { - 1 } \\theta = 1 \\} ) =$ $\\pi { \\sqrt { \\operatorname* { d e t } ( H ) } }$ . The geometric average retains the volume of the original ellipses, while the volume of $H _ { A + B }$ is 25 times bigger. This magnification indicates that landscape $A$ is not consistent with landscape $B$ . " + ], + "image_footnote": [], + "bbox": [ + 302, + 99, + 694, + 252 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Before showing the proof, we note that the proposition gives a lower bound on the consistency. That is, it provides a pessimistic estimate. Yet, as we motivated, this estimate has a nice geometric interpretation. However, as we outline in a remark after the proof, this estimate is tight in two important limit cases. ", + "bbox": [ + 173, + 342, + 825, + 398 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. In this setting, Equation 1 gives ", + "bbox": [ + 176, + 412, + 431, + 428 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/596eea8a741ad942279debd2491d52a92aa05a586de7895e17c14840da2aca24.jpg", + "text": "$$\n\\mathcal { T } ^ { \\epsilon } ( \\theta ^ { * } ) : = \\operatorname* { m a x } \\left\\{ \\operatorname* { m a x } _ { \\mathcal { L } _ { A } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { B } ( \\theta ) , \\operatorname* { m a x } _ { \\mathcal { L } _ { B } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { A } ( \\theta ) \\right\\} .\n$$", + "text_format": "latex", + "bbox": [ + 328, + 429, + 668, + 464 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Recall that ", + "bbox": [ + 173, + 467, + 246, + 481 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/21a976a5e879ee9209ca5c9b59e1bfb2a24f04e35507e178a276fae55ffbed6c.jpg", + "text": "$$\n\\mathcal { L } _ { A } ( \\theta ) = \\frac { 1 } { 2 } \\theta ^ { \\top } H _ { A } \\theta = \\frac { 1 } { 2 } \\sum _ { i } \\lambda _ { i } ^ { A } \\theta _ { i } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 476, + 612, + 511 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Hence, this is a simple quadratic program with quadratic constraints, and ", + "bbox": [ + 174, + 512, + 651, + 526 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9ce2abf2826fae4e7102aa015c3798909f10f9eaec54c02d6a0174b775da1797.jpg", + "text": "$$\n\\operatorname* { m a x } _ { \\mathcal { L } _ { A } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { B } ( \\theta ) = \\operatorname* { m a x } _ { \\frac { 1 } { 2 } \\sum _ { i } \\lambda _ { i } ^ { A } \\theta _ { i } ^ { 2 } \\leqslant \\epsilon } \\frac { 1 } { 2 } \\sum _ { i } \\lambda _ { i } ^ { B } \\theta _ { i } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 361, + 529, + 635, + 564 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Further, we can change variables and introduce $\\tilde { \\theta } _ { i } = \\theta _ { i } \\sqrt { \\lambda _ { i } ^ { A } / 2 }$ . The problem gets even simpler: ", + "bbox": [ + 178, + 568, + 799, + 585 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/5996b71bb797117f5f8891e620a6ae7ce636799ca473d5cbb01bc4494c39431f.jpg", + "text": "$$\n\\operatorname* { m a x } _ { \\mathcal { L } _ { A } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { B } ( \\theta ) = \\operatorname* { m a x } _ { \\| \\tilde { \\theta } \\| ^ { 2 } \\leqslant \\epsilon } \\sum _ { i } \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { i } ^ { A } } \\tilde { \\theta } _ { i } ^ { 2 } = \\epsilon \\cdot \\operatorname* { m a x } _ { i } \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { i } ^ { A } } .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 588, + 661, + 626 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "All in all, we get ", + "bbox": [ + 174, + 627, + 285, + 642 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/b85f2cb6ec08708cf012c47c2835017a3b4f8aaf8af8794f0f384d01eb7476e9.jpg", + "text": "$$\n\\begin{array} { r l } { T ^ { \\epsilon } ( \\theta ^ { * } ) = \\epsilon \\operatorname* { m a x } \\Bigg \\{ \\underset { i } { \\operatorname* { m a x } } \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { i } ^ { A } } , \\underset { i } { \\operatorname* { m a x } } \\frac { \\lambda _ { i } ^ { A } } { \\lambda _ { i } ^ { B } } \\Bigg \\} } & { } \\\\ { = \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\operatorname* { m a x } \\left\\{ \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { i } ^ { A } } , \\frac { \\lambda _ { i } ^ { A } } { \\lambda _ { B } ^ { B } } \\right\\} } & { } \\\\ { \\leqslant \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\left( \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { A } ^ { A } } + \\frac { \\lambda _ { i } ^ { A } } { \\lambda _ { B } ^ { B } } \\right) } & { } \\\\ { = \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\left\\{ \\frac { ( \\lambda _ { i } ^ { B } ) ^ { 2 } + ( \\lambda _ { i } ^ { A } ) ^ { 2 } } { \\lambda _ { B } ^ { B } \\lambda _ { i } ^ { A } } \\right\\} } & { } \\\\ { \\leqslant \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\left\\{ \\frac { ( \\lambda _ { i } ^ { B } + \\lambda _ { i } ^ { A } ) ^ { 2 } } { \\lambda _ { B } ^ { B } \\lambda _ { i } ^ { A } } \\right\\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 366, + 645, + 629, + 829 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "This means ", + "bbox": [ + 173, + 828, + 251, + 842 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/948472fd43398d65d233b985eedf5e69d21a6131c16dd6d8beffc2221761a3ef.jpg", + "text": "$$\n\\sqrt { { \\cal T } ^ { \\epsilon } ( \\theta ^ { * } ) } \\leqslant \\epsilon \\operatorname* { m a x } _ { i } \\frac { \\lambda _ { i } ^ { B } + \\lambda _ { i } ^ { A } } { \\sqrt { \\lambda _ { i } ^ { B } \\lambda _ { i } ^ { A } } } = 2 \\epsilon \\operatorname* { m a x } _ { i } \\frac { ( \\lambda _ { i } ^ { B } + \\lambda _ { i } ^ { A } ) / 2 } { \\sqrt { \\lambda _ { i } ^ { B } \\lambda _ { i } ^ { A } } } \\leqslant 2 \\epsilon \\frac { \\prod _ { i } ( \\lambda _ { i } ^ { B } + \\lambda _ { i } ^ { A } ) / 2 } { \\prod _ { i } \\sqrt { \\lambda _ { i } ^ { B } \\lambda _ { i } ^ { A } } } = 2 \\epsilon \\frac { \\operatorname* { d e t } ( H _ { A + B } ) } { \\operatorname* { d e t } ( H _ { A \\wedge B } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 183, + 842, + 813, + 881 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where the first inequality comes from the monotonicity of the square root function, and the second inequality comes from the fact that (i) the geometric mean is always smaller or equal than theś arithmetic mean and (ii) for any sequence of numbers $\\alpha _ { i } > 1$ , $\\begin{array} { r } { \\operatorname* { m a x } _ { i } \\dot { \\alpha _ { i } } \\leqslant \\prod _ { i } \\alpha _ { i } } \\end{array}$ . □ ", + "bbox": [ + 176, + 882, + 823, + 925 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Remark 2 (Sanity check). There are two important cases where we can test the bound above. First, if $H _ { A } = H _ { B }$ , then $T ^ { \\epsilon } ( \\theta ^ { * } ) = \\epsilon$ , and the bound returns $T ^ { \\epsilon } ( \\theta ^ { * } ) \\leqslant 2 \\epsilon$ , since the geometric and arithmetic mean are the same. Next, say $\\lambda _ { i } ^ { A } = 0$ but $\\lambda _ { i } ^ { B } > 0$ ; then, both the bound and the inconsistency score are $\\infty$ (highest possible inconsistency). ", + "bbox": [ + 173, + 103, + 825, + 160 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.3 PROOF OF PROPOSITION 1 ", + "text_level": 1, + "bbox": [ + 176, + 176, + 400, + 190 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In this appendix section we consider the AND-masked GD algorithm, introduced at the end of Section 2. We recall that the masked gradients at iteration $k$ are $\\mathsf { \\bar { \\rho } } m _ { t } ( \\theta ^ { k } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { k } )$ , where $m _ { t } ( \\theta ^ { k } )$ vanishes for any component where there are less than $t \\in \\{ d / 2 + 1 , \\ldots , d \\}$ agreeing gradient signs across environments, and is equal to one otherwise. In a full-batch setting, the algorithm is ", + "bbox": [ + 173, + 202, + 826, + 260 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/367147ae234bf3fb44c2105fa911c059b7d9359111400bf47e02ee1e36df1124.jpg", + "text": "$$\n\\theta ^ { k + 1 } = \\theta ^ { k } - \\eta m _ { t } ( \\theta ^ { k } ) \\odot \\nabla { \\mathcal { L } } ( \\theta ^ { k } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 316, + 265, + 550, + 285 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(AND-masked GD) ", + "bbox": [ + 691, + 268, + 823, + 284 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\eta > 0$ is the learning rate. ", + "bbox": [ + 174, + 291, + 387, + 306 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proposition 1. Let $\\mathcal { L }$ have $L$ -Lipschitz gradients and consider a learning rate $\\eta \\leqslant 1 / L$ . After $k$ iterations, AND-masked GD visits at least once a point $\\theta$ where $\\| m _ { t } ( \\theta ) \\odot \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } \\leqslant \\mathcal { O } ( 1 / k )$ . ", + "bbox": [ + 173, + 310, + 825, + 340 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. Thanks to the component-wise $L$ -smoothness and using a Taylor expansion around $\\theta ^ { i }$ we have ", + "bbox": [ + 166, + 354, + 825, + 383 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/d29dd6557fac18231ce1c10b2d58e5259914511b22f4f71f2cb47d435fd3fc25.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { L } ( \\theta ^ { i + 1 } ) \\leqslant \\mathcal { L } ( \\theta ^ { i } ) - \\eta \\langle \\nabla \\mathcal { L } ( \\theta ^ { i } ) , m _ { t } ( \\theta ^ { i } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { i } ) \\rangle + \\displaystyle \\frac { L \\eta ^ { 2 } } { 2 } \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { i } ) \\| ^ { 2 } } \\\\ { \\displaystyle \\qquad = \\mathcal { L } ( \\theta ^ { i } ) - \\left( \\eta - \\frac { L \\eta ^ { 2 } } { 2 } \\right) \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { i } ) \\| ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 387, + 761, + 457 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "If we seek $\\eta - L \\eta ^ { 2 } / 2 \\geqslant \\eta / 2$ , then $\\begin{array} { r } { \\eta \\leqslant \\frac { 1 } { L } } \\end{array}$ , as we assumed in the proposition statement. Therefore, $\\mathcal { L } ( \\theta ^ { i + 1 } ) \\leqslant \\mathcal { L } ( \\theta ^ { i } ) - ( \\eta / 2 ) \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { i } ) \\| ^ { 2 }$ , for all $i \\geqslant 0$ . Summing over $i$ from 0 to a desired iteration $k$ , we get ", + "bbox": [ + 174, + 481, + 825, + 526 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/e864e9db0e9c856ab8b1d90a30efaf655e30e059a297104548b83e9c853e1c21.jpg", + "text": "$$\n\\sum _ { i = 0 } ^ { k - 1 } ( \\eta / 2 ) \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { i } ) \\| ^ { 2 } \\leqslant \\mathcal { L } ( \\theta ^ { 0 } ) - \\mathcal { L } ( \\theta ^ { k } ) \\leqslant \\mathcal { L } ( \\theta ^ { 0 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 303, + 534, + 694, + 575 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Therefore, ", + "bbox": [ + 173, + 582, + 243, + 597 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/4274fdbefe7f256ba4c3acb6c48a69ee80628b61dd3d9512b3330c93c263db3a.jpg", + "text": "$$\n\\operatorname* { m i n } _ { i = 0 , \\dots , k } \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla { \\mathcal { L } } ( \\theta ^ { i } ) \\| ^ { 2 } \\leqslant \\frac { 1 } { k } \\sum _ { i = 0 } ^ { k - 1 } ( \\eta / 2 ) \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla { \\mathcal { L } } ( \\theta ^ { i } ) \\| ^ { 2 } \\leqslant \\frac { 2 { \\mathcal { L } } ( \\theta ^ { 0 } ) } { \\eta k } .\n$$", + "text_format": "latex", + "bbox": [ + 245, + 594, + 751, + 637 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Hence, there exist an iteration $i ^ { * } \\in \\{ 0 , \\ldots , k \\}$ such that $\\| m _ { t } ( { \\theta ^ { i } } ^ { * } ) \\odot \\nabla { \\mathcal { L } } ( { \\theta ^ { i } } ^ { * } ) \\| ^ { 2 } \\leqslant { \\mathcal { O } } ( 1 / k )$ ", + "bbox": [ + 171, + 645, + 771, + 662 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.4 PROOF OF PROPOSITION 2 ", + "text_level": 1, + "bbox": [ + 176, + 678, + 401, + 693 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Here we fix parameters $\\boldsymbol \\theta \\in \\mathbb { R } ^ { n }$ and assume gradients $\\nabla \\mathcal { L } _ { e } ( \\theta ) \\in \\mathbb { R } ^ { n }$ coming from environments $e \\in \\mathcal { E }$ are drawn independently from a multivariate Gaussian with zero mean and $\\sigma ^ { 2 } I$ covariance. We want to show that, in this random setting, the AND-mask introduced in Section 2.3 decreases the magnitude of the gradient step. ", + "bbox": [ + 173, + 704, + 825, + 761 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proposition 2. Consider the setting we just outlined, with $\\begin{array} { r } { \\mathcal { L } = ( 1 / d ) \\sum _ { e = 1 } ^ { d } \\mathcal { L } _ { e } } \\end{array}$ . While $\\mathbb { E } \\| \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } =$ $\\mathcal { O } ( n / d )$ , we have that $\\forall t \\in \\{ d / 2 + 1 , \\ldots , d \\} , \\exists c \\in ( 1 , 2 ]$ such that $\\mathbb { E } \\| m _ { t } ( \\theta ) \\odot \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } \\leqslant \\mathcal { O } ( n / c ^ { d } )$ ", + "bbox": [ + 174, + 766, + 823, + 800 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. Let us drop the argument $\\theta$ for ease of notation. First, let us consider $\\nabla \\mathcal { L }$ (no gradient AND-mask): ", + "bbox": [ + 174, + 814, + 823, + 842 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/685b3ca0815f9640febde7c769a312b0f4cc724fbc78ed839cffff1164e632d1.jpg", + "text": "$$\n\\mathbb { E } \\left\\| \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla { \\mathcal { L } } _ { e _ { i } } \\right\\| ^ { 2 } = \\frac { 1 } { d ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\mathbb { E } \\| \\nabla { \\mathcal { L } } _ { e _ { i } } \\| ^ { 2 } = \\frac { n \\sigma ^ { 2 } } { d } ,\n$$", + "text_format": "latex", + "bbox": [ + 344, + 840, + 651, + 886 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where in the first equality we used the fact that the $\\nabla { \\mathcal { L } } _ { e _ { i } }$ are uncorrelated and in the second the fact that $\\mathbb { E } [ \\| \\nabla { \\mathcal { L } } _ { e _ { i } } \\| ^ { 2 } ]$ is the trace of the covariance of $\\nabla { \\mathcal { L } } _ { e _ { i } }$ . ", + "bbox": [ + 176, + 895, + 823, + 925 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Next, assume we apply the element-wise AND-mask $m _ { t }$ to the gradients, which puts to zero the components (dimensions) where there are less than $t \\in \\{ d / 2 , \\ldots , d \\}$ equal signs. Since Gaussians are symmetric around zero, the probability of having exactly \\` ˘ \\` ˘ $u$ positive $j$ -th gradient component among $d$ environments is $\\begin{array} { r } { P r ( p _ { j } = u ) = \\left( \\frac { 1 } { 2 } \\right) ^ { d } { \\binom { d } { u } } } \\end{array}$ . Hence, the probability to keep the $j$ -th gradient direction (considering also negative consistency) is ", + "bbox": [ + 174, + 102, + 825, + 178 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/79e42b1d987670fc0a2948d73988378badeb21bbed17e3e6889792b660027157.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { P r } [ [ m _ { t } ] _ { j } = 1 ] = \\sum _ { u = t } ^ { d } \\operatorname* { P r } ( p _ { j } = u ) + \\sum _ { u = 0 } ^ { d - t } \\operatorname* { P r } ( p _ { j } = u ) } \\\\ { \\displaystyle \\quad = \\bigg ( \\frac { 1 } { 2 } \\bigg ) ^ { d } \\sum _ { k = t } ^ { d } \\binom { d } { k } + \\bigg ( \\frac { 1 } { 2 } \\bigg ) ^ { d } \\sum _ { k = 0 } ^ { d - t } \\binom { d } { k } } \\\\ { \\displaystyle \\quad = 2 \\bigg ( \\frac { 1 } { 2 } \\bigg ) ^ { d } \\sum _ { k = t } ^ { d } \\binom { d } { k } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 321, + 184, + 674, + 314 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We would now like to compute $\\begin{array} { r } { \\mathbb { E } \\left\\| m _ { t } \\odot \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla { \\mathcal { L } } _ { e _ { i } } \\right) \\right\\| ^ { 2 } } \\end{array}$ . The difficulty lies in the fact that the event $m _ { t } = 1$ makes gradients conditionally dependent. Indeed, conditioning on both $m _ { t } = 1$ and $[ \\nabla { \\mathcal { L } } _ { e } ] _ { j } > 0$ changes the distribution of $[ \\nabla { \\mathcal { L } } _ { e ^ { \\prime } } ] _ { j }$ : this gradient entry is going to be more likely to be positive or negative, depending on the value of $[ \\nabla { \\mathcal { L } } _ { e } ] _ { j }$ and on the details of the gradient mask. To solve the issue, we our strategy is to reduce the discussion (without loss in generality and with no additional assumption) to the case where gradient entries have all the same sign and hence conditional independence is restored. ", + "bbox": [ + 173, + 319, + 826, + 428 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We consider the following writing for the quantity we are interested in: ", + "bbox": [ + 174, + 434, + 640, + 450 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/9dea7cbe92ff8a2537b0350e26a9cfdaf3a05d8c7a457ed3db9de066bc42fc0c.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathbb { E } \\| m _ { t } \\operatorname { \\wp } ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ) \\| ^ { 2 } = \\displaystyle \\sum _ { j = 1 } ^ { n } \\mathbb { E } [ [ m _ { \\epsilon } ] _ { j } ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } ] } & { } \\\\ { = \\displaystyle \\sum _ { j = 1 } ^ { n } \\sum _ { \\tilde { p } _ { \\tilde { \\imath } } = 0 } ^ { d } \\mathbb { E } [ [ m _ { \\epsilon } ] _ { j } ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \\hat { p } _ { j } ] \\mathbb { P } \\mathrm { r } [ p _ { j } = \\hat { p } _ { j } ] } \\\\ { = \\displaystyle \\sum _ { j = 1 } ^ { n } \\sum _ { \\tilde { p } _ { \\tilde { \\jmath } } = 0 } ^ { d - 1 } \\sum _ { \\tilde { p } = 0 } ^ { d } \\mathbb { E } [ ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \\hat { p } _ { j } ] \\mathbb { P } \\mathrm { r } [ p _ { j } = \\hat { p } _ { j } ] } \\\\ { = \\displaystyle 2 \\sum _ { j = 1 } ^ { n } \\sum _ { \\tilde { p } _ { \\tilde { \\jmath } } = - 1 } ^ { d } \\mathbb { E } [ ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \\hat { p } _ { j } ] ( \\displaystyle \\frac { 1 } { 2 } ) ^ { d } ( \\displaystyle \\frac { d } { \\hat { p } _ { j } } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 194, + 455, + 802, + 659 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "where we used the definition of 2-norm, the law of total expectation, and the symmetry of the problem with respect to positive and negative numbers. Finally, since the gradient components within the same environment are conditionally independent, for any $j \\in \\{ 1 , \\ldots , n \\}$ we can write ", + "bbox": [ + 174, + 680, + 825, + 723 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/a6737b8880753d8398d087807804dfa99ba376c0ada5082513235d64fd61aeb6.jpg", + "text": "$$\n\\mathbb { E } \\left. m _ { t } \\odot \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla \\mathcal { L } _ { e _ { i } } \\right) \\right. ^ { 2 } = 2 n \\sum _ { \\hat { p } _ { j } = t } ^ { d } \\mathbb { E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\bigg | p _ { j } = \\hat { p } _ { j } \\right] \\left( \\frac { 1 } { 2 } \\right) ^ { d } { \\binom { d } { \\hat { p } _ { j } } } .\n$$", + "text_format": "latex", + "bbox": [ + 214, + 728, + 782, + 780 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Finally, we note that the following bound holds: ", + "bbox": [ + 173, + 791, + 488, + 806 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/2a0c3376533f95b021b180970ae8320bdde179fa117addf7c8df5d8ab5ec195c.jpg", + "text": "$$\n\\mathbb { E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\middle | p _ { j } = \\hat { p } _ { j } \\leqslant d \\right] \\leqslant \\mathbb { E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\middle | p _ { j } = d \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 250, + 811, + 748, + 863 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Indeed, if all environments lead to positive (or, symmetrically, negative) and non-interacting gradients in the $j$ -th direction, the average will be the biggest in norm. Moreover — crucially — conditioned on the event $p _ { j } \\ = \\ d$ , gradients coming from different environments are distributed as a positive half-normal distributions. Moreover, they are conditionally independent; this because, since they are all positive, the value of a gradient in one environment cannot influence the value of the gradient in another one. We remark that conditional independence on the right-hand side is therefore not an assumption, but is intrinsic to the upper bound. ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 147 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Putting it all together, we have ", + "bbox": [ + 173, + 152, + 377, + 167 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/b0ea8f99191510776f1eb7f6b94844751c31f9516464951e5b50bcf3b7c4682e.jpg", + "text": "$$\n\\begin{array} { r l r } { \\mathbb { E } \\left\\| m _ { t } \\odot \\left( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla \\mathcal { L } _ { e _ { i } } \\right) \\right\\| ^ { 2 } \\leqslant 2 n \\displaystyle \\sum _ { \\hat { p } _ { j } = t } ^ { d } \\mathbb { E } \\left[ \\left( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\bigg | p _ { j } = d \\right] \\left( \\displaystyle \\frac { 1 } { 2 } \\right) ^ { d } \\left( \\displaystyle \\frac { d } { \\hat { p } _ { j } } \\right) } & \\\\ & { \\leqslant 2 n \\displaystyle \\sum _ { \\hat { p } _ { j } = t } ^ { d } \\sigma ^ { 2 } \\left( \\displaystyle \\frac { 1 } { 2 } \\right) ^ { d } \\left( \\displaystyle \\frac { d } { \\hat { p } _ { j } } \\right) } & \\\\ & { \\leqslant \\sigma ^ { 2 } n ( d - t ) \\Big ( \\displaystyle \\frac { d } { t } \\Big ) \\left( \\displaystyle \\frac { 1 } { 2 } \\right) ^ { d - 1 } , } & \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 220, + 174, + 777, + 310 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where in the second line we bounded the squared average of a sum of half normal distributions: let $\\{ X _ { i } \\} _ { i = 1 } ^ { d }$ be a family of uncorrelated positive half-normal distributions derived from a Gaussians witha mean zero and variance $\\sigma ^ { 2 }$ , we have12 that $\\mathbb { E } [ X _ { i } ] = \\sigma { \\sqrt { 2 / \\pi } }$ and $\\mathbb { E } [ X _ { i } ^ { 2 } ] = \\sigma ^ { 2 }$ . Also, $\\mathbb { E } [ X _ { i } X _ { j } ] =$ $\\mathbb { E } [ X _ { i } ] \\mathbb { E } [ X _ { j } ] \\leqslant \\sigma ^ { 2 }$ . Therefore, ", + "bbox": [ + 173, + 313, + 826, + 375 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/065e30c1ad78f5231b6e27920abb354e3f10351bec2bc65dc220df415ec86c1e.jpg", + "text": "$$\n{ \\mathbb E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } X _ { i } \\right) ^ { 2 } \\right] = \\frac { 1 } { d ^ { 2 } } \\sum _ { i , j = 1 } ^ { d } { \\mathbb E } [ X _ { i } X _ { j } ] \\leqslant \\sigma ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 380, + 656, + 431 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Finally, if we set $r = t / d \\in ( 0 . 5 , 1 ]$ , we have13 ", + "bbox": [ + 173, + 445, + 478, + 462 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/2170d830a87b4de3cf6184af5db587f8b1c174420240916b8a01f4949d1c3472.jpg", + "text": "$$\n{ \\binom { d } { t } } \\sim \\left( { \\frac { 1 } { r ^ { r } ( 1 - r ) ^ { 1 - r } } } \\right) ^ { d }\n$$", + "text_format": "latex", + "bbox": [ + 408, + 467, + 588, + 506 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "as $d \\to \\infty$ (discarding all polynomial terms). Hence $\\binom { d } { t }$ is of the form $q ^ { d }$ , with $1 \\leqslant q < 2$ . So, the quantity σ2npd ´ tq dt 12 d´1 w ill be exponentially decreasing at a rate $O ( n / ( 2 - q ) ^ { d } )$ . Notably, if $t = d / 2$ , then we lose the exponential rate and get back to $\\mathcal { O } ( n / d )$ . □ ", + "bbox": [ + 173, + 512, + 826, + 563 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B APPENDIX TO SECTION 3 ", + "text_level": 1, + "bbox": [ + 176, + 102, + 419, + 118 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We used Pytorch Paszke et al. (2017) to implement all experiments in this paper. Our codebase is publicly available at https://github.com/gibipara92/ learning-explanations-hard-to-vary. ", + "bbox": [ + 174, + 133, + 826, + 175 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.1 SECTION 3.1 ", + "text_level": 1, + "bbox": [ + 174, + 191, + 305, + 207 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/6f0c7c5052a41e9e1ef008540e72190e924615cdadd95c8bd72c9f13df719068.jpg", + "table_caption": [ + "Table 1: Hyperparameter ranges for synthetic data experiments. The regularizers L1 and L2 are never combined; instead, one weight regularization type out of L1, L2 and none is selected and we sample from the respective range afterwards. " + ], + "table_footnote": [], + "table_body": "
HyperparameterRanges
No.hidden units{256,512}
No.hidden layers{3,5}
Batch-size{64,128,256}
Optimizer{Adamβ1 =0.9,β2=0.99, SGD + momentum0.9}
Learning rate{1e-3,1e-2,1e-1}
Batch-normalization{Yes,No}
Dropout{0.0, 0.5}
L2 regularization{1e-5,1e-4,1e-3}
L1 regularization{1e-6, 1e-5, 1e-4}
", + "bbox": [ + 254, + 271, + 738, + 426 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.2 DATASET ", + "text_level": 1, + "bbox": [ + 174, + 453, + 282, + 467 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Here we report more technical details about the synthetic dataset described in Section 3. Each example is constructed as follows: we first choose the label randomly to be either $+ 1$ or $- 1$ , with equal probability. The example is a vector with $d _ { S } + d _ { M }$ entries, consisting of the shortcut and the mechanism. In our experiments, $d _ { M } = 2$ and $d _ { S } = 3 2$ . ", + "bbox": [ + 174, + 478, + 678, + 549 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The Gaussian shortcuts are obtained by first sampling one random vector $\\mathbf { x } _ { s } \\in \\mathbb { R } ^ { d _ { S } }$ per environment. Its components $x _ { s , i }$ are sampled independently from a Normal distribution: $x _ { s , i } \\sim \\mathcal { N } ( 0 , 0 . 1 )$ . We use $\\mathbf { x } _ { s }$ for class 1, and $- \\mathbf { x } _ { s }$ for class -1. In the test set, all shortcut components are sampled i.i.d. from the same Normal distribution. Effectively, each example of the test set belongs to a different domain. The mechanism is implemented as the two interconnected spirals shown in Figure 13 by sampling the radius $r \\sim \\mathrm { U n i f } ( 0 . 0 8 , 1 . 0 )$ and then computing the angle as $\\alpha = 2 \\pi n r$ where $n$ is the number of revolutions of the spiral. We add uniform noise in the range $\\left[ - 0 . 0 2 , 0 . 0 2 \\right]$ to the radii afterwards. ", + "bbox": [ + 174, + 555, + 678, + 654 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/2703df68fff546c00ca1b4b0275d73937f6cdf748f6bd7fabe5dde2e5d70d63d.jpg", + "image_caption": [ + "Figure 13: The spirals used as the mechanism in the synthetic memorization dataset. " + ], + "image_footnote": [], + "bbox": [ + 694, + 482, + 820, + 580 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 654, + 821, + 681 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The training dataset consists of 1280 examples per environment and we use $D = 3 2$ environments unless otherwise mentioned. The training datasets consists of 2000 examples. ", + "bbox": [ + 176, + 688, + 823, + 715 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.3 EXPERIMENT ", + "text_level": 1, + "bbox": [ + 176, + 732, + 310, + 747 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We train all networks for $\\lfloor 3 0 0 0 / D \\rfloor$ epochs, dropping the learning rate by a factor 10 halfway through, and again at three-quarters of training. For computational reason, we stop each trial before completion if the training accuracy exceeds $9 7 \\%$ and the test accuracy is below $60 \\%$ . All networks are MLPs with LeakyReLU activation functions and a cross-entropy loss on the output. We run a hyperparameter search over the ranges shown in Table 1. For IRM and the AND-mask, we select the best-performing run and re-run it 50 times with different random seeds. For DANN and the standard baselines nothing produced results significantly better than chance. ", + "bbox": [ + 174, + 758, + 825, + 856 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.3.1 STANDARD REGULARIZERS AND AND-MASK ", + "text_level": 1, + "bbox": [ + 174, + 871, + 544, + 886 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The networks with the L1, L2, Dropout and Batch-normalization regularizers, have hyperparameters that were randomly selected from Table 1. For the AND-mask we used the very same ranges. The regularizers L1 and L2 are never combined; instead, one weight regularization type out of L1, L2 and none is selected and we sample from the respective range afterwards. The parameters found to work best from the grid search were: agreement threshold of 1, 256 hidden units, 3 hidden layers, batch size 128, Adam with learning rate 1e-2, no batch norm, no dropout, L2-regularization with a coefficient of 1e-4, no L1-regularization. In practice, we often found it helpful to rescale the gradients after masking to compensate for the decreasing overall magnitude. We add the option for gradient rescaling as an additional hyperparameter, as we found it to help in several experiments. It rescales gradient components layer-wise after masking, by multiplying the remaining gradient components by $c$ , where $c$ is the ratio of the number of components in that layer over the number of non-masked components in that layer (i.e. the sum of the binary elements in the mask).14. We speculate that for very large layers, a less extreme normalization scheme or the additional use of gradient clipping might be appropriate. ", + "bbox": [ + 176, + 895, + 823, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 270 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "B.3.2 DOMAIN ADVERSARIAL NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 286, + 547, + 300 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The experiments using DANN follow a similar pattern. The model consists of an embedding network, a classification network, and a “domain discrimination” network. All three modules are two-layer multi-layer perceptrons (MLP). The number of hidden units of all MLPs are sampled from the range specified in Table 1, and we trained 100 models. Both label classifier and domain discriminator are applied to the output of the embedding network. The label classifier is trained to minimize the cross-entropy-loss between the predicted and the true label. Similarly, the domain discriminator is trained to minimize the loss between predicted and true domain-label. The embedding network is trained to minimize the regular task classification loss and at the same time to maximize the the domain-loss achieved by the domain discriminator. ", + "bbox": [ + 174, + 310, + 825, + 435 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "B.3.3 INVARIANT RISK MINIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 452, + 459, + 467 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For the experiments using IRM we used the authors’ PyTorch implementation from https: //github.com/facebookresearch/InvariantRiskMinimization. We perform a random hyperparameter search over with the ranges shown in Table 2 ", + "bbox": [ + 176, + 477, + 823, + 518 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/cff0315a83239b01dd138e0c51da4a8cc4094cf8d5e2370e9466bb8d73216aa9.jpg", + "table_caption": [ + "Table 2: Hyperparameter ranges for IRM. " + ], + "table_footnote": [], + "table_body": "
HyperparameterRanges
No.hidden units{256,512}
No.hidden layers{3,5}
Batch-size{64,128,256}
Optimizer{Adamβ1=0.9,β2=0.999, SGD + momentum0.9}
Batch-normalization{Yes,No}
Penalty weight{10.0,100.0,1000.0}
Number of annealing iterations{0,1,2,4,8}
Learning rate{1e-3,1e-2, 1e-1, 1}
", + "bbox": [ + 222, + 556, + 772, + 699 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "B.3.4 CURVES FOR ALL EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 723, + 455, + 737 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In Figure 14 we show the learning curves of training and test accuracy for the different methods. ", + "bbox": [ + 171, + 747, + 802, + 762 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "B.3.5 CORRELATION PLOTS ", + "text_level": 1, + "bbox": [ + 176, + 777, + 380, + 791 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For the correlation plots in Figure 7 we used a randomly initialized MLP with the following configuration: 3 hidden layers, 256 hidden units. The dataset was using 16 environments and batches of size 1024. The lines in Figure 7 are linear least-squares regressions to the gradient data shown as scatter plots. We repeat the experiment 10 times with different network weight seeds, resulting in the 10 regression lines. Zero gradients are excluded from the regression computation, as most gradients are masked out by the product mask in both cases. ", + "bbox": [ + 174, + 801, + 825, + 886 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/f1c54ea4ad51704480b75ad6c14459bd6db2dfafae7f409ff4810c75f7d4bc0c.jpg", + "image_caption": [ + "Figure 14: Learning curves for the evaluated methods. The top row shows the accuracy on the training set, the bottom row shows the accuracy on the test set. " + ], + "image_footnote": [], + "bbox": [ + 243, + 104, + 753, + 277 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "B.4 FURTHER VISUALIZATIONS AND EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 348, + 542, + 361 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "In Figure 15 we show how many environments need to be present for the baseline without AND-mask to switch the decision boundary from the shortcuts to the mechanism. Under the same experimental condition as in the main paper, the baseline first succeeds at 1024 environments. ", + "bbox": [ + 173, + 372, + 825, + 415 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/4ec4fe862dad3a66e23c73556d1743d289326e46a0506be20524dc68015f4975.jpg", + "image_caption": [ + "Figure 15: Relationship between number of training environments and test accuracy for the AND-mask method compared to the baseline. We show the best performance out of five runs using the settings that were used for the experiment in the main text. " + ], + "image_footnote": [], + "bbox": [ + 174, + 428, + 825, + 502 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "B.5 SECTION 3.2: CIFAR-10 MEMORIZATION AND LABEL NOISE EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 578, + 738, + 593 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Memorization experiment In Figure 16, we report the test performance (dashed lines) corresponding to the curves presented in the main paper for the CIFAR-10 memorization experiment. The test performance with standard labels decreases slower than the training performance as the threshold increases, and they eventually reach the same value. This is consistent with the hypothesis that by training on the consistent directions, the AND-mask selects the invariant patterns and prunes out the signals that are not invariant. ", + "bbox": [ + 174, + 604, + 570, + 729 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/8ca71fe755e2a88b3312a7912c3fb6fe9a5bde54794c3813b8551085371d1d41.jpg", + "image_caption": [ + "Figure 16: Dashed lines show test acc, solid lines show training acc. " + ], + "image_footnote": [], + "bbox": [ + 584, + 608, + 820, + 704 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Network architecture and training details Each trial ", + "text_level": 1, + "bbox": [ + 174, + 746, + 568, + 760 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "trains the ResNet “FastResNet” from the PyTorch-Ignite example15 for 80 epochs on the full CIFAR-10 training set. We use the Adam optimizer with a learning rate of 5e´4, and a 0.1 learning rate decay at epoch 40 and 60. We fix the batch size to 80. We set up 14 trials by evaluating each of the AND-mask-thresholds $\\{ 0 , 0 . 0 5 , 0 . 1 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 \\}$ for two datasets: (a) unchanged CIFAR-10, (b) CIFAR-10 with the training labels replaced by random labels. Note that a threshold of 0 corresponds to not using the AND-mask. Each trial is run twice with separate random seeds. ", + "bbox": [ + 173, + 760, + 825, + 843 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Label noise experiment We trained the same ResNet as for the experiment above, once with and once without the AND-mask. We ran each experiment with three different starting learning rates $\\{ 5 \\mathrm { e } { - } 4 , 1 \\mathrm { e } { - } 3 , 5 \\mathrm { e } { - } 3 \\}$ and a learning rate decay at epoch 60. The baseline worked best with a learning rate of $\\mathrm { 1 e { - 3 } }$ , while the AND-mask with 5e´3, likely to compensate for the masked out gradients. The AND-mask threshold that worked best was 0.2, which is consistent with the results obtain in the experiment above. ", + "bbox": [ + 174, + 858, + 823, + 887 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "B.6 SECTION 3.3: BEHAVIORAL CLONING ON COINRUN ", + "text_level": 1, + "bbox": [ + 174, + 175, + 578, + 190 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The target policy $\\pi ^ { * }$ is obtained by training PPO (Schulman et al., 2017) for 400M time steps using the code16 for the paper Cobbe et al. (2020). This policy is trained on the full distribution of levels in order to maximize its generality. We use $\\pi ^ { * }$ to generate a behavioral cloning (BC) dataset, consisting of pairs $( s , \\pi ^ { * } ( a | s ) )$ , where $s$ are the input-images ( $6 4 \\times 6 4$ RGB) and $\\pi ^ { * } ( a | s )$ is the discrete probability distribution over actions output by $\\pi ^ { * }$ . ", + "bbox": [ + 173, + 200, + 825, + 272 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The states are sampled randomly from trajectories generated by $\\pi ^ { * }$ . In order to test for generalization performance, the BC training dataset is restricted to 64 distinct levels. We generate 1000 examples per training level. The test set consists of 2000 examples, each from a different level which does not appear in the training set. ", + "bbox": [ + 176, + 279, + 823, + 335 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/31904abde0e46daf3cc3ab7cd2a873c086e55035d97a7d88b06b70d71bcc27ed.jpg", + "image_caption": [ + "Figure 17: Screenshots of 6 levels of CoinRun (from OpenAI). " + ], + "image_footnote": [], + "bbox": [ + 241, + 348, + 756, + 407 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "A ResNet- $1 8 ~ \\hat { \\pi } _ { \\boldsymbol { \\theta } }$ is trained to minimize the loss $D _ { \\mathrm { K L } } ( \\pi ^ { * } | | \\hat { \\pi } _ { \\boldsymbol { \\theta } } )$ . We ran two automatic hyperparameter optimization studies using Tree-structured Parzen Estimation (TPE) (Bergstra et al., 2013) of 1024 trials each, with and without the AND-mask. The learning rate was decayed by a factor of 10 half-way at at $3 / _ { 4 }$ of the training epochs. ", + "bbox": [ + 173, + 446, + 826, + 503 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The “temporal” version of the AND-mask used for this experiment is reported in Algorithm 1. ", + "bbox": [ + 168, + 510, + 789, + 525 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Algorithm 1: Temporal AND-mask Adam ", + "text_level": 1, + "bbox": [ + 176, + 542, + 454, + 558 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "1 $\\mathbf { m } \\gets \\beta _ { 1 } \\cdot \\mathbf { m } + ( 1 - \\beta _ { 1 } ) \\cdot \\mathbf { g }$ \n2 $\\mathbf { v } \\beta _ { 2 } \\cdot \\mathbf { v } + ( 1 - \\beta _ { 2 } ) \\cdot ( \\mathbf { g } \\circ \\mathbf { g } )$ \n3 $\\mathbf { a } \\beta _ { 3 } \\cdot \\mathbf { a } + ( 1 - \\beta _ { 3 } ) \\cdot \\mathsf { e l e m w i s e \\_ s i g n } ( \\mathbf { g } )$ \n4 $\\mathbf { b } \\gets \\mathbb { 1 } [ | \\mathbf { a } | \\geqslant \\tau ]$ \n5 $\\theta \\gets \\theta - \\alpha ( \\mathbf { m } \\circ \\mathbf { b } ) \\oslash \\sqrt { \\mathbf { v } + \\epsilon }$ ", + "bbox": [ + 160, + 560, + 486, + 636 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "In blue we highlight the additional lines compared to traditional Adam. The threshold $\\tau$ and $\\beta _ { 3 }$ are hyperparameters that we included in the $1 ^ { \\cdot } 0 2 4$ trials of the search using Tree-structured Parsen Estimators. For the top 10 runs, hyperparameter values that were selected via the TPE search for the AND-mask are the following. ", + "bbox": [ + 173, + 654, + 825, + 710 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/0df6cb3d3ccefcd63cf93d8f8b0fe917f2737c8485f30e8be43ecdf3b990e9fb.jpg", + "table_caption": [ + "Table 3: Hyperparameters for the 5 best runs using the AND-mask, from the TPE search. " + ], + "table_footnote": [], + "table_body": "
Test KL divlrβ1β3Tweight decay
1.652e-20.00780.210.790.360.057
1.656e-20.00720.260.860.400.041
1.662e-20.00800.230.840.410.045
1.665e-20.00680.330.720.470.077
1.672e-20.00630.670.650.470.080
", + "bbox": [ + 194, + 746, + 802, + 847 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "We found that applying weight decay as a second independent update after the AND-mask routine improved performance. To keep the comparison fair, we added this as a switch in the hyperparameter search for the Adam baseline as well, and it improved performance there as well. ", + "bbox": [ + 176, + 859, + 825, + 901 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/58e7efa834f50663cd23c9bf7b8b533a909ff815082f83abdc760db4f08069bd.jpg", + "image_caption": [ + "Figure 18: Learning curves for the behavioral cloning experiment on CoinRun. Training loss is shown on the left, test loss is shown on the right. We show the mean over the top-10 runs for each method. The shaded regions correspond to the $9 5 \\%$ confidence interval of the mean based on bootstrapping. " + ], + "image_footnote": [], + "bbox": [ + 184, + 109, + 813, + 251 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "C APPENDIX TO SECTION 4 ", + "text_level": 1, + "bbox": [ + 176, + 337, + 419, + 352 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "C.1 RELATED WORK IN CAUSAL INFERENCE ", + "text_level": 1, + "bbox": [ + 176, + 368, + 491, + 382 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Causal graphs and causal factorizations The formalization of causality through directed acyclic graphs (Pearl, 2009) is a key element informing our exposition. According to such formalization, a causal model gives rise to each observed distribution. It is thereby possible to exploit properties of the causal factorization of the joint probability distribution over the observed variables. Clearly, there are many ways to factorize a joint distribution into conditionals; a distinguishing feature of the causal factorization is that many of the conditionals, which we can think of as physical mechanisms underlying the statistical dependencies represented, are expected to remain invariant under interventions or changing external conditions. This postulate has appeared in various forms in the literature (Haavelmo, 1943; Simon, 1953; Hurwicz, 1962; Pearl, 2009; Schölkopf et al., 2012).17 ", + "bbox": [ + 174, + 395, + 825, + 520 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Causal models and robust regression Based on this insight, it was proposed that regression based on causal features should presents desirable invariance and robustness properties (Mooij et al., 2009; Schölkopf et al., 2012; Peters et al., 2016; Rojas-Carulla et al., 2018; Heinze-Deml et al., 2018; von Kügelgen et al., 2019; Parascandolo et al., 2018). In this view, the mechanisms can be considered as features of the patterns such that they support stable conditional probabilities. Thus learning the mechanisms may help achieve a stable performance across a number of conditions. Other works connecting causality and learning through invariances are (Subbaswamy et al., 2019; Heinze-Deml and Meinshausen, 2017), and perhaps – most related to our work – (Arjovsky et al., 2019): we presented a comparison with this method in the following section. ", + "bbox": [ + 174, + 536, + 825, + 661 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Causal regularization Recently (Janzing, 2019) showed that biasing learning towards models of lower complexity might in some cases be beneficial for a notion of generalization from observational to interventional regimes. Our proposed solution is however different, in that we only indirectly deal with penalizing model complexity, and rather focus on our proposed notion of consistency. ", + "bbox": [ + 174, + 679, + 825, + 734 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "C.2 LEARNING INVARIANCES IN THE DATA ", + "text_level": 1, + "bbox": [ + 178, + 753, + 482, + 766 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Here we are going to compare ILC to other approaches for learning invariances in the data with neural networks, and in particular to Invariant Risk Minimization (IRM) Arjovsky et al. (2019). The authors of IRM analyze a set up where minimizing training error might lead to models which absorb all the correlations found within the training data, thus failing to recover the relevant causal explanation. They consider a multi-environment setting and focus on the objective of extracting data representations that lead to invariant prediction across environments. ", + "bbox": [ + 174, + 779, + 825, + 863 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "While the high level objective is close to the one we focused on, the differences become clear when considering the definition of invariant predictors presented in Arjovsky et al. (2019): ", + "bbox": [ + 176, + 869, + 823, + 897 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition 1. A data representation $\\Phi : \\mathcal { X } \\to \\mathcal { H }$ elicits an invariant predictor $w \\circ \\Phi$ across environments $\\mathcal { E }$ if there is a classifier $w : \\mathcal { H } \\to \\mathcal { V }$ simultaneously optimal for all environments, i.e., $w \\in \\arg \\operatorname* { m i n } _ { \\bar { w } : \\mathcal { H } y } R ^ { e } ( \\bar { w } \\circ \\Phi ) \\forall e \\in \\mathcal { E }$ . ", + "bbox": [ + 173, + 103, + 826, + 147 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In particular, the objective minimized by IRM is: ", + "bbox": [ + 173, + 156, + 496, + 172 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/d421d612904c053e8e4322cd0bb5720e95ddfd648a9b1728a199ee63b1536984.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\Phi : \\mathcal { X } \\to \\mathcal { V } } \\sum _ { e \\in \\mathcal { E } _ { \\mathrm { t r } } } R ^ { e } ( \\Phi ) + \\lambda \\cdot \\left\\| \\nabla _ { w | w = 1 . 0 } R ^ { e } ( w \\cdot \\Phi ) \\right\\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 334, + 179, + 660, + 213 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where $\\Phi$ are the logits predicted by the neural network and $w$ is a dummy scaling variable (see› Arjovsky et al. (2019)). The relevant part is the penalty term $\\lambda \\cdot \\left\\| \\nabla _ { w | w = 1 . 0 } R ^ { e } ( w \\cdot \\Phi ) \\right\\| ^ { 2 }$ : One way to interpret it, is that the penalty is large on every environment where the distribution outputted by $\\Phi$ could be made ‘closer’ to the distribution of the labels by either sharpening $\\mathit { w } > 1 $ ) or softening it (i.e., closer to uniform $w < 1$ ). ", + "bbox": [ + 173, + 218, + 825, + 292 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Let us consider the example from IRM, where the authors describe two datasets of images that each contain either a cow or a camel: In one of the datasets, there is grass on $80 \\%$ of the images with cows, while in the other dataset there is grass on $90 \\%$ of them. IRM then makes the point that we can learn to ignore grass as a feature, because its correlation with the label cow is inconsistent ( $80 \\%$ vs $90 \\%$ ). The setting we consider in this paper is slightly different: take our example from the CIFAR-10 experiments. Under our concept of invariance, we expect that (depending on the data generating process) even a single dataset where we treat every image as coming from its own ‘environment’ should be sufficient to discover invariances. Drawing a connection to the setting from IRM, we would argue that the second dataset should not be necessary to learn that ‘grass’ is not ‘cow’. If one treats every example as coming from its own environment, there is already sufficient information in the first dataset to realize that cows are not grass: Grass is predictive of cows only in $80 \\%$ of the data, so grass cannot be ‘cow’. The actual cow on the other hand, should be present in $100 \\%$ of the images, and as such it is the invariance we are looking for. Note that this is of course a much more strict definition of invariance: If our dataset contains images labeled as ’cows’ but that have no cows within them, we might start to discard the features of cows as well. 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We show that averaging gradients across examples", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 220, + 470, + 231 + ], + "spans": [ + { + "bbox": [ + 141, + 220, + 233, + 231 + ], + "score": 1.0, + "content": "– akin to a logical OR", + "type": "text" + }, + { + "bbox": [ + 233, + 220, + 248, + 230 + ], + "score": 0.41, + "content": "( \\vee )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 220, + 470, + 231 + ], + "score": 1.0, + "content": "of patterns – can favor memorization and ‘patchwork’", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 230, + 471, + 243 + ], + "spans": [ + { + "bbox": [ + 141, + 230, + 471, + 243 + ], + "score": 1.0, + "content": "solutions that sew together different strategies, instead of identifying invariances.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 241, + 470, + 253 + ], + "spans": [ + { + "bbox": [ + 141, + 241, + 470, + 253 + ], + "score": 1.0, + "content": "To inspect this, we first formalize a notion of consistency for minima of the loss", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 252, + 469, + 265 + ], + "spans": [ + { + "bbox": [ + 141, + 252, + 469, + 265 + ], + "score": 1.0, + "content": "surface, which measures to what extent a minimum appears only when examples", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 142, + 264, + 469, + 275 + ], + "spans": [ + { + "bbox": [ + 142, + 264, + 469, + 275 + ], + "score": 1.0, + "content": "are pooled. We then propose and experimentally validate a simple alternative", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 275, + 469, + 286 + ], + "spans": [ + { + "bbox": [ + 142, + 275, + 284, + 286 + ], + "score": 1.0, + "content": "algorithm based on a logical AND", + "type": "text" + }, + { + "bbox": [ + 285, + 275, + 300, + 285 + ], + "score": 0.6, + "content": "( \\wedge )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 275, + 469, + 286 + ], + "score": 1.0, + "content": ", that focuses on invariances and prevents", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 286, + 469, + 297 + ], + "spans": [ + { + "bbox": [ + 142, + 286, + 469, + 297 + ], + "score": 1.0, + "content": "memorization in a set of real-world tasks. 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Note", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 401, + 330, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 330, + 413 + ], + "score": 1.0, + "content": "the two global minima on the top-right and bottom-left.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 412, + 329, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 242, + 424 + ], + "score": 1.0, + "content": "Depending on the initial values of", + "type": "text" + }, + { + "bbox": [ + 242, + 413, + 249, + 423 + ], + "score": 0.77, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 412, + 329, + 424 + ], + "score": 1.0, + "content": "— marked as white", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 424, + 329, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 329, + 435 + ], + "score": 1.0, + "content": "circles — gradient descent converges to one of the two", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 434, + 330, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 330, + 446 + ], + "score": 1.0, + "content": "minima. 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Averaging gradients forgoes informa-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 335, + 554, + 507, + 566 + ], + "spans": [ + { + "bbox": [ + 335, + 554, + 507, + 566 + ], + "score": 1.0, + "content": "tion that can identify patterns shared across dif-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 336, + 565, + 412, + 574 + ], + "spans": [ + { + "bbox": [ + 336, + 565, + 412, + 574 + ], + "score": 1.0, + "content": "ferent environments.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 53.5 + } + ], + "index": 49.0 + }, + { + "type": "text", + "bbox": [ + 109, + 583, + 502, + 594 + ], + "lines": [], + "index": 56, + "bbox_fs": [ + 107, + 581, + 504, + 596 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "In this paper, we argue that the answer is yes. In particular, we hypothesize that if the goal is to", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "find invariant mechanisms in the data, these can be identified by finding explanations (e.g. model", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 623, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 634 + ], + "score": 1.0, + "content": "parameters) that are hard to vary across examples. A notion of invariance implies something that", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "stays the same, as something else changes. We assume that data comes from different environments:", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "An invariant mechanism is shared across all, generalizes out of distribution (o.o.d.), but might be hard", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "to model; each environment also has spurious explanations that are easy to spot (‘shortcuts’), but do", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "not generalize o.o.d. From the point of view of causal modeling, such invariant mechanisms can be", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "interpreted as conditional distributions of the targets given causal features of the inputs; invariance", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "of such conditionals is expected if they represent causal mechanisms, that is — stable properties of", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "the physical world (see e.g. Hoover (1990)). Generalizing o.o.d. means therefore that the predictor", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "should perform equally well on data coming from different settings, as long as they share the causal", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 720, + 161, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 161, + 732 + ], + "score": 1.0, + "content": "mechanisms.", + "type": "text" + } + ], + "index": 68 + } + ], + "index": 62.5, + "bbox_fs": [ + 105, + 600, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "We formalize a notion of consistency, which characterizes to what extent a minimum of the loss", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "surface appears only when data from different environments are pooled. Minima with low consistency", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "are ‘patchwork’ solutions, which (we hypothesize) sew together different strategies and should not be", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "expected to generalize to new environments. An intuitive description of this principle was proposed", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 444, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 444, + 140 + ], + "score": 1.0, + "content": "by physicist David Deutsch: “good explanations are hard to vary” (Deutsch, 2011).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "Using the notion of consistency, we define Invariant Learning Consistency (ILC), a measure of the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "expected consistency of the solution found by a learning algorithm on a given hypothesis class. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "ILC can be improved by changing the hypothesis class or the learning algorithm, and in the last", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 174, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 104, + 174, + 506, + 191 + ], + "score": 1.0, + "content": "part of the paper we focus on the latter. We then analyse why current practices in deep learning", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "provide little incentive for networks to learn invariances, and show that standard training is instead", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "set up with the explicit objective of greedily maximizing speed of learning, i.e., progress on the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "training loss. When learning “as fast as possible” is not the main objective, we show we can trade-off", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "some “learning speed” for prioritizing learning the invariances. A practical instantiation of ILC leads", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "to o.o.d. generalization on a challenging synthetic task where several established regularizers fail", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "to generalize; moreover, following the memorization task from Zhang et al. (2017), ILC prevents", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "convergence on CIFAR-10 with random labels, as no shared mechanism is present, and similarly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "when a portion of training labels is incorrect. Lastly, we set up a behavioural cloning task based on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 497, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 497, + 287 + ], + "score": 1.0, + "content": "the game CoinRun (Cobbe et al., 2019b), and observe better generalization on new unseen levels.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 111, + 295, + 312, + 397 + ], + "lines": [ + { + "bbox": [ + 110, + 295, + 313, + 307 + ], + "spans": [ + { + "bbox": [ + 110, + 295, + 313, + 307 + ], + "score": 1.0, + "content": "An example. Take these two second-hand books", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 111, + 307, + 313, + 319 + ], + "spans": [ + { + "bbox": [ + 111, + 307, + 313, + 319 + ], + "score": 1.0, + "content": "of chess puzzles. We can learn the two independent", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 110, + 317, + 313, + 330 + ], + "spans": [ + { + "bbox": [ + 110, + 317, + 313, + 330 + ], + "score": 1.0, + "content": "shortcuts (blue arrows for the left book OR hand-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 110, + 328, + 313, + 341 + ], + "spans": [ + { + "bbox": [ + 110, + 328, + 313, + 341 + ], + "score": 1.0, + "content": "written solutions on the right), or actually learn", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 109, + 339, + 313, + 353 + ], + "spans": [ + { + "bbox": [ + 109, + 339, + 313, + 353 + ], + "score": 1.0, + "content": "to play chess (the invariant mechanism). While", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 109, + 350, + 313, + 364 + ], + "spans": [ + { + "bbox": [ + 109, + 350, + 313, + 364 + ], + "score": 1.0, + "content": "both strategies solve other problems from the same", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 110, + 362, + 313, + 374 + ], + "spans": [ + { + "bbox": [ + 110, + 362, + 313, + 374 + ], + "score": 1.0, + "content": "books (i.i.d.), only the latter generalises to new", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 110, + 373, + 313, + 385 + ], + "spans": [ + { + "bbox": [ + 110, + 373, + 313, + 385 + ], + "score": 1.0, + "content": "chess puzzle books (o.o.d.). How to distinguish", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 110, + 383, + 313, + 396 + ], + "spans": [ + { + "bbox": [ + 110, + 383, + 313, + 396 + ], + "score": 1.0, + "content": "the two? We would not have learned about the red", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 25 + }, + { + "type": "image", + "bbox": [ + 319, + 295, + 501, + 390 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 319, + 295, + 501, + 390 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 319, + 295, + 501, + 390 + ], + "spans": [ + { + "bbox": [ + 319, + 295, + 501, + 390 + ], + "score": 0.969, + "type": "image", + "image_path": "4e958bc770fbe8a283a4b825875d313db41b01e0fd8c86aa895fd80d473209b2.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 319, + 295, + 501, + 308.57142857142856 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 319, + 308.57142857142856, + 501, + 322.1428571428571 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 319, + 322.1428571428571, + 501, + 335.71428571428567 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 319, + 335.71428571428567, + 501, + 349.2857142857142 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 319, + 349.2857142857142, + 501, + 362.8571428571428 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 319, + 362.8571428571428, + 501, + 376.42857142857133 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 319, + 376.42857142857133, + 501, + 389.9999999999999 + ], + "spans": [], + "index": 33 + } + ] + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 110, + 394, + 474, + 406 + ], + "lines": [ + { + "bbox": [ + 109, + 394, + 476, + 407 + ], + "spans": [ + { + "bbox": [ + 109, + 394, + 476, + 407 + ], + "score": 1.0, + "content": "arrows had we trained on the book on the right, and vice versa with the hand-written notes.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 107, + 430, + 340, + 442 + ], + "lines": [ + { + "bbox": [ + 104, + 429, + 341, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 341, + 444 + ], + "score": 1.0, + "content": "2 EXPLANATIONS THAT ARE HARD TO VARY", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 104, + 451, + 504, + 466 + ], + "spans": [ + { + "bbox": [ + 104, + 451, + 189, + 466 + ], + "score": 1.0, + "content": "We consider datasets", + "type": "text" + }, + { + "bbox": [ + 190, + 453, + 225, + 465 + ], + "score": 0.93, + "content": "\\{ \\mathcal { D } ^ { e } \\} _ { e \\in \\mathcal { E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 451, + 248, + 466 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 248, + 453, + 279, + 465 + ], + "score": 0.92, + "content": "| \\mathcal { E } | = d", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 451, + 298, + 466 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 299, + 453, + 357, + 465 + ], + "score": 0.84, + "content": "\\mathcal { D } ^ { e } = ( x _ { i } ^ { e } , y _ { i } ^ { e } )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 451, + 360, + 466 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 361, + 453, + 420, + 465 + ], + "score": 0.86, + "content": "i _ { e } = 1 , \\dots , n ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 451, + 445, + 466 + ], + "score": 1.0, + "content": ". Here", + "type": "text" + }, + { + "bbox": [ + 446, + 453, + 504, + 465 + ], + "score": 0.89, + "content": "x _ { i } ^ { e } \\in \\mathcal { X } \\subseteq \\mathbb { R } ^ { m }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 463, + 504, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 463, + 308, + 477 + ], + "score": 1.0, + "content": "is the vector containing the observed inputs, and", + "type": "text" + }, + { + "bbox": [ + 308, + 465, + 363, + 476 + ], + "score": 0.92, + "content": "y _ { i } ^ { e } \\in \\mathcal { V } \\subseteq \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 463, + 479, + 477 + ], + "score": 1.0, + "content": "the targets. The superscript", + "type": "text" + }, + { + "bbox": [ + 480, + 465, + 504, + 474 + ], + "score": 0.84, + "content": "e \\in { \\mathcal { E } }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "indexes some aspect of the data collection process, and can be interpreted as an environment label.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 484, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 245, + 500 + ], + "score": 1.0, + "content": "Our objective is to infer a function", + "type": "text" + }, + { + "bbox": [ + 245, + 486, + 293, + 497 + ], + "score": 0.91, + "content": "f : \\mathcal { X } \\mathcal { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 484, + 493, + 500 + ], + "score": 1.0, + "content": "— which we call mechanism — assigning a target", + "type": "text" + }, + { + "bbox": [ + 493, + 487, + 504, + 498 + ], + "score": 0.87, + "content": "y _ { i } ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 484, + 505, + 500 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 160, + 510 + ], + "score": 1.0, + "content": "to each input", + "type": "text" + }, + { + "bbox": [ + 161, + 497, + 172, + 509 + ], + "score": 0.88, + "content": "\\boldsymbol { x } _ { i } ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 496, + 506, + 510 + ], + "score": 1.0, + "content": "; as explained in the introduction, we assume that such function is shared across all", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 264, + 521 + ], + "score": 1.0, + "content": "environments. For estimation purposes,", + "type": "text" + }, + { + "bbox": [ + 265, + 509, + 272, + 520 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 508, + 506, + 521 + ], + "score": 1.0, + "content": "may be parametrized by a neural network with continuous", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 518, + 480, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 202, + 532 + ], + "score": 1.0, + "content": "activations; for weights", + "type": "text" + }, + { + "bbox": [ + 202, + 519, + 252, + 529 + ], + "score": 0.91, + "content": "\\theta \\in \\Theta \\subseteq \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 518, + 414, + 532 + ], + "score": 1.0, + "content": ", we denote the neural network output at", + "type": "text" + }, + { + "bbox": [ + 414, + 519, + 441, + 529 + ], + "score": 0.9, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 518, + 453, + 532 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 453, + 519, + 477, + 531 + ], + "score": 0.92, + "content": "f _ { \\boldsymbol { \\theta } } ( \\boldsymbol { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 518, + 480, + 532 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 357, + 553 + ], + "score": 1.0, + "content": "Gradient-based optimization. To find an appropriate model", + "type": "text" + }, + { + "bbox": [ + 357, + 541, + 367, + 552 + ], + "score": 0.87, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 539, + 506, + 553 + ], + "score": 1.0, + "content": ", standard optimizers rely on gradi-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 239, + 564 + ], + "score": 1.0, + "content": "ents from a pooled loss function", + "type": "text" + }, + { + "bbox": [ + 239, + 552, + 290, + 562 + ], + "score": 0.91, + "content": "\\mathcal { L } : \\mathbb { R } ^ { n } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 550, + 506, + 564 + ], + "score": 1.0, + "content": ". This function measures the average performance ofř", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 560, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 403, + 578 + ], + "score": 1.0, + "content": "the neural network when predicting data labels, across all environments: ř", + "type": "text" + }, + { + "bbox": [ + 403, + 561, + 501, + 577 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { L } ( \\boldsymbol { \\theta } ) : = \\dot { \\frac { 1 } { | \\mathcal { E } | } } \\sum _ { e \\in \\mathcal { E } } \\mathcal { L } _ { e } ( \\boldsymbol { \\theta } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 560, + 506, + 578 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 574, + 507, + 594 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 126, + 594 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 576, + 296, + 592 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\begin{array} { r } { \\mathcal { L } _ { e } ( \\theta ) : = \\frac { 1 } { | \\mathcal { D } ^ { e } | } \\sum _ { ( x _ { i } ^ { e } , y _ { i } ^ { e } ) \\in \\mathcal { D } ^ { e } } \\ell ( f ( x _ { i } ^ { e } ; \\theta ) , y _ { i } ^ { e } ) } \\end{array} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 574, + 326, + 594 + ], + "score": 1.0, + "content": "; where", + "type": "text" + }, + { + "bbox": [ + 326, + 577, + 424, + 589 + ], + "score": 0.91, + "content": "\\ell : \\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { p } \\to [ 0 , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 574, + 507, + 594 + ], + "score": 1.0, + "content": "is usually chosen to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 133, + 603 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 133, + 591, + 146, + 600 + ], + "score": 0.7, + "content": "L 2", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 588, + 506, + 603 + ], + "score": 1.0, + "content": "loss or the cross-entropy loss. The parameter updates according to gradient descent (GD)", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 101, + 595, + 509, + 622 + ], + "spans": [ + { + "bbox": [ + 101, + 595, + 160, + 622 + ], + "score": 1.0, + "content": "are given by", + "type": "text" + }, + { + "bbox": [ + 161, + 601, + 269, + 615 + ], + "score": 0.92, + "content": "\\theta _ { \\mathrm { G D } } ^ { k + 1 } = \\theta _ { \\mathrm { G D } } ^ { k } - \\eta \\bar { \\nabla } \\bar { \\mathcal { L } } ( \\theta _ { \\mathrm { G D } } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 595, + 301, + 622 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 302, + 603, + 329, + 614 + ], + "score": 0.91, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 595, + 509, + 622 + ], + "score": 1.0, + "content": "is the learning rate. Under some standard", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 614, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 232, + 629 + ], + "score": 1.0, + "content": "assumptions (Lee et al., 2016),", + "type": "text" + }, + { + "bbox": [ + 232, + 615, + 271, + 627 + ], + "score": 0.89, + "content": "( \\theta _ { \\mathrm { G D } } ^ { k } ) _ { k \\geqslant 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 614, + 407, + 629 + ], + "score": 1.0, + "content": "converges to a local minimizer of", + "type": "text" + }, + { + "bbox": [ + 416, + 614, + 505, + 629 + ], + "score": 1.0, + "content": ", with probability one.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 106, + 637, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "When do we not learn invariances? We start by describing what might prevent learning invari-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 649, + 294, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 294, + 661 + ], + "score": 1.0, + "content": "ances in standard gradient-based optimization.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5 + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "(i) Training stops once the loss is low enough. If optimization learned spurious patterns by the time", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "it converged, invariances will not be learned anymore. This depends on the rate at which different", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "patterns are learned. The rates at which invariant patterns emerge (and vice-versa, the spurious", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "patterns do not) can be improved by e.g.: (a) careful architecture design, e.g. as done by hardcoding", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "spatial equivariance in convolutional networks; (b) fine-tuning models pre-trained on large amounts", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 720, + 408, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 408, + 732 + ], + "score": 1.0, + "content": "of data, where strong features already emerged and can be readily selected.", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 54.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 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, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "We formalize a notion of consistency, which characterizes to what extent a minimum of the loss", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "surface appears only when data from different environments are pooled. Minima with low consistency", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "are ‘patchwork’ solutions, which (we hypothesize) sew together different strategies and should not be", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "expected to generalize to new environments. An intuitive description of this principle was proposed", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 444, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 444, + 140 + ], + "score": 1.0, + "content": "by physicist David Deutsch: “good explanations are hard to vary” (Deutsch, 2011).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 83, + 506, + 140 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "Using the notion of consistency, we define Invariant Learning Consistency (ILC), a measure of the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "expected consistency of the solution found by a learning algorithm on a given hypothesis class. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "ILC can be improved by changing the hypothesis class or the learning algorithm, and in the last", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 174, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 104, + 174, + 506, + 191 + ], + "score": 1.0, + "content": "part of the paper we focus on the latter. We then analyse why current practices in deep learning", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "provide little incentive for networks to learn invariances, and show that standard training is instead", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "set up with the explicit objective of greedily maximizing speed of learning, i.e., progress on the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "training loss. When learning “as fast as possible” is not the main objective, we show we can trade-off", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "some “learning speed” for prioritizing learning the invariances. A practical instantiation of ILC leads", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "to o.o.d. generalization on a challenging synthetic task where several established regularizers fail", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "to generalize; moreover, following the memorization task from Zhang et al. (2017), ILC prevents", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "convergence on CIFAR-10 with random labels, as no shared mechanism is present, and similarly", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "when a portion of training labels is incorrect. Lastly, we set up a behavioural cloning task based on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 497, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 497, + 287 + ], + "score": 1.0, + "content": "the game CoinRun (Cobbe et al., 2019b), and observe better generalization on new unseen levels.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 144, + 506, + 287 + ] + }, + { + "type": "text", + "bbox": [ + 111, + 295, + 312, + 397 + ], + "lines": [ + { + "bbox": [ + 110, + 295, + 313, + 307 + ], + "spans": [ + { + "bbox": [ + 110, + 295, + 313, + 307 + ], + "score": 1.0, + "content": "An example. Take these two second-hand books", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 111, + 307, + 313, + 319 + ], + "spans": [ + { + "bbox": [ + 111, + 307, + 313, + 319 + ], + "score": 1.0, + "content": "of chess puzzles. We can learn the two independent", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 110, + 317, + 313, + 330 + ], + "spans": [ + { + "bbox": [ + 110, + 317, + 313, + 330 + ], + "score": 1.0, + "content": "shortcuts (blue arrows for the left book OR hand-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 110, + 328, + 313, + 341 + ], + "spans": [ + { + "bbox": [ + 110, + 328, + 313, + 341 + ], + "score": 1.0, + "content": "written solutions on the right), or actually learn", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 109, + 339, + 313, + 353 + ], + "spans": [ + { + "bbox": [ + 109, + 339, + 313, + 353 + ], + "score": 1.0, + "content": "to play chess (the invariant mechanism). While", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 109, + 350, + 313, + 364 + ], + "spans": [ + { + "bbox": [ + 109, + 350, + 313, + 364 + ], + "score": 1.0, + "content": "both strategies solve other problems from the same", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 110, + 362, + 313, + 374 + ], + "spans": [ + { + "bbox": [ + 110, + 362, + 313, + 374 + ], + "score": 1.0, + "content": "books (i.i.d.), only the latter generalises to new", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 110, + 373, + 313, + 385 + ], + "spans": [ + { + "bbox": [ + 110, + 373, + 313, + 385 + ], + "score": 1.0, + "content": "chess puzzle books (o.o.d.). How to distinguish", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 110, + 383, + 313, + 396 + ], + "spans": [ + { + "bbox": [ + 110, + 383, + 313, + 396 + ], + "score": 1.0, + "content": "the two? 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Here", + "type": "text" + }, + { + "bbox": [ + 446, + 453, + 504, + 465 + ], + "score": 0.89, + "content": "x _ { i } ^ { e } \\in \\mathcal { X } \\subseteq \\mathbb { R } ^ { m }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 463, + 504, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 463, + 308, + 477 + ], + "score": 1.0, + "content": "is the vector containing the observed inputs, and", + "type": "text" + }, + { + "bbox": [ + 308, + 465, + 363, + 476 + ], + "score": 0.92, + "content": "y _ { i } ^ { e } \\in \\mathcal { V } \\subseteq \\mathbb { R } ^ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 463, + 479, + 477 + ], + "score": 1.0, + "content": "the targets. The superscript", + "type": "text" + }, + { + "bbox": [ + 480, + 465, + 504, + 474 + ], + "score": 0.84, + "content": "e \\in { \\mathcal { E } }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "indexes some aspect of the data collection process, and can be interpreted as an environment label.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 484, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 245, + 500 + ], + "score": 1.0, + "content": "Our objective is to infer a function", + "type": "text" + }, + { + "bbox": [ + 245, + 486, + 293, + 497 + ], + "score": 0.91, + "content": "f : \\mathcal { X } \\mathcal { Y }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 484, + 493, + 500 + ], + "score": 1.0, + "content": "— which we call mechanism — assigning a target", + "type": "text" + }, + { + "bbox": [ + 493, + 487, + 504, + 498 + ], + "score": 0.87, + "content": "y _ { i } ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 484, + 505, + 500 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 160, + 510 + ], + "score": 1.0, + "content": "to each input", + "type": "text" + }, + { + "bbox": [ + 161, + 497, + 172, + 509 + ], + "score": 0.88, + "content": "\\boldsymbol { x } _ { i } ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 496, + 506, + 510 + ], + "score": 1.0, + "content": "; as explained in the introduction, we assume that such function is shared across all", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 508, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 264, + 521 + ], + "score": 1.0, + "content": "environments. 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To find an appropriate model", + "type": "text" + }, + { + "bbox": [ + 357, + 541, + 367, + 552 + ], + "score": 0.87, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 539, + 506, + 553 + ], + "score": 1.0, + "content": ", standard optimizers rely on gradi-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 239, + 564 + ], + "score": 1.0, + "content": "ents from a pooled loss function", + "type": "text" + }, + { + "bbox": [ + 239, + 552, + 290, + 562 + ], + "score": 0.91, + "content": "\\mathcal { L } : \\mathbb { R } ^ { n } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 550, + 506, + 564 + ], + "score": 1.0, + "content": ". This function measures the average performance ofř", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 560, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 403, + 578 + ], + "score": 1.0, + "content": "the neural network when predicting data labels, across all environments: ř", + "type": "text" + }, + { + "bbox": [ + 403, + 561, + 501, + 577 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { L } ( \\boldsymbol { \\theta } ) : = \\dot { \\frac { 1 } { | \\mathcal { E } | } } \\sum _ { e \\in \\mathcal { E } } \\mathcal { L } _ { e } ( \\boldsymbol { \\theta } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 560, + 506, + 578 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 574, + 507, + 594 + ], + "spans": [ + { + "bbox": [ + 104, + 574, + 126, + 594 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 576, + 296, + 592 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\begin{array} { r } { \\mathcal { L } _ { e } ( \\theta ) : = \\frac { 1 } { | \\mathcal { D } ^ { e } | } \\sum _ { ( x _ { i } ^ { e } , y _ { i } ^ { e } ) \\in \\mathcal { D } ^ { e } } \\ell ( f ( x _ { i } ^ { e } ; \\theta ) , y _ { i } ^ { e } ) } \\end{array} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 574, + 326, + 594 + ], + "score": 1.0, + "content": "; where", + "type": "text" + }, + { + "bbox": [ + 326, + 577, + 424, + 589 + ], + "score": 0.91, + "content": "\\ell : \\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { p } \\to [ 0 , + \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 574, + 507, + 594 + ], + "score": 1.0, + "content": "is usually chosen to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 133, + 603 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 133, + 591, + 146, + 600 + ], + "score": 0.7, + "content": "L 2", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 588, + 506, + 603 + ], + "score": 1.0, + "content": "loss or the cross-entropy loss. The parameter updates according to gradient descent (GD)", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 101, + 595, + 509, + 622 + ], + "spans": [ + { + "bbox": [ + 101, + 595, + 160, + 622 + ], + "score": 1.0, + "content": "are given by", + "type": "text" + }, + { + "bbox": [ + 161, + 601, + 269, + 615 + ], + "score": 0.92, + "content": "\\theta _ { \\mathrm { G D } } ^ { k + 1 } = \\theta _ { \\mathrm { G D } } ^ { k } - \\eta \\bar { \\nabla } \\bar { \\mathcal { L } } ( \\theta _ { \\mathrm { G D } } ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 595, + 301, + 622 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 302, + 603, + 329, + 614 + ], + "score": 0.91, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 595, + 509, + 622 + ], + "score": 1.0, + "content": "is the learning rate. Under some standard", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 614, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 232, + 629 + ], + "score": 1.0, + "content": "assumptions (Lee et al., 2016),", + "type": "text" + }, + { + "bbox": [ + 232, + 615, + 271, + 627 + ], + "score": 0.89, + "content": "( \\theta _ { \\mathrm { G D } } ^ { k } ) _ { k \\geqslant 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 614, + 407, + 629 + ], + "score": 1.0, + "content": "converges to a local minimizer of", + "type": "text" + }, + { + "bbox": [ + 416, + 614, + 505, + 629 + ], + "score": 1.0, + "content": ", with probability one.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46, + "bbox_fs": [ + 101, + 539, + 509, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 637, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "When do we not learn invariances? We start by describing what might prevent learning invari-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 649, + 294, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 294, + 661 + ], + "score": 1.0, + "content": "ances in standard gradient-based optimization.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 637, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "(i) Training stops once the loss is low enough. If optimization learned spurious patterns by the time", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "it converged, invariances will not be learned anymore. This depends on the rate at which different", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "patterns are learned. The rates at which invariant patterns emerge (and vice-versa, the spurious", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "patterns do not) can be improved by e.g.: (a) careful architecture design, e.g. as done by hardcoding", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "spatial equivariance in convolutional networks; (b) fine-tuning models pre-trained on large amounts", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 720, + 408, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 408, + 732 + ], + "score": 1.0, + "content": "of data, where strong features already emerged and can be readily selected.", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 54.5, + "bbox_fs": [ + 105, + 665, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "(ii) Learning signals: everything looks relevant for a dataset of size 1. Due to the summation in the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 220, + 106 + ], + "score": 1.0, + "content": "definition of the pooled loss", + "type": "text" + }, + { + "bbox": [ + 220, + 94, + 228, + 104 + ], + "score": 0.7, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 93, + 506, + 106 + ], + "score": 1.0, + "content": ", gradients for each example are computed independently. Informally,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "each signal is identical to the one for an equivalent dataset of size 1, where every pattern appears", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "relevant to the task. To find invariant patterns across examples, if we compute our training signals on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 420, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 420, + 140 + ], + "score": 1.0, + "content": "each of them independently, we have to rely on the way these are aggregated.1", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 506, + 233 + ], + "lines": [ + { + "bbox": [ + 104, + 142, + 507, + 156 + ], + "spans": [ + { + "bbox": [ + 104, + 142, + 507, + 156 + ], + "score": 1.0, + "content": "(iii) Aggregating gradients: averaging maximizes learning speed. The default method to pool gradi-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 153, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 153, + 277, + 168 + ], + "score": 1.0, + "content": "ents is the arithmetic mean. 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While speed is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "often desirable, by construction GD ignores one potentially crucial piece of information: The gradient", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 123, + 210 + ], + "score": 0.7, + "content": "\\nabla \\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 199, + 250, + 213 + ], + "score": 1.0, + "content": "is the result of averaging signals", + "type": "text" + }, + { + "bbox": [ + 250, + 200, + 270, + 211 + ], + "score": 0.92, + "content": "\\nabla { \\mathcal { L } } _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 199, + 506, + 213 + ], + "score": 1.0, + "content": ", which correspond to the patterns visible from each environ-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 211, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 223 + ], + "score": 1.0, + "content": "ment at this stage of optimization. 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Repeating this reasoning for all environment pairs, we arrive", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 581, + 254, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 254, + 594 + ], + "score": 1.0, + "content": "at the following inconsistency score:", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47.5 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 604, + 399, + 626 + ], + "lines": [ + { + "bbox": [ + 211, + 604, + 399, + 626 + ], + "spans": [ + { + "bbox": [ + 211, + 604, + 399, + 626 + ], + "score": 0.92, + "content": "\\mathcal { T } ^ { \\epsilon } ( \\theta ^ { * } ) : = \\operatorname* { m a x } _ { ( e , e ^ { \\prime } ) \\in \\mathcal { E } ^ { 2 } } \\operatorname* { m a x } _ { \\theta \\in N _ { e , \\theta ^ { * } } ^ { \\epsilon } } | \\mathcal { L } _ { e ^ { \\prime } } ( \\theta ) - \\mathcal { L } _ { e } ( \\theta ^ { * } ) | .", + "type": "interline_equation", + "image_path": "a69693a9a6f57a5b78c26af8fee718ab41a3bce63580312e3f479ef37f36a8d5.jpg" + } + ] + } + ], + "index": 50, + "virtual_lines": [ + { + "bbox": [ + 211, + 604, + 399, + 626 + ], + "spans": [], + "index": 50 + } + ] + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 105, + 646, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 646, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 119, + 646, + 294, + 659 + ], + "score": 1.0, + "content": "1After computing the gradients for a dataset of", + "type": "text" + }, + { + "bbox": [ + 294, + 648, + 317, + 657 + ], + "score": 0.87, + "content": "n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 646, + 377, + 659 + ], + "score": 1.0, + "content": "examples, if an", + "type": "text" + }, + { + "bbox": [ + 377, + 649, + 384, + 656 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 646, + 505, + 659 + ], + "score": 1.0, + "content": "-th example appeared, we would", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "just compute one more vector of gradients and add it to the sum. A Gaussian Process (Rasmussen, 2003) for", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 666, + 477, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 477, + 678 + ], + "score": 1.0, + "content": "example would require recomputing the entire solution from scratch, as all interactions are considered.ř", + "type": "text" + } + ] + }, + { + "bbox": [ + 117, + 673, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 117, + 673, + 380, + 691 + ], + "score": 1.0, + "content": "2The same reasoning holds for SGD in the finite-sum optimization case", + "type": "text" + }, + { + "bbox": [ + 380, + 677, + 442, + 689 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathcal { L } = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\mathcal { L } _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 673, + 505, + 691 + ], + "score": 1.0, + "content": ", where gradients", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 687, + 488, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 488, + 699 + ], + "score": 1.0, + "content": "from a mini-batch are seen as unbiased estimators of gradients from the pooled loss. (Bottou et al., 2018).", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 696, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 118, + 696, + 169, + 712 + ], + "score": 1.0, + "content": "3Assume that", + "type": "text" + }, + { + "bbox": [ + 170, + 699, + 177, + 708 + ], + "score": 0.75, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 696, + 192, + 712 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 193, + 699, + 200, + 708 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 696, + 415, + 712 + ], + "score": 1.0, + "content": "-Lipschitz gradients (i.e. curvature bounded from above by", + "type": "text" + }, + { + "bbox": [ + 415, + 700, + 422, + 708 + ], + "score": 0.74, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 696, + 497, + 712 + ], + "score": 1.0, + "content": "). Then, at any point", + "type": "text" + }, + { + "bbox": [ + 497, + 698, + 502, + 708 + ], + "score": 0.79, + "content": "\\tilde { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 696, + 506, + 712 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 708, + 415, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 229, + 722 + ], + "score": 1.0, + "content": "we can construct the upper bound", + "type": "text" + }, + { + "bbox": [ + 230, + 709, + 412, + 722 + ], + "score": 0.93, + "content": "\\hat { \\mathcal { L } } _ { \\tilde { \\theta } } ( \\theta ) = \\mathcal { L } ( \\tilde { \\theta } ) + \\nabla \\mathcal { L } ( \\tilde { \\theta } ) ^ { \\top } ( \\theta - \\tilde { \\theta } ) + L \\| \\theta - \\tilde { \\theta } \\| ^ { 2 } / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 708, + 415, + 722 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + }, + { + "bbox": [ + 119, + 720, + 504, + 734 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 504, + 734 + ], + "score": 1.0, + "content": "4Loosely speaking, a sum is large if any of the summands is large, a product is large if all factors are large.", + "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": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "(ii) Learning signals: everything looks relevant for a dataset of size 1. 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For instance, if", + "type": "text" + }, + { + "bbox": [ + 448, + 388, + 458, + 398 + ], + "score": 0.73, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 386, + 506, + 400 + ], + "score": 1.0, + "content": "is gradient", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 312, + 411 + ], + "score": 1.0, + "content": "A descent, the result of Lee et al. 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To each", + "type": "text" + }, + { + "bbox": [ + 171, + 410, + 208, + 422 + ], + "score": 0.92, + "content": "\\theta ^ { \\ast } \\in \\Theta _ { A } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 407, + 507, + 424 + ], + "score": 1.0, + "content": "A , we want to associate a consistency score, quantifying the concept “good", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 419, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 118, + 430 + ], + "score": 0.85, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 419, + 506, + 433 + ], + "score": 1.0, + "content": "Aare hard to vary”. In other words, we would like the score to capture the consistency of the loss", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 431, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 177, + 443 + ], + "score": 1.0, + "content": "landscape around", + "type": "text" + }, + { + "bbox": [ + 177, + 432, + 188, + 442 + ], + "score": 0.87, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 431, + 506, + 443 + ], + "score": 1.0, + "content": "across the different environments. For example, in Fig. 1 the loss landscape near", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 442, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 506, + 454 + ], + "score": 1.0, + "content": "the bottom-left minimizer is consistent across environments, while the top-right minimizer is not.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 374, + 507, + 454 + ] + }, + { + "type": "image", + "bbox": [ + 349, + 456, + 414, + 537 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 349, + 456, + 414, + 537 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 349, + 456, + 414, + 536 + ], + "spans": [ + { + "bbox": [ + 349, + 456, + 414, + 536 + ], + "score": 0.846, + "type": "image", + "image_path": "eb9ca484544d5d5cf5c2980ba2b3999fd126d44ac36d9c5259ef498ccebf057c.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 349, + 456, + 414, + 537 + ], + "spans": [], + "index": 38 + } + ] + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 337, + 546 + ], + "lines": [ + { + "bbox": [ + 106, + 453, + 338, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 270, + 465 + ], + "score": 1.0, + "content": "Let us characterize the landscape around", + "type": "text" + }, + { + "bbox": [ + 271, + 453, + 282, + 463 + ], + "score": 0.86, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 453, + 338, + 465 + ], + "score": 1.0, + "content": "from the per-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 463, + 338, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 236, + 477 + ], + "score": 1.0, + "content": "spective of a fixed environment", + "type": "text" + }, + { + "bbox": [ + 237, + 464, + 260, + 474 + ], + "score": 0.89, + "content": "e \\in { \\mathcal { E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 463, + 338, + 477 + ], + "score": 1.0, + "content": ". We define the set", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 470, + 342, + 494 + ], + "spans": [ + { + "bbox": [ + 107, + 475, + 131, + 489 + ], + "score": 0.92, + "content": "N _ { e , \\theta ^ { * } } ^ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 470, + 342, + 494 + ], + "score": 1.0, + "content": "to be the largest path-connected region of space con-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 487, + 336, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 155, + 501 + ], + "score": 1.0, + "content": "taining both", + "type": "text" + }, + { + "bbox": [ + 156, + 488, + 167, + 498 + ], + "score": 0.87, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 487, + 212, + 501 + ], + "score": 1.0, + "content": "and the set", + "type": "text" + }, + { + "bbox": [ + 213, + 487, + 336, + 500 + ], + "score": 0.71, + "content": "\\{ \\theta \\in \\Theta \\mathrm { s . t . } | \\mathcal { L } _ { e } ( \\theta ) - \\mathcal { L } _ { e } ( \\theta ^ { * } ) | \\leqslant", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 495, + 340, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 119, + 511 + ], + "score": 0.82, + "content": "\\epsilon \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 495, + 144, + 517 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 144, + 500, + 169, + 510 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 495, + 249, + 517 + ], + "score": 1.0, + "content": ". 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We would like to evaluate the elements of this", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 556, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 281, + 568 + ], + "score": 1.0, + "content": "set with respect to a different environment ˇ", + "type": "text" + }, + { + "bbox": [ + 281, + 556, + 308, + 567 + ], + "score": 0.91, + "content": "e ^ { \\prime } \\neq e", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 556, + 381, + 568 + ], + "score": 1.0, + "content": ". We will say that", + "type": "text" + }, + { + "bbox": [ + 382, + 556, + 390, + 566 + ], + "score": 0.85, + "content": "e ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 556, + 465, + 568 + ], + "score": 1.0, + "content": "is consistent with", + "type": "text" + }, + { + "bbox": [ + 465, + 558, + 471, + 566 + ], + "score": 0.73, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 556, + 483, + 568 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 483, + 556, + 495, + 566 + ], + "score": 0.86, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 556, + 506, + 568 + ], + "score": 1.0, + "content": "if", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 107, + 567, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 107, + 567, + 226, + 583 + ], + "score": 0.9, + "content": "\\operatorname* { m a x } _ { \\theta \\in N _ { e , \\theta ^ { * } } ^ { \\epsilon } } \\mathbf { \\bar { | } } \\mathcal L _ { e ^ { \\prime } } ( \\theta ) - \\mathcal L _ { e } ( \\theta ) \\vert", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 567, + 506, + 582 + ], + "score": 1.0, + "content": "is small. 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Finally,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 368, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 354, + 106 + ], + "score": 1.0, + "content": "we can write down an invariant learning consistency score for", + "type": "text" + }, + { + "bbox": [ + 355, + 94, + 364, + 104 + ], + "score": 0.82, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 93, + 368, + 106 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 108, + 399, + 124 + ], + "lines": [ + { + "bbox": [ + 212, + 108, + 399, + 124 + ], + "spans": [ + { + "bbox": [ + 212, + 108, + 399, + 124 + ], + "score": 0.92, + "content": "\\operatorname { I L C } ( \\mathcal { A } , p _ { \\theta ^ { 0 } } ) : = - \\mathbb { E } _ { \\theta ^ { 0 } \\sim p ( \\theta ^ { 0 } ) } \\left[ \\mathcal { T } ^ { \\epsilon } ( \\mathcal { A } _ { \\infty } ( \\theta ^ { 0 } , \\mathcal { E } ) \\right] .", + "type": "interline_equation", + "image_path": "15d53d902ccedfade86b78eb659b805fa5c09679e4e9226207ff64ac2c9b60ad.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 212, + 108, + 399, + 124 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 128, + 506, + 150 + ], + "lines": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "That is, the learning consistency of an algorithm measures the expected consistency across environ-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 340, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 340, + 150 + ], + "score": 1.0, + "content": "ments of the minimizer it converges to on the pooled data.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 153, + 505, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "Example: low consistency of a classic patchwork solution. One-hidden-layer networks with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 507, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 396, + 178 + ], + "score": 1.0, + "content": "sigmoid activations and enough neurons can approximate any function", + "type": "text" + }, + { + "bbox": [ + 396, + 164, + 460, + 176 + ], + "score": 0.92, + "content": "f ^ { * } : [ 0 , \\dot { 1 } ] \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 163, + 507, + 178 + ], + "score": 1.0, + "content": "(Cybenko,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 505, + 188 + ], + "score": 1.0, + "content": "1989). In appendix A.1 we show how the construction used to obtain the weights leads to a maximally", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 186, + 474, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 243, + 199 + ], + "score": 1.0, + "content": "inconsistent solution according to", + "type": "text" + }, + { + "bbox": [ + 243, + 186, + 272, + 198 + ], + "score": 0.93, + "content": "\\mathcal { T } ^ { \\epsilon } ( \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 186, + 474, + 199 + ], + "score": 1.0, + "content": ", which would not be expected to generalize o.o.d.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "title", + "bbox": [ + 106, + 210, + 342, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 210, + 342, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 342, + 223 + ], + "score": 1.0, + "content": "2.2 ILC AS A LOGICAL AND BETWEEN LANDSCAPES", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 388, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 224, + 388, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 388, + 239 + ], + "score": 1.0, + "content": "Here we draw a connection between our definition of inconsistency", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 237, + 388, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 388, + 248 + ], + "score": 1.0, + "content": "and the local geometric properties of the loss landscapes. For the sake", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 247, + 388, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 275, + 259 + ], + "score": 1.0, + "content": "of clarity, we consider two environments", + "type": "text" + }, + { + "bbox": [ + 276, + 248, + 284, + 258 + ], + "score": 0.65, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 247, + 302, + 259 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 248, + 312, + 258 + ], + "score": 0.75, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 247, + 365, + 259 + ], + "score": 1.0, + "content": ") and assume", + "type": "text" + }, + { + "bbox": [ + 365, + 248, + 376, + 258 + ], + "score": 0.86, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 247, + 388, + 259 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 389, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 389, + 271 + ], + "score": 1.0, + "content": "be a local minimizer (with zero loss) for both environments. 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In", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 389, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 305, + 337 + ], + "score": 1.0, + "content": "contrast, the geometric mean, or Karcher mean,", + "type": "text" + }, + { + "bbox": [ + 305, + 325, + 333, + 336 + ], + "score": 0.91, + "content": "H _ { A \\land B }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 325, + 389, + 337 + ], + "score": 1.0, + "content": "(Ando et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 334, + 388, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 388, + 349 + ], + "score": 1.0, + "content": "2004) is affected by the inconsistencies between landscapes: It performs", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 345, + 385, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 357, + 361 + ], + "score": 1.0, + "content": "a “logical AND”. 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We just saw that the consistency of", + "type": "text" + }, + { + "bbox": [ + 370, + 402, + 381, + 412 + ], + "score": 0.88, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "is linked to the geometric mean", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 172, + 426 + ], + "score": 1.0, + "content": "of the Hessians", + "type": "text" + }, + { + "bbox": [ + 172, + 413, + 225, + 425 + ], + "score": 0.93, + "content": "\\{ \\bar { H _ { e } } ( \\theta ^ { * } ) \\} _ { e \\in \\mathcal { E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 412, + 409, + 426 + ], + "score": 1.0, + "content": ". Under the simplifying assumption that each ś", + "type": "text" + }, + { + "bbox": [ + 409, + 413, + 423, + 424 + ], + "score": 0.88, + "content": "H _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "is diagonal6 and allś", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 423, + 508, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 423, + 154, + 441 + ], + "score": 1.0, + "content": "eigenvalues", + "type": "text" + }, + { + "bbox": [ + 154, + 426, + 165, + 438 + ], + "score": 0.89, + "content": "\\lambda _ { i } ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 423, + 307, + 441 + ], + "score": 1.0, + "content": "are positive, their geometric mean is", + "type": "text" + }, + { + "bbox": [ + 308, + 424, + 502, + 438 + ], + "score": 0.88, + "content": "\\begin{array} { r } { H ^ { \\wedge } : = \\mathrm { d i a g } ( ( \\prod _ { e \\in \\mathcal { E } } \\lambda _ { 1 } ^ { e } ) ^ { 1 / | \\varepsilon | } , \\dots , ( \\prod _ { e \\in \\mathcal { E } } \\lambda _ { n } ^ { e } ) ^ { 1 / | \\varepsilon | } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 423, + 508, + 441 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 437, + 507, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 290, + 449 + ], + "score": 1.0, + "content": "The curvature of the corresponding loss in the", + "type": "text" + }, + { + "bbox": [ + 290, + 437, + 295, + 446 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 437, + 507, + 449 + ], + "score": 1.0, + "content": "-th eigendirection depends on how consistent the cur-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 455, + 460 + ], + "score": 1.0, + "content": "vatures of each environment are in that direction. Consider now optimizing from a point ř ř", + "type": "text" + }, + { + "bbox": [ + 456, + 447, + 466, + 457 + ], + "score": 0.85, + "content": "\\theta ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "; gradient", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 103, + 454, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 103, + 454, + 160, + 476 + ], + "score": 1.0, + "content": "descent reads", + "type": "text" + }, + { + "bbox": [ + 161, + 457, + 272, + 470 + ], + "score": 0.91, + "content": "\\theta ^ { k + 1 } = \\theta ^ { k } - \\eta H ^ { + } ( \\theta ^ { k } - \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 454, + 301, + 476 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 302, + 457, + 478, + 473 + ], + "score": 0.91, + "content": "\\begin{array} { r } { H ^ { + } : = \\mathrm { d i a g } \\big ( \\frac { 1 } { | \\mathcal { E } | } \\sum _ { e \\in \\mathcal { E } } \\breve { \\lambda _ { 1 } ^ { e } } , \\dots , \\frac { 1 } { | \\mathcal { E } | } \\sum _ { e \\in \\mathcal { E } } \\lambda _ { n } ^ { e } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 454, + 497, + 476 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 497, + 460, + 504, + 470 + ], + "score": 0.63, + "content": "\\eta", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 103, + 469, + 507, + 491 + ], + "spans": [ + { + "bbox": [ + 103, + 469, + 201, + 491 + ], + "score": 1.0, + "content": "small enough7, we have", + "type": "text" + }, + { + "bbox": [ + 202, + 473, + 373, + 488 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\vert \\theta _ { i } ^ { k + 1 } - \\theta _ { i } ^ { * } \\vert = ( 1 - \\eta \\frac { 1 } { \\vert \\mathcal { E } \\vert } \\sum _ { e \\in \\mathcal { E } } \\lambda _ { i } ^ { e } ) \\vert \\theta _ { i } ^ { k } - \\theta _ { i } ^ { * } \\vert } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 469, + 507, + 491 + ], + "score": 1.0, + "content": ". As noted, this choice maximises", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 214, + 500 + ], + "score": 1.0, + "content": "the speed of convergence to", + "type": "text" + }, + { + "bbox": [ + 215, + 487, + 226, + 497 + ], + "score": 0.85, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 487, + 505, + 500 + ], + "score": 1.0, + "content": ", but does not take into account whether this minimizer is consistent. We", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "score": 1.0, + "content": "can reduce the speed of convergence on directions where landscapes have different curvatures – which", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "would lead to a high inconsistency – by following the gradients from the geometric mean of the land-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 519, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 407, + 533 + ], + "score": 1.0, + "content": "scapes, as opposed to the arithmetic mean. I.e, we substitute the full gradient", + "type": "text" + }, + { + "bbox": [ + 407, + 519, + 504, + 532 + ], + "score": 0.91, + "content": "\\nabla { \\mathcal { L } } ( \\theta ) = H ^ { + } ( \\theta ^ { k } - \\theta ^ { * } )", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 103, + 531, + 507, + 549 + ], + "spans": [ + { + "bbox": [ + 103, + 531, + 127, + 549 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 533, + 231, + 546 + ], + "score": 0.93, + "content": "\\nabla \\mathcal { L } ^ { \\wedge } ( \\theta ) = H ^ { \\wedge } ( \\theta ^ { k } - \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 531, + 317, + 549 + ], + "score": 1.0, + "content": ". Also, we have that8", + "type": "text" + }, + { + "bbox": [ + 318, + 532, + 446, + 546 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\nabla \\mathcal { L } ^ { \\wedge } ( \\theta ) = \\big ( \\prod _ { e \\in \\mathcal { E } } \\nabla \\mathcal { L } _ { e } ( \\theta ) \\big ) ^ { 1 / | \\varepsilon | } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 531, + 507, + 549 + ], + "score": 1.0, + "content": ": to reduce the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "speed of convergence in directions with inconsistency, we can take the element-wise geometric mean", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 556, + 408, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 408, + 568 + ], + "score": 1.0, + "content": "of gradients from different environments (see also Fig. 11 in the appendix).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 106, + 575, + 327, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 575, + 327, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 327, + 587 + ], + "score": 1.0, + "content": "2.3 MASKING GRADIENTS WITH A LOGICAL AND", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 504, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "score": 1.0, + "content": "The element-wise geometric mean of gradients, instead of the arithmetic mean, increases consistency", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 600, + 411, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 411, + 613 + ], + "score": 1.0, + "content": "in the convex quadratic case. However, there are a few practical limitations:", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5 + }, + { + "type": "text", + "bbox": [ + 108, + 613, + 504, + 635 + ], + "lines": [ + { + "bbox": [ + 108, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 108, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "(i) The geometric mean is only defined when all the signs are consistent. It is still to be defined how", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 123, + 624, + 455, + 636 + ], + "spans": [ + { + "bbox": [ + 123, + 624, + 455, + 636 + ], + "score": 1.0, + "content": "sign inconsistencies, which can occur in non-convex settings, should be dealt with.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52.5 + }, + { + "type": "text", + "bbox": [ + 109, + 637, + 504, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 505, + 650 + ], + "score": 1.0, + "content": "(ii) It provides little flexibility for ‘partial’ agreement: Even a single zero gradient component in one", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 122, + 648, + 320, + 660 + ], + "spans": [ + { + "bbox": [ + 122, + 648, + 320, + 660 + ], + "score": 1.0, + "content": "environment stops optimization in that direction.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54.5 + } + ], + "page_idx": 3, + "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": [ + 106, + 667, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 117, + 664, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 117, + 664, + 506, + 681 + ], + "score": 1.0, + "content": "5This provides a useful simplified perspective. 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(2017).)", + "type": "text" + } + ] + }, + { + "bbox": [ + 119, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 119, + 687, + 130, + 698 + ], + "score": 0.32, + "content": "^ 6 \\mathrm { I t }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "was shown in (Becker et al., 1988) and recently in (Adolphs et al., 2019; Singh and Alistarh, 2020) that", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 697, + 449, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 449, + 711 + ], + "score": 1.0, + "content": "neural networks have a strong diagonal dominance of the Hessian matrix at the end of training.", + "type": "text" + } + ] + }, + { + "bbox": [ + 119, + 707, + 468, + 721 + ], + "spans": [ + { + "bbox": [ + 119, + 707, + 170, + 721 + ], + "score": 1.0, + "content": "7Smaller than", + "type": "text" + }, + { + "bbox": [ + 171, + 708, + 196, + 720 + ], + "score": 0.85, + "content": "1 / \\lambda _ { \\operatorname* { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 707, + 199, + 721 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 200, + 709, + 217, + 719 + ], + "score": 0.82, + "content": "\\lambda _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 707, + 468, + 721 + ], + "score": 1.0, + "content": "is the maximum eigenvalue of Hessians from different environments,` ˘", + "type": "text" + } + ] + }, + { + "bbox": [ + 115, + 716, + 446, + 736 + ], + "spans": [ + { + "bbox": [ + 115, + 716, + 170, + 736 + ], + "score": 1.0, + "content": "8This holds if", + "type": "text" + }, + { + "bbox": [ + 171, + 721, + 197, + 731 + ], + "score": 0.9, + "content": "\\theta - \\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 716, + 307, + 736 + ], + "score": 1.0, + "content": "is positive, otherwise we have", + "type": "text" + }, + { + "bbox": [ + 308, + 719, + 441, + 733 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\nabla \\mathcal { L } ^ { \\wedge } ( \\theta ) = - \\left( \\prod _ { e \\in \\mathcal { E } } \\left| \\nabla \\mathcal { L } _ { e } ( \\theta ) \\right| \\right) ^ { 1 / | \\mathcal { E } | } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 716, + 446, + 736 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "This consistency is our formalization of the principle “good explanations are hard to vary”. Finally,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 368, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 354, + 106 + ], + "score": 1.0, + "content": "we can write down an invariant learning consistency score for", + "type": "text" + }, + { + "bbox": [ + 355, + 94, + 364, + 104 + ], + "score": 0.82, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 93, + 368, + 106 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 81, + 506, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 108, + 399, + 124 + ], + "lines": [ + { + "bbox": [ + 212, + 108, + 399, + 124 + ], + "spans": [ + { + "bbox": [ + 212, + 108, + 399, + 124 + ], + "score": 0.92, + "content": "\\operatorname { I L C } ( \\mathcal { A } , p _ { \\theta ^ { 0 } } ) : = - \\mathbb { E } _ { \\theta ^ { 0 } \\sim p ( \\theta ^ { 0 } ) } \\left[ \\mathcal { T } ^ { \\epsilon } ( \\mathcal { A } _ { \\infty } ( \\theta ^ { 0 } , \\mathcal { E } ) \\right] .", + "type": "interline_equation", + "image_path": "15d53d902ccedfade86b78eb659b805fa5c09679e4e9226207ff64ac2c9b60ad.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 212, + 108, + 399, + 124 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 128, + 506, + 150 + ], + "lines": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "That is, the learning consistency of an algorithm measures the expected consistency across environ-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 340, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 340, + 150 + ], + "score": 1.0, + "content": "ments of the minimizer it converges to on the pooled data.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 127, + 506, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 153, + 505, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "Example: low consistency of a classic patchwork solution. One-hidden-layer networks with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 507, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 396, + 178 + ], + "score": 1.0, + "content": "sigmoid activations and enough neurons can approximate any function", + "type": "text" + }, + { + "bbox": [ + 396, + 164, + 460, + 176 + ], + "score": 0.92, + "content": "f ^ { * } : [ 0 , \\dot { 1 } ] \\to \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 163, + 507, + 178 + ], + "score": 1.0, + "content": "(Cybenko,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 174, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 505, + 188 + ], + "score": 1.0, + "content": "1989). In appendix A.1 we show how the construction used to obtain the weights leads to a maximally", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 186, + 474, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 243, + 199 + ], + "score": 1.0, + "content": "inconsistent solution according to", + "type": "text" + }, + { + "bbox": [ + 243, + 186, + 272, + 198 + ], + "score": 0.93, + "content": "\\mathcal { T } ^ { \\epsilon } ( \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 186, + 474, + 199 + ], + "score": 1.0, + "content": ", which would not be expected to generalize o.o.d.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 154, + 507, + 199 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 210, + 342, + 222 + ], + "lines": [ + { + "bbox": [ + 106, + 210, + 342, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 342, + 223 + ], + "score": 1.0, + "content": "2.2 ILC AS A LOGICAL AND BETWEEN LANDSCAPES", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 388, + 373 + ], + "lines": [ + { + "bbox": [ + 105, + 224, + 388, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 388, + 239 + ], + "score": 1.0, + "content": "Here we draw a connection between our definition of inconsistency", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 237, + 388, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 388, + 248 + ], + "score": 1.0, + "content": "and the local geometric properties of the loss landscapes. For the sake", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 247, + 388, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 275, + 259 + ], + "score": 1.0, + "content": "of clarity, we consider two environments", + "type": "text" + }, + { + "bbox": [ + 276, + 248, + 284, + 258 + ], + "score": 0.65, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 247, + 302, + 259 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 302, + 248, + 312, + 258 + ], + "score": 0.75, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 247, + 365, + 259 + ], + "score": 1.0, + "content": ") and assume", + "type": "text" + }, + { + "bbox": [ + 365, + 248, + 376, + 258 + ], + "score": 0.86, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 247, + 388, + 259 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 389, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 389, + 271 + ], + "score": 1.0, + "content": "be a local minimizer (with zero loss) for both environments. Using a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 269, + 389, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 228, + 282 + ], + "score": 1.0, + "content": "Taylor approximation5, we get", + "type": "text" + }, + { + "bbox": [ + 229, + 269, + 373, + 281 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { L } ( \\theta ) \\approx \\frac { 1 } { 2 } ( \\theta - \\theta ^ { * } ) ^ { \\top } H _ { A + B } ( \\theta - \\theta ^ { * } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 269, + 389, + 282 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 279, + 389, + 294 + ], + "spans": [ + { + "bbox": [ + 107, + 280, + 162, + 293 + ], + "score": 0.95, + "content": "\\lVert \\dot { \\theta } - \\theta ^ { * } \\rVert \\approx 0", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 279, + 192, + 294 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 193, + 281, + 295, + 292 + ], + "score": 0.9, + "content": "H _ { A + B } = \\left( H _ { A } + \\mathbf { \\bar { \\mathit { H } } } _ { B } \\right) / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 279, + 389, + 294 + ], + "score": 1.0, + "content": "is the arithmetic mean", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 289, + 387, + 306 + ], + "spans": [ + { + "bbox": [ + 104, + 289, + 173, + 306 + ], + "score": 1.0, + "content": "of the Hessians", + "type": "text" + }, + { + "bbox": [ + 174, + 292, + 252, + 304 + ], + "score": 0.9, + "content": "H _ { A } : = \\nabla ^ { 2 } \\mathcal { L } _ { A } ( \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 289, + 272, + 306 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 273, + 291, + 352, + 303 + ], + "score": 0.87, + "content": "\\Dot { H _ { B } } : = \\nabla ^ { 2 } \\mathcal { L } _ { A } ( \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 289, + 358, + 306 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 358, + 291, + 387, + 303 + ], + "score": 0.86, + "content": "H _ { A + B }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 302, + 387, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 378, + 315 + ], + "score": 1.0, + "content": "does not capture the possibly conflicting geometries of landscape", + "type": "text" + }, + { + "bbox": [ + 378, + 303, + 387, + 312 + ], + "score": 0.69, + "content": "A", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 313, + 388, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 117, + 325 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 118, + 314, + 127, + 324 + ], + "score": 0.75, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 313, + 388, + 325 + ], + "score": 1.0, + "content": ": It performs a “logical OR” on the dominant eigendirections. In", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 325, + 389, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 305, + 337 + ], + "score": 1.0, + "content": "contrast, the geometric mean, or Karcher mean,", + "type": "text" + }, + { + "bbox": [ + 305, + 325, + 333, + 336 + ], + "score": 0.91, + "content": "H _ { A \\land B }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 325, + 389, + 337 + ], + "score": 1.0, + "content": "(Ando et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 334, + 388, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 388, + 349 + ], + "score": 1.0, + "content": "2004) is affected by the inconsistencies between landscapes: It performs", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 345, + 385, + 361 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 357, + 361 + ], + "score": 1.0, + "content": "a “logical AND”. 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This magnification shows", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 394, + 360, + 493, + 370 + ], + "spans": [ + { + "bbox": [ + 394, + 360, + 456, + 370 + ], + "score": 1.0, + "content": "inconsistency of", + "type": "text" + }, + { + "bbox": [ + 457, + 360, + 465, + 369 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 360, + 480, + 370 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 481, + 360, + 489, + 369 + ], + "score": 0.8, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 360, + 493, + 370 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 374, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 284, + 387 + ], + "score": 1.0, + "content": "for the geometric mean of positive numbers,", + "type": "text" + }, + { + "bbox": [ + 285, + 374, + 415, + 385 + ], + "score": 0.91, + "content": "0 \\leqslant \\operatorname* { d e t } ( H _ { A \\wedge B } ) \\leqslant \\operatorname* { d e t } ( H _ { A + B } )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 372, + 506, + 387 + ], + "score": 1.0, + "content": "; thus, inconsistency is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 384, + 485, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 199, + 396 + ], + "score": 1.0, + "content": "lowest when shapes of", + "type": "text" + }, + { + "bbox": [ + 200, + 385, + 208, + 394 + ], + "score": 0.77, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 384, + 226, + 396 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 226, + 385, + 235, + 394 + ], + "score": 0.81, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 384, + 485, + 396 + ], + "score": 1.0, + "content": "are similar – exactly as in the bottom-left minimizer of Fig. 1.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 372, + 506, + 396 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 402, + 506, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 370, + 415 + ], + "score": 1.0, + "content": "From Hessians to gradients. We just saw that the consistency of", + "type": "text" + }, + { + "bbox": [ + 370, + 402, + 381, + 412 + ], + "score": 0.88, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "is linked to the geometric mean", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 172, + 426 + ], + "score": 1.0, + "content": "of the Hessians", + "type": "text" + }, + { + "bbox": [ + 172, + 413, + 225, + 425 + ], + "score": 0.93, + "content": "\\{ \\bar { H _ { e } } ( \\theta ^ { * } ) \\} _ { e \\in \\mathcal { E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 412, + 409, + 426 + ], + "score": 1.0, + "content": ". 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Consider now optimizing from a point ř ř", + "type": "text" + }, + { + "bbox": [ + 456, + 447, + 466, + 457 + ], + "score": 0.85, + "content": "\\theta ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "; gradient", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 103, + 454, + 504, + 476 + ], + "spans": [ + { + "bbox": [ + 103, + 454, + 160, + 476 + ], + "score": 1.0, + "content": "descent reads", + "type": "text" + }, + { + "bbox": [ + 161, + 457, + 272, + 470 + ], + "score": 0.91, + "content": "\\theta ^ { k + 1 } = \\theta ^ { k } - \\eta H ^ { + } ( \\theta ^ { k } - \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 454, + 301, + 476 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 302, + 457, + 478, + 473 + ], + "score": 0.91, + "content": "\\begin{array} { r } { H ^ { + } : = \\mathrm { d i a g } \\big ( \\frac { 1 } { | \\mathcal { E } | } \\sum _ { e \\in \\mathcal { E } } \\breve { \\lambda _ { 1 } ^ { e } } , \\dots , \\frac { 1 } { | \\mathcal { E } | } \\sum _ { e \\in \\mathcal { E } } \\lambda _ { n } ^ { e } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 454, + 497, + 476 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 497, + 460, + 504, + 470 + ], + "score": 0.63, + "content": "\\eta", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 103, + 469, + 507, + 491 + ], + "spans": [ + { + "bbox": [ + 103, + 469, + 201, + 491 + ], + "score": 1.0, + "content": "small enough7, we have", + "type": "text" + }, + { + "bbox": [ + 202, + 473, + 373, + 488 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\vert \\theta _ { i } ^ { k + 1 } - \\theta _ { i } ^ { * } \\vert = ( 1 - \\eta \\frac { 1 } { \\vert \\mathcal { E } \\vert } \\sum _ { e \\in \\mathcal { E } } \\lambda _ { i } ^ { e } ) \\vert \\theta _ { i } ^ { k } - \\theta _ { i } ^ { * } \\vert } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 469, + 507, + 491 + ], + "score": 1.0, + "content": ". As noted, this choice maximises", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 214, + 500 + ], + "score": 1.0, + "content": "the speed of convergence to", + "type": "text" + }, + { + "bbox": [ + 215, + 487, + 226, + 497 + ], + "score": 0.85, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 487, + 505, + 500 + ], + "score": 1.0, + "content": ", but does not take into account whether this minimizer is consistent. We", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 510 + ], + "score": 1.0, + "content": "can reduce the speed of convergence on directions where landscapes have different curvatures – which", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "would lead to a high inconsistency – by following the gradients from the geometric mean of the land-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 519, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 407, + 533 + ], + "score": 1.0, + "content": "scapes, as opposed to the arithmetic mean. I.e, we substitute the full gradient", + "type": "text" + }, + { + "bbox": [ + 407, + 519, + 504, + 532 + ], + "score": 0.91, + "content": "\\nabla { \\mathcal { L } } ( \\theta ) = H ^ { + } ( \\theta ^ { k } - \\theta ^ { * } )", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 103, + 531, + 507, + 549 + ], + "spans": [ + { + "bbox": [ + 103, + 531, + 127, + 549 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 533, + 231, + 546 + ], + "score": 0.93, + "content": "\\nabla \\mathcal { L } ^ { \\wedge } ( \\theta ) = H ^ { \\wedge } ( \\theta ^ { k } - \\theta ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 531, + 317, + 549 + ], + "score": 1.0, + "content": ". Also, we have that8", + "type": "text" + }, + { + "bbox": [ + 318, + 532, + 446, + 546 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\nabla \\mathcal { L } ^ { \\wedge } ( \\theta ) = \\big ( \\prod _ { e \\in \\mathcal { E } } \\nabla \\mathcal { L } _ { e } ( \\theta ) \\big ) ^ { 1 / | \\varepsilon | } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 531, + 507, + 549 + ], + "score": 1.0, + "content": ": to reduce the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "speed of convergence in directions with inconsistency, we can take the element-wise geometric mean", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 556, + 408, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 408, + 568 + ], + "score": 1.0, + "content": "of gradients from different environments (see also Fig. 11 in the appendix).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 41.5, + "bbox_fs": [ + 103, + 401, + 508, + 568 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 575, + 327, + 586 + ], + "lines": [ + { + "bbox": [ + 106, + 575, + 327, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 327, + 587 + ], + "score": 1.0, + "content": "2.3 MASKING GRADIENTS WITH A LOGICAL AND", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 49 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 504, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "score": 1.0, + "content": "The element-wise geometric mean of gradients, instead of the arithmetic mean, increases consistency", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 600, + 411, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 411, + 613 + ], + "score": 1.0, + "content": "in the convex quadratic case. However, there are a few practical limitations:", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 588, + 505, + 613 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 613, + 504, + 635 + ], + "lines": [ + { + "bbox": [ + 108, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 108, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "(i) The geometric mean is only defined when all the signs are consistent. It is still to be defined how", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 123, + 624, + 455, + 636 + ], + "spans": [ + { + "bbox": [ + 123, + 624, + 455, + 636 + ], + "score": 1.0, + "content": "sign inconsistencies, which can occur in non-convex settings, should be dealt with.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52.5, + "bbox_fs": [ + 108, + 612, + 505, + 636 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 637, + 504, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 636, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 505, + 650 + ], + "score": 1.0, + "content": "(ii) It provides little flexibility for ‘partial’ agreement: Even a single zero gradient component in one", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 122, + 648, + 320, + 660 + ], + "spans": [ + { + "bbox": [ + 122, + 648, + 320, + 660 + ], + "score": 1.0, + "content": "environment stops optimization in that direction.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 54.5, + "bbox_fs": [ + 106, + 636, + 505, + 660 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 94 + ], + "lines": [ + { + "bbox": [ + 102, + 81, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 102, + 81, + 507, + 97 + ], + "score": 1.0, + "content": "(iii) For numerical stability, it needs to be computed in log domain (more computationally expensive).", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 98, + 504, + 131 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 505, + 111 + ], + "score": 1.0, + "content": "(iv) Adaptive step-size schemes (e.g. Adam (Kingma and Ba, 2015)) rescale the signal component-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 122, + 108, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 122, + 108, + 505, + 122 + ], + "score": 1.0, + "content": "wise for local curvature adaptation. The exact magnitude of the geometric mean would be ignored", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 122, + 119, + 501, + 133 + ], + "spans": [ + { + "bbox": [ + 122, + 119, + 501, + 133 + ], + "score": 1.0, + "content": "and most of the difference from arithmetic averaging will come from the zero-ed components.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 134, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 133, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 506, + 147 + ], + "score": 1.0, + "content": "(i) can be overcome by treating different signs as zeros, resulting in a geometric mean of 0 if there", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 506, + 158 + ], + "score": 1.0, + "content": "is any sign disagreement across environments for a gradient component. For (ii) we can allow for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 156, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 506, + 169 + ], + "score": 1.0, + "content": "some disagreement (with a hyperparameter), by not masking out if there is a large percentage of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 168, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 505, + 179 + ], + "score": 1.0, + "content": "environments with gradients in that direction. (iii) and (iv) can be addressed together: Since the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 178, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 506, + 190 + ], + "score": 1.0, + "content": "final magnitude will be rescaled except for masked components, i.e. where the geometric mean is 0,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "we can use the average gradients (fast to compute) and mask out the components based on the sign", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 200, + 308, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 308, + 213 + ], + "score": 1.0, + "content": "agreement (computable avoiding the log domain).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "The AND-mask. We translate the reasoning we just presented to a practical algorithm that we will", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "score": 1.0, + "content": "refer to as the AND-mask. In its most simple implementation, we zero out those gradient components", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "score": 1.0, + "content": "with respect to weights that have inconsistent signs across environments. 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For convenience, our implementation of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 503, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 238, + 301 + ], + "score": 1.0, + "content": "the AND-mask uses a threshold", + "type": "text" + }, + { + "bbox": [ + 238, + 288, + 277, + 299 + ], + "score": 0.92, + "content": "\\tau \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 286, + 400, + 301 + ], + "score": 1.0, + "content": "as hyper-parameter instead of ř", + "type": "text" + }, + { + "bbox": [ + 401, + 289, + 405, + 298 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 286, + 448, + 301 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 448, + 287, + 503, + 300 + ], + "score": 0.91, + "content": "\\begin{array} { r } { t = \\frac { d } { 2 } ( \\tau + 1 ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 475, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 259, + 313 + ], + "score": 1.0, + "content": "Mathematically, for every component", + "type": "text" + }, + { + "bbox": [ + 259, + 299, + 284, + 311 + ], + "score": 0.92, + "content": "[ m _ { \\tau } ] _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 298, + 296, + 313 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 297, + 300, + 311, + 310 + ], + "score": 0.81, + "content": "m _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 298, + 315, + 313 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 316, + 298, + 470, + 312 + ], + "score": 0.89, + "content": "\\begin{array} { r } { [ m _ { \\tau } ] _ { j } = \\pmb { 1 } \\left[ \\tau d \\leqslant | \\sum _ { e } \\mathrm { s i g n } ( [ \\nabla \\mathcal { L } _ { e } ] _ { j } ) | \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 298, + 475, + 313 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "Computing the AND-mask has the same time and space complexity of standard gradient descent,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "i.e., linear in the number of examples that we average. Due to its simplicity and computational", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 337, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 351 + ], + "score": 1.0, + "content": "efficiency, this is the algorithm that we will use in the experiment section. As a first result, we show", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "that following the AND-masked gradient leads to convergence in the directions made visible by the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 359, + 318, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 318, + 373 + ], + "score": 1.0, + "content": "AND-mask. The proof is presented in appendix A.3.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 503, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 504, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 186, + 389 + ], + "score": 1.0, + "content": "Proposition 1. Let", + "type": "text" + }, + { + "bbox": [ + 186, + 377, + 195, + 386 + ], + "score": 0.73, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 375, + 218, + 389 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 218, + 377, + 226, + 386 + ], + "score": 0.76, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 375, + 427, + 389 + ], + "score": 1.0, + "content": "-Lipschitz gradients and consider a learning rate", + "type": "text" + }, + { + "bbox": [ + 428, + 376, + 466, + 388 + ], + "score": 0.89, + "content": "\\eta \\leqslant 1 / L", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 375, + 497, + 389 + ], + "score": 1.0, + "content": ". After", + "type": "text" + }, + { + "bbox": [ + 497, + 376, + 504, + 386 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 387, + 490, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 328, + 401 + ], + "score": 1.0, + "content": "iterations, AND-masked GD visits at least once a point", + "type": "text" + }, + { + "bbox": [ + 329, + 388, + 335, + 398 + ], + "score": 0.75, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 387, + 363, + 401 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 363, + 387, + 485, + 399 + ], + "score": 0.91, + "content": "\\| m _ { t } ( \\theta ) \\odot \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } \\leqslant \\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 387, + 490, + 401 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 357, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 356, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 356, + 425 + ], + "score": 1.0, + "content": "Behaviour in the face of randomness. Here we put the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 422, + 357, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 357, + 437 + ], + "score": 1.0, + "content": "AND mask through a theoretical test: For gradients coming", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 434, + 357, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 357, + 447 + ], + "score": 1.0, + "content": "from different environments that are inconsistent (or even ran-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 445, + 357, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 357, + 457 + ], + "score": 1.0, + "content": "dom), how fast does the AND mask reduce the magnitude of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 358, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 358, + 469 + ], + "score": 1.0, + "content": "the step taken in parameter space, compared to standard GD?", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 467, + 356, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 356, + 480 + ], + "score": 1.0, + "content": "In case of inconsistency, the AND mask should quickly make", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 479, + 257, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 257, + 491 + ], + "score": 1.0, + "content": "the gradient steps more conservative.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 355, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 357, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 302, + 508 + ], + "score": 1.0, + "content": "To assess this property, we consider a fixed set of", + "type": "text" + }, + { + "bbox": [ + 302, + 498, + 310, + 505 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 495, + 357, + 508 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 506, + 357, + 518 + ], + "spans": [ + { + "bbox": [ + 107, + 507, + 113, + 516 + ], + "score": 0.77, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 506, + 169, + 518 + ], + "score": 1.0, + "content": "and gradients", + "type": "text" + }, + { + "bbox": [ + 170, + 506, + 190, + 517 + ], + "score": 0.91, + "content": "\\nabla { \\mathcal { L } } _ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 506, + 357, + 518 + ], + "score": 1.0, + "content": "drawn independently from a multivariate", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 517, + 292, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 292, + 529 + ], + "score": 1.0, + "content": "Gaussian with zero mean and unit covariance.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "image", + "bbox": [ + 378, + 412, + 487, + 493 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 378, + 412, + 487, + 493 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 378, + 412, + 487, + 493 + ], + "spans": [ + { + "bbox": [ + 378, + 412, + 487, + 493 + ], + "score": 0.962, + "type": "image", + "image_path": "ad04a04ecd51008a3704226762cc27ef505decdf650968cedaac64cbe9c1e43f.jpg" + } + ] + } + ], + "index": 33.5, + "virtual_lines": [ + { + "bbox": [ + 378, + 412, + 487, + 452.5 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 378, + 452.5, + 487, + 493.0 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 363, + 496, + 505, + 526 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 363, + 496, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 363, + 496, + 506, + 507 + ], + "score": 1.0, + "content": "Figure 4: Magnitude of gradient (aver-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 363, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 363, + 505, + 484, + 518 + ], + "score": 1.0, + "content": "age or masked) on random data", + "type": "text" + }, + { + "bbox": [ + 484, + 506, + 505, + 517 + ], + "score": 0.89, + "content": "( \\left| \\theta \\right| =", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 363, + 515, + 426, + 527 + ], + "spans": [ + { + "bbox": [ + 363, + 515, + 385, + 527 + ], + "score": 1.0, + "content": "3000,", + "type": "text" + }, + { + "bbox": [ + 385, + 516, + 421, + 526 + ], + "score": 0.72, + "content": "t = 0 . 8 d", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 515, + 426, + 527 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + } + ], + "index": 36.25 + }, + { + "type": "text", + "bbox": [ + 107, + 533, + 503, + 559 + ], + "lines": [ + { + "bbox": [ + 102, + 526, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 102, + 526, + 333, + 554 + ], + "score": 1.0, + "content": "Proposition 2. Consider the setting we just outlined, with", + "type": "text" + }, + { + "bbox": [ + 333, + 532, + 414, + 546 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathcal { L } = ( 1 / d ) \\sum _ { e = 1 } ^ { d } \\mathcal { L } _ { e } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 526, + 444, + 554 + ], + "score": 1.0, + "content": ". While", + "type": "text" + }, + { + "bbox": [ + 445, + 533, + 506, + 546 + ], + "score": 0.89, + "content": "\\mathbb { E } \\| \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } =", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 107, + 544, + 503, + 561 + ], + "spans": [ + { + "bbox": [ + 107, + 546, + 139, + 559 + ], + "score": 0.91, + "content": "\\mathcal { O } ( n / d )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 544, + 194, + 561 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + }, + { + "bbox": [ + 194, + 546, + 330, + 559 + ], + "score": 0.86, + "content": "\\forall t \\in \\{ d / 2 + 1 , \\ldots , d \\} , \\exists c \\in ( 1 , 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 544, + 369, + 561 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 369, + 546, + 503, + 559 + ], + "score": 0.92, + "content": "\\mathbb { E } \\| m _ { t } ( \\theta ) \\odot \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } \\leqslant \\mathcal { O } ( n / c ^ { d } )", + "type": "inline_equation" + } + ], + "index": 42 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "The proof is presented in Appendix A.4, and an illustration with numerical verification in Fig. 4 (the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "magnitudes of masked gradients (•) for more than 100 examples were always zero in the numerical", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "verification). Intuitively, in the presence of purely random patterns, the AND-mask has a desirable", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 597, + 475, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 475, + 609 + ], + "score": 1.0, + "content": "property: it decreases the strength of these signals exponentially fast, as opposed to linearly.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + }, + { + "type": "title", + "bbox": [ + 108, + 622, + 200, + 635 + ], + "lines": [ + { + "bbox": [ + 104, + 621, + 201, + 637 + ], + "spans": [ + { + "bbox": [ + 104, + 621, + 201, + 637 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 645, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 657 + ], + "score": 1.0, + "content": "Real-world datasets are generated by (causal) generative processes which share mechanisms (Pearl,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "2009). However, mechanisms and spurious signals are often entangled, making it hard to assess what", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "part of the learning signal is due to either. As the goal of this paper is to dissect these two components", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "to understand how they ultimately contribute to the learning process, we create a simple synthetic", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "dataset that allows us to control the complexity, intensity, and number of shortcuts in the data. After", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "that, we evaluate whether spurious signals can be detected even in high-dimensional networks and", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 711, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 507, + 723 + ], + "score": 1.0, + "content": "datasets by testing the AND-mask on a memorization task similar to the one proposed in Zhang et al.", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 721, + 459, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 459, + 734 + ], + "score": 1.0, + "content": "(2017), and on a behavioral cloning task using the game CoinRun (Cobbe et al., 2019a).", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 51.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "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": [ + 105, + 82, + 505, + 94 + ], + "lines": [ + { + "bbox": [ + 102, + 81, + 507, + 97 + ], + "spans": [ + { + "bbox": [ + 102, + 81, + 507, + 97 + ], + "score": 1.0, + "content": "(iii) For numerical stability, it needs to be computed in log domain (more computationally expensive).", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 102, + 81, + 507, + 97 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 98, + 504, + 131 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 505, + 111 + ], + "score": 1.0, + "content": "(iv) Adaptive step-size schemes (e.g. Adam (Kingma and Ba, 2015)) rescale the signal component-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 122, + 108, + 505, + 122 + ], + "spans": [ + { + "bbox": [ + 122, + 108, + 505, + 122 + ], + "score": 1.0, + "content": "wise for local curvature adaptation. The exact magnitude of the geometric mean would be ignored", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 122, + 119, + 501, + 133 + ], + "spans": [ + { + "bbox": [ + 122, + 119, + 501, + 133 + ], + "score": 1.0, + "content": "and most of the difference from arithmetic averaging will come from the zero-ed components.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 97, + 505, + 133 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 134, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 133, + 506, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 506, + 147 + ], + "score": 1.0, + "content": "(i) can be overcome by treating different signs as zeros, resulting in a geometric mean of 0 if there", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 146, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 506, + 158 + ], + "score": 1.0, + "content": "is any sign disagreement across environments for a gradient component. 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(iii) and (iv) can be addressed together: Since the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 178, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 506, + 190 + ], + "score": 1.0, + "content": "final magnitude will be rescaled except for masked components, i.e. where the geometric mean is 0,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "we can use the average gradients (fast to compute) and mask out the components based on the sign", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 200, + 308, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 308, + 213 + ], + "score": 1.0, + "content": "agreement (computable avoiding the log domain).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 133, + 506, + 213 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "The AND-mask. We translate the reasoning we just presented to a practical algorithm that we will", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 246 + ], + "score": 1.0, + "content": "refer to as the AND-mask. In its most simple implementation, we zero out those gradient components", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 256 + ], + "score": 1.0, + "content": "with respect to weights that have inconsistent signs across environments. 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For convenience, our implementation of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 503, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 238, + 301 + ], + "score": 1.0, + "content": "the AND-mask uses a threshold", + "type": "text" + }, + { + "bbox": [ + 238, + 288, + 277, + 299 + ], + "score": 0.92, + "content": "\\tau \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 286, + 400, + 301 + ], + "score": 1.0, + "content": "as hyper-parameter instead of ř", + "type": "text" + }, + { + "bbox": [ + 401, + 289, + 405, + 298 + ], + "score": 0.7, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 286, + 448, + 301 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 448, + 287, + 503, + 300 + ], + "score": 0.91, + "content": "\\begin{array} { r } { t = \\frac { d } { 2 } ( \\tau + 1 ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 475, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 259, + 313 + ], + "score": 1.0, + "content": "Mathematically, for every component", + "type": "text" + }, + { + "bbox": [ + 259, + 299, + 284, + 311 + ], + "score": 0.92, + "content": "[ m _ { \\tau } ] _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 298, + 296, + 313 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 297, + 300, + 311, + 310 + ], + "score": 0.81, + "content": "m _ { \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 298, + 315, + 313 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 316, + 298, + 470, + 312 + ], + "score": 0.89, + "content": "\\begin{array} { r } { [ m _ { \\tau } ] _ { j } = \\pmb { 1 } \\left[ \\tau d \\leqslant | \\sum _ { e } \\mathrm { s i g n } ( [ \\nabla \\mathcal { L } _ { e } ] _ { j } ) | \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 298, + 475, + 313 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 221, + 506, + 313 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "Computing the AND-mask has the same time and space complexity of standard gradient descent,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 327, + 505, + 339 + ], + "score": 1.0, + "content": "i.e., linear in the number of examples that we average. 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Consider the setting we just outlined, with", + "type": "text" + }, + { + "bbox": [ + 333, + 532, + 414, + 546 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathcal { L } = ( 1 / d ) \\sum _ { e = 1 } ^ { d } \\mathcal { L } _ { e } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 526, + 444, + 554 + ], + "score": 1.0, + "content": ". 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However, mechanisms and spurious signals are often entangled, making it hard to assess what", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "part of the learning signal is due to either. As the goal of this paper is to dissect these two components", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "to understand how they ultimately contribute to the learning process, we create a simple synthetic", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "dataset that allows us to control the complexity, intensity, and number of shortcuts in the data. After", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "that, we evaluate whether spurious signals can be detected even in high-dimensional networks and", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 711, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 507, + 723 + ], + "score": 1.0, + "content": "datasets by testing the AND-mask on a memorization task similar to the one proposed in Zhang et al.", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 721, + 459, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 459, + 734 + ], + "score": 1.0, + "content": "(2017), and on a behavioral cloning task using the game CoinRun (Cobbe et al., 2019a).", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 645, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 132, + 81, + 479, + 128 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 132, + 81, + 479, + 128 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 81, + 479, + 128 + ], + "spans": [ + { + "bbox": [ + 132, + 81, + 479, + 128 + ], + "score": 0.958, + "type": "image", + "image_path": "2c6666a00d98e65f93e0f8cfc43ecdc82f29522ed1a9935aadd3b95f63205d2e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 132, + 81, + 479, + 96.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 132, + 96.66666666666667, + 479, + 112.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 132, + 112.33333333333334, + 479, + 128.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 136, + 504, + 157 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 136, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 147 + ], + "score": 1.0, + "content": "Figure 5: A 4-dimensional instantiation of the synthetic memorization dataset for visualization. Every example", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 145, + 492, + 158 + ], + "spans": [ + { + "bbox": [ + 104, + 145, + 492, + 158 + ], + "score": 1.0, + "content": "is a dot in both circles, and it can be classified by finding either of the “oracle” decision boundaries shown.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 108, + 172, + 311, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 313, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 313, + 184 + ], + "score": 1.0, + "content": "3.1 THE SYNTHETIC MEMORIZATION DATASET", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 190, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 378, + 204 + ], + "score": 1.0, + "content": "We introduce a binary classification task. The input dimensionality is", + "type": "text" + }, + { + "bbox": [ + 378, + 190, + 433, + 201 + ], + "score": 0.91, + "content": "d = d _ { M } + d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 187, + 464, + 204 + ], + "score": 1.0, + "content": ". 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While the mechanism is shared, it needs a highly non-linear decision", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "boundary to classify the data. The shortcuts are not shared across environments, but provide a simple", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "way to classify the data, even when pooling all the environments together. See Figure 5 for a concrete", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 164, + 258 + ], + "score": 1.0, + "content": "example with", + "type": "text" + }, + { + "bbox": [ + 165, + 245, + 180, + 255 + ], + "score": 0.9, + "content": "d _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 245, + 199, + 258 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 200, + 245, + 212, + 255 + ], + "score": 0.87, + "content": "d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 245, + 358, + 258 + ], + "score": 1.0, + "content": "equal to 2, and two environments (", + "type": "text" + }, + { + "bbox": [ + 358, + 245, + 367, + 254 + ], + "score": 0.63, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 245, + 386, + 258 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 386, + 245, + 396, + 255 + ], + "score": 0.77, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 245, + 469, + 258 + ], + "score": 1.0, + "content": "). The spirals (on", + "type": "text" + }, + { + "bbox": [ + 470, + 245, + 485, + 256 + ], + "score": 0.84, + "content": "d _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 245, + 506, + 258 + ], + "score": 1.0, + "content": ") are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 255, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 286, + 268 + ], + "score": 1.0, + "content": "invariant but hard to model. The shortcuts (on", + "type": "text" + }, + { + "bbox": [ + 286, + 256, + 299, + 267 + ], + "score": 0.8, + "content": "d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 255, + 506, + 268 + ], + "score": 1.0, + "content": ") are simple blobs but different in every environment:", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 267, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 117, + 279 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 267, + 125, + 276 + ], + "score": 0.71, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 267, + 363, + 279 + ], + "score": 1.0, + "content": ", linearly separable through a vertical decision boundary, in", + "type": "text" + }, + { + "bbox": [ + 363, + 267, + 372, + 276 + ], + "score": 0.78, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 267, + 506, + 279 + ], + "score": 1.0, + "content": "with a horizontal one. If the two", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "environments are pooled, a new diagonal decision boundary emerges on the shortcut dimensions as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 427, + 301 + ], + "score": 1.0, + "content": "the most ‘natural’ one. While this perfectly classifies data in both environments", + "type": "text" + }, + { + "bbox": [ + 428, + 289, + 436, + 298 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 288, + 454, + 301 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 455, + 289, + 464, + 299 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 288, + 505, + 301 + ], + "score": 1.0, + "content": ", critically", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 349, + 312 + ], + "score": 1.0, + "content": "it would have not been found by training on either partition", + "type": "text" + }, + { + "bbox": [ + 350, + 300, + 358, + 309 + ], + "score": 0.54, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 299, + 371, + 312 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 371, + 300, + 380, + 309 + ], + "score": 0.66, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "alone. The out-of-distribution", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "(o.o.d.) test data has the same mechanism but random shortcuts. Therefore, any method relying", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 322, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 506, + 333 + ], + "score": 1.0, + "content": "exclusively on the shortcuts will have chance-level o.o.d. performance. Details about the dataset,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 333, + 338, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 338, + 345 + ], + "score": 1.0, + "content": "baselines, and training curves are reported in appendix B.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "score": 1.0, + "content": "Despite the apparent simplicity of this dataset, note that it is challenging to find the invariant", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "mechanism. In high dimensions, even with tens of pooled environments, the shortcuts allow for a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "simple classification rule under almost every classical definition of ‘simple’: the boundary is linear, it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 383, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 506, + 394 + ], + "score": 1.0, + "content": "has a large margin, it can be expressed with small weights, it is fast to learn, robust to input noise,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "and has perfect accuracy and no i.i.d. generalization gap. Finding the complex decision boundary of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 404, + 471, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 471, + 416 + ], + "score": 1.0, + "content": "the spirals, instead, is a fiddly process and arguably a much slower path towards small loss.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 422, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 435 + ], + "score": 1.0, + "content": "Baselines. We evaluate several domain-agnostic baselines (all multilayer perceptrons) with some", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 433, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 506, + 445 + ], + "score": 1.0, + "content": "of the most common regularizers used in deep learning — Dropout, L1, L2, Batch normalization.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "We also consider methods that explicitly make use of the environment labels, namely: (i) Domain", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 455, + 504, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 504, + 467 + ], + "score": 1.0, + "content": "Adversarial Neural Networks (DANN) (Ganin et al., 2016), a method specifically designed to address", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "domain adaptation by obfuscating domain information with an adversarial classifier; (ii) Invariant", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "Risk Minimization (IRM) (Arjovsky et al., 2019), discussed in detail in appendix B. The AND-mask", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 488, + 308, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 308, + 499 + ], + "score": 1.0, + "content": "is trained with the same configurations in Table 1.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 325, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 327, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 327, + 519 + ], + "score": 1.0, + "content": "Results. Fig. 6 shows training and test accuracy.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 515, + 326, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 326, + 529 + ], + "score": 1.0, + "content": "DANN fails because it can align the representation-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 527, + 325, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 325, + 540 + ], + "score": 1.0, + "content": "layer distributions from different environments using", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 539, + 324, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 324, + 550 + ], + "score": 1.0, + "content": "only shortcuts, such that they become indistinguishable", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 549, + 325, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 325, + 561 + ], + "score": 1.0, + "content": "to the domain-discriminating classifier. 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In par-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 582, + 325, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 325, + 593 + ], + "score": 1.0, + "content": "ticular, the combination of the AND-mask with L1 or", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 593, + 326, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 326, + 605 + ], + "score": 1.0, + "content": "L2 regularization gave the most robust results overall,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 325, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 325, + 616 + ], + "score": 1.0, + "content": "as they help suppress neurons that at initialization are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 615, + 220, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 220, + 626 + ], + "score": 1.0, + "content": "tuned towards the shortcuts.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39 + }, + { + "type": "image", + "bbox": [ + 340, + 506, + 493, + 608 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 340, + 506, + 493, + 608 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 340, + 506, + 493, + 608 + ], + "spans": [ + { + "bbox": [ + 340, + 506, + 493, + 608 + ], + "score": 0.967, + "type": "image", + "image_path": "1e0e4467ccc660e2ea7f279aa37ed5069560e29245353b3bd2c3a0696908dec2.jpg" + } + ] + } + ], + "index": 40.0, + "virtual_lines": [ + { + "bbox": [ + 340, + 506, + 493, + 557.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 340, + 557.0, + 493, + 608.0 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 343, + 612, + 493, + 622 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 341, + 609, + 496, + 624 + ], + "spans": [ + { + "bbox": [ + 341, + 609, + 496, + 624 + ], + "score": 1.0, + "content": "Figure 6: Results on the synthetic dataset.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + } + ], + "index": 43.0 + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 507, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 507, + 646 + ], + "score": 1.0, + "content": "Correlations between average, memorization and generalization gradients. 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(iii) Randomly permuting over the shortcuts dimensions,", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 720, + 507, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 507, + 733 + ], + "score": 1.0, + "content": "isolating the “generalization” signal of the mechanisms alone. Figure 7 shows the correlation be-", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 51 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "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": "image", + "bbox": [ + 132, + 81, + 479, + 128 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 132, + 81, + 479, + 128 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 81, + 479, + 128 + ], + "spans": [ + { + "bbox": [ + 132, + 81, + 479, + 128 + ], + "score": 0.958, + "type": "image", + "image_path": "2c6666a00d98e65f93e0f8cfc43ecdc82f29522ed1a9935aadd3b95f63205d2e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 132, + 81, + 479, + 96.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 132, + 96.66666666666667, + 479, + 112.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 132, + 112.33333333333334, + 479, + 128.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 136, + 504, + 157 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 136, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 147 + ], + "score": 1.0, + "content": "Figure 5: A 4-dimensional instantiation of the synthetic memorization dataset for visualization. Every example", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 145, + 492, + 158 + ], + "spans": [ + { + "bbox": [ + 104, + 145, + 492, + 158 + ], + "score": 1.0, + "content": "is a dot in both circles, and it can be classified by finding either of the “oracle” decision boundaries shown.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 108, + 172, + 311, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 313, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 313, + 184 + ], + "score": 1.0, + "content": "3.1 THE SYNTHETIC MEMORIZATION DATASET", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 190, + 505, + 344 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 504, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 378, + 204 + ], + "score": 1.0, + "content": "We introduce a binary classification task. The input dimensionality is", + "type": "text" + }, + { + "bbox": [ + 378, + 190, + 433, + 201 + ], + "score": 0.91, + "content": "d = d _ { M } + d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 187, + 464, + 204 + ], + "score": 1.0, + "content": ". While", + "type": "text" + }, + { + "bbox": [ + 464, + 190, + 504, + 202 + ], + "score": 0.91, + "content": "p ( y | x _ { d _ { M } } )", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 200, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 104, + 200, + 348, + 214 + ], + "score": 1.0, + "content": "is the same across all environments (i.e. the mechanism),", + "type": "text" + }, + { + "bbox": [ + 348, + 201, + 394, + 213 + ], + "score": 0.93, + "content": "p ( y | x _ { d _ { S } } , e )", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 200, + 506, + 214 + ], + "score": 1.0, + "content": "is not the same across all", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 212, + 504, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 504, + 224 + ], + "score": 1.0, + "content": "environments (the shortcuts). While the mechanism is shared, it needs a highly non-linear decision", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "boundary to classify the data. The shortcuts are not shared across environments, but provide a simple", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "way to classify the data, even when pooling all the environments together. See Figure 5 for a concrete", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 164, + 258 + ], + "score": 1.0, + "content": "example with", + "type": "text" + }, + { + "bbox": [ + 165, + 245, + 180, + 255 + ], + "score": 0.9, + "content": "d _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 245, + 199, + 258 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 200, + 245, + 212, + 255 + ], + "score": 0.87, + "content": "d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 245, + 358, + 258 + ], + "score": 1.0, + "content": "equal to 2, and two environments (", + "type": "text" + }, + { + "bbox": [ + 358, + 245, + 367, + 254 + ], + "score": 0.63, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 245, + 386, + 258 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 386, + 245, + 396, + 255 + ], + "score": 0.77, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 245, + 469, + 258 + ], + "score": 1.0, + "content": "). The spirals (on", + "type": "text" + }, + { + "bbox": [ + 470, + 245, + 485, + 256 + ], + "score": 0.84, + "content": "d _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 245, + 506, + 258 + ], + "score": 1.0, + "content": ") are", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 255, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 286, + 268 + ], + "score": 1.0, + "content": "invariant but hard to model. The shortcuts (on", + "type": "text" + }, + { + "bbox": [ + 286, + 256, + 299, + 267 + ], + "score": 0.8, + "content": "d _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 255, + 506, + 268 + ], + "score": 1.0, + "content": ") are simple blobs but different in every environment:", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 267, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 117, + 279 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 267, + 125, + 276 + ], + "score": 0.71, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 267, + 363, + 279 + ], + "score": 1.0, + "content": ", linearly separable through a vertical decision boundary, in", + "type": "text" + }, + { + "bbox": [ + 363, + 267, + 372, + 276 + ], + "score": 0.78, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 267, + 506, + 279 + ], + "score": 1.0, + "content": "with a horizontal one. If the two", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 506, + 290 + ], + "score": 1.0, + "content": "environments are pooled, a new diagonal decision boundary emerges on the shortcut dimensions as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 288, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 427, + 301 + ], + "score": 1.0, + "content": "the most ‘natural’ one. While this perfectly classifies data in both environments", + "type": "text" + }, + { + "bbox": [ + 428, + 289, + 436, + 298 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 288, + 454, + 301 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 455, + 289, + 464, + 299 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 288, + 505, + 301 + ], + "score": 1.0, + "content": ", critically", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 349, + 312 + ], + "score": 1.0, + "content": "it would have not been found by training on either partition", + "type": "text" + }, + { + "bbox": [ + 350, + 300, + 358, + 309 + ], + "score": 0.54, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 299, + 371, + 312 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 371, + 300, + 380, + 309 + ], + "score": 0.66, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "alone. The out-of-distribution", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 309, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 309, + 506, + 324 + ], + "score": 1.0, + "content": "(o.o.d.) test data has the same mechanism but random shortcuts. Therefore, any method relying", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 322, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 506, + 333 + ], + "score": 1.0, + "content": "exclusively on the shortcuts will have chance-level o.o.d. performance. Details about the dataset,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 333, + 338, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 338, + 345 + ], + "score": 1.0, + "content": "baselines, and training curves are reported in appendix B.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 187, + 506, + 345 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 505, + 415 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 361 + ], + "score": 1.0, + "content": "Despite the apparent simplicity of this dataset, note that it is challenging to find the invariant", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "mechanism. In high dimensions, even with tens of pooled environments, the shortcuts allow for a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "simple classification rule under almost every classical definition of ‘simple’: the boundary is linear, it", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 383, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 506, + 394 + ], + "score": 1.0, + "content": "has a large margin, it can be expressed with small weights, it is fast to learn, robust to input noise,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "and has perfect accuracy and no i.i.d. generalization gap. Finding the complex decision boundary of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 404, + 471, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 471, + 416 + ], + "score": 1.0, + "content": "the spirals, instead, is a fiddly process and arguably a much slower path towards small loss.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 350, + 506, + 416 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 422, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 435 + ], + "score": 1.0, + "content": "Baselines. We evaluate several domain-agnostic baselines (all multilayer perceptrons) with some", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 433, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 506, + 445 + ], + "score": 1.0, + "content": "of the most common regularizers used in deep learning — Dropout, L1, L2, Batch normalization.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "We also consider methods that explicitly make use of the environment labels, namely: (i) Domain", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 455, + 504, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 504, + 467 + ], + "score": 1.0, + "content": "Adversarial Neural Networks (DANN) (Ganin et al., 2016), a method specifically designed to address", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "domain adaptation by obfuscating domain information with an adversarial classifier; (ii) Invariant", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "Risk Minimization (IRM) (Arjovsky et al., 2019), discussed in detail in appendix B. The AND-mask", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 488, + 308, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 308, + 499 + ], + "score": 1.0, + "content": "is trained with the same configurations in Table 1.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 421, + 506, + 499 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 325, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 327, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 327, + 519 + ], + "score": 1.0, + "content": "Results. Fig. 6 shows training and test accuracy.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 515, + 326, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 326, + 529 + ], + "score": 1.0, + "content": "DANN fails because it can align the representation-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 527, + 325, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 325, + 540 + ], + "score": 1.0, + "content": "layer distributions from different environments using", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 539, + 324, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 324, + 550 + ], + "score": 1.0, + "content": "only shortcuts, such that they become indistinguishable", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 549, + 325, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 325, + 561 + ], + "score": 1.0, + "content": "to the domain-discriminating classifier. The AND-mask", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 326, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 326, + 573 + ], + "score": 1.0, + "content": "was the only method to achieve perfect test accuracy,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 571, + 326, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 326, + 583 + ], + "score": 1.0, + "content": "by fitting the spirals instead of the shortcuts. In par-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 582, + 325, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 325, + 593 + ], + "score": 1.0, + "content": "ticular, the combination of the AND-mask with L1 or", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 593, + 326, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 326, + 605 + ], + "score": 1.0, + "content": "L2 regularization gave the most robust results overall,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 325, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 325, + 616 + ], + "score": 1.0, + "content": "as they help suppress neurons that at initialization are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 615, + 220, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 220, + 626 + ], + "score": 1.0, + "content": "tuned towards the shortcuts.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 504, + 327, + 626 + ] + }, + { + "type": "image", + "bbox": [ + 340, + 506, + 493, + 608 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 340, + 506, + 493, + 608 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 340, + 506, + 493, + 608 + ], + "spans": [ + { + "bbox": [ + 340, + 506, + 493, + 608 + ], + "score": 0.967, + "type": "image", + "image_path": "1e0e4467ccc660e2ea7f279aa37ed5069560e29245353b3bd2c3a0696908dec2.jpg" + } + ] + } + ], + "index": 40.0, + "virtual_lines": [ + { + "bbox": [ + 340, + 506, + 493, + 557.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 340, + 557.0, + 493, + 608.0 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 343, + 612, + 493, + 622 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 341, + 609, + 496, + 624 + ], + "spans": [ + { + "bbox": [ + 341, + 609, + 496, + 624 + ], + "score": 1.0, + "content": "Figure 6: Results on the synthetic dataset.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + } + ], + "index": 43.0 + }, + { + "type": "text", + "bbox": [ + 106, + 633, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 507, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 507, + 646 + ], + "score": 1.0, + "content": "Correlations between average, memorization and generalization gradients. Due to the syn-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "thetic nature of the dataset, we can intervene on its data-generating process in order to examine", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "the learning signals coming from the mechanisms and from the shortcuts. We isolate the two and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "measure their contribution to the average gradients, as we vary the agreement threshold of the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "mask. More precisely, we look at the gradients computed with respect to the weights of a ran-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "domly initialized network for different sets of data: (i) The original data, with mechanisms and", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "shortcuts. (ii) Randomly permuting the dataset over the mechanisms dimensions, thus leaving the", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 104, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "“memorization” signal of the shortcuts. (iii) Randomly permuting over the shortcuts dimensions,", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 720, + 507, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 507, + 733 + ], + "score": 1.0, + "content": "isolating the “generalization” signal of the mechanisms alone. Figure 7 shows the correlation be-", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "tween the components of the original average gradient (i) and the shortcut gradients ((ii), dashed", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "line), and between the original average gradients and the mechanism gradients ((iii), solid line).", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 51, + "bbox_fs": [ + 104, + 632, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "tween the components of the original average gradient (i) and the shortcut gradients ((ii), dashed", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "line), and between the original average gradients and the mechanism gradients ((iii), solid line).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 337, + 225 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 336, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 336, + 116 + ], + "score": 1.0, + "content": "While the signal from the mechanisms is present in the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 338, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 226, + 128 + ], + "score": 1.0, + "content": "original average gradients (i.e.", + "type": "text" + }, + { + "bbox": [ + 226, + 116, + 259, + 127 + ], + "score": 0.9, + "content": "\\rho \\approx 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 115, + 272, + 128 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 273, + 116, + 298, + 126 + ], + "score": 0.88, + "content": "\\tau = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 115, + 338, + 128 + ], + "score": 1.0, + "content": "), its mag-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 337, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 337, + 137 + ], + "score": 1.0, + "content": "nitude is smaller and it is ‘drowned’ by the memorization", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 337, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 337, + 148 + ], + "score": 1.0, + "content": "signal. Instead, increasing the threshold of the AND-mask", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 337, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 337, + 160 + ], + "score": 1.0, + "content": "(right side) suppresses memorization gradients due to the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 337, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 178, + 171 + ], + "score": 1.0, + "content": "shortcuts, and for", + "type": "text" + }, + { + "bbox": [ + 178, + 160, + 203, + 169 + ], + "score": 0.88, + "content": "\\tau \\approx 1", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 159, + 337, + 171 + ], + "score": 1.0, + "content": "most of the gradient components", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 337, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 337, + 181 + ], + "score": 1.0, + "content": "remaining contain signal from the mechanism. On the left", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 338, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 338, + 192 + ], + "score": 1.0, + "content": "side, we test the other side of our hypothesis: An XOR-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 337, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 337, + 204 + ], + "score": 1.0, + "content": "mask zeroes out consistent gradients, preserves those with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 336, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 336, + 215 + ], + "score": 1.0, + "content": "different signs, and results in a sharper decrease of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 276, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 276, + 226 + ], + "score": 1.0, + "content": "correlation with the mechanism gradients.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7 + }, + { + "type": "image", + "bbox": [ + 348, + 114, + 499, + 201 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 348, + 114, + 499, + 201 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 348, + 114, + 499, + 201 + ], + "spans": [ + { + "bbox": [ + 348, + 114, + 499, + 201 + ], + "score": 0.965, + "type": "image", + "image_path": "f6c918dad9067762cd130aed6eaf781a1015531ba0a3e0f3f71834d439bdadd8.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 348, + 114, + 499, + 157.5 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 348, + 157.5, + 499, + 201.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 366, + 204, + 481, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 365, + 203, + 483, + 214 + ], + "spans": [ + { + "bbox": [ + 365, + 203, + 483, + 214 + ], + "score": 1.0, + "content": "Figure 7: Gradient correlations.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + } + ], + "index": 14.25 + }, + { + "type": "title", + "bbox": [ + 108, + 234, + 257, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 258, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 258, + 246 + ], + "score": 1.0, + "content": "3.2 EXPERIMENTS ON CIFAR-10", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 251, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "Memorization in a vision task. Zhang et al. (2017) showed that neural networks trained with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "standard regularizers — like L2 and Dropout — can still memorize large training datasets with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 217, + 286 + ], + "score": 1.0, + "content": "shuffled labels, i.e. reaching", + "type": "text" + }, + { + "bbox": [ + 218, + 274, + 250, + 284 + ], + "score": 0.89, + "content": "\\approx 1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "training accuracy. Their experiments raised significant questions", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 283, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 298 + ], + "score": 1.0, + "content": "about the generalization properties of neural networks and the role of regularizers in constraining", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "the hypothesis class. Our hypothesis is that ILC — for example implemented as the AND-mask", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "— should prevent memorization on a similar task with the shuffled labels, as gradients will tend to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "largely ‘disagree’ in the absence of a shared mechanism. However, when the labels are not shuffled,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 328, + 483, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 483, + 340 + ], + "score": 1.0, + "content": "ILC should have a much weaker effect, as real shared mechanisms are still present in the data.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "To test our hypothesis, we ran an experiment that closely resembles the one in (Zhang et al., 2017) on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 356, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 506, + 368 + ], + "score": 1.0, + "content": "CIFAR-10. We trained a ResNet on CIFAR-10 with random labels, with and without the AND-mask.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "score": 1.0, + "content": "In all experiments we used batch size 80, and treated each example as its own “environment”. Recall", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "that standard gradient averaging is equivalent to an AND-mask with threshold 0. As shown in Figure", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "score": 1.0, + "content": "8, the ResNet with standard average gradients memorized the data, while slightly increasing the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "score": 1.0, + "content": "threshold for the AND-mask quickly prevented memorization (dark blue line). In contrast, training", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "the same networks on the dataset with the original labels resulted in both of them converging and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "generalizing to the test set, confirming that the mask did not significantly affect the generalization", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 434, + 327, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 327, + 446 + ], + "score": 1.0, + "content": "error with a general underlying mechanism in the data.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 357, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 450, + 357, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 357, + 462 + ], + "score": 1.0, + "content": "Note that there is no standard notion of environments in CIFAR-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 460, + 357, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 357, + 473 + ], + "score": 1.0, + "content": "10, which is why we treated every example as coming from", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 473, + 357, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 357, + 483 + ], + "score": 1.0, + "content": "its own environment. This assumption is not unreasonable,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 483, + 358, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 358, + 494 + ], + "score": 1.0, + "content": "as every image in the dataset was literally collected in a dif-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 494, + 357, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 357, + 506 + ], + "score": 1.0, + "content": "ferent physical environment. If anything, it is the standard", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 505, + 357, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 357, + 516 + ], + "score": 1.0, + "content": "i.i.d. assumption that hides this variety behind a notion of a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 516, + 356, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 356, + 528 + ], + "score": 1.0, + "content": "single distribution encompassing all environments. The results", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 527, + 356, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 356, + 539 + ], + "score": 1.0, + "content": "of this experiment further support this interpretation, and can", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 537, + 356, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 356, + 550 + ], + "score": 1.0, + "content": "serve as evidence that — in some cases — we might be able", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 549, + 356, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 356, + 560 + ], + "score": 1.0, + "content": "to identify invariances even without an explicit partition into", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 560, + 357, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 357, + 570 + ], + "score": 1.0, + "content": "environments, as this can be already identified at the level of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 570, + 192, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 192, + 583 + ], + "score": 1.0, + "content": "individual examples.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39.5 + }, + { + "type": "image", + "bbox": [ + 365, + 450, + 501, + 522 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 365, + 450, + 501, + 522 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 365, + 450, + 501, + 522 + ], + "spans": [ + { + "bbox": [ + 365, + 450, + 501, + 522 + ], + "score": 0.959, + "type": "image", + "image_path": "f6190599675b973b7ae2e4d3afc55b69d7b084c707dcc9dcb178991769022b01.jpg" + } + ] + } + ], + "index": 46.5, + "virtual_lines": [ + { + "bbox": [ + 365, + 450, + 501, + 486.0 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 365, + 486.0, + 501, + 522.0 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 363, + 539, + 505, + 570 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 363, + 539, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 363, + 539, + 505, + 550 + ], + "score": 1.0, + "content": "Figure 8: As the AND-mask threshold", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 363, + 550, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 363, + 550, + 505, + 559 + ], + "score": 1.0, + "content": "increases, memorization on CIFAR-10", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 363, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 363, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "with random labels is quickly hindered.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + } + ], + "index": 47.75 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 358, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 358, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 358, + 602 + ], + "score": 1.0, + "content": "Label noise. Following up on this experiment, we test how", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 600, + 358, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 358, + 612 + ], + "score": 1.0, + "content": "the AND-mask performs in the presence of label noise, i.e. when", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 611, + 358, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 358, + 623 + ], + "score": 1.0, + "content": "a portion of the labels in the training set are randomly shuffled", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 622, + 359, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 109, + 634 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 110, + 622, + 129, + 633 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 622, + 359, + 634 + ], + "score": 1.0, + "content": "here). According to our hypothesis, gradients computed", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 633, + 359, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 359, + 645 + ], + "score": 1.0, + "content": "on examples with random labels should disagree and get masked", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 644, + 359, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 359, + 655 + ], + "score": 1.0, + "content": "out by the AND-mask, while signal from correctly labeled data", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 655, + 360, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 360, + 667 + ], + "score": 1.0, + "content": "should contribute to update the model. As shown in Figure 9,", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 666, + 359, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 359, + 678 + ], + "score": 1.0, + "content": "the performance on the incorrectly labeled portion of the dataset", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 676, + 358, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 358, + 690 + ], + "score": 1.0, + "content": "is well below chance for the AND-mask (as it predicts correctly", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 688, + 358, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 358, + 700 + ], + "score": 1.0, + "content": "despite the wrong labels), while the baseline again memorizes", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 699, + 358, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 358, + 711 + ], + "score": 1.0, + "content": "the incorrect labels. On the test set (with untouched labels), the", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 710, + 358, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 358, + 722 + ], + "score": 1.0, + "content": "baseline peaks early then decreases as the model overfits, while", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 720, + 289, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 289, + 734 + ], + "score": 1.0, + "content": "the AND-mask slowly but steadily improves.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 57 + }, + { + "type": "image", + "bbox": [ + 369, + 588, + 502, + 663 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 369, + 588, + 502, + 663 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 369, + 588, + 502, + 663 + ], + "spans": [ + { + "bbox": [ + 369, + 588, + 502, + 663 + ], + "score": 0.962, + "type": "image", + "image_path": "cd69c7bf4b0dcb56f492eb009b3fe37088a338122999723f978315d05295eae2.jpg" + } + ] + } + ], + "index": 64.5, + "virtual_lines": [ + { + "bbox": [ + 369, + 588, + 502, + 625.5 + ], + "spans": [], + "index": 64 + }, + { + "bbox": [ + 369, + 625.5, + 502, + 663.0 + ], + "spans": [], + "index": 65 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 365, + 677, + 505, + 718 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 365, + 677, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 365, + 677, + 505, + 688 + ], + "score": 1.0, + "content": "Figure 9: The AND-mask prevents", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 365, + 687, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 365, + 687, + 505, + 698 + ], + "score": 1.0, + "content": "overfitting to the incorrectly labeled", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 365, + 698, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 365, + 698, + 505, + 708 + ], + "score": 1.0, + "content": "portion of the training set (left) without", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 366, + 707, + 483, + 718 + ], + "spans": [ + { + "bbox": [ + 366, + 707, + 483, + 718 + ], + "score": 1.0, + "content": "hurting the test accuracy (right).", + "type": "text" + } + ], + "index": 69 + } + ], + "index": 67.5 + } + ], + "index": 66.0 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "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": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 506, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 337, + 225 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 336, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 336, + 116 + ], + "score": 1.0, + "content": "While the signal from the mechanisms is present in the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 338, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 226, + 128 + ], + "score": 1.0, + "content": "original average gradients (i.e.", + "type": "text" + }, + { + "bbox": [ + 226, + 116, + 259, + 127 + ], + "score": 0.9, + "content": "\\rho \\approx 0 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 115, + 272, + 128 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 273, + 116, + 298, + 126 + ], + "score": 0.88, + "content": "\\tau = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 115, + 338, + 128 + ], + "score": 1.0, + "content": "), its mag-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 337, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 337, + 137 + ], + "score": 1.0, + "content": "nitude is smaller and it is ‘drowned’ by the memorization", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 337, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 337, + 148 + ], + "score": 1.0, + "content": "signal. Instead, increasing the threshold of the AND-mask", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 337, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 337, + 160 + ], + "score": 1.0, + "content": "(right side) suppresses memorization gradients due to the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 337, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 178, + 171 + ], + "score": 1.0, + "content": "shortcuts, and for", + "type": "text" + }, + { + "bbox": [ + 178, + 160, + 203, + 169 + ], + "score": 0.88, + "content": "\\tau \\approx 1", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 159, + 337, + 171 + ], + "score": 1.0, + "content": "most of the gradient components", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 337, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 337, + 181 + ], + "score": 1.0, + "content": "remaining contain signal from the mechanism. 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Zhang et al. (2017) showed that neural networks trained with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "standard regularizers — like L2 and Dropout — can still memorize large training datasets with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 217, + 286 + ], + "score": 1.0, + "content": "shuffled labels, i.e. reaching", + "type": "text" + }, + { + "bbox": [ + 218, + 274, + 250, + 284 + ], + "score": 0.89, + "content": "\\approx 1 0 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "training accuracy. Their experiments raised significant questions", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 283, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 505, + 298 + ], + "score": 1.0, + "content": "about the generalization properties of neural networks and the role of regularizers in constraining", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "the hypothesis class. Our hypothesis is that ILC — for example implemented as the AND-mask", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "— should prevent memorization on a similar task with the shuffled labels, as gradients will tend to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "largely ‘disagree’ in the absence of a shared mechanism. However, when the labels are not shuffled,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 328, + 483, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 483, + 340 + ], + "score": 1.0, + "content": "ILC should have a much weaker effect, as real shared mechanisms are still present in the data.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 251, + 506, + 340 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "To test our hypothesis, we ran an experiment that closely resembles the one in (Zhang et al., 2017) on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 356, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 506, + 368 + ], + "score": 1.0, + "content": "CIFAR-10. We trained a ResNet on CIFAR-10 with random labels, with and without the AND-mask.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 380 + ], + "score": 1.0, + "content": "In all experiments we used batch size 80, and treated each example as its own “environment”. Recall", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "that standard gradient averaging is equivalent to an AND-mask with threshold 0. As shown in Figure", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 506, + 402 + ], + "score": 1.0, + "content": "8, the ResNet with standard average gradients memorized the data, while slightly increasing the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "score": 1.0, + "content": "threshold for the AND-mask quickly prevented memorization (dark blue line). In contrast, training", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "the same networks on the dataset with the original labels resulted in both of them converging and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "generalizing to the test set, confirming that the mask did not significantly affect the generalization", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 434, + 327, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 327, + 446 + ], + "score": 1.0, + "content": "error with a general underlying mechanism in the data.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 345, + 506, + 446 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 357, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 450, + 357, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 357, + 462 + ], + "score": 1.0, + "content": "Note that there is no standard notion of environments in CIFAR-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 460, + 357, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 357, + 473 + ], + "score": 1.0, + "content": "10, which is why we treated every example as coming from", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 473, + 357, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 357, + 483 + ], + "score": 1.0, + "content": "its own environment. This assumption is not unreasonable,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 483, + 358, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 358, + 494 + ], + "score": 1.0, + "content": "as every image in the dataset was literally collected in a dif-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 494, + 357, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 357, + 506 + ], + "score": 1.0, + "content": "ferent physical environment. If anything, it is the standard", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 505, + 357, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 357, + 516 + ], + "score": 1.0, + "content": "i.i.d. assumption that hides this variety behind a notion of a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 516, + 356, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 356, + 528 + ], + "score": 1.0, + "content": "single distribution encompassing all environments. The results", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 527, + 356, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 356, + 539 + ], + "score": 1.0, + "content": "of this experiment further support this interpretation, and can", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 537, + 356, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 356, + 550 + ], + "score": 1.0, + "content": "serve as evidence that — in some cases — we might be able", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 549, + 356, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 356, + 560 + ], + "score": 1.0, + "content": "to identify invariances even without an explicit partition into", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 560, + 357, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 357, + 570 + ], + "score": 1.0, + "content": "environments, as this can be already identified at the level of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 570, + 192, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 192, + 583 + ], + "score": 1.0, + "content": "individual examples.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 450, + 358, + 583 + ] + }, + { + "type": "image", + "bbox": [ + 365, + 450, + 501, + 522 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 365, + 450, + 501, + 522 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 365, + 450, + 501, + 522 + ], + "spans": [ + { + "bbox": [ + 365, + 450, + 501, + 522 + ], + "score": 0.959, + "type": "image", + "image_path": "f6190599675b973b7ae2e4d3afc55b69d7b084c707dcc9dcb178991769022b01.jpg" + } + ] + } + ], + "index": 46.5, + "virtual_lines": [ + { + "bbox": [ + 365, + 450, + 501, + 486.0 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 365, + 486.0, + 501, + 522.0 + ], + "spans": [], + "index": 47 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 363, + 539, + 505, + 570 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 363, + 539, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 363, + 539, + 505, + 550 + ], + "score": 1.0, + "content": "Figure 8: As the AND-mask threshold", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 363, + 550, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 363, + 550, + 505, + 559 + ], + "score": 1.0, + "content": "increases, memorization on CIFAR-10", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 363, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 363, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "with random labels is quickly hindered.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + } + ], + "index": 47.75 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 358, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 358, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 358, + 602 + ], + "score": 1.0, + "content": "Label noise. Following up on this experiment, we test how", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 600, + 358, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 358, + 612 + ], + "score": 1.0, + "content": "the AND-mask performs in the presence of label noise, i.e. when", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 611, + 358, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 358, + 623 + ], + "score": 1.0, + "content": "a portion of the labels in the training set are randomly shuffled", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 622, + 359, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 109, + 634 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 110, + 622, + 129, + 633 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 622, + 359, + 634 + ], + "score": 1.0, + "content": "here). According to our hypothesis, gradients computed", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 633, + 359, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 359, + 645 + ], + "score": 1.0, + "content": "on examples with random labels should disagree and get masked", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 106, + 644, + 359, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 359, + 655 + ], + "score": 1.0, + "content": "out by the AND-mask, while signal from correctly labeled data", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 655, + 360, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 360, + 667 + ], + "score": 1.0, + "content": "should contribute to update the model. As shown in Figure 9,", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 666, + 359, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 359, + 678 + ], + "score": 1.0, + "content": "the performance on the incorrectly labeled portion of the dataset", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 676, + 358, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 358, + 690 + ], + "score": 1.0, + "content": "is well below chance for the AND-mask (as it predicts correctly", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 688, + 358, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 358, + 700 + ], + "score": 1.0, + "content": "despite the wrong labels), while the baseline again memorizes", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 699, + 358, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 358, + 711 + ], + "score": 1.0, + "content": "the incorrect labels. On the test set (with untouched labels), the", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 710, + 358, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 358, + 722 + ], + "score": 1.0, + "content": "baseline peaks early then decreases as the model overfits, while", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 720, + 289, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 289, + 734 + ], + "score": 1.0, + "content": "the AND-mask slowly but steadily improves.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 57, + "bbox_fs": [ + 105, + 588, + 360, + 734 + ] + }, + { + "type": "image", + "bbox": [ + 369, + 588, + 502, + 663 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 369, + 588, + 502, + 663 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 369, + 588, + 502, + 663 + ], + "spans": [ + { + "bbox": [ + 369, + 588, + 502, + 663 + ], + "score": 0.962, + "type": "image", + "image_path": "cd69c7bf4b0dcb56f492eb009b3fe37088a338122999723f978315d05295eae2.jpg" + } + ] + } + ], + "index": 64.5, + "virtual_lines": [ + { + "bbox": [ + 369, + 588, + 502, + 625.5 + ], + "spans": [], + "index": 64 + }, + { + "bbox": [ + 369, + 625.5, + 502, + 663.0 + ], + "spans": [], + "index": 65 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 365, + 677, + 505, + 718 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 365, + 677, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 365, + 677, + 505, + 688 + ], + "score": 1.0, + "content": "Figure 9: The AND-mask prevents", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 365, + 687, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 365, + 687, + 505, + 698 + ], + "score": 1.0, + "content": "overfitting to the incorrectly labeled", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 365, + 698, + 505, + 708 + ], + "spans": [ + { + "bbox": [ + 365, + 698, + 505, + 708 + ], + "score": 1.0, + "content": "portion of the training set (left) without", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 366, + 707, + 483, + 718 + ], + "spans": [ + { + "bbox": [ + 366, + 707, + 483, + 718 + ], + "score": 1.0, + "content": "hurting the test accuracy (right).", + "type": "text" + } + ], + "index": 69 + } + ], + "index": 67.5 + } + ], + "index": 66.0 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 109, + 82, + 291, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 292, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 292, + 95 + ], + "score": 1.0, + "content": "3.3 BEHAVIORAL CLONING ON COINRUN", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 104, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 103, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 506, + 116 + ], + "score": 1.0, + "content": "CoinRun (Cobbe et al., 2019b) is a game introduced to test how RL agents generalize to novel", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "situations. The agent needs to collect coins, jumping on top of walls and boxes and avoiding", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 507, + 140 + ], + "score": 1.0, + "content": "enemies.9 Each level is procedurally generated — i.e. it has a different combination of sprites,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "background, and layout — but the physics and goals are invariant. Cobbe et al. (2019b) showed that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "score": 1.0, + "content": "state-of-the-art RL algorithms fail to model these invariant mechanisms, performing poorly on new", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 156, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 156, + 505, + 173 + ], + "score": 1.0, + "content": "levels unless trained on thousands of them. To test our hypothesis, we set up a behavioral cloning", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 361, + 183 + ], + "score": 1.0, + "content": "task using CoinRun.10 We start by pre-training a strong policy", + "type": "text" + }, + { + "bbox": [ + 362, + 170, + 374, + 180 + ], + "score": 0.86, + "content": "\\pi ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 168, + 506, + 183 + ], + "score": 1.0, + "content": "using standard PPO (Schulman", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 506, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 193 + ], + "score": 1.0, + "content": "et al., 2017) for 400M steps on the full distribution of levels. We then generate a dataset of pairs", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 107, + 191, + 156, + 203 + ], + "score": 0.9, + "content": "( s , \\pi ^ { * } ( a | s ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "from the on-policy distribution. The training data consists of 1000 states from each", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 201, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 368, + 215 + ], + "score": 1.0, + "content": "of 64 levels, while test data comes from 2000 levels. A ResNet-", + "type": "text" + }, + { + "bbox": [ + 368, + 203, + 392, + 213 + ], + "score": 0.54, + "content": "1 8 ~ \\hat { \\pi } _ { \\boldsymbol { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 201, + 506, + 215 + ], + "score": 1.0, + "content": "is then trained to minimize", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 140, + 226 + ], + "score": 1.0, + "content": "the loss", + "type": "text" + }, + { + "bbox": [ + 141, + 213, + 196, + 226 + ], + "score": 0.93, + "content": "D _ { \\mathrm { K L } } ( \\pi ^ { * } | | \\hat { \\pi } _ { \\boldsymbol { \\theta } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "on the training set. We compare the generalization performance of regular", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "Adam to a version that uses the AND-mask. For each method we ran an automatic hyperparameter", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 236, + 507, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 507, + 248 + ], + "score": 1.0, + "content": "optimization study using Tree-structured Parzen Estimation (Bergstra et al., 2013) of 1024 trials.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 247, + 408, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 409, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 409, + 259 + ], + "score": 1.0, + "content": "Despite the theoretical computational efficiency of computing the AND-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 258, + 409, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 409, + 270 + ], + "score": 1.0, + "content": "mask as presented in Section 2.3 (i.e., linear time and memory in the size of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 268, + 408, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 408, + 281 + ], + "score": 1.0, + "content": "the mini-batch, just like classic SGD), current deep learning frameworks like", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 279, + 408, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 408, + 291 + ], + "score": 1.0, + "content": "PyTorch (Paszke et al., 2017) have optimized routines that sum gradients", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 290, + 408, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 408, + 303 + ], + "score": 1.0, + "content": "across examples in a mini-batch before it is possible to efficiently compute", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 300, + 410, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 410, + 314 + ], + "score": 1.0, + "content": "the AND-mask. We therefore test the AND-mask in a slightly different way.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 408, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 408, + 324 + ], + "score": 1.0, + "content": "In training, in each iteration we sample a batch of data from a randomly", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 323, + 408, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 408, + 335 + ], + "score": 1.0, + "content": "chosen level out of the 64 available (and cycle through them all once per", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 334, + 408, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 408, + 347 + ], + "score": 1.0, + "content": "epoch). We then apply the AND-mask ‘temporally’, only allowing gradients", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 408, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 408, + 357 + ], + "score": 1.0, + "content": "that are consistent across time (and therefore across levels). See Algorithm", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 356, + 408, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 408, + 367 + ], + "score": 1.0, + "content": "1 in appendix B.6 for a detailed description of this alternative formulation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 367, + 408, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 408, + 379 + ], + "score": 1.0, + "content": "of the AND-mask. The figure shows the minimum test loss for the 10 best", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 378, + 409, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 409, + 391 + ], + "score": 1.0, + "content": "runs, supporting the hypothesis that the AND-mask helps identify invariant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 390, + 249, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 249, + 400 + ], + "score": 1.0, + "content": "mechanisms across different levels.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 20.5 + }, + { + "type": "image", + "bbox": [ + 420, + 256, + 498, + 398 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 420, + 256, + 498, + 398 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 420, + 256, + 498, + 398 + ], + "spans": [ + { + "bbox": [ + 420, + 256, + 498, + 398 + ], + "score": 0.932, + "type": "image", + "image_path": "f98c901e8f6899d5e94e32da3effe7ac6f1e48a2f2b95afa4d08c00b4e8b8a48.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 420, + 256, + 498, + 398 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 415, + 211, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 213, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 213, + 430 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 436, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "Generalization and covariate shift. The classic formulation of statistical learning theory (Vapnik)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 447, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 459 + ], + "score": 1.0, + "content": "concerns learning from independent and identically distributed samples. The case where the distribu-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "tion of the covariates at test time differs from the one observed during training is termed covariate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "shift (Sugiyama et al., 2007; Quionero-Candela et al., 2009; Sugiyama and Kawanabe, 2012). Stan-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "dard solutions involve re-weighting of the training examples, but require the additional assumption of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 492, + 315, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 315, + 503 + ], + "score": 1.0, + "content": "overlapping supports for train and test distributions.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 516 + ], + "score": 1.0, + "content": "Causal models and invariances. As we mentioned in the Introduction, causality provides a strong", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "motivation for our work, based on the notion that statistical dependencies are epiphenomena of an", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 524, + 504, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 504, + 535 + ], + "score": 1.0, + "content": "underlying causal model (Pearl, 2009; Peters et al., 2017). The causal description identifies stable", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "elements – e.g. physical mechanisms – connecting causes and effects, which are expected to remain", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "invariant under interventions or changing external conditions (Haavelmo, 1943; Schölkopf et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "2012)). This motivates our notion of invariant mechanisms, and inspired related notions which have", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "been proposed for robust regression (Rojas-Carulla et al., 2018; Heinze-Deml et al., 2018; Arjovsky", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "et al., 2019; Hermann and Lampinen, 2020; Ahuja et al., 2020; Krueger et al., 2020). We discuss this", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 590, + 233, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 233, + 601 + ], + "score": 1.0, + "content": "in more detail in appendix C.1.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "Domain generalization. ILC can be used in a setting of domain generalization (Muandet et al., 2013),", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "but it is not limited to it: as demonstrated in the experiments in Section 3.2, the AND-mask can be", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "applied even if domain labels are not available. In contrast, by treating every example as a single", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "domain, methods relying on domain classifiers (like DANN Ganin et al. (2016) or Balaji et al. (2018))", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 645, + 484, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 389, + 657 + ], + "score": 1.0, + "content": "would require as many output units as there are training examples (i.e.", + "type": "text" + }, + { + "bbox": [ + 389, + 645, + 419, + 655 + ], + "score": 0.86, + "content": "5 0 \\mathrm { ^ { \\circ } 0 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 645, + 484, + 657 + ], + "score": 1.0, + "content": "for CIFAR-10).", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 108, + 656, + 504, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Gradient agreement. Looking at gradient agreement to learn meaningful representations in neural", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "networks has been explored in (Du et al., 2018; Eshratifar et al., 2018; Fort et al., 2019; Zhang et al.,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "2019b). These approaches mainly rely on a measure of cosine similarity between gradients, which", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 700, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 698, + 348, + 714 + ], + "spans": [ + { + "bbox": [ + 118, + 698, + 348, + 714 + ], + "score": 1.0, + "content": "9See Figure 17 in appendix B.6 for a visualization of the game.", + "type": "text" + } + ] + }, + { + "bbox": [ + 115, + 708, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 115, + 708, + 506, + 725 + ], + "score": 1.0, + "content": "10To obtain a robust evaluation, we preferred to approach behavioral cloning instead of the full RL problem, as", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 504, + 733 + ], + "score": 1.0, + "content": "it is a standard supervised learning task and has substantially fewer moving parts than most deep RL algorithms.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 109, + 82, + 291, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 292, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 292, + 95 + ], + "score": 1.0, + "content": "3.3 BEHAVIORAL CLONING ON COINRUN", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 104, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 103, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 506, + 116 + ], + "score": 1.0, + "content": "CoinRun (Cobbe et al., 2019b) is a game introduced to test how RL agents generalize to novel", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "situations. The agent needs to collect coins, jumping on top of walls and boxes and avoiding", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 507, + 140 + ], + "score": 1.0, + "content": "enemies.9 Each level is procedurally generated — i.e. it has a different combination of sprites,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "background, and layout — but the physics and goals are invariant. Cobbe et al. (2019b) showed that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "score": 1.0, + "content": "state-of-the-art RL algorithms fail to model these invariant mechanisms, performing poorly on new", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 156, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 156, + 505, + 173 + ], + "score": 1.0, + "content": "levels unless trained on thousands of them. To test our hypothesis, we set up a behavioral cloning", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 361, + 183 + ], + "score": 1.0, + "content": "task using CoinRun.10 We start by pre-training a strong policy", + "type": "text" + }, + { + "bbox": [ + 362, + 170, + 374, + 180 + ], + "score": 0.86, + "content": "\\pi ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 168, + 506, + 183 + ], + "score": 1.0, + "content": "using standard PPO (Schulman", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 506, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 193 + ], + "score": 1.0, + "content": "et al., 2017) for 400M steps on the full distribution of levels. We then generate a dataset of pairs", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 107, + 191, + 156, + 203 + ], + "score": 0.9, + "content": "( s , \\pi ^ { * } ( a | s ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "from the on-policy distribution. The training data consists of 1000 states from each", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 201, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 368, + 215 + ], + "score": 1.0, + "content": "of 64 levels, while test data comes from 2000 levels. A ResNet-", + "type": "text" + }, + { + "bbox": [ + 368, + 203, + 392, + 213 + ], + "score": 0.54, + "content": "1 8 ~ \\hat { \\pi } _ { \\boldsymbol { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 201, + 506, + 215 + ], + "score": 1.0, + "content": "is then trained to minimize", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 140, + 226 + ], + "score": 1.0, + "content": "the loss", + "type": "text" + }, + { + "bbox": [ + 141, + 213, + 196, + 226 + ], + "score": 0.93, + "content": "D _ { \\mathrm { K L } } ( \\pi ^ { * } | | \\hat { \\pi } _ { \\boldsymbol { \\theta } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "on the training set. We compare the generalization performance of regular", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 237 + ], + "score": 1.0, + "content": "Adam to a version that uses the AND-mask. For each method we ran an automatic hyperparameter", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 236, + 507, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 507, + 248 + ], + "score": 1.0, + "content": "optimization study using Tree-structured Parzen Estimation (Bergstra et al., 2013) of 1024 trials.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 103, + 507, + 248 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 247, + 408, + 399 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 409, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 409, + 259 + ], + "score": 1.0, + "content": "Despite the theoretical computational efficiency of computing the AND-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 258, + 409, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 409, + 270 + ], + "score": 1.0, + "content": "mask as presented in Section 2.3 (i.e., linear time and memory in the size of", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 268, + 408, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 408, + 281 + ], + "score": 1.0, + "content": "the mini-batch, just like classic SGD), current deep learning frameworks like", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 279, + 408, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 408, + 291 + ], + "score": 1.0, + "content": "PyTorch (Paszke et al., 2017) have optimized routines that sum gradients", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 290, + 408, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 408, + 303 + ], + "score": 1.0, + "content": "across examples in a mini-batch before it is possible to efficiently compute", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 300, + 410, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 410, + 314 + ], + "score": 1.0, + "content": "the AND-mask. We therefore test the AND-mask in a slightly different way.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 312, + 408, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 408, + 324 + ], + "score": 1.0, + "content": "In training, in each iteration we sample a batch of data from a randomly", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 323, + 408, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 408, + 335 + ], + "score": 1.0, + "content": "chosen level out of the 64 available (and cycle through them all once per", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 334, + 408, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 408, + 347 + ], + "score": 1.0, + "content": "epoch). We then apply the AND-mask ‘temporally’, only allowing gradients", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 345, + 408, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 408, + 357 + ], + "score": 1.0, + "content": "that are consistent across time (and therefore across levels). See Algorithm", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 356, + 408, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 408, + 367 + ], + "score": 1.0, + "content": "1 in appendix B.6 for a detailed description of this alternative formulation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 367, + 408, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 408, + 379 + ], + "score": 1.0, + "content": "of the AND-mask. The figure shows the minimum test loss for the 10 best", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 378, + 409, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 409, + 391 + ], + "score": 1.0, + "content": "runs, supporting the hypothesis that the AND-mask helps identify invariant", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 390, + 249, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 249, + 400 + ], + "score": 1.0, + "content": "mechanisms across different levels.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 246, + 410, + 400 + ] + }, + { + "type": "image", + "bbox": [ + 420, + 256, + 498, + 398 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 420, + 256, + 498, + 398 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 420, + 256, + 498, + 398 + ], + "spans": [ + { + "bbox": [ + 420, + 256, + 498, + 398 + ], + "score": 0.932, + "type": "image", + "image_path": "f98c901e8f6899d5e94e32da3effe7ac6f1e48a2f2b95afa4d08c00b4e8b8a48.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 420, + 256, + 498, + 398 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 415, + 211, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 213, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 213, + 430 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 436, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "Generalization and covariate shift. The classic formulation of statistical learning theory (Vapnik)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 447, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 459 + ], + "score": 1.0, + "content": "concerns learning from independent and identically distributed samples. The case where the distribu-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "tion of the covariates at test time differs from the one observed during training is termed covariate", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "shift (Sugiyama et al., 2007; Quionero-Candela et al., 2009; Sugiyama and Kawanabe, 2012). Stan-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "dard solutions involve re-weighting of the training examples, but require the additional assumption of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 492, + 315, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 315, + 503 + ], + "score": 1.0, + "content": "overlapping supports for train and test distributions.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 435, + 506, + 503 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 505, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 516 + ], + "score": 1.0, + "content": "Causal models and invariances. As we mentioned in the Introduction, causality provides a strong", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "motivation for our work, based on the notion that statistical dependencies are epiphenomena of an", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 524, + 504, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 504, + 535 + ], + "score": 1.0, + "content": "underlying causal model (Pearl, 2009; Peters et al., 2017). The causal description identifies stable", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "elements – e.g. physical mechanisms – connecting causes and effects, which are expected to remain", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 559 + ], + "score": 1.0, + "content": "invariant under interventions or changing external conditions (Haavelmo, 1943; Schölkopf et al.,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 569 + ], + "score": 1.0, + "content": "2012)). This motivates our notion of invariant mechanisms, and inspired related notions which have", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "been proposed for robust regression (Rojas-Carulla et al., 2018; Heinze-Deml et al., 2018; Arjovsky", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "et al., 2019; Hermann and Lampinen, 2020; Ahuja et al., 2020; Krueger et al., 2020). We discuss this", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 590, + 233, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 233, + 601 + ], + "score": 1.0, + "content": "in more detail in appendix C.1.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 500, + 506, + 601 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 612 + ], + "score": 1.0, + "content": "Domain generalization. ILC can be used in a setting of domain generalization (Muandet et al., 2013),", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "but it is not limited to it: as demonstrated in the experiments in Section 3.2, the AND-mask can be", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "applied even if domain labels are not available. In contrast, by treating every example as a single", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "domain, methods relying on domain classifiers (like DANN Ganin et al. (2016) or Balaji et al. (2018))", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 645, + 484, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 389, + 657 + ], + "score": 1.0, + "content": "would require as many output units as there are training examples (i.e.", + "type": "text" + }, + { + "bbox": [ + 389, + 645, + 419, + 655 + ], + "score": 0.86, + "content": "5 0 \\mathrm { ^ { \\circ } 0 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 645, + 484, + 657 + ], + "score": 1.0, + "content": "for CIFAR-10).", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 600, + 506, + 657 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 656, + 504, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Gradient agreement. Looking at gradient agreement to learn meaningful representations in neural", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "networks has been explored in (Du et al., 2018; Eshratifar et al., 2018; Fort et al., 2019; Zhang et al.,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "2019b). These approaches mainly rely on a measure of cosine similarity between gradients, which", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 301, + 95 + ], + "score": 1.0, + "content": "we did not consider here for two main reasons:", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 301, + 83, + 312, + 94 + ], + "score": 0.35, + "content": "( i )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 313, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "It is a ‘global’ property of the gradients, and it", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 471, + 105 + ], + "score": 1.0, + "content": "would not allow us to extract precise information about different patterns in the network;", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 472, + 94, + 486, + 105 + ], + "score": 0.3, + "content": "( i i )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 486, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "It is", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "unclear how to extend it beyond pairs of vectors, and for pairwise interactions its computational cost", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 304, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 304, + 127 + ], + "score": 1.0, + "content": "scales quadratic in the number of examples used.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 51, + "bbox_fs": [ + 106, + 655, + 506, + 690 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 301, + 95 + ], + "score": 1.0, + "content": "we did not consider here for two main reasons:", + "type": "text" + }, + { + "bbox": [ + 301, + 83, + 312, + 94 + ], + "score": 0.35, + "content": "( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "It is a ‘global’ property of the gradients, and it", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 471, + 105 + ], + "score": 1.0, + "content": "would not allow us to extract precise information about different patterns in the network;", + "type": "text" + }, + { + "bbox": [ + 472, + 94, + 486, + 105 + ], + "score": 0.3, + "content": "( i i )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "It is", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "unclear how to extend it beyond pairs of vectors, and for pairwise interactions its computational cost", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 304, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 304, + 127 + ], + "score": 1.0, + "content": "scales quadratic in the number of examples used.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 138, + 201, + 151 + ], + "lines": [ + { + "bbox": [ + 104, + 136, + 203, + 154 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 203, + 154 + ], + "score": 1.0, + "content": "5 CONCLUSIONS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 162, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 507, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 507, + 175 + ], + "score": 1.0, + "content": "Generalizing out of distribution is one of the most significant open challenges in machine learning,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 506, + 185 + ], + "score": 1.0, + "content": "and relying on invariances across environments or examples may be key in certain contexts. In", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "this paper we analyzed how neural networks trained by averaging gradients across examples might", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 207 + ], + "score": 1.0, + "content": "converge to solutions that ignore the invariances, especially if these are harder to learn than spurious", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 218 + ], + "score": 1.0, + "content": "patterns. We argued that if learning signals are collected on one example at the time — as it is the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 217, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 229 + ], + "score": 1.0, + "content": "case for gradients, e.g., computed with backpropagation — the way these signals are aggregated", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "score": 1.0, + "content": "can play a significant role in the patterns that will ultimately be expressed: Averaging gradients in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 238, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 308, + 250 + ], + "score": 1.0, + "content": "particular can be too permissive, acting as a logical", + "type": "text" + }, + { + "bbox": [ + 309, + 239, + 323, + 248 + ], + "score": 0.39, + "content": "O R", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 238, + 506, + 250 + ], + "score": 1.0, + "content": "of a collection of distinct patterns, and lead to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 248, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 248, + 506, + 263 + ], + "score": 1.0, + "content": "a ‘patchwork’ solution. We introduced and formalized the concept of Invariant Learning Consistency,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "and showed how to learn invariances even in the face of alternative explanations that — although", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "spurious — fulfill most characteristics of a good solution. The AND-mask is but one of multiple", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 295 + ], + "score": 1.0, + "content": "possible ways to improve consistency, and it is unlikely to be a practical algorithm for all applications.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "However, we believe this should not distract from the general idea which we are trying to put forward", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "— namely, that it is worthwhile to study learning of explanations that are hard to vary, with the longer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 460, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 460, + 327 + ], + "score": 1.0, + "content": "term goal of advancing our understanding of learning, memorization and generalization.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 344, + 218, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 343, + 220, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 220, + 359 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 369, + 505, + 424 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "We wish to thank Sebastian Gomez, Luca Biggio, Julius von Kügelgen, Paolo Penna, Ioannis", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "Anagno, Ricards Marcinkevics, Sidak Pal Singh, Damien Teney for feedback on the manuscript, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "thank Nando de Freitas for fruitful discussions in the early stage of this project. We also thank the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "Max Planck ETH Center for Learning Systems for supporting Giambattista Parascandolo, and the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 412, + 502, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 502, + 426 + ], + "score": 1.0, + "content": "International Max Planck Research School for Intelligent Systems for supporting Alexander Neitz.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 108, + 442, + 175, + 454 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 176, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 176, + 455 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 105, + 461, + 507, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "L. 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Any reasonable", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 166, + 552 + ], + "score": 1.0, + "content": "modification on", + "type": "text" + }, + { + "bbox": [ + 166, + 540, + 176, + 550 + ], + "score": 0.85, + "content": "\\theta _ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 540, + 194, + 552 + ], + "score": 1.0, + "content": "(say", + "type": "text" + }, + { + "bbox": [ + 194, + 541, + 208, + 550 + ], + "score": 0.77, + "content": "\\pm 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 540, + 506, + 552 + ], + "score": 1.0, + "content": ") leaves the performance on environment A unchanged, while the performance on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 550, + 227, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 227, + 561 + ], + "score": 1.0, + "content": "environment B quickly degrades.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 505, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 244, + 586 + ], + "score": 1.0, + "content": "However, it is easy to realize that", + "type": "text" + }, + { + "bbox": [ + 245, + 574, + 256, + 583 + ], + "score": 0.83, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "— while being a solution which can be returned by gradient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 584, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 225, + 598 + ], + "score": 1.0, + "content": "descent using the pooled data", + "type": "text" + }, + { + "bbox": [ + 226, + 585, + 247, + 595 + ], + "score": 0.81, + "content": "\\mathbf { A } { + } \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 584, + 505, + 598 + ], + "score": 1.0, + "content": "— is not consistent (formal definition given in the main paper in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 282, + 609 + ], + "score": 1.0, + "content": "Section 2). Indeed, it is possible to modify", + "type": "text" + }, + { + "bbox": [ + 282, + 595, + 293, + 607 + ], + "score": 0.87, + "content": "\\tilde { \\theta } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "such that the loss in environment A remains almost", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "unchanged, while the loss in environment B gets larger. 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Any reasonable", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 166, + 552 + ], + "score": 1.0, + "content": "modification on", + "type": "text" + }, + { + "bbox": [ + 166, + 540, + 176, + 550 + ], + "score": 0.85, + "content": "\\theta _ { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 540, + 194, + 552 + ], + "score": 1.0, + "content": "(say", + "type": "text" + }, + { + "bbox": [ + 194, + 541, + 208, + 550 + ], + "score": 0.77, + "content": "\\pm 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 540, + 506, + 552 + ], + "score": 1.0, + "content": ") leaves the performance on environment A unchanged, while the performance on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 550, + 227, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 227, + 561 + ], + "score": 1.0, + "content": "environment B quickly degrades.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 573, + 505, + 658 + ], + "lines": [ + { + "bbox": [ + 106, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 244, + 586 + ], + "score": 1.0, + "content": "However, it is easy to realize that", + "type": "text" + }, + { + "bbox": [ + 245, + 574, + 256, + 583 + ], + "score": 0.83, + "content": "\\theta ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "— while being a solution which can be returned by gradient", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 584, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 225, + 598 + ], + "score": 1.0, + "content": "descent using the pooled data", + "type": "text" + }, + { + "bbox": [ + 226, + 585, + 247, + 595 + ], + "score": 0.81, + "content": "\\mathbf { A } { + } \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 584, + 505, + 598 + ], + "score": 1.0, + "content": "— is not consistent (formal definition given in the main paper in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 282, + 609 + ], + "score": 1.0, + "content": "Section 2). Indeed, it is possible to modify", + "type": "text" + }, + { + "bbox": [ + 282, + 595, + 293, + 607 + ], + "score": 0.87, + "content": "\\tilde { \\theta } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "such that the loss in environment A remains almost", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 506, + 621 + ], + "score": 1.0, + "content": "unchanged, while the loss in environment B gets larger. In particular, on the right panel of Figure 10,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 618, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 163, + 635 + ], + "score": 1.0, + "content": "we show that", + "type": "text" + }, + { + "bbox": [ + 163, + 619, + 308, + 632 + ], + "score": 0.9, + "content": "\\tilde { \\theta } ^ { * } = ( 1 0 0 , - 5 0 , 1 0 0 , - 7 5 , 1 , - 0 . 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 618, + 358, + 635 + ], + "score": 1.0, + "content": "is such that", + "type": "text" + }, + { + "bbox": [ + 359, + 619, + 454, + 632 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { A } ( \\theta ^ { * } ) \\leqslant \\bar { \\mathcal { L } _ { A } ( \\tilde { \\theta } ^ { * } ) } + \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 618, + 478, + 635 + ], + "score": 1.0, + "content": "(with", + "type": "text" + }, + { + "bbox": [ + 478, + 622, + 484, + 630 + ], + "score": 0.61, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 618, + 506, + 635 + ], + "score": 1.0, + "content": "very", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 149, + 646 + ], + "score": 1.0, + "content": "small) but", + "type": "text" + }, + { + "bbox": [ + 149, + 632, + 227, + 645 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { B } ( \\theta ^ { * } ) \\ll \\mathcal { L } _ { B } ( \\tilde { \\theta } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 633, + 505, + 646 + ], + "score": 1.0, + "content": ". According to our definition in Equation 1 (see main paper), we have", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 645, + 408, + 659 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 232, + 658 + ], + "score": 0.89, + "content": "{ \\cal T } ^ { \\epsilon } ( \\theta ^ { * } ) \\leq | { \\mathcal L } _ { B } ( \\theta ^ { * } ) - { \\mathcal L } _ { B } ( \\tilde { \\theta } ^ { * } ) |", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 645, + 408, + 659 + ], + "score": 1.0, + "content": "— that is a large number (low consistency).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 573, + 506, + 659 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "Remark 1 (Connection to out of distribution generalization). The main point of this analysis was to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "score": 1.0, + "content": "show an example of where our measure of consistency behaves according to expectations: A typical", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "implementation of the universal approximation theorem — which one would not expect to generalize", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 691, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 505, + 705 + ], + "score": 1.0, + "content": "out of distribution, due to its ‘patchwork’ behavior — leads indeed to a very low consistency score.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 659, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 234, + 114 + ], + "score": 1.0, + "content": "Geometric mean of matrices.", + "type": "text" + }, + { + "bbox": [ + 239, + 99, + 281, + 118 + ], + "score": 1.0, + "content": "Given an", + "type": "text" + }, + { + "bbox": [ + 281, + 105, + 288, + 113 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 99, + 324, + 118 + ], + "score": 1.0, + "content": "-tuple of", + "type": "text" + }, + { + "bbox": [ + 324, + 103, + 348, + 114 + ], + "score": 0.9, + "content": "d \\times d", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 99, + 452, + 118 + ], + "score": 1.0, + "content": "positive definite matrices", + "type": "text" + }, + { + "bbox": [ + 452, + 103, + 486, + 117 + ], + "score": 0.93, + "content": "( A _ { j } ) _ { j = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 99, + 506, + 118 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 198, + 113, + 510, + 149 + ], + "spans": [ + { + "bbox": [ + 198, + 113, + 433, + 149 + ], + "score": 1.0, + "content": "ean Ando et al. (2004) is the unique positive definite solution, where log is the matrix logarithm. This matrix average he it relevant to signal processing and medical imaging. 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In the setting we just described, the consistency score in Equation 1 can be estimated", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 698, + 153, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 153, + 711 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 685, + 505, + 711 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 706, + 369, + 736 + ], + "lines": [ + { + "bbox": [ + 241, + 706, + 369, + 736 + ], + "spans": [ + { + "bbox": [ + 241, + 706, + 369, + 736 + ], + "score": 0.94, + "content": "{ \\mathcal { T } } ^ { \\epsilon } ( \\theta ^ { * } ) \\leqslant 2 \\epsilon \\left( { \\frac { \\operatorname* { d e t } \\bigl ( H _ { A + B } \\bigr ) } { \\operatorname* { d e t } \\bigl ( H _ { A \\wedge B } \\bigr ) } } \\right) ^ { 2 } .", + "type": "interline_equation", + "image_path": "b7adb9e04414281e3352f0794331547efa43be680e37fb2969ea051af4c4c98f.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 241, + 706, + 369, + 721.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 241, + 721.0, + 369, + 736.0 + ], + "spans": [], + "index": 29 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 185, + 79, + 425, + 200 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 185, + 79, + 425, + 200 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 185, + 79, + 425, + 200 + ], + "spans": [ + { + "bbox": [ + 185, + 79, + 425, + 200 + ], + "score": 0.969, + "type": "image", + "image_path": "18902103c5165a84f4ea044ec9a748571660c465a1a120f77d810e8def45c95e.jpg" + } + ] + } + ], + "index": 3.5, + "virtual_lines": [ + { + "bbox": [ + 185, + 79, + 425, + 94.125 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 185, + 94.125, + 425, + 109.25 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 185, + 109.25, + 425, + 124.375 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 185, + 124.375, + 425, + 139.5 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 185, + 139.5, + 425, + 154.625 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 185, + 154.625, + 425, + 169.75 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 185, + 169.75, + 425, + 184.875 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 185, + 184.875, + 425, + 200.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 505, + 253 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 242, + 222 + ], + "score": 1.0, + "content": "Figure 12: Plotted are contour lines", + "type": "text" + }, + { + "bbox": [ + 243, + 208, + 296, + 219 + ], + "score": 0.92, + "content": "\\theta ^ { \\top } H ^ { - 1 } \\theta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 208, + 311, + 222 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 311, + 209, + 388, + 221 + ], + "score": 0.91, + "content": "H _ { A } = \\mathrm { d i a g } ( 0 . 0 1 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 208, + 405, + 222 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 406, + 209, + 482, + 220 + ], + "score": 0.88, + "content": "H _ { B } = \\mathrm { d i a g } ( 1 , 0 . 0 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 208, + 506, + 222 + ], + "score": 1.0, + "content": ". It is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 415, + 232 + ], + "score": 1.0, + "content": "convenient to provide this visualization because it is linked to the matrix determinant: a", + "type": "text" + }, + { + "bbox": [ + 415, + 220, + 505, + 231 + ], + "score": 0.88, + "content": "\\mathrm { V o l } ( \\{ \\theta ^ { \\top } H ^ { - 1 } \\theta = 1 \\} ) =", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 107, + 230, + 151, + 244 + ], + "score": 0.92, + "content": "\\pi { \\sqrt { \\operatorname* { d e t } ( H ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 231, + 459, + 244 + ], + "score": 1.0, + "content": ". The geometric average retains the volume of the original ellipses, while the volume of", + "type": "text" + }, + { + "bbox": [ + 459, + 232, + 484, + 243 + ], + "score": 0.9, + "content": "H _ { A + B }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "is 25", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 242, + 450, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 313, + 253 + ], + "score": 1.0, + "content": "times bigger. This magnification indicates that landscape", + "type": "text" + }, + { + "bbox": [ + 313, + 243, + 321, + 251 + ], + "score": 0.74, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 242, + 438, + 253 + ], + "score": 1.0, + "content": "is not consistent with landscape", + "type": "text" + }, + { + "bbox": [ + 438, + 243, + 446, + 251 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 242, + 450, + 253 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 505, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 507, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 507, + 284 + ], + "score": 1.0, + "content": "Before showing the proof, we note that the proposition gives a lower bound on the consistency.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "That is, it provides a pessimistic estimate. Yet, as we motivated, this estimate has a nice geometric", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 294, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 305 + ], + "score": 1.0, + "content": "interpretation. However, as we outline in a remark after the proof, this estimate is tight in two", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 305, + 195, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 195, + 317 + ], + "score": 1.0, + "content": "important limit cases.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 108, + 327, + 264, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 265, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 265, + 342 + ], + "score": 1.0, + "content": "Proof. In this setting, Equation 1 gives", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 340, + 409, + 368 + ], + "lines": [ + { + "bbox": [ + 201, + 340, + 409, + 368 + ], + "spans": [ + { + "bbox": [ + 201, + 340, + 409, + 368 + ], + "score": 0.91, + "content": "\\mathcal { T } ^ { \\epsilon } ( \\theta ^ { * } ) : = \\operatorname* { m a x } \\left\\{ \\operatorname* { m a x } _ { \\mathcal { L } _ { A } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { B } ( \\theta ) , \\operatorname* { m a x } _ { \\mathcal { L } _ { B } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { A } ( \\theta ) \\right\\} .", + "type": "interline_equation", + "image_path": "596eea8a741ad942279debd2491d52a92aa05a586de7895e17c14840da2aca24.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 201, + 340, + 409, + 368 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 151, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 153, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 153, + 381 + ], + "score": 1.0, + "content": "Recall that", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 377, + 375, + 405 + ], + "lines": [ + { + "bbox": [ + 235, + 377, + 375, + 405 + ], + "spans": [ + { + "bbox": [ + 235, + 377, + 375, + 405 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { A } ( \\theta ) = \\frac { 1 } { 2 } \\theta ^ { \\top } H _ { A } \\theta = \\frac { 1 } { 2 } \\sum _ { i } \\lambda _ { i } ^ { A } \\theta _ { i } ^ { 2 } .", + "type": "interline_equation", + "image_path": "21a976a5e879ee9209ca5c9b59e1bfb2a24f04e35507e178a276fae55ffbed6c.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 235, + 377, + 375, + 405 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 399, + 417 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 400, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 400, + 419 + ], + "score": 1.0, + "content": "Hence, this is a simple quadratic program with quadratic constraints, and", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 419, + 389, + 447 + ], + "lines": [ + { + "bbox": [ + 221, + 419, + 389, + 447 + ], + "spans": [ + { + "bbox": [ + 221, + 419, + 389, + 447 + ], + "score": 0.94, + "content": "\\operatorname* { m a x } _ { \\mathcal { L } _ { A } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { B } ( \\theta ) = \\operatorname* { m a x } _ { \\frac { 1 } { 2 } \\sum _ { i } \\lambda _ { i } ^ { A } \\theta _ { i } ^ { 2 } \\leqslant \\epsilon } \\frac { 1 } { 2 } \\sum _ { i } \\lambda _ { i } ^ { B } \\theta _ { i } ^ { 2 } .", + "type": "interline_equation", + "image_path": "9ce2abf2826fae4e7102aa015c3798909f10f9eaec54c02d6a0174b775da1797.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 221, + 419, + 389, + 447 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 450, + 489, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 492, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 297, + 466 + ], + "score": 1.0, + "content": "Further, we can change variables and introduce", + "type": "text" + }, + { + "bbox": [ + 297, + 450, + 359, + 464 + ], + "score": 0.93, + "content": "\\tilde { \\theta } _ { i } = \\theta _ { i } \\sqrt { \\lambda _ { i } ^ { A } / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 448, + 492, + 466 + ], + "score": 1.0, + "content": ". The problem gets even simpler:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 466, + 405, + 496 + ], + "lines": [ + { + "bbox": [ + 205, + 466, + 405, + 496 + ], + "spans": [ + { + "bbox": [ + 205, + 466, + 405, + 496 + ], + "score": 0.91, + "content": "\\operatorname* { m a x } _ { \\mathcal { L } _ { A } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { B } ( \\theta ) = \\operatorname* { m a x } _ { \\| \\tilde { \\theta } \\| ^ { 2 } \\leqslant \\epsilon } \\sum _ { i } \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { i } ^ { A } } \\tilde { \\theta } _ { i } ^ { 2 } = \\epsilon \\cdot \\operatorname* { m a x } _ { i } \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { i } ^ { A } } .", + "type": "interline_equation", + "image_path": "5996b71bb797117f5f8891e620a6ae7ce636799ca473d5cbb01bc4494c39431f.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 205, + 466, + 405, + 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{ i } ^ { B } } \\Bigg \\} } & { } \\\\ { = \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\operatorname* { m a x } \\left\\{ \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { i } ^ { A } } , \\frac { \\lambda _ { i } ^ { A } } { \\lambda _ { B } ^ { B } } \\right\\} } & { } \\\\ { \\leqslant \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\left( \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { A } ^ { A } } + \\frac { \\lambda _ { i } ^ { A } } { \\lambda _ { B } ^ { B } } \\right) } & { } \\\\ { = \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\left\\{ \\frac { ( \\lambda _ { i } ^ { B } ) ^ { 2 } + ( \\lambda _ { i } ^ { A } ) ^ { 2 } } { \\lambda _ { B } ^ { B } \\lambda _ { i } ^ { A } } \\right\\} } & { } \\\\ { \\leqslant \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\left\\{ \\frac { ( \\lambda _ { i } ^ { B } + \\lambda _ { i } ^ { A } ) ^ { 2 } } { \\lambda _ { B } ^ { B } \\lambda _ { i } ^ { A } } \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "b85f2cb6ec08708cf012c47c2835017a3b4f8aaf8af8794f0f384d01eb7476e9.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 224, + 511, + 385, + 525.6 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 224, + 525.6, + 385, + 540.2 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 224, + 540.2, + 385, + 554.8000000000001 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 224, + 554.8000000000001, + 385, + 569.4000000000001 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 224, + 569.4000000000001, + 385, + 584.0000000000001 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 224, + 584.0000000000001, + 385, + 598.6000000000001 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 224, + 598.6000000000001, + 385, + 613.2000000000002 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 224, + 613.2000000000002, + 385, + 627.8000000000002 + ], + "spans": 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\\lambda _ { i } ^ { A } } } = 2 \\epsilon \\operatorname* { m a x } _ { i } \\frac { ( \\lambda _ { i } ^ { B } + \\lambda _ { i } ^ { A } ) / 2 } { \\sqrt { \\lambda _ { i } ^ { B } \\lambda _ { i } ^ { A } } } \\leqslant 2 \\epsilon \\frac { \\prod _ { i } ( \\lambda _ { i } ^ { B } + \\lambda _ { i } ^ { A } ) / 2 } { \\prod _ { i } \\sqrt { \\lambda _ { i } ^ { B } \\lambda _ { i } ^ { A } } } = 2 \\epsilon \\frac { \\operatorname* { d e t } ( H _ { A + B } ) } { \\operatorname* { d e t } ( H _ { A \\wedge B } ) } ,", + "type": "interline_equation", + "image_path": "948472fd43398d65d233b985eedf5e69d21a6131c16dd6d8beffc2221761a3ef.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 112, + 667, + 498, + 677.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 112, + 677.3333333333334, + 498, + 687.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 112, + 687.6666666666667, + 498, + 698.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "where the first inequality comes from the monotonicity of the square root function, and the second", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "inequality comes from the fact that (i) the geometric mean is always smaller or equal than theś", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 719, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 322, + 735 + ], + "score": 1.0, + "content": "arithmetic mean and (ii) for any sequence of numbers", + "type": "text" + }, + { + "bbox": [ + 322, + 721, + 351, + 732 + ], + "score": 0.86, + "content": "\\alpha _ { i } > 1", + "type": 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1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 311, + 209, + 388, + 221 + ], + "score": 0.91, + "content": "H _ { A } = \\mathrm { d i a g } ( 0 . 0 1 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 208, + 405, + 222 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 406, + 209, + 482, + 220 + ], + "score": 0.88, + "content": "H _ { B } = \\mathrm { d i a g } ( 1 , 0 . 0 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 208, + 506, + 222 + ], + "score": 1.0, + "content": ". It is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 415, + 232 + ], + "score": 1.0, + "content": "convenient to provide this visualization because it is linked to the matrix determinant: a", + "type": "text" + }, + { + "bbox": [ + 415, + 220, + 505, + 231 + ], + "score": 0.88, + "content": "\\mathrm { V o l } ( \\{ \\theta ^ { \\top } H ^ { - 1 } \\theta = 1 \\} ) =", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 107, + 230, + 151, + 244 + ], + "score": 0.92, + "content": "\\pi { \\sqrt { \\operatorname* { d e t } ( H ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 231, + 459, + 244 + ], + "score": 1.0, + "content": ". The geometric average retains the volume of the original ellipses, while the volume of", + "type": "text" + }, + { + "bbox": [ + 459, + 232, + 484, + 243 + ], + "score": 0.9, + "content": "H _ { A + B }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "is 25", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 242, + 450, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 313, + 253 + ], + "score": 1.0, + "content": "times bigger. This magnification indicates that landscape", + "type": "text" + }, + { + "bbox": [ + 313, + 243, + 321, + 251 + ], + "score": 0.74, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 242, + 438, + 253 + ], + "score": 1.0, + "content": "is not consistent with landscape", + "type": "text" + }, + { + "bbox": [ + 438, + 243, + 446, + 251 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 242, + 450, + 253 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 271, + 505, + 316 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 507, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 507, + 284 + ], + "score": 1.0, + "content": "Before showing the proof, we note that the proposition gives a lower bound on the consistency.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "That is, it provides a pessimistic estimate. Yet, as we motivated, this estimate has a nice geometric", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 294, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 305 + ], + "score": 1.0, + "content": "interpretation. However, as we outline in a remark after the proof, this estimate is tight in two", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 305, + 195, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 195, + 317 + ], + "score": 1.0, + "content": "important limit cases.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 270, + 507, + 317 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 327, + 264, + 339 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 265, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 265, + 342 + ], + "score": 1.0, + "content": "Proof. In this setting, Equation 1 gives", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 325, + 265, + 342 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 340, + 409, + 368 + ], + "lines": [ + { + "bbox": [ + 201, + 340, + 409, + 368 + ], + "spans": [ + { + "bbox": [ + 201, + 340, + 409, + 368 + ], + "score": 0.91, + "content": "\\mathcal { T } ^ { \\epsilon } ( \\theta ^ { * } ) : = \\operatorname* { m a x } \\left\\{ \\operatorname* { m a x } _ { \\mathcal { L } _ { A } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { B } ( \\theta ) , \\operatorname* { m a x } _ { \\mathcal { L } _ { B } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { A } ( \\theta ) \\right\\} .", + "type": "interline_equation", + "image_path": "596eea8a741ad942279debd2491d52a92aa05a586de7895e17c14840da2aca24.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 201, + 340, + 409, + 368 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 151, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 153, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 153, + 381 + ], + "score": 1.0, + "content": "Recall that", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 106, + 369, + 153, + 381 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 377, + 375, + 405 + ], + "lines": [ + { + "bbox": [ + 235, + 377, + 375, + 405 + ], + "spans": [ + { + "bbox": [ + 235, + 377, + 375, + 405 + ], + "score": 0.92, + "content": "\\mathcal { L } _ { A } ( \\theta ) = \\frac { 1 } { 2 } \\theta ^ { \\top } H _ { A } \\theta = \\frac { 1 } { 2 } \\sum _ { i } \\lambda _ { i } ^ { A } \\theta _ { i } ^ { 2 } .", + "type": "interline_equation", + "image_path": "21a976a5e879ee9209ca5c9b59e1bfb2a24f04e35507e178a276fae55ffbed6c.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 235, + 377, + 375, + 405 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 406, + 399, + 417 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 400, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 400, + 419 + ], + "score": 1.0, + "content": "Hence, this is a simple quadratic program with quadratic constraints, and", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 405, + 400, + 419 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 419, + 389, + 447 + ], + "lines": [ + { + "bbox": [ + 221, + 419, + 389, + 447 + ], + "spans": [ + { + "bbox": [ + 221, + 419, + 389, + 447 + ], + "score": 0.94, + "content": "\\operatorname* { m a x } _ { \\mathcal { L } _ { A } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { B } ( \\theta ) = \\operatorname* { m a x } _ { \\frac { 1 } { 2 } \\sum _ { i } \\lambda _ { i } ^ { A } \\theta _ { i } ^ { 2 } \\leqslant \\epsilon } \\frac { 1 } { 2 } \\sum _ { i } \\lambda _ { i } ^ { B } \\theta _ { i } ^ { 2 } .", + "type": "interline_equation", + "image_path": "9ce2abf2826fae4e7102aa015c3798909f10f9eaec54c02d6a0174b775da1797.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 221, + 419, + 389, + 447 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 450, + 489, + 464 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 492, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 297, + 466 + ], + "score": 1.0, + "content": "Further, we can change variables and introduce", + "type": "text" + }, + { + "bbox": [ + 297, + 450, + 359, + 464 + ], + "score": 0.93, + "content": "\\tilde { \\theta } _ { i } = \\theta _ { i } \\sqrt { \\lambda _ { i } ^ { A } / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 448, + 492, + 466 + ], + "score": 1.0, + "content": ". The problem gets even simpler:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 448, + 492, + 466 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 466, + 405, + 496 + ], + "lines": [ + { + "bbox": [ + 205, + 466, + 405, + 496 + ], + "spans": [ + { + "bbox": [ + 205, + 466, + 405, + 496 + ], + "score": 0.91, + "content": "\\operatorname* { m a x } _ { \\mathcal { L } _ { A } ( \\theta ) \\leqslant \\epsilon } \\mathcal { L } _ { B } ( \\theta ) = \\operatorname* { m a x } _ { \\| \\tilde { \\theta } \\| ^ { 2 } \\leqslant \\epsilon } \\sum _ { i } \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { i } ^ { A } } \\tilde { \\theta } _ { i } ^ { 2 } = \\epsilon \\cdot \\operatorname* { m a x } _ { i } \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { i } ^ { A } } .", + "type": "interline_equation", + "image_path": "5996b71bb797117f5f8891e620a6ae7ce636799ca473d5cbb01bc4494c39431f.jpg" + } + ] + } + ], + "index": 23.5, + 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{ A } } , \\underset { i } { \\operatorname* { m a x } } \\frac { \\lambda _ { i } ^ { A } } { \\lambda _ { i } ^ { B } } \\Bigg \\} } & { } \\\\ { = \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\operatorname* { m a x } \\left\\{ \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { i } ^ { A } } , \\frac { \\lambda _ { i } ^ { A } } { \\lambda _ { B } ^ { B } } \\right\\} } & { } \\\\ { \\leqslant \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\left( \\frac { \\lambda _ { i } ^ { B } } { \\lambda _ { A } ^ { A } } + \\frac { \\lambda _ { i } ^ { A } } { \\lambda _ { B } ^ { B } } \\right) } & { } \\\\ { = \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\left\\{ \\frac { ( \\lambda _ { i } ^ { B } ) ^ { 2 } + ( \\lambda _ { i } ^ { A } ) ^ { 2 } } { \\lambda _ { B } ^ { B } \\lambda _ { i } ^ { A } } \\right\\} } & { } \\\\ { \\leqslant \\epsilon \\cdot \\underset { i } { \\operatorname* { m a x } } \\left\\{ \\frac { ( \\lambda _ { i } ^ { B } + \\lambda _ { i } ^ { A } ) ^ { 2 } } { \\lambda _ { B } ^ { B } \\lambda _ { i } ^ { A } } \\right\\} . } \\end{array}", + "type": "interline_equation", + "image_path": "b85f2cb6ec08708cf012c47c2835017a3b4f8aaf8af8794f0f384d01eb7476e9.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 224, + 511, + 385, + 525.6 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 224, + 525.6, + 385, + 540.2 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 224, + 540.2, + 385, + 554.8000000000001 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 224, + 554.8000000000001, + 385, + 569.4000000000001 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 224, + 569.4000000000001, + 385, + 584.0000000000001 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 224, + 584.0000000000001, + 385, + 598.6000000000001 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 224, + 598.6000000000001, + 385, + 613.2000000000002 + ], + "spans": [], + 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\\leqslant \\epsilon \\operatorname* { m a x } _ { i } \\frac { \\lambda _ { i } ^ { B } + \\lambda _ { i } ^ { A } } { \\sqrt { \\lambda _ { i } ^ { B } \\lambda _ { i } ^ { A } } } = 2 \\epsilon \\operatorname* { m a x } _ { i } \\frac { ( \\lambda _ { i } ^ { B } + \\lambda _ { i } ^ { A } ) / 2 } { \\sqrt { \\lambda _ { i } ^ { B } \\lambda _ { i } ^ { A } } } \\leqslant 2 \\epsilon \\frac { \\prod _ { i } ( \\lambda _ { i } ^ { B } + \\lambda _ { i } ^ { A } ) / 2 } { \\prod _ { i } \\sqrt { \\lambda _ { i } ^ { B } \\lambda _ { i } ^ { A } } } = 2 \\epsilon \\frac { \\operatorname* { d e t } ( H _ { A + B } ) } { \\operatorname* { d e t } ( H _ { A \\wedge B } ) } ,", + "type": "interline_equation", + "image_path": "948472fd43398d65d233b985eedf5e69d21a6131c16dd6d8beffc2221761a3ef.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 112, + 667, + 498, + 677.3333333333334 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 112, + 677.3333333333334, + 498, + 687.6666666666667 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 112, + 687.6666666666667, + 498, + 698.0000000000001 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "where the first inequality comes from the monotonicity of the square root function, and the second", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "inequality comes from the fact that (i) the geometric mean is always smaller or equal than theś", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 719, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 322, + 735 + ], + "score": 1.0, + "content": "arithmetic mean and (ii) for any sequence of numbers", + "type": "text" + }, + { + "bbox": [ + 322, + 721, + 351, + 732 + ], + "score": 0.86, + "content": "\\alpha _ { i } > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 719, + 356, + 735 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 356, + 720, + 428, + 733 + ], + "score": 0.73, + "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { i } \\dot { \\alpha _ { i } } \\leqslant \\prod _ { i } \\alpha _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 719, + 432, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 493, + 720, + 506, + 732 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 698, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 94 + ], + "score": 1.0, + "content": "Remark 2 (Sanity check). There are two important cases where we can test the bound above. First, if", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 151, + 105 + ], + "score": 0.92, + "content": "H _ { A } = H _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 93, + 173, + 105 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 174, + 93, + 221, + 105 + ], + "score": 0.92, + "content": "T ^ { \\epsilon } ( \\theta ^ { * } ) = \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 93, + 312, + 105 + ], + "score": 1.0, + "content": ", and the bound returns", + "type": "text" + }, + { + "bbox": [ + 312, + 93, + 363, + 105 + ], + "score": 0.92, + "content": "T ^ { \\epsilon } ( \\theta ^ { * } ) \\leqslant 2 \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 93, + 505, + 105 + ], + "score": 1.0, + "content": ", since the geometric and arithmetic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 225, + 117 + ], + "score": 1.0, + "content": "mean are the same. Next, say", + "type": "text" + }, + { + "bbox": [ + 226, + 104, + 257, + 117 + ], + "score": 0.93, + "content": "\\lambda _ { i } ^ { A } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 103, + 273, + 117 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 274, + 104, + 306, + 117 + ], + "score": 0.93, + "content": "\\lambda _ { i } ^ { B } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "; then, both the bound and the inconsistency score", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 264, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 121, + 128 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 122, + 117, + 131, + 125 + ], + "score": 0.78, + "content": "\\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 115, + 264, + 128 + ], + "score": 1.0, + "content": "(highest possible inconsistency).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 140, + 245, + 151 + ], + "lines": [ + { + "bbox": [ + 106, + 140, + 246, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 246, + 153 + ], + "score": 1.0, + "content": "A.3 PROOF OF PROPOSITION 1", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 506, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "score": 1.0, + "content": "In this appendix section we consider the AND-masked GD algorithm, introduced at the end of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 504, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 343, + 184 + ], + "score": 1.0, + "content": "Section 2. We recall that the masked gradients at iteration", + "type": "text" + }, + { + "bbox": [ + 343, + 173, + 349, + 182 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 171, + 366, + 184 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 366, + 172, + 442, + 184 + ], + "score": 0.91, + "content": "\\mathsf { \\bar { \\rho } } m _ { t } ( \\theta ^ { k } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 171, + 473, + 184 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 473, + 172, + 504, + 184 + ], + "score": 0.92, + "content": "m _ { t } ( \\theta ^ { k } )", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 182, + 324, + 196 + ], + "score": 1.0, + "content": "vanishes for any component where there are less than", + "type": "text" + }, + { + "bbox": [ + 325, + 183, + 407, + 195 + ], + "score": 0.92, + "content": "t \\in \\{ d / 2 + 1 , \\ldots , d \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 182, + 506, + 196 + ], + "score": 1.0, + "content": "agreeing gradient signs", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 471, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 471, + 207 + ], + "score": 1.0, + "content": "across environments, and is equal to one otherwise. In a full-batch setting, the algorithm is", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 210, + 337, + 226 + ], + "lines": [ + { + "bbox": [ + 194, + 210, + 337, + 226 + ], + "spans": [ + { + "bbox": [ + 194, + 210, + 337, + 226 + ], + "score": 0.93, + "content": "\\theta ^ { k + 1 } = \\theta ^ { k } - \\eta m _ { t } ( \\theta ^ { k } ) \\odot \\nabla { \\mathcal { L } } ( \\theta ^ { k } ) ,", + "type": "interline_equation", + "image_path": "367147ae234bf3fb44c2105fa911c059b7d9359111400bf47e02ee1e36df1124.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 194, + 210, + 337, + 226 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 423, + 213, + 504, + 225 + ], + "lines": [ + { + "bbox": [ + 423, + 212, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 423, + 212, + 505, + 227 + ], + "score": 1.0, + "content": "(AND-masked GD)", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 231, + 237, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 230, + 238, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 133, + 244 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 232, + 158, + 243 + ], + "score": 0.91, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 230, + 238, + 244 + ], + "score": 1.0, + "content": "is the learning rate.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 246, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 504, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 186, + 259 + ], + "score": 1.0, + "content": "Proposition 1. Let", + "type": "text" + }, + { + "bbox": [ + 186, + 247, + 195, + 256 + ], + "score": 0.77, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 245, + 218, + 259 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 218, + 246, + 226, + 256 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 245, + 428, + 259 + ], + "score": 1.0, + "content": "-Lipschitz gradients and consider a learning rate", + "type": "text" + }, + { + "bbox": [ + 428, + 246, + 466, + 257 + ], + "score": 0.9, + "content": "\\eta \\leqslant 1 / L", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 245, + 497, + 259 + ], + "score": 1.0, + "content": ". After", + "type": "text" + }, + { + "bbox": [ + 497, + 247, + 504, + 256 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 489, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 328, + 271 + ], + "score": 1.0, + "content": "iterations, AND-masked GD visits at least once a point", + "type": "text" + }, + { + "bbox": [ + 329, + 258, + 335, + 267 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 255, + 363, + 271 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 363, + 257, + 485, + 270 + ], + "score": 0.91, + "content": "\\| m _ { t } ( \\theta ) \\odot \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } \\leqslant \\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 255, + 489, + 271 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 102, + 281, + 505, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 265, + 295 + ], + "score": 1.0, + "content": "Proof. 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Therefore,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 297, + 407 + ], + "score": 0.9, + "content": "\\mathcal { L } ( \\theta ^ { i + 1 } ) \\leqslant \\mathcal { L } ( \\theta ^ { i } ) - ( \\eta / 2 ) \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { i } ) \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 393, + 328, + 408 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 328, + 395, + 352, + 406 + ], + "score": 0.89, + "content": "i \\geqslant 0", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 393, + 419, + 408 + ], + "score": 1.0, + "content": ". 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We", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 581, + 504, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 504, + 593 + ], + "score": 1.0, + "content": "want to show that, in this random setting, the AND-mask introduced in Section 2.3 decreases the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 591, + 233, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 233, + 605 + ], + "score": 1.0, + "content": "magnitude of the gradient step.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 504, + 634 + ], + "lines": [ + { + "bbox": [ + 102, + 602, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 102, + 602, + 333, + 628 + ], + "score": 1.0, + "content": "Proposition 2. Consider the setting we just outlined, with", + "type": "text" + }, + { + "bbox": [ + 333, + 606, + 414, + 621 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { L } = ( 1 / d ) \\sum _ { e = 1 } ^ { d } \\mathcal { L } _ { e } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 602, + 444, + 628 + ], + "score": 1.0, + "content": ". While", + "type": "text" + }, + { + "bbox": [ + 445, + 608, + 506, + 621 + ], + "score": 0.9, + "content": "\\mathbb { E } \\| \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 619, + 503, + 635 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 138, + 633 + ], + "score": 0.91, + "content": "\\mathcal { O } ( n / d )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 619, + 194, + 635 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + }, + { + "bbox": [ + 195, + 621, + 329, + 633 + ], + "score": 0.8, + "content": "\\forall t \\in \\{ d / 2 + 1 , \\ldots , d \\} , \\exists c \\in ( 1 , 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 619, + 369, + 635 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 369, + 621, + 503, + 633 + ], + "score": 0.92, + "content": "\\mathbb { E } \\| m _ { t } ( \\theta ) \\odot \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } \\leqslant \\mathcal { O } ( n / c ^ { d } )", + "type": "inline_equation" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 504, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 245, + 658 + ], + "score": 1.0, + "content": "Proof. Let us drop the argument", + "type": "text" + }, + { + "bbox": [ + 246, + 646, + 252, + 655 + ], + "score": 0.83, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 644, + 433, + 658 + ], + "score": 1.0, + "content": "for ease of notation. First, let us consider", + "type": "text" + }, + { + "bbox": [ + 433, + 646, + 450, + 656 + ], + "score": 0.83, + "content": "\\nabla \\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "(no gradient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 654, + 161, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 161, + 669 + ], + "score": 1.0, + "content": "AND-mask):", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 666, + 399, + 702 + ], + "lines": [ + { + "bbox": [ + 211, + 666, + 399, + 702 + ], + "spans": [ + { + "bbox": [ + 211, + 666, + 399, + 702 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left\\| \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla { \\mathcal { L } } _ { e _ { i } } \\right\\| ^ { 2 } = \\frac { 1 } { d ^ { 2 } } \\sum _ { i = 1 } ^ { d } \\mathbb { E } \\| \\nabla { \\mathcal { L } } _ { e _ { i } } \\| ^ { 2 } = \\frac { n \\sigma ^ { 2 } } { d } ,", + "type": "interline_equation", + "image_path": "685b3ca0815f9640febde7c769a312b0f4cc724fbc78ed839cffff1164e632d1.jpg" + } + ] + } + ], + "index": 38.5, + "virtual_lines": [ + { + "bbox": [ + 211, + 666, + 399, + 684.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 211, + 684.0, + 399, + 702.0 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 309, + 722 + ], + "score": 1.0, + "content": "where in the first equality we used the fact that the", + "type": "text" + }, + { + "bbox": [ + 309, + 710, + 333, + 721 + ], + "score": 0.91, + "content": "\\nabla { \\mathcal { L } } _ { e _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 708, + 506, + 722 + ], + "score": 1.0, + "content": "are uncorrelated and in the second the fact", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 718, + 330, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 124, + 735 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 720, + 174, + 732 + ], + "score": 0.93, + "content": "\\mathbb { E } [ \\| \\nabla { \\mathcal { L } } _ { e _ { i } } \\| ^ { 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 718, + 303, + 735 + ], + "score": 1.0, + "content": "is the trace of the covariance of", + "type": "text" + }, + { + "bbox": [ + 303, + 722, + 326, + 732 + ], + "score": 0.89, + "content": "\\nabla { \\mathcal { L } } _ { e _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 718, + 330, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 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": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "16", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 513, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 495, + 514, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 495, + 514, + 505, + 524 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 94 + ], + "score": 1.0, + "content": "Remark 2 (Sanity check). There are two important cases where we can test the bound above. First, if", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 151, + 105 + ], + "score": 0.92, + "content": "H _ { A } = H _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 93, + 173, + 105 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 174, + 93, + 221, + 105 + ], + "score": 0.92, + "content": "T ^ { \\epsilon } ( \\theta ^ { * } ) = \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 93, + 312, + 105 + ], + "score": 1.0, + "content": ", and the bound returns", + "type": "text" + }, + { + "bbox": [ + 312, + 93, + 363, + 105 + ], + "score": 0.92, + "content": "T ^ { \\epsilon } ( \\theta ^ { * } ) \\leqslant 2 \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 93, + 505, + 105 + ], + "score": 1.0, + "content": ", since the geometric and arithmetic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 225, + 117 + ], + "score": 1.0, + "content": "mean are the same. Next, say", + "type": "text" + }, + { + "bbox": [ + 226, + 104, + 257, + 117 + ], + "score": 0.93, + "content": "\\lambda _ { i } ^ { A } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 103, + 273, + 117 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 274, + 104, + 306, + 117 + ], + "score": 0.93, + "content": "\\lambda _ { i } ^ { B } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 103, + 506, + 117 + ], + "score": 1.0, + "content": "; then, both the bound and the inconsistency score", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 264, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 121, + 128 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 122, + 117, + 131, + 125 + ], + "score": 0.78, + "content": "\\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 115, + 264, + 128 + ], + "score": 1.0, + "content": "(highest possible inconsistency).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 128 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 140, + 245, + 151 + ], + "lines": [ + { + "bbox": [ + 106, + 140, + 246, + 153 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 246, + 153 + ], + "score": 1.0, + "content": "A.3 PROOF OF PROPOSITION 1", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 506, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 172 + ], + "score": 1.0, + "content": "In this appendix section we consider the AND-masked GD algorithm, introduced at the end of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 171, + 504, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 343, + 184 + ], + "score": 1.0, + "content": "Section 2. We recall that the masked gradients at iteration", + "type": "text" + }, + { + "bbox": [ + 343, + 173, + 349, + 182 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 171, + 366, + 184 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 366, + 172, + 442, + 184 + ], + "score": 0.91, + "content": "\\mathsf { \\bar { \\rho } } m _ { t } ( \\theta ^ { k } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 171, + 473, + 184 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 473, + 172, + 504, + 184 + ], + "score": 0.92, + "content": "m _ { t } ( \\theta ^ { k } )", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 182, + 324, + 196 + ], + "score": 1.0, + "content": "vanishes for any component where there are less than", + "type": "text" + }, + { + "bbox": [ + 325, + 183, + 407, + 195 + ], + "score": 0.92, + "content": "t \\in \\{ d / 2 + 1 , \\ldots , d \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 182, + 506, + 196 + ], + "score": 1.0, + "content": "agreeing gradient signs", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 471, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 471, + 207 + ], + "score": 1.0, + "content": "across environments, and is equal to one otherwise. In a full-batch setting, the algorithm is", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 160, + 506, + 207 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 210, + 337, + 226 + ], + "lines": [ + { + "bbox": [ + 194, + 210, + 337, + 226 + ], + "spans": [ + { + "bbox": [ + 194, + 210, + 337, + 226 + ], + "score": 0.93, + "content": "\\theta ^ { k + 1 } = \\theta ^ { k } - \\eta m _ { t } ( \\theta ^ { k } ) \\odot \\nabla { \\mathcal { L } } ( \\theta ^ { k } ) ,", + "type": "interline_equation", + "image_path": "367147ae234bf3fb44c2105fa911c059b7d9359111400bf47e02ee1e36df1124.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 194, + 210, + 337, + 226 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 423, + 213, + 504, + 225 + ], + "lines": [ + { + "bbox": [ + 423, + 212, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 423, + 212, + 505, + 227 + ], + "score": 1.0, + "content": "(AND-masked GD)", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 423, + 212, + 505, + 227 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 231, + 237, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 230, + 238, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 133, + 244 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 232, + 158, + 243 + ], + "score": 0.91, + "content": "\\eta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 230, + 238, + 244 + ], + "score": 1.0, + "content": "is the learning rate.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 230, + 238, + 244 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 246, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 504, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 186, + 259 + ], + "score": 1.0, + "content": "Proposition 1. Let", + "type": "text" + }, + { + "bbox": [ + 186, + 247, + 195, + 256 + ], + "score": 0.77, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 245, + 218, + 259 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 218, + 246, + 226, + 256 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 245, + 428, + 259 + ], + "score": 1.0, + "content": "-Lipschitz gradients and consider a learning rate", + "type": "text" + }, + { + "bbox": [ + 428, + 246, + 466, + 257 + ], + "score": 0.9, + "content": "\\eta \\leqslant 1 / L", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 245, + 497, + 259 + ], + "score": 1.0, + "content": ". After", + "type": "text" + }, + { + "bbox": [ + 497, + 247, + 504, + 256 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 255, + 489, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 328, + 271 + ], + "score": 1.0, + "content": "iterations, AND-masked GD visits at least once a point", + "type": "text" + }, + { + "bbox": [ + 329, + 258, + 335, + 267 + ], + "score": 0.8, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 255, + 363, + 271 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 363, + 257, + 485, + 270 + ], + "score": 0.91, + "content": "\\| m _ { t } ( \\theta ) \\odot \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } \\leqslant \\mathcal { O } ( 1 / k )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 255, + 489, + 271 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 245, + 504, + 271 + ] + }, + { + "type": "text", + "bbox": [ + 102, + 281, + 505, + 304 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 265, + 295 + ], + "score": 1.0, + "content": "Proof. Thanks to the component-wise", + "type": "text" + }, + { + "bbox": [ + 265, + 282, + 273, + 291 + ], + "score": 0.81, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 280, + 479, + 295 + ], + "score": 1.0, + "content": "-smoothness and using a Taylor expansion around", + "type": "text" + }, + { + "bbox": [ + 480, + 281, + 489, + 291 + ], + "score": 0.87, + "content": "\\theta ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 280, + 506, + 295 + ], + "score": 1.0, + "content": "we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 291, + 129, + 306 + ], + "spans": [ + { + "bbox": [ + 104, + 291, + 129, + 306 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 280, + 506, + 306 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 307, + 466, + 362 + ], + "lines": [ + { + "bbox": [ + 144, + 307, + 466, + 362 + ], + "spans": [ + { + "bbox": [ + 144, + 307, + 466, + 362 + ], + "score": 0.93, + "content": "\\begin{array} { l } { \\displaystyle \\mathcal { L } ( \\theta ^ { i + 1 } ) \\leqslant \\mathcal { L } ( \\theta ^ { i } ) - \\eta \\langle \\nabla \\mathcal { L } ( \\theta ^ { i } ) , m _ { t } ( \\theta ^ { i } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { i } ) \\rangle + \\displaystyle \\frac { L \\eta ^ { 2 } } { 2 } \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { i } ) \\| ^ { 2 } } \\\\ { \\displaystyle \\qquad = \\mathcal { L } ( \\theta ^ { i } ) - \\left( \\eta - \\frac { L \\eta ^ { 2 } } { 2 } \\right) \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { i } ) \\| ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "d29dd6557fac18231ce1c10b2d58e5259914511b22f4f71f2cb47d435fd3fc25.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 144, + 307, + 466, + 325.3333333333333 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 144, + 325.3333333333333, + 466, + 343.66666666666663 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 144, + 343.66666666666663, + 466, + 361.99999999999994 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 507, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 151, + 396 + ], + "score": 1.0, + "content": "If we seek", + "type": "text" + }, + { + "bbox": [ + 151, + 381, + 223, + 394 + ], + "score": 0.93, + "content": "\\eta - L \\eta ^ { 2 } / 2 \\geqslant \\eta / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 381, + 247, + 396 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 247, + 381, + 275, + 395 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\eta \\leqslant \\frac { 1 } { L } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 381, + 507, + 396 + ], + "score": 1.0, + "content": ", as we assumed in the proposition statement. Therefore,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 297, + 407 + ], + "score": 0.9, + "content": "\\mathcal { L } ( \\theta ^ { i + 1 } ) \\leqslant \\mathcal { L } ( \\theta ^ { i } ) - ( \\eta / 2 ) \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { i } ) \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 393, + 328, + 408 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 328, + 395, + 352, + 406 + ], + "score": 0.89, + "content": "i \\geqslant 0", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 393, + 419, + 408 + ], + "score": 1.0, + "content": ". Summing over", + "type": "text" + }, + { + "bbox": [ + 419, + 396, + 424, + 405 + ], + "score": 0.72, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "from 0 to a desired", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 405, + 182, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 142, + 419 + ], + "score": 1.0, + "content": "iteration", + "type": "text" + }, + { + "bbox": [ + 142, + 407, + 149, + 416 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 405, + 182, + 419 + ], + "score": 1.0, + "content": ", we get", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 381, + 507, + 419 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 423, + 425, + 456 + ], + "lines": [ + { + "bbox": [ + 186, + 423, + 425, + 456 + ], + "spans": [ + { + "bbox": [ + 186, + 423, + 425, + 456 + ], + "score": 0.92, + "content": "\\sum _ { i = 0 } ^ { k - 1 } ( \\eta / 2 ) \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla \\mathcal { L } ( \\theta ^ { i } ) \\| ^ { 2 } \\leqslant \\mathcal { L } ( \\theta ^ { 0 } ) - \\mathcal { L } ( \\theta ^ { k } ) \\leqslant \\mathcal { L } ( \\theta ^ { 0 } ) .", + "type": "interline_equation", + "image_path": "e864e9db0e9c856ab8b1d90a30efaf655e30e059a297104548b83e9c853e1c21.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 186, + 423, + 425, + 439.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 186, + 439.5, + 425, + 456.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 461, + 149, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 460, + 151, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 151, + 474 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 460, + 151, + 474 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 471, + 460, + 505 + ], + "lines": [ + { + "bbox": [ + 150, + 471, + 460, + 505 + ], + "spans": [ + { + "bbox": [ + 150, + 471, + 460, + 505 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { i = 0 , \\dots , k } \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla { \\mathcal { L } } ( \\theta ^ { i } ) \\| ^ { 2 } \\leqslant \\frac { 1 } { k } \\sum _ { i = 0 } ^ { k - 1 } ( \\eta / 2 ) \\| m _ { t } ( \\theta ^ { i } ) \\odot \\nabla { \\mathcal { L } } ( \\theta ^ { i } ) \\| ^ { 2 } \\leqslant \\frac { 2 { \\mathcal { L } } ( \\theta ^ { 0 } ) } { \\eta k } .", + "type": "interline_equation", + "image_path": "4274fdbefe7f256ba4c3acb6c48a69ee80628b61dd3d9512b3330c93c263db3a.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 150, + 471, + 460, + 482.3333333333333 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 150, + 482.3333333333333, + 460, + 493.66666666666663 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 150, + 493.66666666666663, + 460, + 504.99999999999994 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 511, + 472, + 525 + ], + "lines": [ + { + "bbox": [ + 104, + 509, + 470, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 228, + 528 + ], + "score": 1.0, + "content": "Hence, there exist an iteration", + "type": "text" + }, + { + "bbox": [ + 228, + 513, + 290, + 525 + ], + "score": 0.93, + "content": "i ^ { * } \\in \\{ 0 , \\ldots , k \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 509, + 330, + 528 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 330, + 511, + 470, + 525 + ], + "score": 0.93, + "content": "\\| m _ { t } ( { \\theta ^ { i } } ^ { * } ) \\odot \\nabla { \\mathcal { L } } ( { \\theta ^ { i } } ^ { * } ) \\| ^ { 2 } \\leqslant { \\mathcal { O } } ( 1 / k )", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 104, + 509, + 470, + 528 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 537, + 246, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 537, + 246, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 246, + 550 + ], + "score": 1.0, + "content": "A.4 PROOF OF PROPOSITION 2", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 205, + 572 + ], + "score": 1.0, + "content": "Here we fix parameters", + "type": "text" + }, + { + "bbox": [ + 206, + 559, + 237, + 569 + ], + "score": 0.91, + "content": "\\boldsymbol \\theta \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 558, + 330, + 572 + ], + "score": 1.0, + "content": "and assume gradients", + "type": "text" + }, + { + "bbox": [ + 331, + 559, + 389, + 571 + ], + "score": 0.93, + "content": "\\nabla \\mathcal { L } _ { e } ( \\theta ) \\in \\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 558, + 505, + 572 + ], + "score": 1.0, + "content": "coming from environments", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 107, + 570, + 129, + 580 + ], + "score": 0.88, + "content": "e \\in \\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 570, + 424, + 581 + ], + "score": 1.0, + "content": "are drawn independently from a multivariate Gaussian with zero mean and", + "type": "text" + }, + { + "bbox": [ + 424, + 569, + 442, + 580 + ], + "score": 0.89, + "content": "\\sigma ^ { 2 } I", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 570, + 505, + 581 + ], + "score": 1.0, + "content": "covariance. We", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 581, + 504, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 504, + 593 + ], + "score": 1.0, + "content": "want to show that, in this random setting, the AND-mask introduced in Section 2.3 decreases the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 591, + 233, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 233, + 605 + ], + "score": 1.0, + "content": "magnitude of the gradient step.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 558, + 505, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 504, + 634 + ], + "lines": [ + { + "bbox": [ + 102, + 602, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 102, + 602, + 333, + 628 + ], + "score": 1.0, + "content": "Proposition 2. Consider the setting we just outlined, with", + "type": "text" + }, + { + "bbox": [ + 333, + 606, + 414, + 621 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { L } = ( 1 / d ) \\sum _ { e = 1 } ^ { d } \\mathcal { L } _ { e } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 602, + 444, + 628 + ], + "score": 1.0, + "content": ". While", + "type": "text" + }, + { + "bbox": [ + 445, + 608, + 506, + 621 + ], + "score": 0.9, + "content": "\\mathbb { E } \\| \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 619, + 503, + 635 + ], + "spans": [ + { + "bbox": [ + 107, + 621, + 138, + 633 + ], + "score": 0.91, + "content": "\\mathcal { O } ( n / d )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 619, + 194, + 635 + ], + "score": 1.0, + "content": ", we have that", + "type": "text" + }, + { + "bbox": [ + 195, + 621, + 329, + 633 + ], + "score": 0.8, + "content": "\\forall t \\in \\{ d / 2 + 1 , \\ldots , d \\} , \\exists c \\in ( 1 , 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 619, + 369, + 635 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 369, + 621, + 503, + 633 + ], + "score": 0.92, + "content": "\\mathbb { E } \\| m _ { t } ( \\theta ) \\odot \\nabla \\mathcal { L } ( \\theta ) \\| ^ { 2 } \\leqslant \\mathcal { O } ( n / c ^ { d } )", + "type": "inline_equation" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 102, + 602, + 506, + 635 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 504, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 245, + 658 + ], + "score": 1.0, + "content": "Proof. Let us drop the argument", + "type": "text" + }, + { + "bbox": [ + 246, + 646, + 252, + 655 + ], + "score": 0.83, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 644, + 433, + 658 + ], + "score": 1.0, + "content": "for ease of notation. 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The difficulty lies in the fact that the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 130, + 285 + ], + "score": 1.0, + "content": "event", + "type": "text" + }, + { + "bbox": [ + 131, + 273, + 162, + 284 + ], + "score": 0.9, + "content": "m _ { t } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 272, + 455, + 285 + ], + "score": 1.0, + "content": "makes gradients conditionally dependent. 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To", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "solve the issue, we our strategy is to reduce the discussion (without loss in generality and with no", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "score": 1.0, + "content": "additional assumption) to the case where gradient entries have all the same sign and hence conditional", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 328, + 209, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 209, + 340 + ], + "score": 1.0, + "content": "independence is restored.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 392, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 391, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 391, + 358 + ], + "score": 1.0, + "content": "We consider the following writing for the quantity we are interested in:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 361, + 491, + 522 + ], + "lines": [ + { + "bbox": [ + 119, + 361, + 491, + 522 + ], + "spans": [ + { + "bbox": [ + 119, + 361, + 491, + 522 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { \\mathbb { E } \\| m _ { t } \\operatorname { \\wp } ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ) \\| ^ { 2 } = \\displaystyle \\sum _ { j = 1 } ^ { n } \\mathbb { E } [ [ m _ { \\epsilon } ] _ { j } ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } ] } & { } \\\\ { = \\displaystyle \\sum _ { j = 1 } ^ { n } \\sum _ { \\tilde { p } _ { \\tilde { \\imath } } = 0 } ^ { d } \\mathbb { E } [ [ m _ { \\epsilon } ] _ { j } ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \\hat { p } _ { j } ] \\mathbb { P } \\mathrm { r } [ p _ { j } = \\hat { p } _ { j } ] } \\\\ { = \\displaystyle \\sum _ { j = 1 } ^ { n } \\sum _ { \\tilde { p } _ { \\tilde { \\jmath } } = 0 } ^ { d - 1 } \\sum _ { \\tilde { p } = 0 } ^ { d } \\mathbb { E } [ ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \\hat { p } _ { j } ] \\mathbb { P } \\mathrm { r } [ p _ { j } = \\hat { p } _ { j } ] } \\\\ { = \\displaystyle 2 \\sum _ { j = 1 } ^ { n } \\sum _ { \\tilde { p } _ { \\tilde { \\jmath } } = - 1 } ^ { d } \\mathbb { E } [ ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \\hat { p } _ { j } ] ( \\displaystyle \\frac { 1 } { 2 } ) ^ { d } ( \\displaystyle \\frac { d } { \\hat { p } _ { j } } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "9dea7cbe92ff8a2537b0350e26a9cfdaf3a05d8c7a457ed3db9de066bc42fc0c.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 119, + 361, + 491, + 414.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 119, + 414.6666666666667, + 491, + 468.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 119, + 468.33333333333337, + 491, + 522.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "where we used the definition of 2-norm, the law of total expectation, and the symmetry of the problem", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 550, + 504, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 504, + 562 + ], + "score": 1.0, + "content": "with respect to positive and negative numbers. Finally, since the gradient components within the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 561, + 450, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 336, + 574 + ], + "score": 1.0, + "content": "same environment are conditionally independent, for any", + "type": "text" + }, + { + "bbox": [ + 336, + 561, + 395, + 573 + ], + "score": 0.93, + "content": "j \\in \\{ 1 , \\ldots , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 561, + 450, + 574 + ], + "score": 1.0, + "content": "we can write", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 577, + 479, + 618 + ], + "lines": [ + { + "bbox": [ + 131, + 577, + 479, + 618 + ], + "spans": [ + { + "bbox": [ + 131, + 577, + 479, + 618 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left. m _ { t } \\odot \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla \\mathcal { L } _ { e _ { i } } \\right) \\right. ^ { 2 } = 2 n \\sum _ { \\hat { p } _ { j } = t } ^ { d } \\mathbb { E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\bigg | p _ { j } = \\hat { p } _ { j } \\right] \\left( \\frac { 1 } { 2 } \\right) ^ { d } { \\binom { d } { \\hat { p } _ { j } } } .", + "type": "interline_equation", + "image_path": "a6737b8880753d8398d087807804dfa99ba376c0ada5082513235d64fd61aeb6.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 131, + 577, + 479, + 590.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 131, + 590.6666666666666, + 479, + 604.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 131, + 604.3333333333333, + 479, + 617.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 299, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 299, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 299, + 640 + ], + "score": 1.0, + "content": "Finally, we note that the following bound holds:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 643, + 458, + 684 + ], + "lines": [ + { + "bbox": [ + 153, + 643, + 458, + 684 + ], + "spans": [ + { + "bbox": [ + 153, + 643, + 458, + 684 + ], + "score": 0.94, + "content": "\\mathbb { E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\middle | p _ { j } = \\hat { p } _ { j } \\leqslant d \\right] \\leqslant \\mathbb { E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\middle | p _ { j } = d \\right] .", + "type": "interline_equation", + "image_path": "2a0c3376533f95b021b180970ae8320bdde179fa117addf7c8df5d8ab5ec195c.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 153, + 643, + 458, + 656.6666666666666 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 153, + 656.6666666666666, + 458, + 670.3333333333333 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 153, + 670.3333333333333, + 458, + 683.9999999999999 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "Indeed, if all environments lead to positive (or, symmetrically, negative) and non-interacting gradients", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 131, + 712 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 132, + 699, + 137, + 711 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "-th direction, the average will be the biggest in norm. 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The difficulty lies in the fact that the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 130, + 285 + ], + "score": 1.0, + "content": "event", + "type": "text" + }, + { + "bbox": [ + 131, + 273, + 162, + 284 + ], + "score": 0.9, + "content": "m _ { t } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 272, + 455, + 285 + ], + "score": 1.0, + "content": "makes gradients conditionally dependent. 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To", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 318 + ], + "score": 1.0, + "content": "solve the issue, we our strategy is to reduce the discussion (without loss in generality and with no", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 330 + ], + "score": 1.0, + "content": "additional assumption) to the case where gradient entries have all the same sign and hence conditional", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 328, + 209, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 209, + 340 + ], + "score": 1.0, + "content": "independence is restored.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15, + "bbox_fs": [ + 102, + 251, + 509, + 340 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 392, + 357 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 391, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 391, + 358 + ], + "score": 1.0, + "content": "We consider the following writing for the quantity we are interested in:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 343, + 391, + 358 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 361, + 491, + 522 + ], + "lines": [ + { + "bbox": [ + 119, + 361, + 491, + 522 + ], + "spans": [ + { + "bbox": [ + 119, + 361, + 491, + 522 + ], + "score": 0.95, + "content": "\\begin{array} { r l } { \\mathbb { E } \\| m _ { t } \\operatorname { \\wp } ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ) \\| ^ { 2 } = \\displaystyle \\sum _ { j = 1 } ^ { n } \\mathbb { E } [ [ m _ { \\epsilon } ] _ { j } ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } ] } & { } \\\\ { = \\displaystyle \\sum _ { j = 1 } ^ { n } \\sum _ { \\tilde { p } _ { \\tilde { \\imath } } = 0 } ^ { d } \\mathbb { E } [ [ m _ { \\epsilon } ] _ { j } ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \\hat { p } _ { j } ] \\mathbb { P } \\mathrm { r } [ p _ { j } = \\hat { p } _ { j } ] } \\\\ { = \\displaystyle \\sum _ { j = 1 } ^ { n } \\sum _ { \\tilde { p } _ { \\tilde { \\jmath } } = 0 } ^ { d - 1 } \\sum _ { \\tilde { p } = 0 } ^ { d } \\mathbb { E } [ ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \\hat { p } _ { j } ] \\mathbb { P } \\mathrm { r } [ p _ { j } = \\hat { p } _ { j } ] } \\\\ { = \\displaystyle 2 \\sum _ { j = 1 } ^ { n } \\sum _ { \\tilde { p } _ { \\tilde { \\jmath } } = - 1 } ^ { d } \\mathbb { E } [ ( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla { \\mathcal { L } } _ { \\epsilon _ { \\epsilon } } ] _ { i } ) ^ { 2 } | p _ { j } = \\hat { p } _ { j } ] ( \\displaystyle \\frac { 1 } { 2 } ) ^ { d } ( \\displaystyle \\frac { d } { \\hat { p } _ { j } } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "9dea7cbe92ff8a2537b0350e26a9cfdaf3a05d8c7a457ed3db9de066bc42fc0c.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 119, + 361, + 491, + 414.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 119, + 414.6666666666667, + 491, + 468.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 119, + 468.33333333333337, + 491, + 522.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "where we used the definition of 2-norm, the law of total expectation, and the symmetry of the problem", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 550, + 504, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 504, + 562 + ], + "score": 1.0, + "content": "with respect to positive and negative numbers. Finally, since the gradient components within the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 561, + 450, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 336, + 574 + ], + "score": 1.0, + "content": "same environment are conditionally independent, for any", + "type": "text" + }, + { + "bbox": [ + 336, + 561, + 395, + 573 + ], + "score": 0.93, + "content": "j \\in \\{ 1 , \\ldots , n \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 561, + 450, + 574 + ], + "score": 1.0, + "content": "we can write", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 538, + 505, + 574 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 577, + 479, + 618 + ], + "lines": [ + { + "bbox": [ + 131, + 577, + 479, + 618 + ], + "spans": [ + { + "bbox": [ + 131, + 577, + 479, + 618 + ], + "score": 0.93, + "content": "\\mathbb { E } \\left. m _ { t } \\odot \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla \\mathcal { L } _ { e _ { i } } \\right) \\right. ^ { 2 } = 2 n \\sum _ { \\hat { p } _ { j } = t } ^ { d } \\mathbb { E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\bigg | p _ { j } = \\hat { p } _ { j } \\right] \\left( \\frac { 1 } { 2 } \\right) ^ { d } { \\binom { d } { \\hat { p } _ { j } } } .", + "type": "interline_equation", + "image_path": "a6737b8880753d8398d087807804dfa99ba376c0ada5082513235d64fd61aeb6.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 131, + 577, + 479, + 590.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 131, + 590.6666666666666, + 479, + 604.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 131, + 604.3333333333333, + 479, + 617.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 299, + 639 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 299, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 299, + 640 + ], + "score": 1.0, + "content": "Finally, we note that the following bound holds:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 627, + 299, + 640 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 643, + 458, + 684 + ], + "lines": [ + { + "bbox": [ + 153, + 643, + 458, + 684 + ], + "spans": [ + { + "bbox": [ + 153, + 643, + 458, + 684 + ], + "score": 0.94, + "content": "\\mathbb { E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\middle | p _ { j } = \\hat { p } _ { j } \\leqslant d \\right] \\leqslant \\mathbb { E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\middle | p _ { j } = d \\right] .", + "type": "interline_equation", + "image_path": "2a0c3376533f95b021b180970ae8320bdde179fa117addf7c8df5d8ab5ec195c.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 153, + 643, + 458, + 656.6666666666666 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 153, + 656.6666666666666, + 458, + 670.3333333333333 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 153, + 670.3333333333333, + 458, + 683.9999999999999 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "Indeed, if all environments lead to positive (or, symmetrically, negative) and non-interacting gradients", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 131, + 712 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 132, + 699, + 137, + 711 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "-th direction, the average will be the biggest in norm. Moreover — crucially — conditioned", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 160, + 722 + ], + "score": 1.0, + "content": "on the event", + "type": "text" + }, + { + "bbox": [ + 160, + 710, + 192, + 722 + ], + "score": 0.91, + "content": "p _ { j } \\ = \\ d", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 711, + 505, + 722 + ], + "score": 1.0, + "content": ", gradients coming from different environments are distributed as a positive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "half-normal distributions. Moreover, they are conditionally independent; this because, since they are", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "all positive, the value of a gradient in one environment cannot influence the value of the gradient", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "in another one. We remark that conditional independence on the right-hand side is therefore not an", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 296, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 296, + 116 + ], + "score": 1.0, + "content": "assumption, but is intrinsic to the upper bound.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "all positive, the value of a gradient in one environment cannot influence the value of the gradient", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "in another one. We remark that conditional independence on the right-hand side is therefore not an", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 296, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 296, + 116 + ], + "score": 1.0, + "content": "assumption, but is intrinsic to the upper bound.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 231, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 231, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 231, + 135 + ], + "score": 1.0, + "content": "Putting it all together, we have", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 138, + 476, + 246 + ], + "lines": [ + { + "bbox": [ + 135, + 138, + 476, + 246 + ], + "spans": [ + { + "bbox": [ + 135, + 138, + 476, + 246 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } { \\mathbb { E } \\left\\| m _ { t } \\odot \\left( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla \\mathcal { L } _ { e _ { i } } \\right) \\right\\| ^ { 2 } \\leqslant 2 n \\displaystyle \\sum _ { \\hat { p } _ { j } = t } ^ { d } \\mathbb { E } \\left[ \\left( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\bigg | p _ { j } = d \\right] \\left( \\displaystyle \\frac { 1 } { 2 } \\right) ^ { d } \\left( \\displaystyle \\frac { d } { \\hat { p } _ { j } } \\right) } & \\\\ & { \\leqslant 2 n \\displaystyle \\sum _ { \\hat { p } _ { j } = t } ^ { d } \\sigma ^ { 2 } \\left( \\displaystyle \\frac { 1 } { 2 } \\right) ^ { d } \\left( \\displaystyle \\frac { d } { \\hat { p } _ { j } } \\right) } & \\\\ & { \\leqslant \\sigma ^ { 2 } n ( d - t ) \\Big ( \\displaystyle \\frac { d } { t } \\Big ) \\left( \\displaystyle \\frac { 1 } { 2 } \\right) ^ { d - 1 } , } & \\end{array}", + "type": "interline_equation", + "image_path": "b0ea8f99191510776f1eb7f6b94844751c31f9516464951e5b50bcf3b7c4682e.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 135, + 138, + 476, + 174.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 135, + 174.0, + 476, + 210.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 135, + 210.0, + 476, + 246.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 506, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "where in the second line we bounded the squared average of a sum of half normal distributions: let", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 258, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 107, + 259, + 141, + 272 + ], + "score": 0.92, + "content": "\\{ X _ { i } \\} _ { i = 1 } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 258, + 506, + 274 + ], + "score": 1.0, + "content": "be a family of uncorrelated positive half-normal distributions derived from a Gaussians witha", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 205, + 285 + ], + "score": 1.0, + "content": "mean zero and variance", + "type": "text" + }, + { + "bbox": [ + 205, + 271, + 217, + 282 + ], + "score": 0.86, + "content": "\\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 271, + 282, + 285 + ], + "score": 1.0, + "content": ", we have12 that", + "type": "text" + }, + { + "bbox": [ + 283, + 271, + 354, + 285 + ], + "score": 0.93, + "content": "\\mathbb { E } [ X _ { i } ] = \\sigma { \\sqrt { 2 / \\pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 271, + 372, + 285 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 372, + 271, + 425, + 284 + ], + "score": 0.92, + "content": "\\mathbb { E } [ X _ { i } ^ { 2 } ] = \\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 271, + 454, + 285 + ], + "score": 1.0, + "content": ". Also,", + "type": "text" + }, + { + "bbox": [ + 454, + 271, + 505, + 285 + ], + "score": 0.91, + "content": "\\mathbb { E } [ X _ { i } X _ { j } ] =", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 284, + 231, + 297 + ], + "spans": [ + { + "bbox": [ + 107, + 284, + 182, + 297 + ], + "score": 0.93, + "content": "\\mathbb { E } [ X _ { i } ] \\mathbb { E } [ X _ { j } ] \\leqslant \\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 284, + 231, + 297 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 301, + 402, + 342 + ], + "lines": [ + { + "bbox": [ + 208, + 301, + 402, + 342 + ], + "spans": [ + { + "bbox": [ + 208, + 301, + 402, + 342 + ], + "score": 0.94, + "content": "{ \\mathbb E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } X _ { i } \\right) ^ { 2 } \\right] = \\frac { 1 } { d ^ { 2 } } \\sum _ { i , j = 1 } ^ { d } { \\mathbb E } [ X _ { i } X _ { j } ] \\leqslant \\sigma ^ { 2 } .", + "type": "interline_equation", + "image_path": "065e30c1ad78f5231b6e27920abb354e3f10351bec2bc65dc220df415ec86c1e.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 208, + 301, + 402, + 321.5 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 208, + 321.5, + 402, + 342.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 293, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 294, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 175, + 367 + ], + "score": 1.0, + "content": "Finally, if we set", + "type": "text" + }, + { + "bbox": [ + 176, + 353, + 248, + 366 + ], + "score": 0.91, + "content": "r = t / d \\in ( 0 . 5 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 352, + 294, + 367 + ], + "score": 1.0, + "content": ", we have13", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 370, + 360, + 401 + ], + "lines": [ + { + "bbox": [ + 250, + 370, + 360, + 401 + ], + "spans": [ + { + "bbox": [ + 250, + 370, + 360, + 401 + ], + "score": 0.93, + "content": "{ \\binom { d } { t } } \\sim \\left( { \\frac { 1 } { r ^ { r } ( 1 - r ) ^ { 1 - r } } } \\right) ^ { d }", + "type": "interline_equation", + "image_path": "2170d830a87b4de3cf6184af5db587f8b1c174420240916b8a01f4949d1c3472.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 370, + 360, + 385.5 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 250, + 385.5, + 360, + 401.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 506, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 117, + 421 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 408, + 148, + 417 + ], + "score": 0.89, + "content": "d \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 405, + 320, + 421 + ], + "score": 1.0, + "content": "(discarding all polynomial terms). Hence", + "type": "text" + }, + { + "bbox": [ + 320, + 406, + 333, + 421 + ], + "score": 0.88, + "content": "\\binom { d } { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 405, + 392, + 421 + ], + "score": 1.0, + "content": "is of the form", + "type": "text" + }, + { + "bbox": [ + 392, + 407, + 403, + 419 + ], + "score": 0.89, + "content": "q ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 405, + 427, + 421 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 428, + 408, + 470, + 419 + ], + "score": 0.9, + "content": "1 \\leqslant q < 2", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 405, + 506, + 421 + ], + "score": 1.0, + "content": ". So, the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 103, + 417, + 507, + 440 + ], + "spans": [ + { + "bbox": [ + 103, + 417, + 239, + 440 + ], + "score": 1.0, + "content": "quantity σ2npd ´ tq dt 12 d´1 w", + "type": "text" + }, + { + "bbox": [ + 232, + 419, + 396, + 437 + ], + "score": 1.0, + "content": "ill be exponentially decreasing at a rate", + "type": "text" + }, + { + "bbox": [ + 396, + 421, + 456, + 435 + ], + "score": 0.92, + "content": "O ( n / ( 2 - q ) ^ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 419, + 507, + 437 + ], + "score": 1.0, + "content": ". Notably, if", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 139, + 446 + ], + "score": 0.91, + "content": "t = d / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 433, + 339, + 447 + ], + "score": 1.0, + "content": ", then we lose the exponential rate and get back to", + "type": "text" + }, + { + "bbox": [ + 339, + 434, + 371, + 446 + ], + "score": 0.93, + "content": "\\mathcal { O } ( n / d )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 433, + 376, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 434, + 505, + 445 + ], + "score": 0.996, + "content": "□", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 701, + 506, + 731 + ], + "lines": [ + { + "bbox": [ + 115, + 697, + 417, + 714 + ], + "spans": [ + { + "bbox": [ + 115, + 697, + 417, + 714 + ], + "score": 1.0, + "content": "12https://en.wikipedia.org/wiki/Half-normal_distribution", + "type": "text" + } + ] + }, + { + "bbox": [ + 114, + 707, + 508, + 725 + ], + "spans": [ + { + "bbox": [ + 114, + 707, + 508, + 725 + ], + "score": 1.0, + "content": "13Theorem 1 in Buric, Tomislav, and Neven Elezovi ´ c. “Asymptotic expansions of the binomial coefficients.” ´", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 366, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 366, + 733 + ], + "score": 1.0, + "content": "Journal of applied mathematics and computing 46.1-2 (2014): 135-145.", + "type": "text" + } + ] + } + ] + }, + { + "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": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 117 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 116 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 231, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 231, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 231, + 135 + ], + "score": 1.0, + "content": "Putting it all together, we have", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 119, + 231, + 135 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 138, + 476, + 246 + ], + "lines": [ + { + "bbox": [ + 135, + 138, + 476, + 246 + ], + "spans": [ + { + "bbox": [ + 135, + 138, + 476, + 246 + ], + "score": 0.94, + "content": "\\begin{array} { r l r } { \\mathbb { E } \\left\\| m _ { t } \\odot \\left( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } \\nabla \\mathcal { L } _ { e _ { i } } \\right) \\right\\| ^ { 2 } \\leqslant 2 n \\displaystyle \\sum _ { \\hat { p } _ { j } = t } ^ { d } \\mathbb { E } \\left[ \\left( \\displaystyle \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } [ \\nabla \\mathcal { L } _ { e _ { i } } ] _ { j } \\right) ^ { 2 } \\bigg | p _ { j } = d \\right] \\left( \\displaystyle \\frac { 1 } { 2 } \\right) ^ { d } \\left( \\displaystyle \\frac { d } { \\hat { p } _ { j } } \\right) } & \\\\ & { \\leqslant 2 n \\displaystyle \\sum _ { \\hat { p } _ { j } = t } ^ { d } \\sigma ^ { 2 } \\left( \\displaystyle \\frac { 1 } { 2 } \\right) ^ { d } \\left( \\displaystyle \\frac { d } { \\hat { p } _ { j } } \\right) } & \\\\ & { \\leqslant \\sigma ^ { 2 } n ( d - t ) \\Big ( \\displaystyle \\frac { d } { t } \\Big ) \\left( \\displaystyle \\frac { 1 } { 2 } \\right) ^ { d - 1 } , } & \\end{array}", + "type": "interline_equation", + "image_path": "b0ea8f99191510776f1eb7f6b94844751c31f9516464951e5b50bcf3b7c4682e.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 135, + 138, + 476, + 174.0 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 135, + 174.0, + 476, + 210.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 135, + 210.0, + 476, + 246.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 506, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "where in the second line we bounded the squared average of a sum of half normal distributions: let", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 258, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 107, + 259, + 141, + 272 + ], + "score": 0.92, + "content": "\\{ X _ { i } \\} _ { i = 1 } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 258, + 506, + 274 + ], + "score": 1.0, + "content": "be a family of uncorrelated positive half-normal distributions derived from a Gaussians witha", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 104, + 271, + 205, + 285 + ], + "score": 1.0, + "content": "mean zero and variance", + "type": "text" + }, + { + "bbox": [ + 205, + 271, + 217, + 282 + ], + "score": 0.86, + "content": "\\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 271, + 282, + 285 + ], + "score": 1.0, + "content": ", we have12 that", + "type": "text" + }, + { + "bbox": [ + 283, + 271, + 354, + 285 + ], + "score": 0.93, + "content": "\\mathbb { E } [ X _ { i } ] = \\sigma { \\sqrt { 2 / \\pi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 271, + 372, + 285 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 372, + 271, + 425, + 284 + ], + "score": 0.92, + "content": "\\mathbb { E } [ X _ { i } ^ { 2 } ] = \\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 271, + 454, + 285 + ], + "score": 1.0, + "content": ". Also,", + "type": "text" + }, + { + "bbox": [ + 454, + 271, + 505, + 285 + ], + "score": 0.91, + "content": "\\mathbb { E } [ X _ { i } X _ { j } ] =", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 284, + 231, + 297 + ], + "spans": [ + { + "bbox": [ + 107, + 284, + 182, + 297 + ], + "score": 0.93, + "content": "\\mathbb { E } [ X _ { i } ] \\mathbb { E } [ X _ { j } ] \\leqslant \\sigma ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 284, + 231, + 297 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 248, + 506, + 297 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 301, + 402, + 342 + ], + "lines": [ + { + "bbox": [ + 208, + 301, + 402, + 342 + ], + "spans": [ + { + "bbox": [ + 208, + 301, + 402, + 342 + ], + "score": 0.94, + "content": "{ \\mathbb E } \\left[ \\left( \\frac { 1 } { d } \\sum _ { i = 1 } ^ { d } X _ { i } \\right) ^ { 2 } \\right] = \\frac { 1 } { d ^ { 2 } } \\sum _ { i , j = 1 } ^ { d } { \\mathbb E } [ X _ { i } X _ { j } ] \\leqslant \\sigma ^ { 2 } .", + "type": "interline_equation", + "image_path": "065e30c1ad78f5231b6e27920abb354e3f10351bec2bc65dc220df415ec86c1e.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 208, + 301, + 402, + 321.5 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 208, + 321.5, + 402, + 342.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 353, + 293, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 294, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 175, + 367 + ], + "score": 1.0, + "content": "Finally, if we set", + "type": "text" + }, + { + "bbox": [ + 176, + 353, + 248, + 366 + ], + "score": 0.91, + "content": "r = t / d \\in ( 0 . 5 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 352, + 294, + 367 + ], + "score": 1.0, + "content": ", we have13", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 352, + 294, + 367 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 370, + 360, + 401 + ], + "lines": [ + { + "bbox": [ + 250, + 370, + 360, + 401 + ], + "spans": [ + { + "bbox": [ + 250, + 370, + 360, + 401 + ], + "score": 0.93, + "content": "{ \\binom { d } { t } } \\sim \\left( { \\frac { 1 } { r ^ { r } ( 1 - r ) ^ { 1 - r } } } \\right) ^ { d }", + "type": "interline_equation", + "image_path": "2170d830a87b4de3cf6184af5db587f8b1c174420240916b8a01f4949d1c3472.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 370, + 360, + 385.5 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 250, + 385.5, + 360, + 401.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 506, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 117, + 421 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 408, + 148, + 417 + ], + "score": 0.89, + "content": "d \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 405, + 320, + 421 + ], + "score": 1.0, + "content": "(discarding all polynomial terms). Hence", + "type": "text" + }, + { + "bbox": [ + 320, + 406, + 333, + 421 + ], + "score": 0.88, + "content": "\\binom { d } { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 405, + 392, + 421 + ], + "score": 1.0, + "content": "is of the form", + "type": "text" + }, + { + "bbox": [ + 392, + 407, + 403, + 419 + ], + "score": 0.89, + "content": "q ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 405, + 427, + 421 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 428, + 408, + 470, + 419 + ], + "score": 0.9, + "content": "1 \\leqslant q < 2", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 405, + 506, + 421 + ], + "score": 1.0, + "content": ". So, the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 103, + 417, + 507, + 440 + ], + "spans": [ + { + "bbox": [ + 103, + 417, + 239, + 440 + ], + "score": 1.0, + "content": "quantity σ2npd ´ tq dt 12 d´1 w", + "type": "text" + }, + { + "bbox": [ + 232, + 419, + 396, + 437 + ], + "score": 1.0, + "content": "ill be exponentially decreasing at a rate", + "type": "text" + }, + { + "bbox": [ + 396, + 421, + 456, + 435 + ], + "score": 0.92, + "content": "O ( n / ( 2 - q ) ^ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 419, + 507, + 437 + ], + "score": 1.0, + "content": ". Notably, if", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 139, + 446 + ], + "score": 0.91, + "content": "t = d / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 433, + 339, + 447 + ], + "score": 1.0, + "content": ", then we lose the exponential rate and get back to", + "type": "text" + }, + { + "bbox": [ + 339, + 434, + 371, + 446 + ], + "score": 0.93, + "content": "\\mathcal { O } ( n / d )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 433, + 376, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 434, + 505, + 445 + ], + "score": 0.996, + "content": "□", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 103, + 405, + 507, + 447 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 257, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 259, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 259, + 96 + ], + "score": 1.0, + "content": "B APPENDIX TO SECTION 3", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 506, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 119 + ], + "score": 1.0, + "content": "We used Pytorch Paszke et al. (2017) to implement all experiments in this pa-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 125, + 128 + ], + "score": 1.0, + "content": "per.", + "type": "text" + }, + { + "bbox": [ + 138, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "Our codebase is publicly available at https://github.com/gibipara92/", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 127, + 317, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 317, + 141 + ], + "score": 1.0, + "content": "learning-explanations-hard-to-vary.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 107, + 152, + 187, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 189, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 189, + 165 + ], + "score": 1.0, + "content": "B.1 SECTION 3.1", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "table", + "bbox": [ + 156, + 215, + 452, + 338 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 175, + 505, + 206 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 186 + ], + "score": 1.0, + "content": "Table 1: Hyperparameter ranges for synthetic data experiments. The regularizers L1 and L2 are never combined;", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 186, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 505, + 196 + ], + "score": 1.0, + "content": "instead, one weight regularization type out of L1, L2 and none is selected and we sample from the respective", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 196, + 171, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 171, + 207 + ], + "score": 1.0, + "content": "range afterwards.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "table_body", + "bbox": [ + 156, + 215, + 452, + 338 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 156, + 215, + 452, + 338 + ], + "spans": [ + { + "bbox": [ + 156, + 215, + 452, + 338 + ], + "score": 0.982, + "html": "
HyperparameterRanges
No.hidden units{256,512}
No.hidden layers{3,5}
Batch-size{64,128,256}
Optimizer{Adamβ1 =0.9,β2=0.99, SGD + momentum0.9}
Learning rate{1e-3,1e-2,1e-1}
Batch-normalization{Yes,No}
Dropout{0.0, 0.5}
L2 regularization{1e-5,1e-4,1e-3}
L1 regularization{1e-6, 1e-5, 1e-4}
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The example is a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 412, + 416, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 153, + 425 + ], + "score": 1.0, + "content": "vector with", + "type": "text" + }, + { + "bbox": [ + 153, + 412, + 191, + 424 + ], + "score": 0.93, + "content": "d _ { S } + d _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 412, + 416, + 425 + ], + "score": 1.0, + "content": "entries, consisting of the shortcut and the mechanism. In", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 423, + 267, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 176, + 435 + ], + "score": 1.0, + "content": "our experiments,", + "type": "text" + }, + { + "bbox": [ + 176, + 424, + 209, + 434 + ], + "score": 0.91, + "content": "d _ { M } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 423, + 227, + 435 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 227, + 424, + 263, + 434 + ], + "score": 0.91, + "content": "d _ { S } = 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 423, + 267, + 435 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 415, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 416, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 416, + 452 + ], + "score": 1.0, + "content": "The Gaussian shortcuts are obtained by first sampling one random vector", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 450, + 416, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 146, + 462 + ], + "score": 0.92, + "content": "\\mathbf { x } _ { s } \\in \\mathbb { R } ^ { d _ { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 450, + 286, + 464 + ], + "score": 1.0, + "content": "per environment. 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We use", + "type": "text" + }, + { + "bbox": [ + 321, + 464, + 333, + 473 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 462, + 395, + 475 + ], + "score": 1.0, + "content": "for class 1, and", + "type": "text" + }, + { + "bbox": [ + 396, + 464, + 414, + 474 + ], + "score": 0.86, + "content": "- \\mathbf { x } _ { s }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 473, + 416, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 416, + 484 + ], + "score": 1.0, + "content": "for class -1. In the test set, all shortcut components are sampled i.i.d. from the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 483, + 416, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 416, + 496 + ], + "score": 1.0, + "content": "same Normal distribution. Effectively, each example of the test set belongs to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 495, + 416, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 416, + 507 + ], + "score": 1.0, + "content": "a different domain. 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For computational reason, we stop each trial before completion", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 231, + 635 + ], + "score": 1.0, + "content": "if the training accuracy exceeds", + "type": "text" + }, + { + "bbox": [ + 231, + 623, + 250, + 634 + ], + "score": 0.87, + "content": "9 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 623, + 369, + 635 + ], + "score": 1.0, + "content": "and the test accuracy is below", + "type": "text" + }, + { + "bbox": [ + 369, + 623, + 388, + 634 + ], + "score": 0.87, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 623, + 505, + 635 + ], + "score": 1.0, + "content": ". All networks are MLPs with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "LeakyReLU activation functions and a cross-entropy loss on the output. We run a hyperparameter", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "search over the ranges shown in Table 1. For IRM and the AND-mask, we select the best-performing", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "run and re-run it 50 times with different random seeds. For DANN and the standard baselines nothing", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 667, + 304, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 304, + 680 + ], + "score": 1.0, + "content": "produced results significantly better than chance.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38 + }, + { + "type": "title", + "bbox": [ + 107, + 690, + 333, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 690, + 334, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 334, + 703 + ], + "score": 1.0, + "content": "B.3.1 STANDARD REGULARIZERS AND AND-MASK", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "The networks with the L1, L2, Dropout and Batch-normalization regularizers, have hyperparameters", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "that were randomly selected from Table 1. For the AND-mask we used the very same ranges. The", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "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": "19", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 257, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 259, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 259, + 96 + ], + "score": 1.0, + "content": "B APPENDIX TO SECTION 3", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 506, + 139 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 119 + ], + "score": 1.0, + "content": "We used Pytorch Paszke et al. (2017) to implement all experiments in this pa-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 125, + 128 + ], + "score": 1.0, + "content": "per.", + "type": "text" + }, + { + "bbox": [ + 138, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "Our codebase is publicly available at https://github.com/gibipara92/", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 127, + 317, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 317, + 141 + ], + "score": 1.0, + "content": "learning-explanations-hard-to-vary.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 104, + 506, + 141 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 152, + 187, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 189, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 189, + 165 + ], + "score": 1.0, + "content": "B.1 SECTION 3.1", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "table", + "bbox": [ + 156, + 215, + 452, + 338 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 175, + 505, + 206 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 186 + ], + "score": 1.0, + "content": "Table 1: Hyperparameter ranges for synthetic data experiments. The regularizers L1 and L2 are never combined;", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 186, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 505, + 196 + ], + "score": 1.0, + "content": "instead, one weight regularization type out of L1, L2 and none is selected and we sample from the respective", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 196, + 171, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 171, + 207 + ], + "score": 1.0, + "content": "range afterwards.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "table_body", + "bbox": [ + 156, + 215, + 452, + 338 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 156, + 215, + 452, + 338 + ], + "spans": [ + { + "bbox": [ + 156, + 215, + 452, + 338 + ], + "score": 0.982, + "html": "
HyperparameterRanges
No.hidden units{256,512}
No.hidden layers{3,5}
Batch-size{64,128,256}
Optimizer{Adamβ1 =0.9,β2=0.99, SGD + momentum0.9}
Learning rate{1e-3,1e-2,1e-1}
Batch-normalization{Yes,No}
Dropout{0.0, 0.5}
L2 regularization{1e-5,1e-4,1e-3}
L1 regularization{1e-6, 1e-5, 1e-4}
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The example is a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 412, + 416, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 153, + 425 + ], + "score": 1.0, + "content": "vector with", + "type": "text" + }, + { + "bbox": [ + 153, + 412, + 191, + 424 + ], + "score": 0.93, + "content": "d _ { S } + d _ { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 412, + 416, + 425 + ], + "score": 1.0, + "content": "entries, consisting of the shortcut and the mechanism. In", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 423, + 267, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 176, + 435 + ], + "score": 1.0, + "content": "our experiments,", + "type": "text" + }, + { + "bbox": [ + 176, + 424, + 209, + 434 + ], + "score": 0.91, + "content": "d _ { M } = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 423, + 227, + 435 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 227, + 424, + 263, + 434 + ], + "score": 0.91, + "content": "d _ { S } = 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 423, + 267, + 435 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 379, + 416, + 435 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 415, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 416, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 416, + 452 + ], + "score": 1.0, + "content": "The Gaussian shortcuts are obtained by first sampling one random vector", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 450, + 416, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 146, + 462 + ], + "score": 0.92, + "content": "\\mathbf { x } _ { s } \\in \\mathbb { R } ^ { d _ { S } }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 450, + 286, + 464 + ], + "score": 1.0, + "content": "per environment. Its components", + "type": "text" + }, + { + "bbox": [ + 286, + 453, + 303, + 463 + ], + "score": 0.88, + "content": "x _ { s , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 450, + 416, + 464 + ], + "score": 1.0, + "content": "are sampled independently", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 462, + 414, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 217, + 475 + ], + "score": 1.0, + "content": "from a Normal distribution:", + "type": "text" + }, + { + "bbox": [ + 218, + 462, + 286, + 474 + ], + "score": 0.92, + "content": "x _ { s , i } \\sim \\mathcal { N } ( 0 , 0 . 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 462, + 321, + 475 + ], + "score": 1.0, + "content": ". We use", + "type": "text" + }, + { + "bbox": [ + 321, + 464, + 333, + 473 + ], + "score": 0.85, + "content": "\\mathbf { x } _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 462, + 395, + 475 + ], + "score": 1.0, + "content": "for class 1, and", + "type": "text" + }, + { + "bbox": [ + 396, + 464, + 414, + 474 + ], + "score": 0.86, + "content": "- \\mathbf { x } _ { s }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 473, + 416, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 416, + 484 + ], + "score": 1.0, + "content": "for class -1. In the test set, all shortcut components are sampled i.i.d. from the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 483, + 416, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 416, + 496 + ], + "score": 1.0, + "content": "same Normal distribution. Effectively, each example of the test set belongs to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 495, + 416, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 416, + 507 + ], + "score": 1.0, + "content": "a different domain. 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The training datasets consists of 2000 examples.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 106, + 545, + 505, + 568 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 580, + 190, + 592 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 192, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 192, + 593 + ], + "score": 1.0, + "content": "B.3 EXPERIMENT", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 205, + 615 + ], + "score": 1.0, + "content": "We train all networks for", + "type": "text" + }, + { + "bbox": [ + 205, + 601, + 246, + 613 + ], + "score": 0.9, + "content": "\\lfloor 3 0 0 0 / D \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "epochs, dropping the learning rate by a factor 10 halfway through,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "and again at three-quarters of training. For computational reason, we stop each trial before completion", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 231, + 635 + ], + "score": 1.0, + "content": "if the training accuracy exceeds", + "type": "text" + }, + { + "bbox": [ + 231, + 623, + 250, + 634 + ], + "score": 0.87, + "content": "9 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 623, + 369, + 635 + ], + "score": 1.0, + "content": "and the test accuracy is below", + "type": "text" + }, + { + "bbox": [ + 369, + 623, + 388, + 634 + ], + "score": 0.87, + "content": "60 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 623, + 505, + 635 + ], + "score": 1.0, + "content": ". All networks are MLPs with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 647 + ], + "score": 1.0, + "content": "LeakyReLU activation functions and a cross-entropy loss on the output. We run a hyperparameter", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "search over the ranges shown in Table 1. For IRM and the AND-mask, we select the best-performing", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "run and re-run it 50 times with different random seeds. For DANN and the standard baselines nothing", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 667, + 304, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 304, + 680 + ], + "score": 1.0, + "content": "produced results significantly better than chance.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 600, + 506, + 680 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 690, + 333, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 690, + 334, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 334, + 703 + ], + "score": 1.0, + "content": "B.3.1 STANDARD REGULARIZERS AND AND-MASK", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "The networks with the L1, L2, Dropout and Batch-normalization regularizers, have hyperparameters", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "that were randomly selected from Table 1. For the AND-mask we used the very same ranges. The", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "regularizers L1 and L2 are never combined; instead, one weight regularization type out of L1, L2", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "and none is selected and we sample from the respective range afterwards. The parameters found to", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 507, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 507, + 118 + ], + "score": 1.0, + "content": "work best from the grid search were: agreement threshold of 1, 256 hidden units, 3 hidden layers,", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "batch size 128, Adam with learning rate 1e-2, no batch norm, no dropout, L2-regularization with a", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "coefficient of 1e-4, no L1-regularization. In practice, we often found it helpful to rescale the gradients", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "after masking to compensate for the decreasing overall magnitude. We add the option for gradient", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 150, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 504, + 160 + ], + "score": 1.0, + "content": "rescaling as an additional hyperparameter, as we found it to help in several experiments. It rescales", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "gradient components layer-wise after masking, by multiplying the remaining gradient components", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 119, + 182 + ], + "score": 1.0, + "content": "by", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 119, + 173, + 124, + 180 + ], + "score": 0.63, + "content": "c", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 125, + 171, + 155, + 182 + ], + "score": 1.0, + "content": ", where", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 155, + 172, + 161, + 180 + ], + "score": 0.7, + "content": "c", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 161, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "is the ratio of the number of components in that layer over the number of non-masked", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "components in that layer (i.e. the sum of the binary elements in the mask).14. We speculate that for", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 206 + ], + "score": 1.0, + "content": "very large layers, a less extreme normalization scheme or the additional use of gradient clipping", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 204, + 195, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 195, + 216 + ], + "score": 1.0, + "content": "might be appropriate.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 43.5, + "bbox_fs": [ + 106, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "regularizers L1 and L2 are never combined; instead, one weight regularization type out of L1, L2", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "and none is selected and we sample from the respective range afterwards. The parameters found to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 507, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 507, + 118 + ], + "score": 1.0, + "content": "work best from the grid search were: agreement threshold of 1, 256 hidden units, 3 hidden layers,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "batch size 128, Adam with learning rate 1e-2, no batch norm, no dropout, L2-regularization with a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "coefficient of 1e-4, no L1-regularization. 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It rescales", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "gradient components layer-wise after masking, by multiplying the remaining gradient components", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 119, + 182 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 173, + 124, + 180 + ], + "score": 0.63, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 171, + 155, + 182 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 155, + 172, + 161, + 180 + ], + "score": 0.7, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "is the ratio of the number of components in that layer over the number of non-masked", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "components in that layer (i.e. the sum of the binary elements in the mask).14. We speculate that for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 206 + ], + "score": 1.0, + "content": "very large layers, a less extreme normalization scheme or the additional use of gradient clipping", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 204, + 195, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 195, + 216 + ], + "score": 1.0, + "content": "might be appropriate.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 107, + 227, + 335, + 238 + ], + "lines": [ + { + "bbox": [ + 106, + 226, + 336, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 336, + 240 + ], + "score": 1.0, + "content": "B.3.2 DOMAIN ADVERSARIAL NEURAL NETWORKS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 246, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "The experiments using DANN follow a similar pattern. 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HyperparameterRanges
No.hidden units{256,512}
No.hidden layers{3,5}
Batch-size{64,128,256}
Optimizer{Adamβ1=0.9,β2=0.999, SGD + momentum0.9}
Batch-normalization{Yes,No}
Penalty weight{10.0,100.0,1000.0}
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HyperparameterRanges
No.hidden units{256,512}
No.hidden layers{3,5}
Batch-size{64,128,256}
Optimizer{Adamβ1=0.9,β2=0.999, SGD + momentum0.9}
Batch-normalization{Yes,No}
Penalty weight{10.0,100.0,1000.0}
Number of annealing iterations{0,1,2,4,8}
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The top row shows the accuracy on the training set, the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 242, + 276, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 276, + 254 + ], + "score": 1.0, + "content": "bottom row shows the accuracy on the test set.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 107, + 276, + 332, + 286 + ], + "lines": [ + { + "bbox": [ + 106, + 275, + 333, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 333, + 287 + ], + "score": 1.0, + "content": "B.4 FURTHER VISUALIZATIONS AND EXPERIMENTS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "In Figure 15 we show how many environments need to be present for the baseline without AND-mask", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "to switch the decision boundary from the shortcuts to the mechanism. 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We show the best performance out of five runs using the settings that were used for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 426, + 223, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 223, + 439 + ], + "score": 1.0, + "content": "the experiment in the main text.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 458, + 452, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 455, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 455, + 471 + ], + "score": 1.0, + "content": "B.5 SECTION 3.2: CIFAR-10 MEMORIZATION AND LABEL NOISE EXPERIMENTS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 349, + 578 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 349, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 349, + 492 + ], + "score": 1.0, + "content": "Memorization experiment In Figure 16, we report the test", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 490, + 350, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 350, + 502 + ], + "score": 1.0, + "content": "performance (dashed lines) corresponding to the curves pre-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 501, + 349, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 349, + 513 + ], + "score": 1.0, + "content": "sented in the main paper for the CIFAR-10 memorization", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 513, + 349, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 349, + 523 + ], + "score": 1.0, + "content": "experiment. The test performance with standard labels de-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 523, + 349, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 349, + 534 + ], + "score": 1.0, + "content": "creases slower than the training performance as the threshold", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 534, + 348, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 348, + 545 + ], + "score": 1.0, + "content": "increases, and they eventually reach the same value. This is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 545, + 349, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 349, + 556 + ], + "score": 1.0, + "content": "consistent with the hypothesis that by training on the consis-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 555, + 349, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 349, + 568 + ], + "score": 1.0, + "content": "tent directions, the AND-mask selects the invariant patterns", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 567, + 299, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 299, + 578 + ], + "score": 1.0, + "content": "and prunes out the signals that are not invariant.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 478, + 350, + 578 + ] + }, + { + "type": "image", + "bbox": [ + 358, + 482, + 502, + 558 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 358, + 482, + 502, + 558 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 358, + 482, + 502, + 558 + ], + "spans": [ + { + "bbox": [ + 358, + 482, + 502, + 558 + ], + "score": 0.954, + "type": "image", + "image_path": "8ca71fe755e2a88b3312a7912c3fb6fe9a5bde54794c3813b8551085371d1d41.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 358, + 482, + 502, + 520.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 358, + 520.0, + 502, + 558.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 356, + 568, + 505, + 588 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 355, + 567, + 507, + 580 + ], + "spans": [ + { + "bbox": [ + 355, + 567, + 507, + 580 + ], + "score": 1.0, + "content": "Figure 16: Dashed lines show test acc,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 356, + 577, + 463, + 590 + ], + "spans": [ + { + "bbox": [ + 356, + 577, + 463, + 590 + ], + "score": 1.0, + "content": "solid lines show training acc.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 591, + 348, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 349, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 349, + 603 + ], + "score": 1.0, + "content": "Network architecture and training details Each trial", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 602, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "trains the ResNet “FastResNet” from the PyTorch-Ignite example15 for 80 epochs on the full", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 611, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 626 + ], + "score": 1.0, + "content": "CIFAR-10 training set. 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Note that a threshold of 0 corresponds", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 656, + 426, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 426, + 669 + ], + "score": 1.0, + "content": "to not using the AND-mask. Each trial is run twice with separate random seeds.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 601, + 506, + 669 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 680, + 504, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 680, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 505, + 693 + ], + "score": 1.0, + "content": "Label noise experiment We trained the same ResNet as for the experiment above, once with and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 690, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 506, + 704 + ], + "score": 1.0, + "content": "once without the AND-mask. 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The shaded regions", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 235, + 393, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 171, + 248 + ], + "score": 1.0, + "content": "correspond to the", + "type": "text" + }, + { + "bbox": [ + 171, + 236, + 189, + 245 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 235, + 393, + 248 + ], + "score": 1.0, + "content": "confidence interval of the mean based on bootstrapping.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 267, + 257, + 279 + ], + "lines": [ + { + "bbox": [ + 106, + 266, + 258, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 258, + 281 + ], + "score": 1.0, + "content": "C APPENDIX TO SECTION 4", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 292, + 301, + 303 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 303, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 303, + 305 + ], + "score": 1.0, + "content": "C.1 RELATED WORK IN CAUSAL INFERENCE", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "Causal graphs and causal factorizations The formalization of causality through directed acyclic", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "graphs (Pearl, 2009) is a key element informing our exposition. According to such formalization,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "a causal model gives rise to each observed distribution. It is thereby possible to exploit properties", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 344, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 360 + ], + "score": 1.0, + "content": "of the causal factorization of the joint probability distribution over the observed variables. Clearly,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "there are many ways to factorize a joint distribution into conditionals; a distinguishing feature", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "of the causal factorization is that many of the conditionals, which we can think of as physical", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "mechanisms underlying the statistical dependencies represented, are expected to remain invariant", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "under interventions or changing external conditions. This postulate has appeared in various forms in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "score": 1.0, + "content": "the literature (Haavelmo, 1943; Simon, 1953; Hurwicz, 1962; Pearl, 2009; Schölkopf et al., 2012).17", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "Causal models and robust regression Based on this insight, it was proposed that regression based", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 437, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 506, + 448 + ], + "score": 1.0, + "content": "on causal features should presents desirable invariance and robustness properties (Mooij et al., 2009;", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "Schölkopf et al., 2012; Peters et al., 2016; Rojas-Carulla et al., 2018; Heinze-Deml et al., 2018; von", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "Kügelgen et al., 2019; Parascandolo et al., 2018). In this view, the mechanisms can be considered", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 483 + ], + "score": 1.0, + "content": "as features of the patterns such that they support stable conditional probabilities. Thus learning the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 481, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 492 + ], + "score": 1.0, + "content": "mechanisms may help achieve a stable performance across a number of conditions. 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The shaded regions", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 235, + 393, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 171, + 248 + ], + "score": 1.0, + "content": "correspond to the", + "type": "text" + }, + { + "bbox": [ + 171, + 236, + 189, + 245 + ], + "score": 0.86, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 235, + 393, + 248 + ], + "score": 1.0, + "content": "confidence interval of the mean based on bootstrapping.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 267, + 257, + 279 + ], + "lines": [ + { + "bbox": [ + 106, + 266, + 258, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 258, + 281 + ], + "score": 1.0, + "content": "C APPENDIX TO SECTION 4", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 292, + 301, + 303 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 303, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 303, + 305 + ], + "score": 1.0, + "content": "C.1 RELATED WORK IN CAUSAL INFERENCE", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "Causal graphs and causal factorizations The formalization of causality through directed acyclic", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "graphs (Pearl, 2009) is a key element informing our exposition. According to such formalization,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "a causal model gives rise to each observed distribution. It is thereby possible to exploit properties", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 344, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 360 + ], + "score": 1.0, + "content": "of the causal factorization of the joint probability distribution over the observed variables. Clearly,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "there are many ways to factorize a joint distribution into conditionals; a distinguishing feature", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "of the causal factorization is that many of the conditionals, which we can think of as physical", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "mechanisms underlying the statistical dependencies represented, are expected to remain invariant", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "under interventions or changing external conditions. This postulate has appeared in various forms in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 414 + ], + "score": 1.0, + "content": "the literature (Haavelmo, 1943; Simon, 1953; Hurwicz, 1962; Pearl, 2009; Schölkopf et al., 2012).17", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 312, + 506, + 414 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "Causal models and robust regression Based on this insight, it was proposed that regression based", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 437, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 506, + 448 + ], + "score": 1.0, + "content": "on causal features should presents desirable invariance and robustness properties (Mooij et al., 2009;", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "Schölkopf et al., 2012; Peters et al., 2016; Rojas-Carulla et al., 2018; Heinze-Deml et al., 2018; von", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "Kügelgen et al., 2019; Parascandolo et al., 2018). 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IRM then makes the point that we can learn", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 448, + 283 + ], + "score": 1.0, + "content": "to ignore grass as a feature, because its correlation with the label cow is inconsistent (", + "type": "text" + }, + { + "bbox": [ + 448, + 271, + 468, + 281 + ], + "score": 0.86, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 271, + 480, + 283 + ], + "score": 1.0, + "content": "vs", + "type": "text" + }, + { + "bbox": [ + 480, + 271, + 500, + 281 + ], + "score": 0.86, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 271, + 506, + 283 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 295 + ], + "score": 1.0, + "content": "The setting we consider in this paper is slightly different: take our example from the CIFAR-10", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "score": 1.0, + "content": "experiments. Under our concept of invariance, we expect that (depending on the data generating", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 316 + ], + "score": 1.0, + "content": "process) even a single dataset where we treat every image as coming from its own ‘environment’", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "should be sufficient to discover invariances. Drawing a connection to the setting from IRM, we would", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "argue that the second dataset should not be necessary to learn that ‘grass’ is not ‘cow’. If one treats", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 348 + ], + "score": 1.0, + "content": "every example as coming from its own environment, there is already sufficient information in the first", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 405, + 360 + ], + "score": 1.0, + "content": "dataset to realize that cows are not grass: Grass is predictive of cows only in", + "type": "text" + }, + { + "bbox": [ + 405, + 347, + 424, + 357 + ], + "score": 0.87, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 346, + 506, + 360 + ], + "score": 1.0, + "content": "of the data, so grass", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 394, + 371 + ], + "score": 1.0, + "content": "cannot be ‘cow’. The actual cow on the other hand, should be present in", + "type": "text" + }, + { + "bbox": [ + 394, + 358, + 418, + 369 + ], + "score": 0.86, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "of the images, and as", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "such it is the invariance we are looking for. Note that this is of course a much more strict definition of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 381, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 392 + ], + "score": 1.0, + "content": "invariance: If our dataset contains images labeled as ’cows’ but that have no cows within them, we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 391, + 308, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 308, + 402 + ], + "score": 1.0, + "content": "might start to discard the features of cows as well.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 238, + 506, + 402 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/hb1sDDSLbV/hb1sDDSLbV_model.json b/parse/train/hb1sDDSLbV/hb1sDDSLbV_model.json new file mode 100644 index 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HyperparameterRanges
No.hidden units{256,512}
No.hidden layers{3,5}
Batch-size{64,128,256}
Optimizer{Adamβ1 =0.9,β2=0.99, SGD + momentum0.9}
Learning rate{1e-3,1e-2,1e-1}
Batch-normalization{Yes,No}
Dropout{0.0, 0.5}
L2 regularization{1e-5,1e-4,1e-3}
L1 regularization{1e-6, 1e-5, 1e-4}
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HyperparameterRanges
No.hidden units{256,512}
No.hidden layers{3,5}
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Number of annealing iterations{0,1,2,4,8}
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Test KL divlrβ1β3Tweight decay
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Saxe, Yamini Bansal, Joel Dapello, Madhu Advani +Harvard University +{asaxe,madvani}@fas.harvard.edu,{ybansal,dapello}@g.harvard.edu + +Artemy Kolchinsky, Brendan D. Tracey + +Santa Fe Institute {artemyk,tracey.brendan}@gmail.com + +David D. Cox +Harvard University +MIT-IBM Watson AI Lab +davidcox@fas.harvard.edu +david.d.cox@ibm.com + +# ABSTRACT + +The practical successes of deep neural networks have not been matched by theoretical progress that satisfyingly explains their behavior. In this work, we study the information bottleneck (IB) theory of deep learning, which makes three specific claims: first, that deep networks undergo two distinct phases consisting of an initial fitting phase and a subsequent compression phase; second, that the compression phase is causally related to the excellent generalization performance of deep networks; and third, that the compression phase occurs due to the diffusion-like behavior of stochastic gradient descent. Here we show that none of these claims hold true in the general case. Through a combination of analytical results and simulation, we demonstrate that the information plane trajectory is predominantly a function of the neural nonlinearity employed: double-sided saturating nonlinearities like tanh yield a compression phase as neural activations enter the saturation regime, but linear activation functions and single-sided saturating nonlinearities like the widely used ReLU in fact do not. Moreover, we find that there is no evident causal connection between compression and generalization: networks that do not compress are still capable of generalization, and vice versa. Next, we show that the compression phase, when it exists, does not arise from stochasticity in training by demonstrating that we can replicate the IB findings using full batch gradient descent rather than stochastic gradient descent. Finally, we show that when an input domain consists of a subset of task-relevant and task-irrelevant information, hidden representations do compress the task-irrelevant information, although the overall information about the input may monotonically increase with training time, and that this compression happens concurrently with the fitting process rather than during a subsequent compression period. + +# 1 INTRODUCTION + +Deep neural networks (Schmidhuber, 2015; LeCun et al., 2015) are the tool of choice for real-world tasks ranging from visual object recognition (Krizhevsky et al., 2012), to unsupervised learning (Goodfellow et al., 2014; Lotter et al., 2016) and reinforcement learning (Silver et al., 2016). These practical successes have spawned many attempts to explain the performance of deep learning systems (Kadmon & Sompolinsky, 2016), mostly in terms of the properties and dynamics of the optimization problem in the space of weights (Saxe et al., 2014; Choromanska et al., 2015; Advani & Saxe, 2017), or the classes of functions that can be efficiently represented by deep networks (Montufar et al., 2014; Poggio et al., 2017). This paper analyzes a recent inventive proposal to study the dynamics of learning through the lens of information theory (Tishby & Zaslavsky, 2015; Shwartz-Ziv & Tishby, 2017). In this view, deep learning is a question of representation learning: each layer of a deep neural network can be seen as a set of summary statistics which contain some but not all of the information present in the input, while retaining as much information about the target output as possible. The amount of information in a hidden layer regarding the input and output can then be measured over the course of learning, yielding a picture of the optimization process in the information plane. Crucially, this method holds the promise to serve as a general analysis that can be used to compare different architectures, using the common currency of mutual information. Moreover, the elegant information bottleneck (IB) theory provides a fundamental bound on the amount of input compression and target output information that any representation can achieve (Tishby et al., 1999). The IB bound thus serves as a method-agnostic ideal to which different architectures and algorithms may be compared. + +A preliminary empirical exploration of these ideas in deep neural networks has yielded striking findings (Shwartz-Ziv & Tishby, 2017). Most saliently, trajectories in the information plane appear to consist of two distinct phases: an initial “fitting” phase where mutual information between the hidden layers and both the input and output increases, and a subsequent “compression” phase where mutual information between the hidden layers and the input decreases. It has been hypothesized that this compression phase is responsible for the excellent generalization performance of deep networks, and further, that this compression phase occurs due to the random diffusion-like behavior of stochastic gradient descent. + +Here we study these phenomena using a combination of analytical methods and simulation. In Section 2, we show that the compression observed by Shwartz-Ziv & Tishby (2017) arises primarily due to the double-saturating tanh activation function used. Using simple models, we elucidate the effect of neural nonlinearity on the compression phase. Importantly, we demonstrate that the ReLU activation function, often the nonlinearity of choice in practice, does not exhibit a compression phase. We discuss how this compression via nonlinearity is related to the assumption of binning or noise in the hidden layer representation. To better understand the dynamics of learning in the information plane, in Section 3 we study deep linear networks in a tractable setting where the mutual information can be calculated exactly. We find that deep linear networks do not compress over the course of training for the setting we examine. Further, we show a dissociation between generalization and compression. In Section 4, we investigate whether stochasticity in the training process causes compression in the information plane. We train networks with full batch gradient descent, and compare the results to those obtained with stochastic gradient descent. We find comparable compression in both cases, indicating that the stochasticity of SGD is not a primary factor in the observed compression phase. Moreover, we show that the two phases of SGD occur even in networks that do not compress, demonstrating that the phases are not causally related to compression. These results may seem difficult to reconcile with the intuition that compression can be necessary to attain good performance: if some input channels primarily convey noise, good generalization requires excluding them. Therefore, in Section 5 we study a situation with explicitly task-relevant and task-irrelevant input dimensions. We show that the hidden-layer mutual information with the task-irrelevant subspace does indeed drop during training, though the overall information with the input increases. However, instead of a secondary compression phase, this task-irrelevant information is compressed at the same time that the taskrelevant information is boosted. Our results highlight the importance of noise assumptions in applying information theoretic analyses to deep learning systems, and put in doubt the generality of the IB theory of deep learning as an explanation of generalization performance in deep architectures. + +# 2 COMPRESSION AND NEURAL NONLINEARITIES + +The starting point for our analysis is the observation that changing the activation function can markedly change the trajectory of a network in the information plane. In Figure 1A, we show our replication of the result reported by Shwartz-Ziv & Tishby (2017) for networks with the tanh nonlinearity.1 This replication was performed with the code supplied by the authors of Shwartz-Ziv & Tishby (2017), and closely follows the experimental setup described therein. Briefly, a neural network with 7 fully connected hidden layers of width 12-10-7-5-4-3-2 is trained with stochastic gradient descent to produce a binary classification from a 12-dimensional input. In our replication we used 256 randomly selected samples per batch. The mutual information of the network layers with respect to the input and output variables is calculated by binning the neuron’s tanh output activations into 30 equal intervals between -1 and 1. Discretized values for each neuron in each layer are then used to directly calculate the joint distributions, over the 4096 equally likely input patterns and true output labels. In line with prior work (Shwartz-Ziv & Tishby, 2017), the dynamics in Fig. 1 show a transition between an initial fitting phase, during which information about the input increases, and a subsequent compression phase, during which information about the input decreases. + +![](images/ae057461c9856b2cafbb8dc433b0cc3cacf699acd8de55f5ace722373b08a275.jpg) +Figure 1: Information plane dynamics and neural nonlinearities. (A) Replication of Shwartz-Ziv & Tishby (2017) for a network with tanh nonlinearities (except for the final classification layer which contains two sigmoidal neurons). The $\mathbf { X }$ -axis plots information between each layer and the input, while the y-axis plots information between each layer and the output. The color scale indicates training time in epochs. Each of the six layers produces a curve in the information plane with the input layer at far right, output layer at the far left. Different layers at the same epoch are connected by fine lines. (B) Information plane dynamics with ReLU nonlinearities (except for the final layer of 2 sigmoidal neurons). Here no compression phase is visible in the ReLU layers. For learning curves of both networks, see Appendix A. (C) Information plane dynamics for a tanh network of size $7 8 4 - 1 0 2 4 - 2 0 - 2 0 - 2 0 - 1 0$ trained on MNIST, estimated using the non-parametric kernel density mutual information estimator of Kolchinsky & Tracey (2017); Kolchinsky et al. (2017), no compression is observed except in the final classification layer with sigmoidal neurons. See Appendix B for the KDE MI method applied to the original Tishby dataset; additional results using a second popular nonparametric $\mathbf { k }$ -NN-based method (Kraskov et al., 2004); and results for other neural nonlinearities. + +We then modified the code to train deep networks using rectified linear activation functions $( f ( x ) =$ $\operatorname* { m a x } ( 0 , x ) )$ . While the activities of tanh networks are bounded in the range $[ - 1 , 1 ]$ , ReLU networks have potentially unbounded positive activities. To calculate mutual information, we first trained the ReLU networks, next identified their largest activity value over the course of training, and finally chose 100 evenly spaced bins between the minimum and maximum activity values to discretize the hidden layer activity. The resulting information plane dynamics are shown in Fig. 1B. The mutual information with the input monotonically increases in all ReLU layers, with no apparent compression phase. To see whether our results were an artifact of the small network size, toy dataset, or simple binning-based mutual information estimator we employed, we also trained larger networks on the MNIST dataset and computed mutual information using a state-of-the-art nonparametric kernel density estimator which assumes hidden activity is distributed as a mixture of Gaussians (see Appendix B for details). Fig. C-D show that, again, tanh networks compressed but ReLU networks did not. Appendix B shows that similar results also obtain with the popular nonparametric $\mathbf { k }$ -nearest-neighbor estimator of Kraskov et al. (2004), and for other neural nonlinearities. Thus, the choice of nonlinearity substantively affects the dynamics in the information plane. + +To understand the impact of neural nonlinearity on the mutual information dynamics, we develop a minimal model that exhibits this phenomenon. In particular, consider the simple three neuron network shown in Fig. 2A. We assume a scalar Gaussian input distribution $X \sim \mathcal { N } ( 0 , 1 )$ , which is fed through the scalar first layer weight $w _ { 1 }$ , and passed through a neural nonlinearity $f ( \cdot )$ , yielding the hidden unit activity $h = f ( w _ { 1 } X )$ . To calculate the mutual information with the input, this hidden unit activity is then binned yielding the new discrete variable $T = \dot { \mathbf { b i n } } ( h )$ (for instance, into 30 evenly spaced bins from $^ { - 1 }$ to 1 for the tanh nonlinearity). This binning process is depicted in Fig. 2B. In this simple setting, the mutual information $I ( T ; X )$ between the binned hidden layer activity $T$ and the input $X$ can be calculated exactly. In particular, + +$$ +\begin{array} { l c l } { { { \cal I } ( T ; X ) } } & { { = } } & { { { \cal H } ( T ) - { \cal H } ( T | X ) } } \\ { { } } & { { = } } & { { { \cal H } ( T ) } } \\ { { } } & { { = } } & { { - \displaystyle \sum _ { i = 1 } ^ { N } p _ { i } \log p _ { i } } } \end{array} +$$ + +where $H ( \cdot )$ denotes entropy, and we have used the fact that $H ( T | X ) = 0$ since $T$ is a deterministic function of $X$ . Here the probabilities $p _ { i } = P ( h \ge b _ { i }$ and $h < b _ { i + 1 }$ ) are simply the probability that an input $X$ produces a hidden unit activity that lands in bin $i$ , defined by lower and upper bin limits $b _ { i }$ and $b _ { i + 1 }$ respectively. This probability can be calculated exactly for monotonic nonlinearities $f ( \cdot )$ using the cumulative density of $X$ , + +$$ +p _ { i } = P ( X \geq f ^ { - 1 } ( b _ { i } ) / w _ { 1 } \mathrm { a n d } X < f ^ { - 1 } ( b _ { i + 1 } ) / w _ { 1 } ) , +$$ + +where $f ^ { - 1 } ( \cdot )$ is the inverse function of $f ( \cdot )$ . + +As shown in Fig. 2C-D, as a function of the weight $w _ { 1 }$ , mutual information with the input first increases and then decreases for the tanh nonlinearity, but always increases for the ReLU nonlinearity. Intuitively, for small weights $w _ { 1 } \approx 0$ , neural activities lie near zero on the approximately linear part of the tanh function. Therefore $f ( w _ { 1 } X ) \approx w _ { 1 } X$ , yielding a rescaled Gaussian with information that grows with the size of the weights. However for very large weights $w _ { 1 } \to \infty$ , the tanh hidden unit nearly always saturates, yielding a discrete variable that concentrates in just two bins. This is more or less a coin flip, containing mutual information with the input of approximately 1 bit. Hence the distribution of $T$ collapses to a much lower entropy distribution, yielding compression for large weight values. With the ReLU nonlinearity, half of the inputs are negative and land in the bin containing a hidden activity of zero. The other half are Gaussian distributed, and thus have entropy that increases with the size of the weight. + +Hence double-saturating nonlinearities can lead to compression of information about the input, as hidden units enter their saturation regime, due to the binning procedure used to calculate mutual information. The crux of the issue is that the actual $I ( h ; X )$ is infinite, unless the network itself adds noise to the hidden layers. In particular, without added noise, the transformation from $X$ to the continuous hidden activity $h$ is deterministic and the mutual information $I ( h ; X )$ would generally be infinite (see Appendix C for extended discussion). Networks that include noise in their processing (e.g., Kolchinsky et al. (2017)) can have finite $I ( T ; X )$ . Otherwise, to obtain a finite MI, one must compute mutual information as though there were binning or added noise in the activations. But this binning/noise is not actually a part of the operation of the network, and is therefore somewhat arbitrary (different binning schemes can result in different mutual information with the input, as shown in Fig. 14 of Appendix C). + +![](images/b7f23d4571d8a39c6d37d9f1a3d373f749c5196988ea773d81a640c8c050f81d.jpg) +Figure 2: Nonlinear compression in a minimal model. (A) A three neuron nonlinear network which receives Gaussian inputs $x$ , multiplies by weight $w _ { 1 }$ , and maps through neural nonlinearity $f ( \cdot )$ to produce hidden unit activity $h$ . (B) The continuous activity $h$ is binned into a discrete variable $T$ for the purpose of calculating mutual information. Blue: continuous tanh nonlinear activation function. Grey: Bin borders for 30 bins evenly spaced between $^ { - 1 }$ and 1. Because of the saturation in the sigmoid, a wide range of large magnitude net input values map to the same bin. (C) Mutual information with the input as a function of weight size $w _ { 1 }$ for a tanh nonlinearity. Information increases for small $w _ { 1 }$ and then decreases for large $w _ { 1 }$ as all inputs land in one of the two bins corresponding to the saturation regions. (D) Mutual information with the input for the ReLU nonlinearity increases without bound. Half of all inputs land in the bin corresponding to zero activity, while the other half have information that scales with the size of the weights. + +We note that the binning procedure can be viewed as implicitly adding noise to the hidden layer activity: a range of $X$ values map to a single bin, such that the mapping between $X$ and $T$ is no longer perfectly invertible (Laughlin, 1981). The binning procedure is therefore crucial to obtaining a finite MI value, and corresponds approximately to a model where noise enters the system after the calculation of $h$ , that is, $T = h + \epsilon$ , where $\epsilon$ is noise of fixed variance independent from $h$ and $X$ . This approach is common in information theoretic analyses of deterministic systems, and can serve as a measure of the complexity of a system’s representation (see Sec 2.4 of Shwartz-Ziv & Tishby (2017)). However, neither binning nor noise is present in the networks that Shwartz-Ziv & Tishby (2017) considered, nor the ones in Fig. 2, either during training or testing. It therefore remains unclear whether robustness of a representation to this sort of noise in fact influences generalization performance in deep learning systems. + +Furthermore, the addition of noise means that different architectures may no longer be compared in a common currency of mutual information: the binning/noise structure is arbitrary, and architectures that implement an identical input-output map can nevertheless have different robustness to noise added in their internal representation. For instance, Appendix C describes a family of linear networks that compute exactly the same input-output map and therefore generalize identically, but yield different mutual information with respect to the input. Finally, we note that approaches which view the weights obtained from the training process as the random variables of interest may sidestep this issue (Achille & Soatto, 2017). + +![](images/b90f7481a85a83f3aa2ebfdd7fb9a3e639595486fdd68886cf2cb42a1f704c6f.jpg) +Figure 3: Generalization and information plane dynamics in deep linear networks. (A) A linear teacher network generates a dataset by passing Gaussian inputs $X$ through its weights and adding noise. (B) A deep linear student network is trained on the dataset (here the network has 1 hidden layer to allow comparison with Fig. 4A, see Supplementary Figure 18 for a deeper network). (C) Training and testing error over time. (D) Information plane dynamics. No compression is observed. + +Hence when a tanh network is initialized with small weights and over the course of training comes to saturate its nonlinear units (as it must to compute most functions of practical interest, see discussion in Appendix D), it will enter a compression period where mutual information decreases. Figures 16-17 of Appendix E show histograms of neural activity over the course of training, demonstrating that activities in the tanh network enter the saturation regime during training. This nonlinearity-based compression furnishes another explanation for the observation that training slows down as tanh networks enter their compression phase (Shwartz-Ziv & Tishby, 2017): some fraction of inputs have saturated the nonlinearities, reducing backpropagated error gradients. + +# 3 INFORMATION PLANE DYNAMICS IN DEEP LINEAR NETWORKS + +The preceding section investigates the role of nonlinearity in the observed compression behavior, tracing the source to double-saturating nonlinearities and the binning methodology used to calculate mutual information. However, other mechanisms could lead to compression as well. Even without nonlinearity, neurons could converge to highly correlated activations, or project out irrelevant directions of the input. These phenomena are not possible to observe in our simple three neuron minimal model, as they require multiple inputs and hidden layer activities. To search for these mechanisms, we turn to a tractable model system: deep linear neural networks (Baldi & Hornik (1989); Fukumizu (1998); Saxe et al. (2014)). In particular, we exploit recent results on the generalization dynamics in simple linear networks trained in a student-teacher setup (Seung et al., 1992; Advani & Saxe, 2017). In a student-teacher setting, one “student” neural network learns to approximate the output of another “teacher” neural network. This setting is a way of generating a dataset with interesting structure that nevertheless allows exact calculation of the generalization performance of the network, exact calculation of the mutual information of the representation (without any binning procedure), and, though we do not do so here, direct comparison to the IB bound which is already known for linear Gaussian problems (Chechik et al., 2005). + +We consider a scenario where a linear teacher neural network generates input and output examples which are then fed to a deep linear student network to learn (Fig. 3A). Following the formulation of (Advani & Saxe, 2017), we assume multivariate Gaussian inputs $\begin{array} { r } { X \sim \mathcal { N } ( 0 , \frac { 1 } { N _ { i } } \overline { { I } } _ { N _ { i } } ) } \end{array}$ and a scalar output $Y$ . The output is generated by the teacher network according to $Y = \dot { W } _ { 0 } X + \epsilon _ { o }$ , where $\epsilon _ { o } \stackrel { - } { \sim } \mathcal { N } ( 0 , \sigma _ { o } ^ { 2 } )$ represents aspects of the target function which cannot be represented by a neural network (that is, the approximation error or bias in statistical learning theory), and the teacher weights $W _ { o }$ are drawn independently from ${ \mathcal N } ( 0 , \sigma _ { w } ^ { 2 } )$ . Here, the weights of the teacher define the rule to be learned. The signal to noise ratio $\mathrm { S N R } = \sigma _ { w } ^ { 2 } / \sigma _ { o } ^ { 2 }$ determines the strength of the rule linking inputs to outputs relative to the inevitable approximation error. We emphasize that the “noise” added to the teacher’s output is fundamentally different from the noise added for the purpose of calculating mutual information: $\epsilon _ { o }$ models the approximation error for the task–even the best possible neural network may still make errors because the target function is not representable exactly as a neural network–and is part of the construction of the dataset, not part of the analysis of the student network. + +To train the student network, a dataset of $P$ examples is generated using the teacher. The student network is then trained to minimize the mean squared error between its output and the target output using standard (batch or stochastic) gradient descent on this dataset. Here the student is a deep linear neural network consisting of potentially many layers, but where the the activation function of each neuron is simply $f ( u ) = u$ . That is, a depth $D$ deep linear network computes the output $\hat { Y } = W _ { D + 1 } W _ { D } \cdot \cdot \cdot W _ { 2 } W _ { 1 } X$ . While linear activation functions stop the network from computing complex nonlinear functions of the input, deep linear networks nevertheless show complicated nonlinear learning trajectories (Saxe et al., 2014), the optimization problem remains nonconvex (Baldi & Hornik, 1989), and the generalization dynamics can exhibit substantial overtraining (Fukumizu, 1998; Advani & Saxe, 2017). + +Importantly, because of the simplified setting considered here, the true generalization error is easily shown to be + +$$ +E _ { g } ( t ) = | | W _ { o } - W _ { t o t } ( t ) | | _ { F } ^ { 2 } + \sigma _ { o } ^ { 2 } +$$ + +where $W _ { t o t } ( t )$ is the overall linear map implemented by the network at training epoch $t$ (that is, $W _ { t o t } = W _ { D + 1 } W _ { D } \cdot \cdot \cdot W _ { 2 } W _ { 1 } )$ . + +Furthermore, the mutual information with the input and output may be calculated exactly, because the distribution of the activity of any hidden layer is Gaussian. Let $T$ be the activity of a specific hidden layer, and let $\bar { W }$ be the linear map from the input to this activity (that is, for layer $l$ , $\bar { W } =$ $W _ { l } \cdots W _ { 2 } W _ { 1 } )$ . Since $T = { \bar { W } } X$ , the mutual information of $X$ and $T$ calculated using differential entropy is infinite. For the purpose of calculating the mutual information, therefore, we assume that Gaussian noise is added to the hidden layer activity, $T = \bar { W } X + \epsilon _ { M I }$ , with mean 0 and variance $\sigma _ { M I } ^ { 2 } = 1 . 0$ . This allows the analysis to apply to networks of any size, including overcomplete layers, but as before we emphasize that we do not add this noise either during training or testing. With these assumptions, $T$ and $X$ are jointly Gaussian and we have + +$$ +I ( T ; X ) = \log \lvert \bar { W } \bar { W } ^ { T } + \sigma _ { M I } ^ { 2 } I _ { N _ { h } } \rvert - \log \lvert \sigma _ { M I } ^ { 2 } I _ { N _ { h } } \rvert +$$ + +where $\left| \cdot \right|$ denotes the determinant of a matrix. Finally the mutual information with the output $Y$ , also jointly Gaussian, can be calculated similarly (see Eqns. (22)-(25) of Appendix G). + +Fig. 3 shows example training and test dynamics over the course of learning in panel C, and the information plane dynamics in panel D. Here the network has an input layer of 100 units, 1 hidden layer of 100 units each and one output unit. The network was trained with batch gradient descent on a dataset of 100 examples drawn from the teacher with signal to noise ratio of 1.0. The linear network behaves qualitatively like the ReLU network, and does not exhibit compression. Nevertheless, it learns a map that generalizes well on this task and shows minimal overtraining. Hence, in the setting we study here, generalization performance can be acceptable without any compression phase. + +The results in (Advani & Saxe (2017)) show that, for the case of linear networks, overtraining is worst when the number of inputs matches the number of training samples, and is reduced by making the number of samples smaller or larger. Fig. 4 shows learning dynamics with the number of samples matched to the size of the network. Here overfitting is substantial, and again no compression is seen in the information plane. Comparing to the result in Fig. 3D, both networks exhibit similar information dynamics with respect to the input (no compression), but yield different generalization performance. + +Hence, in this linear analysis of a generic setting, there do not appear to be additional mechanisms that cause compression over the course of learning; and generalization behavior can be widely different for networks with the same dynamics of information compression regarding the input. We note that, in the setting considered here, all input dimensions have the same variance, and the weights of the teacher are drawn independently. Because of this, there are no special directions in the input, and each subspace of the input contains as much information as any other. It is possible that, in real world tasks, higher variance inputs are also the most likely to be relevant to the task (here, have large weights in the teacher). We have not investigated this possibility here. + +![](images/22b7932d2a586116824d010622a5076fd367f6f9658cd0958fd6f2a3a8c1187a.jpg) +Figure 4: Overtraining and information plane dynamics. (A) Average training and test mean square error for a deep linear network trained with SGD. Overtraining is substantial. Other parameters: $N _ { i } =$ 100, $\mathrm { P } = 1 0 0$ , Number of hidden units $= 1 0 0$ , Batch size $= 5$ (B) Information plane dynamics. No compression is observed, and information about the labels is lost during overtraining. (C) Average train and test accuracy $\%$ correct) for nonlinear tanh networks exhibiting modest overfitting $N = 8$ ). (D) Information plane dynamics. Overfitting occurs despite continued compression. + +![](images/6fb26702a9042ab8cf86aa3fd585a3fceb5522ab227a6c80568e9d5d17d27d7b.jpg) +Figure 5: Stochastic training and the information plane. (A) tanh network trained with SGD. (B) tanh network trained with BGD. (C) ReLU network trained with SGD. (D) ReLU network trained with BGD. Both random and non-random training procedures show similar information plane dynamics. + +To see whether similar behavior arises in nonlinear networks, we trained tanh networks in the same setting as Section 2, but with $30 \%$ of the data, which we found to lead to modest overtraining. Fig. 4C-D shows the resulting train, test, and information plane dynamics. Here the tanh networks show substantial compression, despite exhibiting overtraining. This establishes a dissociation between behavior in the information plane and generalization dynamics: networks that compress may (Fig. 1A) or may not (Fig. 4C-D) generalize well, and networks that do not compress may (Figs.1B, 3A-B) or may not (Fig. 4A-B) generalize well. + +# 4 COMPRESSION IN BATCH GRADIENT DESCENT AND SGD + +Next, we test a core theoretical claim of the information bottleneck theory of deep learning, namely that randomness in stochastic gradient descent is responsible for the compression phase. In particular, because the choice of input samples in SGD is random, the weights evolve in a stochastic way during training. + +Shwartz-Ziv & Tishby (2017) distinguish two phases of SGD optimization: in the first “drift” phase, the mean of the gradients over training samples is large relative to the standard deviation of the gradients; in the second “diffusion” phase, the mean becomes smaller than the standard deviation of the gradients. The authors propose that compression should commence following the transition from a high to a low gradient signal-to-noise ratio (SNR), i.e., the onset of the diffusion phase. The proposed mechanism behind this diffusion-driven compression is as follows. The authors state that during the diffusion phase, the stochastic evolution of the weights can be described as a Fokker-Planck equation under the constraint of small training error. Then, the stationary distribution over weights for this process will have maximum entropy, again subject to the training error constraint. Finally, the authors claim that weights drawn from this stationary distribution will maximize the entropy of inputs given hidden layer activity, $H ( X | T )$ , subject to a training error constraint, and that this training error constraint is equivalent to a constraint on the mutual information $I ( T ; Y )$ for small training error. Since the entropy of the input, $H ( X )$ , is fixed, the result of the diffusion dynamics will be to minimize $I ( X ; T ) \mathrel { \mathop : } = H ( X ) - \bar { H } ( X | \dot { T } )$ for a given value of $I ( T ; Y )$ reached at the end of the drift phase. + +However, this explanation does not hold up to either theoretical or empirical investigation. Let us assume that the diffusion phase does drive the distribution of weights to a maximum entropy distribution subject to a training error constraint. Note that this distribution reflects stochasticity of weights across different training runs. There is no general reason that a given set of weights sampled from this distribution (i.e., the weight parameters found in one particular training run) will maximize $H ( X | T )$ , the entropy of inputs given hidden layer activity. In particular, $H ( X | T )$ reflects (conditional) uncertainty about inputs drawn from the data-generating distribution, rather than uncertainty about any kind of distribution across different training runs. + +We also show empirically that the stochasticity of the SGD is not necessary for compression. To do so, we consider two distinct training procedures: offline stochastic gradient descent (SGD), which learns from a fixed-size dataset, and updates weights by repeatedly sampling a single example from the dataset and calculating the gradient of the error with respect to that single sample (the typical procedure used in practice); and batch gradient descent (BGD), which learns from a fixed-size dataset, and updates weights using the gradient of the total error across all examples. Batch gradient descent uses the full training dataset and, crucially, therefore has no randomness or diffusion-like behavior in its updates. + +We trained tanh and ReLU networks with SGD and BGD and compare their information plane dynamics in Fig. 5 (see Appendix H for a linear network). We find largely consistent information dynamics in both instances, with robust compression in tanh networks for both methods. Thus randomness in the training process does not appear to contribute substantially to compression of information about the input. This finding is consistent with the view presented in Section 2 that compression arises predominantly from the double saturating nonlinearity. + +Finally, we look at the gradient signal-to-noise ratio (SNR) to analyze the relationship between compression and the transition from high to low gradient SNR. Fig. 20 of Appendix I shows the gradient SNR over training, which in all cases shows a phase transition during learning. Hence the gradient SNR transition is a general phenomenon, but is not causally related to compression. Appendix I offers an extended discussion and shows gradient SNR transitions without compression on the MNIST dataset and for linear networks. + +# 5 SIMULTANEOUS FITTING AND COMPRESSION + +Our finding that generalization can occur without compression may seem difficult to reconcile with the intuition that certain tasks involve suppressing irrelevant directions of the input. In the extreme, if certain inputs contribute nothing but noise, then good generalization requires ignoring them. To study this, we consider a variant on the linear student-teacher setup of Section 3: we partition the input $X$ into a set of task-relevant inputs $X _ { r e l }$ and a set of task-irrelevant inputs $X _ { i r r e l }$ , and alter the teacher network so that the teacher’s weights to the task-irrelevant inputs are all zero. Hence the inputs $X _ { i r r e l }$ contribute only noise, while the $X _ { r e l }$ contain signal. We then calculate the information plane dynamics for the whole layer, and for the task-relevant and task-irrelevant inputs separately. Fig. 6 shows information plane dynamics for a deep linear neural network trained using SGD (5 samples/batch) on a task with 30 task-relevant inputs and 70 task-irrelevant inputs. While the overall dynamics show no compression phase, the information specifically about the task-irrelevant subspace does compress over the course of training. This compression process occurs at the same time as the fitting to the task-relevant information. Thus, when a task requires ignoring some inputs, the information with these inputs specifically will indeed be reduced; but overall mutual information with the input in general may still increase. + +![](images/3f982bb2a95db7292dd21c09073b2ac2ac7ccf908216d9ad8ed4dcda065e419e.jpg) +Figure 6: Simultaneous fitting and compression. (A) For a task with a large task-irrelevant subspace in the input, a linear network shows no overall compression of information about the input. (B) The information with the task-relevant subspace increases robustly over training. (C) However, the information specifically about the task-irrelevant subspace does compress after initially growing as the network is trained. + +# 6 DISCUSSION + +Our results suggest that compression dynamics in the information plane are not a general feature of deep networks, but are critically influenced by the nonlinearities employed by the network. Doublesaturating nonlinearities lead to compression, if mutual information is estimated by binning activations or by adding homoscedastic noise, while single-sided saturating nonlinearities like ReLUs do not compress in general. Consistent with this view, we find that stochasticity in the training process does not contribute to compression in the cases we investigate. Furthermore, we have found instances where generalization performance does not clearly track information plane behavior, questioning the causal link between compression and generalization. Hence information compression may parallel the situation with sharp minima: although empirical evidence has shown a correlation with generalization error in certain settings and architectures, further theoretical analysis has shown that sharp minima can in fact generalize well (Dinh et al., 2017). We emphasize that compression still may occur within a subset of the input dimensions if the task demands it. This compression, however, is interleaved rather than in a secondary phase and may not be visible by information metrics that track the overall information between a hidden layer and the input. Finally, we note that our results address the specific claims of one scheme to link the information bottleneck principle with current practice in deep networks. The information bottleneck principle itself is more general and may yet offer important insights into deep networks (Achille & Soatto, 2017). Moreover, the information bottleneck principle could yield fundamentally new training algorithms for networks that are inherently stochastic and where compression is explicitly encouraged with appropriate regularization terms (Chalk et al., 2016; Alemi et al., 2017; Kolchinsky et al., 2017). + +# ACKNOWLEDGMENTS + +We thank Ariel Herbert-Voss for useful discussions. This work was supported by grant numbers IIS 1409097 and CHE 1648973 from the US National Science Foundation, and by IARPA contract #D16PC00002. Andrew Saxe and Madhu Advani thank the Swartz Program in Theoretical Theoretical Neuroscience at Harvard University. Artemy Kolchinsky and Brendan Tracey would like to thank the Santa Fe Institute for helping to support this research. Artemy Kolchinsky was supported by Grant No. FQXi-RFP-1622 from the FQXi foundation and Grant No. CHE-1648973 from the US National Science Foundation. Brendan Tracey was supported by AFOSR MURI on Multi-Information Sources of Multi-Physics Systems under Award Number FA9550-15-1-0038. + +# REFERENCES + +A. Achille and S. Soatto. On the Emergence of Invariance and Disentangling in Deep Representations. arXiv preprint arXiv:1706.01350, 2017. + +M.S. Advani and A.M. Saxe. High-dimensional dynamics of generalization error in neural networks. arXiv preprint arXiv:1710.03667, 2017. +A.A. Alemi, I. Fischer, J.V. 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Proceedings of the 37-th Annual Allerton Conference on Communication, Control and Computing, pp. 368–377, 1999. + +# A LEARNING CURVES FOR tanh AND RELU NETWORKS + +Supplementary Figure 7 shows the learning curves for tanh and ReLU networks depicted in Fig. 1. + +![](images/ce0355c43f7ded6cd043397ccb1f76e1f4c9361219e3e513122d676b64034395.jpg) +Figure 7: Learning curves for (A) tanh neural network in 1 A and (B) ReLU neural network in $1 \textbf { B }$ . Both networks show good generalization with regards to the test data. + +# B ROBUSTNESS OF FINDINGS TO MI ESTIMATION METHOD AND NEURAL ACTIVATION FUNCTIONS + +This Appendix investigates the generality of the finding that compression is not observed in neural network layers with certain activation functions. Figure 1 of the main text shows example results using a binning-based MI estimator and a nonparametric KDE estimator, for both the tanh and ReLU activation functions. Here we describe the KDE MI estimator in detail, and present extended results on other datasets. We also show results for other activation functions. Finally, we provide entropy estimates based on another nonparametric estimator, the popular $\mathbf { k }$ -nearest neighbor approach of Kraskov et al. (2004). Our findings consistently show that double-saturating nonlinearities can yield compression, while single-sided nonlinearities do not. + +# B.1 KERNEL DENSITY ESTIMATION OF MI + +The KDE approach of Kolchinsky & Tracey (2017); Kolchinsky et al. (2017) estimates the mutual information between the input and the hidden layer activity by assuming that the hidden activity is distributed as a mixture of Gaussians. This assumption is well-suited to the present setting under the following interpretation: we take the input activity to be distributed as delta functions at each example in the dataset, corresponding to a uniform distribution over these specific samples. In other words, we assume that the empirical distribution of input samples is the true distribution. Next, the hidden layer activity $h$ is a deterministic function of the input. As mentioned in the main text and discussed in more detail in Appendix C, without the assumption of noise, this would have infinite mutual information with the input. We therefore assume for the purposes of analysis that Gaussian noise of variance $\sigma ^ { 2 }$ is added, that is, $T = h + \epsilon$ where $\epsilon \sim \mathcal { N } ( \bar { 0 } , \bar { \sigma ^ { 2 } } I )$ . Under these assumptions, the distribution of $T$ is genuinely a mixture of Gaussians, with a Gaussian centered on the hidden activity corresponding to each input sample. We emphasize again that the noise $\epsilon$ is added solely for the purposes of analysis, and is not present during training or testing the network. In this setting, an upper bound for the mutual information with the input is (Kolchinsky & Tracey, 2017; Kolchinsky et al., 2017) + +$$ +I ( T ; X ) \leq - { \frac { 1 } { P } } \sum _ { i } \log { \frac { 1 } { P } } \sum _ { j } \exp \left( - { \frac { 1 } { 2 } } { \frac { \vert \vert h _ { i } - h _ { j } \vert \vert _ { 2 } ^ { 2 } } { \sigma ^ { 2 } } } \right) +$$ + +where $P$ is the number of training samples and $h _ { i }$ denotes the hidden activity vector in response to input sample $i$ . Similarly, the mutual information with respect to the output can be calculated as + +$$ +\begin{array} { r c l } { I ( T ; Y ) } & { = } & { { \displaystyle H ( T ) - H ( T | Y ) } } \\ & { \leq } & { \displaystyle - \frac { 1 } { P } \sum _ { i } \log \frac { 1 } { P } \sum _ { j } \exp \left( - \frac { 1 } { 2 } \frac { \| h _ { i } - h _ { j } \| _ { 2 } ^ { 2 } } { \sigma ^ { 2 } } \right) } \\ & & { \displaystyle - \sum _ { l } ^ { L } p _ { l } \left[ - \frac { 1 } { P _ { l } } \sum _ { i , Y _ { i } = l } \log \frac { 1 } { P _ { l } } \sum _ { j , Y _ { j } = l } \exp \left( - \frac { 1 } { 2 } \frac { \| h _ { i } - h _ { j } \| _ { 2 } ^ { 2 } } { \sigma ^ { 2 } } \right) \right] } \end{array} +$$ + +where $L$ is the number of output labels, $P _ { l }$ denotes the number of data samples with output label $l$ , $p _ { l } = P _ { l } / P$ denotes the probability of output label $l$ , and the sums over $i , Y _ { i } = l$ indicate a sum over all examples with output label $l$ . + +![](images/f89ebb8650e008588c11ffbb01a6605fa94820e9aa6ba83b3408ba661f14ce97.jpg) +Figure 8: Information plane dynamics for the network architecture and training dataset of ShwartzZiv & Tishby (2017), estimated with the nonparametric KDE method of Kolchinsky & Tracey (2017); Kolchinsky et al. (2017) and averaged over 50 repetitions. (A) tanh neural network layers show compression. (B) ReLU neural network layers show no compression. (C) The soft-sign activation function, a double-saturating nonlinearity that saturates more gently than tanh, shows modest compression. (D) The soft-plus activation function, a smoothed version of the ReLU, exhibits no compression. Hence double-saturating nonlinearities exhibit the compression effect while singlesaturating nonlinearities do not. + +Figure 8A-B shows the result of applying this MI estimation method on the dataset and network architecture of Shwartz-Ziv & Tishby (2017), with MI estimated on the full dataset and averaged over 50 repetitions. Mutual information was estimated using data samples from the test set, and we took the noise variance $\sigma ^ { 2 } = 0 . 1$ . These results look similar to the estimate derived from binning, with compression in tanh networks but no compression in ReLU networks. Relative to the binning estimate, it appears that compression is less pronounced in the KDE method. + +Figure 1C-D of the main text shows the results of this estimation technique applied to a neural network of size $7 8 4 - 1 0 2 4 - 2 0 - 2 0 - 2 0 - 1 0$ on the MNIST handwritten digit classification dataset. The network was trained using SGD with minibatches of size 128. As before, mutual information was estimated using data samples from the test set, and we took the noise variance $\sigma ^ { 2 } = 0 . 1$ . The smaller layer sizes in the top three hidden layers were selected to ensure the quality of the kernel density estimator given the amount of data in the test set, since the estimates are more accurate for smaller-dimensional data. Because of computational expense, the MNIST results are from a single training run. + +More detailed results for the MNIST dataset are provided in Figure 9 for the tanh activation function, and in Figure 10 for the ReLU activation function. In these figures, the first row shows the evolution of the cross entropy loss (on both training and testing data sets) during training. The second row shows the mutual information between input and the activity of different hidden layers, using the nonparametric KDE estimator described above. The blue region in the second row shows the range of possible MI values, ranging from the upper bound described above (Eq. 10) to the following lower + +bound (Kolchinsky & Tracey, 2017), + +$$ +\begin{array} { r c l } { { I ( T ; Y ) } } & { { \geq } } & { { \displaystyle - \frac { 1 } { P } \sum _ { i } \log \frac { 1 } { P } \sum _ { j } \exp \left( - \frac { 1 } { 2 } \frac { \left\| h _ { i } - h _ { j } \right\| _ { 2 } ^ { 2 } } { 4 \sigma ^ { 2 } } \right) } } \\ { { } } & { { } } & { { \displaystyle - \sum _ { l } ^ { L } p _ { l } \left[ - \frac { 1 } { P _ { l } } \sum _ { i , Y _ { i } = l } \log \frac { 1 } { P _ { l } } \sum _ { j , Y _ { j } = l } \exp \left( - \frac { 1 } { 2 } \frac { \left\| h _ { i } - h _ { j } \right\| _ { 2 } ^ { 2 } } { 4 \sigma ^ { 2 } } \right) \right] . } } \end{array} +$$ + +The third row shows the mutual information between input and activity of different hidden layers, estimated using the binning method (here, the activity of each neuron was discretized into bins of size 0.5). For both the second and third rows, we also plot the entropy of the inputs, $H ( X )$ , as a dashed line. $H ( X )$ is an upper bound on the mutual information $I ( X ; T )$ , and is computed using the assumption of a uniform distribution over the 10,000 testing points in the MNIST dataset, giving $H ( X ) = \log _ { 2 } { 1 0 0 0 0 }$ . + +Finally, the fourth row visualizes the dynamics of the SGD updates during training. For each layer and epoch, the green line shows the $\ell _ { 2 }$ norm of the weights. We also compute the vector of mean updates across SGD minibatches (this vector has one dimension for each weight parameter), as well as the vector of the standard deviation of the updates across SGD minibatches. The $\ell _ { 2 }$ norm of the mean update vector is shown in blue, and the $\ell _ { 2 }$ norm of the standard deviation vector is shown in orange. The gradient SNR, computed as the ratio of the norm of the mean vector to the norm of the standard deviation vector, is shown in red. For both the tanh and ReLU networks, the gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases. Importantly, this phase transition occurs despite a lack of compression in the ReLU network, indicating that noise in SGD updates does not yield compression in this setting. + +![](images/cc8295ba9fbcdf4553fd120007b118b080b408782fb4c7b756ca91d6a943022a.jpg) +Figure 9: Detailed tanh activation function results on MNIST. Row 1: Loss over training. Row 2: Upper and lower bounds for the mutual information $I ( X ; T )$ between the input $( X )$ and each layer’s activity $( T )$ , using the nonparametric KDE estimator (Kolchinsky & Tracey, 2017; Kolchinsky et al., 2017). Dotted line indicates $H ( X ) = \log _ { 2 } { 1 0 0 0 0 }$ , the entropy of a uniform distribution over 10,000 testing samples. Row 3: Binning-based estimate of the mutual information $I ( X ; T )$ , with each neuron’s activity discretized using a bin size of 0.5. Row 4: Gradient SNR and weight norm dynamics. The gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases. + +![](images/6b5d4819da8c875280c0636b21620dd6a2ea91f5c10ae7ab67fc192db6fbb0d4.jpg) +Figure 10: Detailed ReLU activation function results on MNIST. Row 1: Loss over training. Row 2: Upper and lower bounds for the mutual information $I ( X ; T )$ between the input $( X )$ and each layer’s activity $( T )$ , using the nonparametric KDE estimator (Kolchinsky & Tracey, 2017; Kolchinsky et al., 2017). Dotted line indicates $H ( X ) = \log _ { 2 } { 1 0 0 0 0 }$ , the entropy of a uniform distribution over 10,000 testing samples. Row 3: Binning-based estimate of the mutual information $I ( X ; T )$ , with each neuron’s activity discretized using a bin size of 0.5. Row 4: Gradient SNR and weight norm dynamics. The gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases. Importantly, this phase transition occurs despite a lack of compression in the ReLU network, indicating that noise in SGD updates does not yield compression in this setting. + +![](images/6c1149a7423a2791869e903f30e00c9268d38906bea86674e43fe62d02d3186f.jpg) +Figure 11: Alternative activation functions. + +# B.2 OTHER ACTIVATION FUNCTIONS + +Next, in Fig. 8C-D, we show results from the kernel MI estimator from two additional nonlinear activation functions, the softsign function + +$$ +f ( x ) = { \frac { x } { 1 + | x | } } , +$$ + +and the softplus function + +$$ +f ( x ) = \ln ( 1 + e ^ { x } ) . +$$ + +These functions are plotted next to tanh and ReLU in Fig. 11. The softsign function is similar to tanh but saturates more slowly, and yields less compression than tanh. The softplus function is a smoothed version of the ReLU, and yields similar dynamics with no compression. Because softplus never saturates fully to zero, it retains more information with respect to the input than ReLUs in general. + +# B.3 KRASKOV ESTIMATOR + +We additionally investigated the widely-used nonparametric MI estimator of Kraskov et al. (2004). This estimator uses nearest neighbor distances between samples to compute an estimate of the entropy of a continuous random variable. Here we focused for simplicity only on the compression phenomenon in the mutual information between the input and hidden layer activity, leaving aside the information with respect to the output (as this is not relevant to the compression phenomenon). Again, without additional noise assumptions, the MI between the hidden representation and the input would be infinite because the mapping is deterministic. Rather than make specific noise assumptions, we instead use the Kraskov method to estimate the entropy of the hidden representations $T$ . Note that the entropy of $T$ is the mutual information up to an unknown constant so long as the noise assumption is homoscedastic, that is, $T = h + Z$ where the random variable $Z$ is independent of $X$ . To see this, note that + +$$ +\begin{array} { l l l } { { I ( T ; X ) } } & { { = } } & { { H ( T ) - H ( T | X ) } } \\ { { } } & { { = } } & { { H ( T ) - H ( Z ) } } \\ { { } } & { { = } } & { { H ( T ) - c } } \end{array} +$$ + +where the constant $c = H ( Z )$ . Hence observing compression in the layer entropy $H ( T )$ is enough to establish that compression occurs in the mutual information. + +The Kraskov estimate is given by + +$$ +\frac { d } { P } \sum _ { i = 1 } ^ { P } \log ( r _ { i } + \epsilon ) + \frac { d } { 2 } \log ( \pi ) - \log \Gamma ( d / 2 + 1 ) + \psi ( P ) - \psi ( k ) +$$ + +where $d$ is the dimension of the hidden representation, $P$ is the number of samples, $r _ { i }$ is the distance to the $k$ -th nearest neighbor of sample $i$ , $\epsilon$ is a small constant for numerical stability, $\Gamma ( \cdot )$ is the + +Gamma function, and $\psi ( \cdot )$ is the digamma function. Here the parameter $\epsilon$ prevents infinite terms when the nearest neighbor distance ri = 0 for some sample. We took  = 10−16. + +Figure 12 shows the entropy over training for tanh and ReLU networks trained on the dataset of and with the network architecture in Shwartz-Ziv & Tishby (2017), averaged over 50 repeats. In these experiments, we used $k = 2$ . Compression would correspond to decreasing entropy over the course of training, while a lack of compression would correspond to increasing entropy. Several tanh layers exhibit compression, while the ReLU layers do not. Hence qualitatively, the Kraskov estimator returns similar results to the binning and KDE strategies. + +![](images/d35dbe7c6e40d832895cc4e561a74abc5f2ec6a2b6e899768df8078d5351eb42.jpg) +Figure 12: Entropy dynamics over training for the network architecture and training dataset of Shwartz-Ziv & Tishby (2017), estimated with the nonparametric $\mathbf { k }$ -nearest-neighbor-based method of Kraskov et al. (2004). Here the $\mathbf { X } ^ { } -$ -axis is epochs of training time, and the y-axis plots the entropy of the hidden representation, as calculated using nearest-neighbor distances. Note that in this setting, if $T$ is considered to be the hidden activity plus independent noise, the entropy is equal to the mutual information up to a constant (see derivation in text). Layers 0-4 correspond to the hidden layers of size 10-7-5-4-3. (A) tanh neural network layers can show compression over the course of training. (B) ReLU neural network layers show no compression. + +# C NOISE ASSUMPTIONS AND DISCRETE VS CONTINUOUS ENTROPY + +A recurring theme in the results reported in this paper is the necessity of noise assumptions to yield a nontrivial information theoretic analysis. Here we give an extended discussion of this phenomenon, and of issues relating to discrete entropy as opposed to continuous (differential) entropy. + +The activity of a neural network is often a continuous deterministic function of its input. That is, in response to an input $X$ , a specific hidden layer might produce activity $h = f ( X )$ for some function $f$ . The mutual information between $h$ and $X$ is given by + +$$ +\begin{array} { l l l } { { I ( h ; X ) } } & { { = } } & { { H ( h ) - H ( h | X ) . } } \end{array} +$$ + +If $h$ were a discrete variable, then the entropy would be given by + +$$ +H ( h ) = - \sum _ { i = 1 } ^ { N } p _ { i } \log p _ { i } +$$ + +where $p _ { i }$ is the probability of the discrete symbol $i$ , as mentioned in the main text. Then $H ( h | X ) = 0$ because the mapping is deterministic and we have $I ( h ; X ) = H ( h )$ . + +However $h$ is typically continuous. The continuous entropy, defined for a continuous random variable $Z$ with density $p _ { Z }$ by analogy to Eqn. (18) as + +$$ +H ( Z ) = - \int p _ { Z } ( z ) \log p _ { Z } ( z ) d z , +$$ + +can be negative and possibly infinite. In particular, note that if $p _ { Z }$ is a delta function, then $H ( Z ) =$ $- \infty$ . The mutual information between hidden layer activity $h$ and the input $X$ for continuous $h , X$ is + +$$ +I ( h ; X ) = H ( h ) - H ( h | X ) . +$$ + +Now $H ( h | X ) = - \infty$ since given the input $X$ , the hidden activity $h$ is distributed as a delta function at $f ( X )$ . The mutual information is thus generally infinite, so long as the hidden layer activity has finite entropy $H ( h )$ is finite). + +![](images/db291cd24878a5a577925a9f57c17f1e80ff0653e022e4f8350fe21e76b70f02.jpg) +Figure 13: Effect of binning strategy on minimal three neuron model. Mutual information for the simple three neuron model shown in Fig. 2 with bin edges $b _ { i } \in \mathrm { t a n h } ( \operatorname* { l i n s p a c e } ( - 5 0 , 5 0 , N ) )$ . In contrast to linear binning, the mutual information continues to increase as weights grow. + +To yield a finite mutual information, some noise in the mapping is required such that $H ( h | X )$ remains finite. A common choice (and one adopted here for the linear network, the nonparametric kernel density estimator, and the $\mathbf { k }$ -nearest neighbor estimator) is to analyze a new variable with additive noise, $T = h + Z$ , where $Z$ is a random variable independent of $X$ . Then $H ( T | X ) = H ( Z )$ which allows the overall information $I ( T ; X ) = H ( T ) - H ( Z )$ to remain finite. This noise assumption is not present in the actual neural networks either during training or testing, and is made solely for the purpose of calculating the mutual information. + +Another strategy is to partition the continuous variable $h$ into a discrete variable $T$ , for instance by binning the values (the approach taken in Shwartz-Ziv & Tishby (2017)). This allows use of the discrete entropy, which remains finite. Again, however, in practice the network does not operate on the binned variables $T$ but on the continuous variables $h$ , and the binning is solely for the purpose of calculating the mutual information. Moreover, there are many possible binning strategies, which yield different discrete random variables, and different mutual information with respect to the input. The choice of binning strategy is an assumption analogous to choosing a type of noise to add to the representation in the continuous case: because there is in fact no binning in the operation of the network, there is no clear choice for binning methodology. The strategy we use in binning-based experiments reported here is the following: for bounded activations like the tanh activation, we use evenly spaced bins between the minimum and maximum limits of the function. For unbounded activations like ReLU, we first train the network completely; next identify the minimum and maximum hidden activation over all units and all training epochs; and finally bin into equally spaced bins between these minimum and maximum values. We note that this procedure places no restriction on the magnitude that the unbounded activation function can take during training, and yields the same MI estimate as using infinite equally spaced bins (because bins for activities larger than the maximum are never seen during training). + +As an example of another binning strategy that can yield markedly different results, we consider evenly spaced bins in a neuron’s net input, rather than its activity. That is, instead of evenly spaced bins in the neural activity, we determine the bin edges by mapping a set of evenly spaced values through the neural nonlinearity. For tanh, for instance, this spaces bins more tightly in the saturation region as compared to the linear region. Figure 13 shows the results of applying this binning strategy to the minimal three neuron model with tanh activations. This binning scheme captures more information as the weights of the network grow larger. Figure 14 shows information plane dynamics for this binning structure. The tanh network no longer exhibits compression. (We note that the broken DPI in this example is an artifact of performing binning only for analysis, as discussed below). + +![](images/12c7d31d8a03d347bac1cbe449138df659dc492793257cc0c61d13e2608ea4fa.jpg) +Figure 14: Effect of binning strategy on information plane dynamics. Results for the same tanh network and training regime as 1A, but with bin edges $b _ { i } \in$ tanh(linspace $( - 5 0 , 5 0 , N )$ ). Measured with this binning structure, there is no compression in most layers. + +Any implementation of a neural network on digital hardware is ultimately of finite precision, and hence is a binned, discrete representation. However, it is a very high resolution binning compared to that used here or by Shwartz-Ziv & Tishby (2017): single precision would correspond to using roughly $2 ^ { 3 2 }$ bins to discretize each hidden unit’s activity, as compared to the 30-100 used here. If the binning is fine-grained enough that each input $X$ yields a different binned activity pattern $h$ , then $H ( h ) \stackrel { - } { = } \log ( P )$ where $P$ is the number of examples in the dataset, and there will be little to no change in information during training. As an example, we show in Fig. 15 the result of binning at full machine precision. + +Finally, we note two consequences of the assumption of noise/binning for the purposes of analysis. First, this means that the data processing inequality (DPI) does not apply to the noisy/binned mutual information estimates. The DPI states that information can only be destroyed through successive transformations, that is, if $X h _ { 1 } h _ { 2 }$ form a Markov chain, then $I ( X ; \bar { h _ { 1 } } ) \geq I ( \bar { X ; } h _ { 2 } )$ (see, eg, Tishby & Zaslavsky (2015)). Because noise is added only for the purpose of analysis, however, this does not apply here. In particular, for the DPI to apply, the noise added at lower layers would have to propagate through the network to higher layers. That is, if the transformation from hidden layer 1 to hidden layer 2 is $h _ { 2 } = f ( h _ { 1 } )$ and $T _ { 1 } = h _ { 1 } + Z _ { 1 }$ is the hidden layer activity after adding noise, then the DPI would hold for the variable $\tilde { T } _ { 2 } = f ( T _ { 1 } ) + Z _ { 2 } = f ( h _ { 1 } + Z _ { 1 } ) + Z _ { 2 }$ , not the quantity + +![](images/e2187148ec2ae4d72ade118f8c38e42693609c24edb83c7050bd489b4d6120ca.jpg) +Figure 15: Effect of binning at full machine precision. (A) ReLU network. (B) tanh network. Information in most layers stays pinned to $\log _ { 2 } ( P ) = 1 2$ . Compression is only observed in the highest and smallest layers near the very end of training, when the saturation of tanh is strong enough to saturate machine precision. + +$T _ { 2 } = h _ { 2 } + Z _ { 2 } = f ( h _ { 1 } ) + Z _ { 2 }$ used in the analysis. Said another way, the Markov chain for $T _ { 2 }$ is $X h _ { 1 } h _ { 2 } T _ { 2 }$ , not $X h _ { 1 } T _ { 1 } T _ { 2 }$ , so the DPI states only that $I ( X ; h _ { 1 } ) \ge I ( X ; T _ { 2 } )$ . + +A second consequence of the noise assumption is the fact that the mutual information is no longer invariant to invertible transformations of the hidden activity $h$ . A potentially attractive feature of a theory based on mutual information is that it can allow for comparisons between different architectures: mutual information is invariant to any invertible transformation of the variables, so two hidden representations could be very different in detail but yield identical mutual information with respect to the input. However, once noise is added to a hidden representation, this is no longer the case: the variable $T = h + Z$ is not invariant to reparametrizations of $h$ . As a simple example, consider a minimal linear network with scalar weights $w _ { 1 }$ and $w _ { 2 }$ that computes the output ${ \hat { y } } = w _ { 2 } w _ { 1 } X$ . The hidden activity is $h = w _ { 1 } X$ . Now consider the family of networks in which we scale down $w _ { 1 }$ and scale up $w _ { 2 }$ by a factor $c \neq 0$ , that is, these networks have weights $\tilde { w } _ { 1 } = w _ { 1 } / c$ and $\tilde { w } _ { 2 } = c w _ { 2 }$ , yielding the exact same input-output map $\hat { y } = \tilde { w } _ { 2 } \tilde { w } _ { 1 } X = c w _ { 2 } ( w _ { 1 } / c ) X = w _ { 2 } w _ { 1 } X$ . Because they compute the same function, they necessarily generalize identically. However after introducing the noise assumption the mutual information is + +$$ +\begin{array} { l l l } { { I ( T ; X ) } } & { { = } } & { { \log \left( w _ { 1 } ^ { 2 } / c ^ { 2 } + \sigma _ { M I } ^ { 2 } \right) - \log \left( \sigma _ { M I } ^ { 2 } \right) } } \end{array} +$$ + +where we have taken the setting in Section 3 in which $X$ is normal Gaussian, and independent Gaussian noise of variance $\sigma _ { M I } ^ { 2 }$ is added for the purpose of MI computation. Clearly, the mutual information is now dependent on the scaling $c$ of the internal layer, even though this is an invertible linear transformation of the representation. Moreover, this shows that networks which generalize identically can nevertheless have very different mutual information with respect to the input when it is measured in this way. + +# D WEIGHT NORMS OVER TRAINING + +Our argument relating neural saturation to compression in mutual information relies on the notion that in typical training regimes, weights begin small and increase in size over the course of training. We note that this is a virtual necessity for a nonlinearity like tanh, which is linear around the origin: when initialized with small weights, the activity of a tanh network will be in this linear regime and the network can only compute a linear function of its input. Hence a real world nonlinear task can only be learned by increasing the norm of the weights so as to engage the tanh nonlinearity on some examples. This point can also be appreciated from norm-based capacity bounds on neural networks, which show that, for instance, the Rademacher complexity of a neural network with small weights must be low (Bartlett & Mendelson, 2002; Neyshabur et al., 2015). Finally, as an empirical matter, the networks trained in this paper do in fact increase the norm of their weights over the course of training, as shown by the green lines in Figure 20 for tanh and ReLU networks in the training setting of Shwartz-Ziv & Tishby (2017); Figures 9 and 10 for the MNIST networks; and Figure 21 for a linear network. + +# E HISTOGRAMS OF NEURAL ACTIVATIONS + +Supplementary Figures 16 and 17 show histograms of neural activities over the course of training in tanh and ReLU networks respectively. + +![](images/8bbc07b14318e40e4fb277a40f930af47fb6bee0d59af7119d1a65cf2931ed0c.jpg) +Figure 16: Histogram of neural activities in a tanh network during training. The final three layers eventually saturate in the top and bottom bins corresponding to the saturation limits of the tanh activation function, explaining the compression observed in tanh. x-axis: training time in epochs. y-axis: Hidden activity bin values from lowest to highest. Colormap: density of hidden layer activities across all input examples. + +![](images/5f209aa29a04ff3253199dfe5ae51898ad1ec81c4d32a5a079b30809f462dc92.jpg) +Figure 17: Histogram of neural activities in a ReLU network during training. ReLU layers 1-5 have a roughly constant fraction of activities at zero, corresponding to instances where the ReLU is off; the nonzero activities disperse over the course of training without bound, yielding higher entropy distributions. The sigmoid output layer 6 converges to its saturation limits, and is the only layer that compresses during training (c.f. Fig. 1B). $\mathbf { X }$ -axis: training time in epochs. y-axis: Hidden activity value. Colormap: density of hidden layer activities across all input examples. + +# F INFORMATION PLANE DYNAMICS IN DEEPER LINEAR NETWORKS + +Supplementary Figure 18 shows information plane dynamics for a deep neural network with five hidden layers each containing 50 hidden units. + +![](images/cba1fe5de44777e480b97d883918063f6902a3728f471a1abd20879b524f24ab.jpg) +Figure 18: Information plane dynamics in a deep linear neural network. (A) Train and test error during learning. (B) Information plane dynamics. No compression is visible. + +# G LINEAR MUTUAL INFORMATION CALCULATION + +For the linear setting considered here, the mutual information between a hidden representation $T$ and the output $Y$ may be calculated using the relations + +$$ +\begin{array} { c } { { { \cal H } ( Y ) = \displaystyle \frac { N _ { o } } { 2 } \log ( 2 \pi e ) + \displaystyle \frac { 1 } { 2 } \log | W _ { o } W _ { o } ^ { T } + \sigma _ { o } ^ { 2 } I _ { N _ { o } } | , } } \\ { { { \cal H } ( T ) = \displaystyle \frac { N _ { h } } { 2 } \log ( 2 \pi e ) + \displaystyle \frac { 1 } { 2 } \log | \bar { W } \bar { W } ^ { T } + \sigma _ { M I } ^ { 2 } I _ { N _ { h } } | , } } \\ { { { \cal H } ( Y ; T ) = \displaystyle \frac { N _ { o } + N _ { h } } { 2 } \log ( 2 \pi e ) + \displaystyle \frac { 1 } { 2 } \log | \bar { W } \bar { W } ^ { T } + \sigma _ { M I } ^ { 2 } I _ { N _ { h } } \quad \quad \bar { W } W _ { o } ^ { T } , } } \\ { { { \cal I } ( Y ; T ) = { \cal H } ( Y ) + { \cal H } ( T ) - { \cal H } ( Y ; T ) . } } \end{array} +$$ + +# H STOCHASTIC VS BATCH TRAINING + +![](images/547f68d75db31823ea92f3a340ffd0af1f2555a96902c610b5d4b54c25c651f4.jpg) +Figure 19 shows information plane dynamics for stochastic and batch gradient descent learning in a linear network. Randomness in the training process does not dramatically alter the information plane dynamics. +Figure 19: Effect of stochastic training in linear networks. (A) Information plane dynamics for stochastic gradient descent in a linear network (same setting as Fig. 4). (B) Information plane dynamics for batch gradient descent. + +# I GRADIENT SNR PHASE TRANSITION + +The proposed mechanism of compression in Shwartz-Ziv & Tishby (2017) is noise arising from stochastic gradient descent training. The results in Section 4 of the main text show that compression still occurs under batch gradient descent learning, suggesting that in fact noise in the gradient updates is not the cause of compression. Here we investigate a related claim, namely that during training, networks switch between two phases. These phases are defined by the ratio of the mean of the gradient to the standard deviation of the gradient across training examples, called the gradient signal-to-noise ratio. In the first “drift” phase, the SNR is high, while in the second “diffusion” phase the SNR is low. Shwartz-Ziv & Tishby (2017) hypothesize that the drift phase corresponds to movement toward the minimum with no compression, while the diffusion phase corresponds to a constrained diffusion in weight configurations that attain the optimal loss, during which representations compress. However, two phases of gradient descent have been described more generally, sometimes known as the transient and stochastic phases or search and convergence phases (Murata, 1998; Chee & Toulis, 2017), suggesting that these phases might not be related specifically to compression behavior. + +In Fig. 20 we plot the gradient SNR over the course of training for the tanh and ReLU networks in the standard setting of Shwartz-Ziv & Tishby (2017). In particular, for each layer $l$ we calculate the mean and standard deviation as + +$$ +\begin{array} { r c l } { { m _ { l } } } & { { = } } & { { \displaystyle \left\| \left. \frac { \partial E } { \partial W _ { l } } \right. \right\| _ { F } } } \\ { { s _ { l } } } & { { = } } & { { \displaystyle \left\| \mathrm { S T D } \left( \frac { \partial E } { \partial W _ { l } } \right) \right\| _ { F } } } \end{array} +$$ + +where $\langle \cdot \rangle$ denotes the mean and $S T D ( \cdot )$ denotes the element-wise standard deviation across all training samples, and $\left\| \cdot \right\| _ { F }$ denotes the Frobenius norm. The gradient SNR is then the ratio $m _ { l } / s _ { l }$ We additionally plot the norm of the weights $\| W _ { l } \| _ { F }$ over the course of training. + +Both tanh and ReLU networks yield a similar qualitative pattern, with SNR undergoing a step-like transition to a lower value during training. Figures 9 and 10, fourth row, show similar plots for MNIST-trained networks. Again, SNR undergoes a transition from high to low over training. Hence the two phase nature of gradient descent appears to hold across the settings that we examine here. Crucially, this finding shows that the SNR transition is not related to the compression phenomenon because ReLU networks, which show the gradient SNR phase transition, do not compress. + +Finally, to show the generality of the two-phase gradient SNR behavior and its independence from compression, we develop a minimal model of this phenomenon in a three neuron linear network. We consider the student-teacher setting of Fig. 3 but with $N _ { i } = N _ { h } = 1$ , such that the input and hidden layers have just a single neuron (as in the setting of Fig. 2). Here, with just a single hidden neuron, clearly there can be no compression so long as the first layer weight increases over the course of training. Figure 21AC shows that even in this simple setting, the SNR shows the phase transition but the weight norm increases over training. Hence again, the two phases of the gradient are present even though there is no compression. To intuitively understand the source of this behavior, note that the weights are initialized to be small and hence early in learning all must be increased in magnitude, yielding a consistent mean gradient. Once the network reaches the vicinity of the minimum, the mean weight change across all samples by definition goes to zero. The standard deviation remains finite, however, because on some specific examples error could be improved by increasing or decreasing the weights–even though across the whole dataset the mean error has been minimized. + +Hence overall, our results show that a two-phase structure in the gradient SNR occurs in all settings we consider, even though compression occurs only in a subset. The gradient SNR behavior is therefore not causally related to compression dynamics, consistent with the view that saturating nonlinearities are the primary source of compression. + +![](images/fb8ca7128e6032058471eb13f01a0a5dba496e6d717ed509113be8ea4d25e22c.jpg) +Figure 20: Gradient SNR phase transition. (A) tanh networks trained in the standard setting of Shwartz-Ziv & Tishby (2017) show a phase transition in every layer. (B) ReLU networks also show a phase transition in every layer, despite exhibiting no compression. + +![](images/a9a3a5f252886534d662bb947459c62130ced799c330bd5e27586470c9beb20b.jpg) +Figure 21: Minimal model exhibiting gradient SNR phase transition. Here a three neuron linear network (architecture $1 - 1 - 1 )$ learns to approximate a teacher. Other parameters are teacher $S N R = 1$ , number of training samples $P = 1 0 0$ , learning rate .001. Left column: (A) The loss over training with SGD (minibatch size 1). (C) The resulting gradient SNR dynamics. Right column: (B) The loss over training with BGD. (D) The resulting gradient SNR dynamics averaging over all training samples (not minibatches, see text). \ No newline at end of file diff --git a/parse/train/ry_WPG-A-/ry_WPG-A-_content_list.json b/parse/train/ry_WPG-A-/ry_WPG-A-_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..77c4d87236f280caeaedf87f5084e03191f2b1f0 --- /dev/null +++ b/parse/train/ry_WPG-A-/ry_WPG-A-_content_list.json @@ -0,0 +1,1861 @@ +[ + { + "type": "text", + "text": "ON THE INFORMATION BOTTLENECK THEORY OF DEEP LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 99, + 620, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Andrew M. Saxe, Yamini Bansal, Joel Dapello, Madhu Advani \nHarvard University \n{asaxe,madvani}@fas.harvard.edu,{ybansal,dapello}@g.harvard.edu ", + "bbox": [ + 183, + 169, + 802, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Artemy Kolchinsky, Brendan D. Tracey ", + "bbox": [ + 186, + 233, + 459, + 247 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Santa Fe Institute {artemyk,tracey.brendan}@gmail.com ", + "bbox": [ + 186, + 250, + 519, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "David D. Cox \nHarvard University \nMIT-IBM Watson AI Lab \ndavidcox@fas.harvard.edu \ndavid.d.cox@ibm.com ", + "bbox": [ + 555, + 233, + 794, + 303 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 340, + 544, + 354 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The practical successes of deep neural networks have not been matched by theoretical progress that satisfyingly explains their behavior. In this work, we study the information bottleneck (IB) theory of deep learning, which makes three specific claims: first, that deep networks undergo two distinct phases consisting of an initial fitting phase and a subsequent compression phase; second, that the compression phase is causally related to the excellent generalization performance of deep networks; and third, that the compression phase occurs due to the diffusion-like behavior of stochastic gradient descent. Here we show that none of these claims hold true in the general case. Through a combination of analytical results and simulation, we demonstrate that the information plane trajectory is predominantly a function of the neural nonlinearity employed: double-sided saturating nonlinearities like tanh yield a compression phase as neural activations enter the saturation regime, but linear activation functions and single-sided saturating nonlinearities like the widely used ReLU in fact do not. Moreover, we find that there is no evident causal connection between compression and generalization: networks that do not compress are still capable of generalization, and vice versa. Next, we show that the compression phase, when it exists, does not arise from stochasticity in training by demonstrating that we can replicate the IB findings using full batch gradient descent rather than stochastic gradient descent. Finally, we show that when an input domain consists of a subset of task-relevant and task-irrelevant information, hidden representations do compress the task-irrelevant information, although the overall information about the input may monotonically increase with training time, and that this compression happens concurrently with the fitting process rather than during a subsequent compression period. ", + "bbox": [ + 232, + 371, + 766, + 702 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 726, + 336, + 742 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep neural networks (Schmidhuber, 2015; LeCun et al., 2015) are the tool of choice for real-world tasks ranging from visual object recognition (Krizhevsky et al., 2012), to unsupervised learning (Goodfellow et al., 2014; Lotter et al., 2016) and reinforcement learning (Silver et al., 2016). These practical successes have spawned many attempts to explain the performance of deep learning systems (Kadmon & Sompolinsky, 2016), mostly in terms of the properties and dynamics of the optimization problem in the space of weights (Saxe et al., 2014; Choromanska et al., 2015; Advani & Saxe, 2017), or the classes of functions that can be efficiently represented by deep networks (Montufar et al., 2014; Poggio et al., 2017). This paper analyzes a recent inventive proposal to study the dynamics of learning through the lens of information theory (Tishby & Zaslavsky, 2015; Shwartz-Ziv & Tishby, 2017). In this view, deep learning is a question of representation learning: each layer of a deep neural network can be seen as a set of summary statistics which contain some but not all of the information present in the input, while retaining as much information about the target output as possible. The amount of information in a hidden layer regarding the input and output can then be measured over the course of learning, yielding a picture of the optimization process in the information plane. Crucially, this method holds the promise to serve as a general analysis that can be used to compare different architectures, using the common currency of mutual information. Moreover, the elegant information bottleneck (IB) theory provides a fundamental bound on the amount of input compression and target output information that any representation can achieve (Tishby et al., 1999). The IB bound thus serves as a method-agnostic ideal to which different architectures and algorithms may be compared. ", + "bbox": [ + 174, + 757, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 200 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A preliminary empirical exploration of these ideas in deep neural networks has yielded striking findings (Shwartz-Ziv & Tishby, 2017). Most saliently, trajectories in the information plane appear to consist of two distinct phases: an initial “fitting” phase where mutual information between the hidden layers and both the input and output increases, and a subsequent “compression” phase where mutual information between the hidden layers and the input decreases. It has been hypothesized that this compression phase is responsible for the excellent generalization performance of deep networks, and further, that this compression phase occurs due to the random diffusion-like behavior of stochastic gradient descent. ", + "bbox": [ + 174, + 208, + 825, + 319 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Here we study these phenomena using a combination of analytical methods and simulation. In Section 2, we show that the compression observed by Shwartz-Ziv & Tishby (2017) arises primarily due to the double-saturating tanh activation function used. Using simple models, we elucidate the effect of neural nonlinearity on the compression phase. Importantly, we demonstrate that the ReLU activation function, often the nonlinearity of choice in practice, does not exhibit a compression phase. We discuss how this compression via nonlinearity is related to the assumption of binning or noise in the hidden layer representation. To better understand the dynamics of learning in the information plane, in Section 3 we study deep linear networks in a tractable setting where the mutual information can be calculated exactly. We find that deep linear networks do not compress over the course of training for the setting we examine. Further, we show a dissociation between generalization and compression. In Section 4, we investigate whether stochasticity in the training process causes compression in the information plane. We train networks with full batch gradient descent, and compare the results to those obtained with stochastic gradient descent. We find comparable compression in both cases, indicating that the stochasticity of SGD is not a primary factor in the observed compression phase. Moreover, we show that the two phases of SGD occur even in networks that do not compress, demonstrating that the phases are not causally related to compression. These results may seem difficult to reconcile with the intuition that compression can be necessary to attain good performance: if some input channels primarily convey noise, good generalization requires excluding them. Therefore, in Section 5 we study a situation with explicitly task-relevant and task-irrelevant input dimensions. We show that the hidden-layer mutual information with the task-irrelevant subspace does indeed drop during training, though the overall information with the input increases. However, instead of a secondary compression phase, this task-irrelevant information is compressed at the same time that the taskrelevant information is boosted. Our results highlight the importance of noise assumptions in applying information theoretic analyses to deep learning systems, and put in doubt the generality of the IB theory of deep learning as an explanation of generalization performance in deep architectures. ", + "bbox": [ + 174, + 327, + 825, + 672 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 COMPRESSION AND NEURAL NONLINEARITIES ", + "text_level": 1, + "bbox": [ + 176, + 695, + 598, + 712 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The starting point for our analysis is the observation that changing the activation function can markedly change the trajectory of a network in the information plane. In Figure 1A, we show our replication of the result reported by Shwartz-Ziv & Tishby (2017) for networks with the tanh nonlinearity.1 This replication was performed with the code supplied by the authors of Shwartz-Ziv & Tishby (2017), and closely follows the experimental setup described therein. Briefly, a neural network with 7 fully connected hidden layers of width 12-10-7-5-4-3-2 is trained with stochastic gradient descent to produce a binary classification from a 12-dimensional input. In our replication we used 256 randomly selected samples per batch. The mutual information of the network layers with respect to the input and output variables is calculated by binning the neuron’s tanh output activations into 30 equal intervals between -1 and 1. Discretized values for each neuron in each layer are then used to directly calculate the joint distributions, over the 4096 equally likely input patterns and true output labels. In line with prior work (Shwartz-Ziv & Tishby, 2017), the dynamics in Fig. 1 show a transition between an initial fitting phase, during which information about the input increases, and a subsequent compression phase, during which information about the input decreases. ", + "bbox": [ + 174, + 728, + 825, + 895 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/ae057461c9856b2cafbb8dc433b0cc3cacf699acd8de55f5ace722373b08a275.jpg", + "image_caption": [ + "Figure 1: Information plane dynamics and neural nonlinearities. (A) Replication of Shwartz-Ziv & Tishby (2017) for a network with tanh nonlinearities (except for the final classification layer which contains two sigmoidal neurons). The $\\mathbf { X }$ -axis plots information between each layer and the input, while the y-axis plots information between each layer and the output. The color scale indicates training time in epochs. Each of the six layers produces a curve in the information plane with the input layer at far right, output layer at the far left. Different layers at the same epoch are connected by fine lines. (B) Information plane dynamics with ReLU nonlinearities (except for the final layer of 2 sigmoidal neurons). Here no compression phase is visible in the ReLU layers. For learning curves of both networks, see Appendix A. (C) Information plane dynamics for a tanh network of size $7 8 4 - 1 0 2 4 - 2 0 - 2 0 - 2 0 - 1 0$ trained on MNIST, estimated using the non-parametric kernel density mutual information estimator of Kolchinsky & Tracey (2017); Kolchinsky et al. (2017), no compression is observed except in the final classification layer with sigmoidal neurons. See Appendix B for the KDE MI method applied to the original Tishby dataset; additional results using a second popular nonparametric $\\mathbf { k }$ -NN-based method (Kraskov et al., 2004); and results for other neural nonlinearities. " + ], + "image_footnote": [], + "bbox": [ + 173, + 194, + 826, + 601 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We then modified the code to train deep networks using rectified linear activation functions $( f ( x ) =$ $\\operatorname* { m a x } ( 0 , x ) )$ . While the activities of tanh networks are bounded in the range $[ - 1 , 1 ]$ , ReLU networks have potentially unbounded positive activities. To calculate mutual information, we first trained the ReLU networks, next identified their largest activity value over the course of training, and finally chose 100 evenly spaced bins between the minimum and maximum activity values to discretize the hidden layer activity. The resulting information plane dynamics are shown in Fig. 1B. The mutual information with the input monotonically increases in all ReLU layers, with no apparent compression phase. To see whether our results were an artifact of the small network size, toy dataset, or simple binning-based mutual information estimator we employed, we also trained larger networks on the MNIST dataset and computed mutual information using a state-of-the-art nonparametric kernel density estimator which assumes hidden activity is distributed as a mixture of Gaussians (see Appendix B for details). Fig. C-D show that, again, tanh networks compressed but ReLU networks did not. Appendix B shows that similar results also obtain with the popular nonparametric $\\mathbf { k }$ -nearest-neighbor estimator of Kraskov et al. (2004), and for other neural nonlinearities. Thus, the choice of nonlinearity substantively affects the dynamics in the information plane. ", + "bbox": [ + 174, + 138, + 825, + 347 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To understand the impact of neural nonlinearity on the mutual information dynamics, we develop a minimal model that exhibits this phenomenon. In particular, consider the simple three neuron network shown in Fig. 2A. We assume a scalar Gaussian input distribution $X \\sim \\mathcal { N } ( 0 , 1 )$ , which is fed through the scalar first layer weight $w _ { 1 }$ , and passed through a neural nonlinearity $f ( \\cdot )$ , yielding the hidden unit activity $h = f ( w _ { 1 } X )$ . To calculate the mutual information with the input, this hidden unit activity is then binned yielding the new discrete variable $T = \\dot { \\mathbf { b i n } } ( h )$ (for instance, into 30 evenly spaced bins from $^ { - 1 }$ to 1 for the tanh nonlinearity). This binning process is depicted in Fig. 2B. In this simple setting, the mutual information $I ( T ; X )$ between the binned hidden layer activity $T$ and the input $X$ can be calculated exactly. In particular, ", + "bbox": [ + 173, + 353, + 825, + 479 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/cdd26f477326af51295a645420ee626947e9a8e138f5006e24fdda417dbbc27f.jpg", + "text": "$$\n\\begin{array} { l c l } { { { \\cal I } ( T ; X ) } } & { { = } } & { { { \\cal H } ( T ) - { \\cal H } ( T | X ) } } \\\\ { { } } & { { = } } & { { { \\cal H } ( T ) } } \\\\ { { } } & { { = } } & { { - \\displaystyle \\sum _ { i = 1 } ^ { N } p _ { i } \\log p _ { i } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 385, + 484, + 611, + 566 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $H ( \\cdot )$ denotes entropy, and we have used the fact that $H ( T | X ) = 0$ since $T$ is a deterministic function of $X$ . Here the probabilities $p _ { i } = P ( h \\ge b _ { i }$ and $h < b _ { i + 1 }$ ) are simply the probability that an input $X$ produces a hidden unit activity that lands in bin $i$ , defined by lower and upper bin limits $b _ { i }$ and $b _ { i + 1 }$ respectively. This probability can be calculated exactly for monotonic nonlinearities $f ( \\cdot )$ using the cumulative density of $X$ , ", + "bbox": [ + 174, + 571, + 825, + 642 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/af66e5c0522c9c7604634cf8f308e61b27609419bed5b8c877a971a4a87b2e43.jpg", + "text": "$$\np _ { i } = P ( X \\geq f ^ { - 1 } ( b _ { i } ) / w _ { 1 } \\mathrm { a n d } X < f ^ { - 1 } ( b _ { i + 1 } ) / w _ { 1 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 323, + 647, + 671, + 666 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $f ^ { - 1 } ( \\cdot )$ is the inverse function of $f ( \\cdot )$ . ", + "bbox": [ + 176, + 672, + 464, + 688 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As shown in Fig. 2C-D, as a function of the weight $w _ { 1 }$ , mutual information with the input first increases and then decreases for the tanh nonlinearity, but always increases for the ReLU nonlinearity. Intuitively, for small weights $w _ { 1 } \\approx 0$ , neural activities lie near zero on the approximately linear part of the tanh function. Therefore $f ( w _ { 1 } X ) \\approx w _ { 1 } X$ , yielding a rescaled Gaussian with information that grows with the size of the weights. However for very large weights $w _ { 1 } \\to \\infty$ , the tanh hidden unit nearly always saturates, yielding a discrete variable that concentrates in just two bins. This is more or less a coin flip, containing mutual information with the input of approximately 1 bit. Hence the distribution of $T$ collapses to a much lower entropy distribution, yielding compression for large weight values. With the ReLU nonlinearity, half of the inputs are negative and land in the bin containing a hidden activity of zero. The other half are Gaussian distributed, and thus have entropy that increases with the size of the weight. ", + "bbox": [ + 173, + 693, + 825, + 847 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Hence double-saturating nonlinearities can lead to compression of information about the input, as hidden units enter their saturation regime, due to the binning procedure used to calculate mutual information. The crux of the issue is that the actual $I ( h ; X )$ is infinite, unless the network itself adds noise to the hidden layers. In particular, without added noise, the transformation from $X$ to the continuous hidden activity $h$ is deterministic and the mutual information $I ( h ; X )$ would generally be infinite (see Appendix C for extended discussion). Networks that include noise in their processing (e.g., Kolchinsky et al. (2017)) can have finite $I ( T ; X )$ . Otherwise, to obtain a finite MI, one must compute mutual information as though there were binning or added noise in the activations. But this binning/noise is not actually a part of the operation of the network, and is therefore somewhat arbitrary (different binning schemes can result in different mutual information with the input, as shown in Fig. 14 of Appendix C). ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/b7f23d4571d8a39c6d37d9f1a3d373f749c5196988ea773d81a640c8c050f81d.jpg", + "image_caption": [ + "Figure 2: Nonlinear compression in a minimal model. (A) A three neuron nonlinear network which receives Gaussian inputs $x$ , multiplies by weight $w _ { 1 }$ , and maps through neural nonlinearity $f ( \\cdot )$ to produce hidden unit activity $h$ . (B) The continuous activity $h$ is binned into a discrete variable $T$ for the purpose of calculating mutual information. Blue: continuous tanh nonlinear activation function. Grey: Bin borders for 30 bins evenly spaced between $^ { - 1 }$ and 1. Because of the saturation in the sigmoid, a wide range of large magnitude net input values map to the same bin. (C) Mutual information with the input as a function of weight size $w _ { 1 }$ for a tanh nonlinearity. Information increases for small $w _ { 1 }$ and then decreases for large $w _ { 1 }$ as all inputs land in one of the two bins corresponding to the saturation regions. (D) Mutual information with the input for the ReLU nonlinearity increases without bound. Half of all inputs land in the bin corresponding to zero activity, while the other half have information that scales with the size of the weights. " + ], + "image_footnote": [], + "bbox": [ + 277, + 102, + 718, + 335 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 561, + 825, + 645 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We note that the binning procedure can be viewed as implicitly adding noise to the hidden layer activity: a range of $X$ values map to a single bin, such that the mapping between $X$ and $T$ is no longer perfectly invertible (Laughlin, 1981). The binning procedure is therefore crucial to obtaining a finite MI value, and corresponds approximately to a model where noise enters the system after the calculation of $h$ , that is, $T = h + \\epsilon$ , where $\\epsilon$ is noise of fixed variance independent from $h$ and $X$ . This approach is common in information theoretic analyses of deterministic systems, and can serve as a measure of the complexity of a system’s representation (see Sec 2.4 of Shwartz-Ziv & Tishby (2017)). However, neither binning nor noise is present in the networks that Shwartz-Ziv & Tishby (2017) considered, nor the ones in Fig. 2, either during training or testing. It therefore remains unclear whether robustness of a representation to this sort of noise in fact influences generalization performance in deep learning systems. ", + "bbox": [ + 173, + 652, + 825, + 805 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Furthermore, the addition of noise means that different architectures may no longer be compared in a common currency of mutual information: the binning/noise structure is arbitrary, and architectures that implement an identical input-output map can nevertheless have different robustness to noise added in their internal representation. For instance, Appendix C describes a family of linear networks that compute exactly the same input-output map and therefore generalize identically, but yield different mutual information with respect to the input. Finally, we note that approaches which view the weights obtained from the training process as the random variables of interest may sidestep this issue (Achille & Soatto, 2017). ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/b90f7481a85a83f3aa2ebfdd7fb9a3e639595486fdd68886cf2cb42a1f704c6f.jpg", + "image_caption": [ + "Figure 3: Generalization and information plane dynamics in deep linear networks. (A) A linear teacher network generates a dataset by passing Gaussian inputs $X$ through its weights and adding noise. (B) A deep linear student network is trained on the dataset (here the network has 1 hidden layer to allow comparison with Fig. 4A, see Supplementary Figure 18 for a deeper network). (C) Training and testing error over time. (D) Information plane dynamics. No compression is observed. " + ], + "image_footnote": [], + "bbox": [ + 272, + 101, + 725, + 334 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Hence when a tanh network is initialized with small weights and over the course of training comes to saturate its nonlinear units (as it must to compute most functions of practical interest, see discussion in Appendix D), it will enter a compression period where mutual information decreases. Figures 16-17 of Appendix E show histograms of neural activity over the course of training, demonstrating that activities in the tanh network enter the saturation regime during training. This nonlinearity-based compression furnishes another explanation for the observation that training slows down as tanh networks enter their compression phase (Shwartz-Ziv & Tishby, 2017): some fraction of inputs have saturated the nonlinearities, reducing backpropagated error gradients. ", + "bbox": [ + 174, + 430, + 825, + 542 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3 INFORMATION PLANE DYNAMICS IN DEEP LINEAR NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 563, + 730, + 579 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The preceding section investigates the role of nonlinearity in the observed compression behavior, tracing the source to double-saturating nonlinearities and the binning methodology used to calculate mutual information. However, other mechanisms could lead to compression as well. Even without nonlinearity, neurons could converge to highly correlated activations, or project out irrelevant directions of the input. These phenomena are not possible to observe in our simple three neuron minimal model, as they require multiple inputs and hidden layer activities. To search for these mechanisms, we turn to a tractable model system: deep linear neural networks (Baldi & Hornik (1989); Fukumizu (1998); Saxe et al. (2014)). In particular, we exploit recent results on the generalization dynamics in simple linear networks trained in a student-teacher setup (Seung et al., 1992; Advani & Saxe, 2017). In a student-teacher setting, one “student” neural network learns to approximate the output of another “teacher” neural network. This setting is a way of generating a dataset with interesting structure that nevertheless allows exact calculation of the generalization performance of the network, exact calculation of the mutual information of the representation (without any binning procedure), and, though we do not do so here, direct comparison to the IB bound which is already known for linear Gaussian problems (Chechik et al., 2005). ", + "bbox": [ + 174, + 595, + 826, + 804 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We consider a scenario where a linear teacher neural network generates input and output examples which are then fed to a deep linear student network to learn (Fig. 3A). Following the formulation of (Advani & Saxe, 2017), we assume multivariate Gaussian inputs $\\begin{array} { r } { X \\sim \\mathcal { N } ( 0 , \\frac { 1 } { N _ { i } } \\overline { { I } } _ { N _ { i } } ) } \\end{array}$ and a scalar output $Y$ . The output is generated by the teacher network according to $Y = \\dot { W } _ { 0 } X + \\epsilon _ { o }$ , where $\\epsilon _ { o } \\stackrel { - } { \\sim } \\mathcal { N } ( 0 , \\sigma _ { o } ^ { 2 } )$ represents aspects of the target function which cannot be represented by a neural network (that is, the approximation error or bias in statistical learning theory), and the teacher weights $W _ { o }$ are drawn independently from ${ \\mathcal N } ( 0 , \\sigma _ { w } ^ { 2 } )$ . Here, the weights of the teacher define the rule to be learned. The signal to noise ratio $\\mathrm { S N R } = \\sigma _ { w } ^ { 2 } / \\sigma _ { o } ^ { 2 }$ determines the strength of the rule linking inputs to outputs relative to the inevitable approximation error. We emphasize that the “noise” added to the teacher’s output is fundamentally different from the noise added for the purpose of calculating mutual information: $\\epsilon _ { o }$ models the approximation error for the task–even the best possible neural network may still make errors because the target function is not representable exactly as a neural network–and is part of the construction of the dataset, not part of the analysis of the student network. ", + "bbox": [ + 174, + 810, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To train the student network, a dataset of $P$ examples is generated using the teacher. The student network is then trained to minimize the mean squared error between its output and the target output using standard (batch or stochastic) gradient descent on this dataset. Here the student is a deep linear neural network consisting of potentially many layers, but where the the activation function of each neuron is simply $f ( u ) = u$ . That is, a depth $D$ deep linear network computes the output $\\hat { Y } = W _ { D + 1 } W _ { D } \\cdot \\cdot \\cdot W _ { 2 } W _ { 1 } X$ . While linear activation functions stop the network from computing complex nonlinear functions of the input, deep linear networks nevertheless show complicated nonlinear learning trajectories (Saxe et al., 2014), the optimization problem remains nonconvex (Baldi & Hornik, 1989), and the generalization dynamics can exhibit substantial overtraining (Fukumizu, 1998; Advani & Saxe, 2017). ", + "bbox": [ + 173, + 180, + 825, + 321 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Importantly, because of the simplified setting considered here, the true generalization error is easily shown to be ", + "bbox": [ + 173, + 328, + 823, + 356 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/321c56854718da12e7f1927283f3e5bb80e2ef3af1ea37f630feb0175fb7ef5f.jpg", + "text": "$$\nE _ { g } ( t ) = | | W _ { o } - W _ { t o t } ( t ) | | _ { F } ^ { 2 } + \\sigma _ { o } ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 390, + 353, + 607, + 373 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $W _ { t o t } ( t )$ is the overall linear map implemented by the network at training epoch $t$ (that is, $W _ { t o t } = W _ { D + 1 } W _ { D } \\cdot \\cdot \\cdot W _ { 2 } W _ { 1 } )$ . ", + "bbox": [ + 173, + 376, + 825, + 405 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Furthermore, the mutual information with the input and output may be calculated exactly, because the distribution of the activity of any hidden layer is Gaussian. Let $T$ be the activity of a specific hidden layer, and let $\\bar { W }$ be the linear map from the input to this activity (that is, for layer $l$ , $\\bar { W } =$ $W _ { l } \\cdots W _ { 2 } W _ { 1 } )$ . Since $T = { \\bar { W } } X$ , the mutual information of $X$ and $T$ calculated using differential entropy is infinite. For the purpose of calculating the mutual information, therefore, we assume that Gaussian noise is added to the hidden layer activity, $T = \\bar { W } X + \\epsilon _ { M I }$ , with mean 0 and variance $\\sigma _ { M I } ^ { 2 } = 1 . 0$ . This allows the analysis to apply to networks of any size, including overcomplete layers, but as before we emphasize that we do not add this noise either during training or testing. With these assumptions, $T$ and $X$ are jointly Gaussian and we have ", + "bbox": [ + 173, + 410, + 825, + 536 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/986b2b1744c46424503da00a715308fcc981ab1a8272cebe74c3deedeb7eb198.jpg", + "text": "$$\nI ( T ; X ) = \\log \\lvert \\bar { W } \\bar { W } ^ { T } + \\sigma _ { M I } ^ { 2 } I _ { N _ { h } } \\rvert - \\log \\lvert \\sigma _ { M I } ^ { 2 } I _ { N _ { h } } \\rvert\n$$", + "text_format": "latex", + "bbox": [ + 331, + 542, + 666, + 561 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\left| \\cdot \\right|$ denotes the determinant of a matrix. Finally the mutual information with the output $Y$ , also jointly Gaussian, can be calculated similarly (see Eqns. (22)-(25) of Appendix G). ", + "bbox": [ + 173, + 568, + 823, + 597 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Fig. 3 shows example training and test dynamics over the course of learning in panel C, and the information plane dynamics in panel D. Here the network has an input layer of 100 units, 1 hidden layer of 100 units each and one output unit. The network was trained with batch gradient descent on a dataset of 100 examples drawn from the teacher with signal to noise ratio of 1.0. The linear network behaves qualitatively like the ReLU network, and does not exhibit compression. Nevertheless, it learns a map that generalizes well on this task and shows minimal overtraining. Hence, in the setting we study here, generalization performance can be acceptable without any compression phase. ", + "bbox": [ + 173, + 603, + 825, + 702 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The results in (Advani & Saxe (2017)) show that, for the case of linear networks, overtraining is worst when the number of inputs matches the number of training samples, and is reduced by making the number of samples smaller or larger. Fig. 4 shows learning dynamics with the number of samples matched to the size of the network. Here overfitting is substantial, and again no compression is seen in the information plane. Comparing to the result in Fig. 3D, both networks exhibit similar information dynamics with respect to the input (no compression), but yield different generalization performance. ", + "bbox": [ + 174, + 707, + 825, + 805 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Hence, in this linear analysis of a generic setting, there do not appear to be additional mechanisms that cause compression over the course of learning; and generalization behavior can be widely different for networks with the same dynamics of information compression regarding the input. We note that, in the setting considered here, all input dimensions have the same variance, and the weights of the teacher are drawn independently. Because of this, there are no special directions in the input, and each subspace of the input contains as much information as any other. It is possible that, in real world tasks, higher variance inputs are also the most likely to be relevant to the task (here, have large weights in the teacher). We have not investigated this possibility here. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/22b7932d2a586116824d010622a5076fd367f6f9658cd0958fd6f2a3a8c1187a.jpg", + "image_caption": [ + "Figure 4: Overtraining and information plane dynamics. (A) Average training and test mean square error for a deep linear network trained with SGD. Overtraining is substantial. Other parameters: $N _ { i } =$ 100, $\\mathrm { P } = 1 0 0$ , Number of hidden units $= 1 0 0$ , Batch size $= 5$ (B) Information plane dynamics. No compression is observed, and information about the labels is lost during overtraining. (C) Average train and test accuracy $\\%$ correct) for nonlinear tanh networks exhibiting modest overfitting $N = 8$ ). (D) Information plane dynamics. Overfitting occurs despite continued compression. " + ], + "image_footnote": [], + "bbox": [ + 272, + 101, + 730, + 342 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/6fb26702a9042ab8cf86aa3fd585a3fceb5522ab227a6c80568e9d5d17d27d7b.jpg", + "image_caption": [ + "Figure 5: Stochastic training and the information plane. (A) tanh network trained with SGD. (B) tanh network trained with BGD. (C) ReLU network trained with SGD. (D) ReLU network trained with BGD. Both random and non-random training procedures show similar information plane dynamics. " + ], + "image_footnote": [], + "bbox": [ + 179, + 455, + 823, + 566 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To see whether similar behavior arises in nonlinear networks, we trained tanh networks in the same setting as Section 2, but with $30 \\%$ of the data, which we found to lead to modest overtraining. Fig. 4C-D shows the resulting train, test, and information plane dynamics. Here the tanh networks show substantial compression, despite exhibiting overtraining. This establishes a dissociation between behavior in the information plane and generalization dynamics: networks that compress may (Fig. 1A) or may not (Fig. 4C-D) generalize well, and networks that do not compress may (Figs.1B, 3A-B) or may not (Fig. 4A-B) generalize well. ", + "bbox": [ + 173, + 655, + 825, + 753 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 COMPRESSION IN BATCH GRADIENT DESCENT AND SGD ", + "text_level": 1, + "bbox": [ + 176, + 773, + 684, + 790 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Next, we test a core theoretical claim of the information bottleneck theory of deep learning, namely that randomness in stochastic gradient descent is responsible for the compression phase. In particular, because the choice of input samples in SGD is random, the weights evolve in a stochastic way during training. ", + "bbox": [ + 174, + 804, + 825, + 861 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Shwartz-Ziv & Tishby (2017) distinguish two phases of SGD optimization: in the first “drift” phase, the mean of the gradients over training samples is large relative to the standard deviation of the gradients; in the second “diffusion” phase, the mean becomes smaller than the standard deviation of the gradients. The authors propose that compression should commence following the transition from a high to a low gradient signal-to-noise ratio (SNR), i.e., the onset of the diffusion phase. The proposed mechanism behind this diffusion-driven compression is as follows. The authors state that during the diffusion phase, the stochastic evolution of the weights can be described as a Fokker-Planck equation under the constraint of small training error. Then, the stationary distribution over weights for this process will have maximum entropy, again subject to the training error constraint. Finally, the authors claim that weights drawn from this stationary distribution will maximize the entropy of inputs given hidden layer activity, $H ( X | T )$ , subject to a training error constraint, and that this training error constraint is equivalent to a constraint on the mutual information $I ( T ; Y )$ for small training error. Since the entropy of the input, $H ( X )$ , is fixed, the result of the diffusion dynamics will be to minimize $I ( X ; T ) \\mathrel { \\mathop : } = H ( X ) - \\bar { H } ( X | \\dot { T } )$ for a given value of $I ( T ; Y )$ reached at the end of the drift phase. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 256 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "However, this explanation does not hold up to either theoretical or empirical investigation. Let us assume that the diffusion phase does drive the distribution of weights to a maximum entropy distribution subject to a training error constraint. Note that this distribution reflects stochasticity of weights across different training runs. There is no general reason that a given set of weights sampled from this distribution (i.e., the weight parameters found in one particular training run) will maximize $H ( X | T )$ , the entropy of inputs given hidden layer activity. In particular, $H ( X | T )$ reflects (conditional) uncertainty about inputs drawn from the data-generating distribution, rather than uncertainty about any kind of distribution across different training runs. ", + "bbox": [ + 174, + 263, + 825, + 375 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We also show empirically that the stochasticity of the SGD is not necessary for compression. To do so, we consider two distinct training procedures: offline stochastic gradient descent (SGD), which learns from a fixed-size dataset, and updates weights by repeatedly sampling a single example from the dataset and calculating the gradient of the error with respect to that single sample (the typical procedure used in practice); and batch gradient descent (BGD), which learns from a fixed-size dataset, and updates weights using the gradient of the total error across all examples. Batch gradient descent uses the full training dataset and, crucially, therefore has no randomness or diffusion-like behavior in its updates. ", + "bbox": [ + 174, + 382, + 825, + 492 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We trained tanh and ReLU networks with SGD and BGD and compare their information plane dynamics in Fig. 5 (see Appendix H for a linear network). We find largely consistent information dynamics in both instances, with robust compression in tanh networks for both methods. Thus randomness in the training process does not appear to contribute substantially to compression of information about the input. This finding is consistent with the view presented in Section 2 that compression arises predominantly from the double saturating nonlinearity. ", + "bbox": [ + 174, + 500, + 825, + 583 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, we look at the gradient signal-to-noise ratio (SNR) to analyze the relationship between compression and the transition from high to low gradient SNR. Fig. 20 of Appendix I shows the gradient SNR over training, which in all cases shows a phase transition during learning. Hence the gradient SNR transition is a general phenomenon, but is not causally related to compression. Appendix I offers an extended discussion and shows gradient SNR transitions without compression on the MNIST dataset and for linear networks. ", + "bbox": [ + 174, + 590, + 825, + 674 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 SIMULTANEOUS FITTING AND COMPRESSION ", + "text_level": 1, + "bbox": [ + 174, + 704, + 581, + 722 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Our finding that generalization can occur without compression may seem difficult to reconcile with the intuition that certain tasks involve suppressing irrelevant directions of the input. In the extreme, if certain inputs contribute nothing but noise, then good generalization requires ignoring them. To study this, we consider a variant on the linear student-teacher setup of Section 3: we partition the input $X$ into a set of task-relevant inputs $X _ { r e l }$ and a set of task-irrelevant inputs $X _ { i r r e l }$ , and alter the teacher network so that the teacher’s weights to the task-irrelevant inputs are all zero. Hence the inputs $X _ { i r r e l }$ contribute only noise, while the $X _ { r e l }$ contain signal. We then calculate the information plane dynamics for the whole layer, and for the task-relevant and task-irrelevant inputs separately. Fig. 6 shows information plane dynamics for a deep linear neural network trained using SGD (5 samples/batch) on a task with 30 task-relevant inputs and 70 task-irrelevant inputs. While the overall dynamics show no compression phase, the information specifically about the task-irrelevant subspace does compress over the course of training. This compression process occurs at the same time as the fitting to the task-relevant information. Thus, when a task requires ignoring some inputs, the information with these inputs specifically will indeed be reduced; but overall mutual information with the input in general may still increase. ", + "bbox": [ + 174, + 743, + 825, + 922 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/3f982bb2a95db7292dd21c09073b2ac2ac7ccf908216d9ad8ed4dcda065e419e.jpg", + "image_caption": [ + "Figure 6: Simultaneous fitting and compression. (A) For a task with a large task-irrelevant subspace in the input, a linear network shows no overall compression of information about the input. (B) The information with the task-relevant subspace increases robustly over training. (C) However, the information specifically about the task-irrelevant subspace does compress after initially growing as the network is trained. " + ], + "image_footnote": [], + "bbox": [ + 176, + 101, + 820, + 213 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 330, + 823, + 358 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 381, + 310, + 397 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Our results suggest that compression dynamics in the information plane are not a general feature of deep networks, but are critically influenced by the nonlinearities employed by the network. Doublesaturating nonlinearities lead to compression, if mutual information is estimated by binning activations or by adding homoscedastic noise, while single-sided saturating nonlinearities like ReLUs do not compress in general. Consistent with this view, we find that stochasticity in the training process does not contribute to compression in the cases we investigate. Furthermore, we have found instances where generalization performance does not clearly track information plane behavior, questioning the causal link between compression and generalization. Hence information compression may parallel the situation with sharp minima: although empirical evidence has shown a correlation with generalization error in certain settings and architectures, further theoretical analysis has shown that sharp minima can in fact generalize well (Dinh et al., 2017). We emphasize that compression still may occur within a subset of the input dimensions if the task demands it. This compression, however, is interleaved rather than in a secondary phase and may not be visible by information metrics that track the overall information between a hidden layer and the input. Finally, we note that our results address the specific claims of one scheme to link the information bottleneck principle with current practice in deep networks. The information bottleneck principle itself is more general and may yet offer important insights into deep networks (Achille & Soatto, 2017). Moreover, the information bottleneck principle could yield fundamentally new training algorithms for networks that are inherently stochastic and where compression is explicitly encouraged with appropriate regularization terms (Chalk et al., 2016; Alemi et al., 2017; Kolchinsky et al., 2017). ", + "bbox": [ + 174, + 414, + 825, + 693 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 712, + 326, + 723 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We thank Ariel Herbert-Voss for useful discussions. This work was supported by grant numbers IIS 1409097 and CHE 1648973 from the US National Science Foundation, and by IARPA contract #D16PC00002. Andrew Saxe and Madhu Advani thank the Swartz Program in Theoretical Theoretical Neuroscience at Harvard University. Artemy Kolchinsky and Brendan Tracey would like to thank the Santa Fe Institute for helping to support this research. Artemy Kolchinsky was supported by Grant No. FQXi-RFP-1622 from the FQXi foundation and Grant No. CHE-1648973 from the US National Science Foundation. Brendan Tracey was supported by AFOSR MURI on Multi-Information Sources of Multi-Physics Systems under Award Number FA9550-15-1-0038. ", + "bbox": [ + 174, + 736, + 825, + 847 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 871, + 285, + 885 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A. Achille and S. Soatto. On the Emergence of Invariance and Disentangling in Deep Representations. arXiv preprint arXiv:1706.01350, 2017. ", + "bbox": [ + 173, + 895, + 823, + 922 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "M.S. Advani and A.M. Saxe. 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", + "bbox": [ + 166, + 65, + 828, + 928 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 103, + 828, + 482 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A LEARNING CURVES FOR tanh AND RELU NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 507, + 651, + 523 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Supplementary Figure 7 shows the learning curves for tanh and ReLU networks depicted in Fig. 1. ", + "bbox": [ + 176, + 537, + 821, + 554 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/ce0355c43f7ded6cd043397ccb1f76e1f4c9361219e3e513122d676b64034395.jpg", + "image_caption": [ + "Figure 7: Learning curves for (A) tanh neural network in 1 A and (B) ReLU neural network in $1 \\textbf { B }$ . Both networks show good generalization with regards to the test data. " + ], + "image_footnote": [], + "bbox": [ + 274, + 563, + 718, + 696 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B ROBUSTNESS OF FINDINGS TO MI ESTIMATION METHOD AND NEURAL ACTIVATION FUNCTIONS ", + "text_level": 1, + "bbox": [ + 178, + 762, + 789, + 796 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "This Appendix investigates the generality of the finding that compression is not observed in neural network layers with certain activation functions. Figure 1 of the main text shows example results using a binning-based MI estimator and a nonparametric KDE estimator, for both the tanh and ReLU activation functions. Here we describe the KDE MI estimator in detail, and present extended results on other datasets. We also show results for other activation functions. Finally, we provide entropy estimates based on another nonparametric estimator, the popular $\\mathbf { k }$ -nearest neighbor approach of Kraskov et al. (2004). Our findings consistently show that double-saturating nonlinearities can yield compression, while single-sided nonlinearities do not. ", + "bbox": [ + 173, + 811, + 825, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B.1 KERNEL DENSITY ESTIMATION OF MI ", + "text_level": 1, + "bbox": [ + 174, + 103, + 483, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The KDE approach of Kolchinsky & Tracey (2017); Kolchinsky et al. (2017) estimates the mutual information between the input and the hidden layer activity by assuming that the hidden activity is distributed as a mixture of Gaussians. This assumption is well-suited to the present setting under the following interpretation: we take the input activity to be distributed as delta functions at each example in the dataset, corresponding to a uniform distribution over these specific samples. In other words, we assume that the empirical distribution of input samples is the true distribution. Next, the hidden layer activity $h$ is a deterministic function of the input. As mentioned in the main text and discussed in more detail in Appendix C, without the assumption of noise, this would have infinite mutual information with the input. We therefore assume for the purposes of analysis that Gaussian noise of variance $\\sigma ^ { 2 }$ is added, that is, $T = h + \\epsilon$ where $\\epsilon \\sim \\mathcal { N } ( \\bar { 0 } , \\bar { \\sigma ^ { 2 } } I )$ . Under these assumptions, the distribution of $T$ is genuinely a mixture of Gaussians, with a Gaussian centered on the hidden activity corresponding to each input sample. We emphasize again that the noise $\\epsilon$ is added solely for the purposes of analysis, and is not present during training or testing the network. In this setting, an upper bound for the mutual information with the input is (Kolchinsky & Tracey, 2017; Kolchinsky et al., 2017) ", + "bbox": [ + 173, + 159, + 825, + 367 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/fbba9039fce17d693d329e06bce9443a5d04f665dd8f72721a5f3cec7ef6374e.jpg", + "text": "$$\nI ( T ; X ) \\leq - { \\frac { 1 } { P } } \\sum _ { i } \\log { \\frac { 1 } { P } } \\sum _ { j } \\exp \\left( - { \\frac { 1 } { 2 } } { \\frac { \\vert \\vert h _ { i } - h _ { j } \\vert \\vert _ { 2 } ^ { 2 } } { \\sigma ^ { 2 } } } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 316, + 460, + 679, + 501 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $P$ is the number of training samples and $h _ { i }$ denotes the hidden activity vector in response to input sample $i$ . Similarly, the mutual information with respect to the output can be calculated as ", + "bbox": [ + 173, + 599, + 826, + 628 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/cb2fe5a380eba03a05ab84e25bc76b6aae3a086fa63278aa4248b13502ce70b9.jpg", + "text": "$$\n\\begin{array} { r c l } { I ( T ; Y ) } & { = } & { { \\displaystyle H ( T ) - H ( T | Y ) } } \\\\ & { \\leq } & { \\displaystyle - \\frac { 1 } { P } \\sum _ { i } \\log \\frac { 1 } { P } \\sum _ { j } \\exp \\left( - \\frac { 1 } { 2 } \\frac { \\| h _ { i } - h _ { j } \\| _ { 2 } ^ { 2 } } { \\sigma ^ { 2 } } \\right) } \\\\ & & { \\displaystyle - \\sum _ { l } ^ { L } p _ { l } \\left[ - \\frac { 1 } { P _ { l } } \\sum _ { i , Y _ { i } = l } \\log \\frac { 1 } { P _ { l } } \\sum _ { j , Y _ { j } = l } \\exp \\left( - \\frac { 1 } { 2 } \\frac { \\| h _ { i } - h _ { j } \\| _ { 2 } ^ { 2 } } { \\sigma ^ { 2 } } \\right) \\right] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 248, + 696, + 746, + 813 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $L$ is the number of output labels, $P _ { l }$ denotes the number of data samples with output label $l$ , $p _ { l } = P _ { l } / P$ denotes the probability of output label $l$ , and the sums over $i , Y _ { i } = l$ indicate a sum over all examples with output label $l$ . ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/f89ebb8650e008588c11ffbb01a6605fa94820e9aa6ba83b3408ba661f14ce97.jpg", + "image_caption": [ + "Figure 8: Information plane dynamics for the network architecture and training dataset of ShwartzZiv & Tishby (2017), estimated with the nonparametric KDE method of Kolchinsky & Tracey (2017); Kolchinsky et al. (2017) and averaged over 50 repetitions. (A) tanh neural network layers show compression. (B) ReLU neural network layers show no compression. (C) The soft-sign activation function, a double-saturating nonlinearity that saturates more gently than tanh, shows modest compression. (D) The soft-plus activation function, a smoothed version of the ReLU, exhibits no compression. Hence double-saturating nonlinearities exhibit the compression effect while singlesaturating nonlinearities do not. " + ], + "image_footnote": [], + "bbox": [ + 269, + 99, + 728, + 421 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Figure 8A-B shows the result of applying this MI estimation method on the dataset and network architecture of Shwartz-Ziv & Tishby (2017), with MI estimated on the full dataset and averaged over 50 repetitions. Mutual information was estimated using data samples from the test set, and we took the noise variance $\\sigma ^ { 2 } = 0 . 1$ . These results look similar to the estimate derived from binning, with compression in tanh networks but no compression in ReLU networks. Relative to the binning estimate, it appears that compression is less pronounced in the KDE method. ", + "bbox": [ + 173, + 559, + 825, + 643 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Figure 1C-D of the main text shows the results of this estimation technique applied to a neural network of size $7 8 4 - 1 0 2 4 - 2 0 - 2 0 - 2 0 - 1 0$ on the MNIST handwritten digit classification dataset. The network was trained using SGD with minibatches of size 128. As before, mutual information was estimated using data samples from the test set, and we took the noise variance $\\sigma ^ { 2 } = 0 . 1$ . The smaller layer sizes in the top three hidden layers were selected to ensure the quality of the kernel density estimator given the amount of data in the test set, since the estimates are more accurate for smaller-dimensional data. Because of computational expense, the MNIST results are from a single training run. ", + "bbox": [ + 173, + 650, + 825, + 762 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "More detailed results for the MNIST dataset are provided in Figure 9 for the tanh activation function, and in Figure 10 for the ReLU activation function. In these figures, the first row shows the evolution of the cross entropy loss (on both training and testing data sets) during training. The second row shows the mutual information between input and the activity of different hidden layers, using the nonparametric KDE estimator described above. The blue region in the second row shows the range of possible MI values, ranging from the upper bound described above (Eq. 10) to the following lower ", + "bbox": [ + 173, + 768, + 825, + 853 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "bound (Kolchinsky & Tracey, 2017), ", + "bbox": [ + 174, + 103, + 416, + 118 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/2b4d855811c068dfd8a5a6373771f34f8fdd4071c1a630b953dc7e58a0914035.jpg", + "text": "$$\n\\begin{array} { r c l } { { I ( T ; Y ) } } & { { \\geq } } & { { \\displaystyle - \\frac { 1 } { P } \\sum _ { i } \\log \\frac { 1 } { P } \\sum _ { j } \\exp \\left( - \\frac { 1 } { 2 } \\frac { \\left\\| h _ { i } - h _ { j } \\right\\| _ { 2 } ^ { 2 } } { 4 \\sigma ^ { 2 } } \\right) } } \\\\ { { } } & { { } } & { { \\displaystyle - \\sum _ { l } ^ { L } p _ { l } \\left[ - \\frac { 1 } { P _ { l } } \\sum _ { i , Y _ { i } = l } \\log \\frac { 1 } { P _ { l } } \\sum _ { j , Y _ { j } = l } \\exp \\left( - \\frac { 1 } { 2 } \\frac { \\left\\| h _ { i } - h _ { j } \\right\\| _ { 2 } ^ { 2 } } { 4 \\sigma ^ { 2 } } \\right) \\right] . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 125, + 751, + 220 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The third row shows the mutual information between input and activity of different hidden layers, estimated using the binning method (here, the activity of each neuron was discretized into bins of size 0.5). For both the second and third rows, we also plot the entropy of the inputs, $H ( X )$ , as a dashed line. $H ( X )$ is an upper bound on the mutual information $I ( X ; T )$ , and is computed using the assumption of a uniform distribution over the 10,000 testing points in the MNIST dataset, giving $H ( X ) = \\log _ { 2 } { 1 0 0 0 0 }$ . ", + "bbox": [ + 173, + 224, + 825, + 309 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Finally, the fourth row visualizes the dynamics of the SGD updates during training. For each layer and epoch, the green line shows the $\\ell _ { 2 }$ norm of the weights. We also compute the vector of mean updates across SGD minibatches (this vector has one dimension for each weight parameter), as well as the vector of the standard deviation of the updates across SGD minibatches. The $\\ell _ { 2 }$ norm of the mean update vector is shown in blue, and the $\\ell _ { 2 }$ norm of the standard deviation vector is shown in orange. The gradient SNR, computed as the ratio of the norm of the mean vector to the norm of the standard deviation vector, is shown in red. For both the tanh and ReLU networks, the gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases. Importantly, this phase transition occurs despite a lack of compression in the ReLU network, indicating that noise in SGD updates does not yield compression in this setting. ", + "bbox": [ + 173, + 315, + 826, + 455 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/cc8295ba9fbcdf4553fd120007b118b080b408782fb4c7b756ca91d6a943022a.jpg", + "image_caption": [ + "Figure 9: Detailed tanh activation function results on MNIST. Row 1: Loss over training. Row 2: Upper and lower bounds for the mutual information $I ( X ; T )$ between the input $( X )$ and each layer’s activity $( T )$ , using the nonparametric KDE estimator (Kolchinsky & Tracey, 2017; Kolchinsky et al., 2017). Dotted line indicates $H ( X ) = \\log _ { 2 } { 1 0 0 0 0 }$ , the entropy of a uniform distribution over 10,000 testing samples. Row 3: Binning-based estimate of the mutual information $I ( X ; T )$ , with each neuron’s activity discretized using a bin size of 0.5. Row 4: Gradient SNR and weight norm dynamics. The gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases. " + ], + "image_footnote": [], + "bbox": [ + 176, + 97, + 820, + 560 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/6b5d4819da8c875280c0636b21620dd6a2ea91f5c10ae7ab67fc192db6fbb0d4.jpg", + "image_caption": [ + "Figure 10: Detailed ReLU activation function results on MNIST. Row 1: Loss over training. Row 2: Upper and lower bounds for the mutual information $I ( X ; T )$ between the input $( X )$ and each layer’s activity $( T )$ , using the nonparametric KDE estimator (Kolchinsky & Tracey, 2017; Kolchinsky et al., 2017). Dotted line indicates $H ( X ) = \\log _ { 2 } { 1 0 0 0 0 }$ , the entropy of a uniform distribution over 10,000 testing samples. Row 3: Binning-based estimate of the mutual information $I ( X ; T )$ , with each neuron’s activity discretized using a bin size of 0.5. Row 4: Gradient SNR and weight norm dynamics. The gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases. Importantly, this phase transition occurs despite a lack of compression in the ReLU network, indicating that noise in SGD updates does not yield compression in this setting. " + ], + "image_footnote": [], + "bbox": [ + 176, + 98, + 820, + 559 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/6c1149a7423a2791869e903f30e00c9268d38906bea86674e43fe62d02d3186f.jpg", + "image_caption": [ + "Figure 11: Alternative activation functions. " + ], + "image_footnote": [], + "bbox": [ + 352, + 104, + 638, + 273 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.2 OTHER ACTIVATION FUNCTIONS ", + "text_level": 1, + "bbox": [ + 174, + 323, + 441, + 337 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Next, in Fig. 8C-D, we show results from the kernel MI estimator from two additional nonlinear activation functions, the softsign function ", + "bbox": [ + 173, + 348, + 825, + 377 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/dad4d588c208e612df65eed8f8fbb3d99f3777d4303bb3bbf635e10b8ec79bfc.jpg", + "text": "$$\nf ( x ) = { \\frac { x } { 1 + | x | } } ,\n$$", + "text_format": "latex", + "bbox": [ + 442, + 383, + 553, + 415 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "and the softplus function ", + "bbox": [ + 174, + 421, + 338, + 435 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/ed095b104e589ed18d34efef57c8ac25b4d0e00571c34a6136a01f8c8298b2df.jpg", + "text": "$$\nf ( x ) = \\ln ( 1 + e ^ { x } ) .\n$$", + "text_format": "latex", + "bbox": [ + 433, + 434, + 563, + 452 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "These functions are plotted next to tanh and ReLU in Fig. 11. The softsign function is similar to tanh but saturates more slowly, and yields less compression than tanh. The softplus function is a smoothed version of the ReLU, and yields similar dynamics with no compression. Because softplus never saturates fully to zero, it retains more information with respect to the input than ReLUs in general. ", + "bbox": [ + 174, + 454, + 825, + 525 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.3 KRASKOV ESTIMATOR ", + "text_level": 1, + "bbox": [ + 176, + 542, + 372, + 556 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We additionally investigated the widely-used nonparametric MI estimator of Kraskov et al. (2004). This estimator uses nearest neighbor distances between samples to compute an estimate of the entropy of a continuous random variable. Here we focused for simplicity only on the compression phenomenon in the mutual information between the input and hidden layer activity, leaving aside the information with respect to the output (as this is not relevant to the compression phenomenon). Again, without additional noise assumptions, the MI between the hidden representation and the input would be infinite because the mapping is deterministic. Rather than make specific noise assumptions, we instead use the Kraskov method to estimate the entropy of the hidden representations $T$ . Note that the entropy of $T$ is the mutual information up to an unknown constant so long as the noise assumption is homoscedastic, that is, $T = h + Z$ where the random variable $Z$ is independent of $X$ . To see this, note that ", + "bbox": [ + 173, + 568, + 826, + 720 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/ae64ad5f1f4dfb84bbbc558b10a8d6538e1aa3c4bdf44df517831ae47582c452.jpg", + "text": "$$\n\\begin{array} { l l l } { { I ( T ; X ) } } & { { = } } & { { H ( T ) - H ( T | X ) } } \\\\ { { } } & { { = } } & { { H ( T ) - H ( Z ) } } \\\\ { { } } & { { = } } & { { H ( T ) - c } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 387, + 727, + 611, + 784 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where the constant $c = H ( Z )$ . Hence observing compression in the layer entropy $H ( T )$ is enough to establish that compression occurs in the mutual information. ", + "bbox": [ + 173, + 787, + 826, + 816 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The Kraskov estimate is given by ", + "bbox": [ + 174, + 824, + 393, + 838 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/063bc3c5963df32d1711d536fa9f9448266875271f28eef6b7368b3fe23b0943.jpg", + "text": "$$\n\\frac { d } { P } \\sum _ { i = 1 } ^ { P } \\log ( r _ { i } + \\epsilon ) + \\frac { d } { 2 } \\log ( \\pi ) - \\log \\Gamma ( d / 2 + 1 ) + \\psi ( P ) - \\psi ( k )\n$$", + "text_format": "latex", + "bbox": [ + 284, + 844, + 714, + 888 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "where $d$ is the dimension of the hidden representation, $P$ is the number of samples, $r _ { i }$ is the distance to the $k$ -th nearest neighbor of sample $i$ , $\\epsilon$ is a small constant for numerical stability, $\\Gamma ( \\cdot )$ is the ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Gamma function, and $\\psi ( \\cdot )$ is the digamma function. Here the parameter $\\epsilon$ prevents infinite terms when the nearest neighbor distance ri = 0 for some sample. We took \u000f = 10−16. ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Figure 12 shows the entropy over training for tanh and ReLU networks trained on the dataset of and with the network architecture in Shwartz-Ziv & Tishby (2017), averaged over 50 repeats. In these experiments, we used $k = 2$ . Compression would correspond to decreasing entropy over the course of training, while a lack of compression would correspond to increasing entropy. Several tanh layers exhibit compression, while the ReLU layers do not. Hence qualitatively, the Kraskov estimator returns similar results to the binning and KDE strategies. ", + "bbox": [ + 173, + 138, + 825, + 223 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/d35dbe7c6e40d832895cc4e561a74abc5f2ec6a2b6e899768df8078d5351eb42.jpg", + "image_caption": [ + "Figure 12: Entropy dynamics over training for the network architecture and training dataset of Shwartz-Ziv & Tishby (2017), estimated with the nonparametric $\\mathbf { k }$ -nearest-neighbor-based method of Kraskov et al. (2004). Here the $\\mathbf { X } ^ { } -$ -axis is epochs of training time, and the y-axis plots the entropy of the hidden representation, as calculated using nearest-neighbor distances. Note that in this setting, if $T$ is considered to be the hidden activity plus independent noise, the entropy is equal to the mutual information up to a constant (see derivation in text). Layers 0-4 correspond to the hidden layers of size 10-7-5-4-3. (A) tanh neural network layers can show compression over the course of training. (B) ReLU neural network layers show no compression. " + ], + "image_footnote": [], + "bbox": [ + 194, + 236, + 789, + 409 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "C NOISE ASSUMPTIONS AND DISCRETE VS CONTINUOUS ENTROPY ", + "text_level": 1, + "bbox": [ + 176, + 560, + 740, + 575 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A recurring theme in the results reported in this paper is the necessity of noise assumptions to yield a nontrivial information theoretic analysis. Here we give an extended discussion of this phenomenon, and of issues relating to discrete entropy as opposed to continuous (differential) entropy. ", + "bbox": [ + 174, + 590, + 826, + 633 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The activity of a neural network is often a continuous deterministic function of its input. That is, in response to an input $X$ , a specific hidden layer might produce activity $h = f ( X )$ for some function $f$ . The mutual information between $h$ and $X$ is given by ", + "bbox": [ + 173, + 640, + 823, + 681 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/2ddea48d352590005b744757054e3dc800ffa2ab153f41eb772b09d691a581d7.jpg", + "text": "$$\n\\begin{array} { l l l } { { I ( h ; X ) } } & { { = } } & { { H ( h ) - H ( h | X ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 388, + 685, + 609, + 704 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "If $h$ were a discrete variable, then the entropy would be given by ", + "bbox": [ + 171, + 707, + 598, + 722 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/5462ea43985fc5e5b892174c0e7260549818918a5de68060a98efad4810d7fe3.jpg", + "text": "$$\nH ( h ) = - \\sum _ { i = 1 } ^ { N } p _ { i } \\log p _ { i }\n$$", + "text_format": "latex", + "bbox": [ + 403, + 724, + 562, + 768 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "where $p _ { i }$ is the probability of the discrete symbol $i$ , as mentioned in the main text. Then $H ( h | X ) = 0$ because the mapping is deterministic and we have $I ( h ; X ) = H ( h )$ . ", + "bbox": [ + 174, + 772, + 825, + 801 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "However $h$ is typically continuous. The continuous entropy, defined for a continuous random variable $Z$ with density $p _ { Z }$ by analogy to Eqn. (18) as ", + "bbox": [ + 174, + 808, + 821, + 837 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/eef458855d2852432f80cd0e820eab3f8d131b9f4c42ffc6c1f1916f1e85895e.jpg", + "text": "$$\nH ( Z ) = - \\int p _ { Z } ( z ) \\log p _ { Z } ( z ) d z ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 839, + 611, + 872 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "can be negative and possibly infinite. In particular, note that if $p _ { Z }$ is a delta function, then $H ( Z ) =$ $- \\infty$ . The mutual information between hidden layer activity $h$ and the input $X$ for continuous $h , X$ is ", + "bbox": [ + 173, + 876, + 823, + 905 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/80e3b108e389875786004688dedee7b92797c1a0c6b6f9f1c580f719cf08800e.jpg", + "text": "$$\nI ( h ; X ) = H ( h ) - H ( h | X ) .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 907, + 580, + 925 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Now $H ( h | X ) = - \\infty$ since given the input $X$ , the hidden activity $h$ is distributed as a delta function at $f ( X )$ . The mutual information is thus generally infinite, so long as the hidden layer activity has finite entropy $H ( h )$ is finite). ", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/db291cd24878a5a577925a9f57c17f1e80ff0653e022e4f8350fe21e76b70f02.jpg", + "image_caption": [ + "Figure 13: Effect of binning strategy on minimal three neuron model. Mutual information for the simple three neuron model shown in Fig. 2 with bin edges $b _ { i } \\in \\mathrm { t a n h } ( \\operatorname* { l i n s p a c e } ( - 5 0 , 5 0 , N ) )$ . In contrast to linear binning, the mutual information continues to increase as weights grow. " + ], + "image_footnote": [], + "bbox": [ + 303, + 181, + 686, + 417 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "To yield a finite mutual information, some noise in the mapping is required such that $H ( h | X )$ remains finite. A common choice (and one adopted here for the linear network, the nonparametric kernel density estimator, and the $\\mathbf { k }$ -nearest neighbor estimator) is to analyze a new variable with additive noise, $T = h + Z$ , where $Z$ is a random variable independent of $X$ . Then $H ( T | X ) = H ( Z )$ which allows the overall information $I ( T ; X ) = H ( T ) - H ( Z )$ to remain finite. This noise assumption is not present in the actual neural networks either during training or testing, and is made solely for the purpose of calculating the mutual information. ", + "bbox": [ + 173, + 506, + 825, + 603 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Another strategy is to partition the continuous variable $h$ into a discrete variable $T$ , for instance by binning the values (the approach taken in Shwartz-Ziv & Tishby (2017)). This allows use of the discrete entropy, which remains finite. Again, however, in practice the network does not operate on the binned variables $T$ but on the continuous variables $h$ , and the binning is solely for the purpose of calculating the mutual information. Moreover, there are many possible binning strategies, which yield different discrete random variables, and different mutual information with respect to the input. The choice of binning strategy is an assumption analogous to choosing a type of noise to add to the representation in the continuous case: because there is in fact no binning in the operation of the network, there is no clear choice for binning methodology. The strategy we use in binning-based experiments reported here is the following: for bounded activations like the tanh activation, we use evenly spaced bins between the minimum and maximum limits of the function. For unbounded activations like ReLU, we first train the network completely; next identify the minimum and maximum hidden activation over all units and all training epochs; and finally bin into equally spaced bins between these minimum and maximum values. We note that this procedure places no restriction on the magnitude that the unbounded activation function can take during training, and yields the same MI estimate as using infinite equally spaced bins (because bins for activities larger than the maximum are never seen during training). ", + "bbox": [ + 174, + 611, + 825, + 848 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "As an example of another binning strategy that can yield markedly different results, we consider evenly spaced bins in a neuron’s net input, rather than its activity. That is, instead of evenly spaced bins in the neural activity, we determine the bin edges by mapping a set of evenly spaced values through the neural nonlinearity. For tanh, for instance, this spaces bins more tightly in the saturation region as compared to the linear region. Figure 13 shows the results of applying this binning strategy to the minimal three neuron model with tanh activations. This binning scheme captures more information as the weights of the network grow larger. Figure 14 shows information plane dynamics for this binning structure. The tanh network no longer exhibits compression. (We note that the broken DPI in this example is an artifact of performing binning only for analysis, as discussed below). ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/12c7d31d8a03d347bac1cbe449138df659dc492793257cc0c61d13e2608ea4fa.jpg", + "image_caption": [ + "Figure 14: Effect of binning strategy on information plane dynamics. Results for the same tanh network and training regime as 1A, but with bin edges $b _ { i } \\in$ tanh(linspace $( - 5 0 , 5 0 , N )$ ). Measured with this binning structure, there is no compression in most layers. " + ], + "image_footnote": [], + "bbox": [ + 245, + 103, + 758, + 508 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 614, + 825, + 671 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Any implementation of a neural network on digital hardware is ultimately of finite precision, and hence is a binned, discrete representation. However, it is a very high resolution binning compared to that used here or by Shwartz-Ziv & Tishby (2017): single precision would correspond to using roughly $2 ^ { 3 2 }$ bins to discretize each hidden unit’s activity, as compared to the 30-100 used here. If the binning is fine-grained enough that each input $X$ yields a different binned activity pattern $h$ , then $H ( h ) \\stackrel { - } { = } \\log ( P )$ where $P$ is the number of examples in the dataset, and there will be little to no change in information during training. As an example, we show in Fig. 15 the result of binning at full machine precision. ", + "bbox": [ + 173, + 678, + 825, + 790 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Finally, we note two consequences of the assumption of noise/binning for the purposes of analysis. First, this means that the data processing inequality (DPI) does not apply to the noisy/binned mutual information estimates. The DPI states that information can only be destroyed through successive transformations, that is, if $X h _ { 1 } h _ { 2 }$ form a Markov chain, then $I ( X ; \\bar { h _ { 1 } } ) \\geq I ( \\bar { X ; } h _ { 2 } )$ (see, eg, Tishby & Zaslavsky (2015)). Because noise is added only for the purpose of analysis, however, this does not apply here. In particular, for the DPI to apply, the noise added at lower layers would have to propagate through the network to higher layers. That is, if the transformation from hidden layer 1 to hidden layer 2 is $h _ { 2 } = f ( h _ { 1 } )$ and $T _ { 1 } = h _ { 1 } + Z _ { 1 }$ is the hidden layer activity after adding noise, then the DPI would hold for the variable $\\tilde { T } _ { 2 } = f ( T _ { 1 } ) + Z _ { 2 } = f ( h _ { 1 } + Z _ { 1 } ) + Z _ { 2 }$ , not the quantity ", + "bbox": [ + 173, + 796, + 826, + 925 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/e2187148ec2ae4d72ade118f8c38e42693609c24edb83c7050bd489b4d6120ca.jpg", + "image_caption": [ + "Figure 15: Effect of binning at full machine precision. (A) ReLU network. (B) tanh network. Information in most layers stays pinned to $\\log _ { 2 } ( P ) = 1 2$ . Compression is only observed in the highest and smallest layers near the very end of training, when the saturation of tanh is strong enough to saturate machine precision. " + ], + "image_footnote": [], + "bbox": [ + 243, + 102, + 745, + 292 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "$T _ { 2 } = h _ { 2 } + Z _ { 2 } = f ( h _ { 1 } ) + Z _ { 2 }$ used in the analysis. Said another way, the Markov chain for $T _ { 2 }$ is $X h _ { 1 } h _ { 2 } T _ { 2 }$ , not $X h _ { 1 } T _ { 1 } T _ { 2 }$ , so the DPI states only that $I ( X ; h _ { 1 } ) \\ge I ( X ; T _ { 2 } )$ . ", + "bbox": [ + 174, + 398, + 823, + 429 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "A second consequence of the noise assumption is the fact that the mutual information is no longer invariant to invertible transformations of the hidden activity $h$ . A potentially attractive feature of a theory based on mutual information is that it can allow for comparisons between different architectures: mutual information is invariant to any invertible transformation of the variables, so two hidden representations could be very different in detail but yield identical mutual information with respect to the input. However, once noise is added to a hidden representation, this is no longer the case: the variable $T = h + Z$ is not invariant to reparametrizations of $h$ . As a simple example, consider a minimal linear network with scalar weights $w _ { 1 }$ and $w _ { 2 }$ that computes the output ${ \\hat { y } } = w _ { 2 } w _ { 1 } X$ . The hidden activity is $h = w _ { 1 } X$ . Now consider the family of networks in which we scale down $w _ { 1 }$ and scale up $w _ { 2 }$ by a factor $c \\neq 0$ , that is, these networks have weights $\\tilde { w } _ { 1 } = w _ { 1 } / c$ and $\\tilde { w } _ { 2 } = c w _ { 2 }$ , yielding the exact same input-output map $\\hat { y } = \\tilde { w } _ { 2 } \\tilde { w } _ { 1 } X = c w _ { 2 } ( w _ { 1 } / c ) X = w _ { 2 } w _ { 1 } X$ . Because they compute the same function, they necessarily generalize identically. However after introducing the noise assumption the mutual information is ", + "bbox": [ + 174, + 434, + 825, + 616 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/0c9da91e389db59f31738e85823c42564aea9f459391b32abdf8ec6fba0aab0e.jpg", + "text": "$$\n\\begin{array} { l l l } { { I ( T ; X ) } } & { { = } } & { { \\log \\left( w _ { 1 } ^ { 2 } / c ^ { 2 } + \\sigma _ { M I } ^ { 2 } \\right) - \\log \\left( \\sigma _ { M I } ^ { 2 } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 336, + 621, + 660, + 642 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where we have taken the setting in Section 3 in which $X$ is normal Gaussian, and independent Gaussian noise of variance $\\sigma _ { M I } ^ { 2 }$ is added for the purpose of MI computation. Clearly, the mutual information is now dependent on the scaling $c$ of the internal layer, even though this is an invertible linear transformation of the representation. Moreover, this shows that networks which generalize identically can nevertheless have very different mutual information with respect to the input when it is measured in this way. ", + "bbox": [ + 174, + 647, + 825, + 732 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "D WEIGHT NORMS OVER TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 753, + 485, + 768 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Our argument relating neural saturation to compression in mutual information relies on the notion that in typical training regimes, weights begin small and increase in size over the course of training. We note that this is a virtual necessity for a nonlinearity like tanh, which is linear around the origin: when initialized with small weights, the activity of a tanh network will be in this linear regime and the network can only compute a linear function of its input. Hence a real world nonlinear task can only be learned by increasing the norm of the weights so as to engage the tanh nonlinearity on some examples. This point can also be appreciated from norm-based capacity bounds on neural networks, which show that, for instance, the Rademacher complexity of a neural network with small weights must be low (Bartlett & Mendelson, 2002; Neyshabur et al., 2015). Finally, as an empirical matter, the networks trained in this paper do in fact increase the norm of their weights over the course of training, as shown by the green lines in Figure 20 for tanh and ReLU networks in the training setting of Shwartz-Ziv & Tishby (2017); Figures 9 and 10 for the MNIST networks; and Figure 21 for a linear network. ", + "bbox": [ + 173, + 784, + 825, + 924 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "E HISTOGRAMS OF NEURAL ACTIVATIONS ", + "text_level": 1, + "bbox": [ + 174, + 174, + 539, + 189 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Supplementary Figures 16 and 17 show histograms of neural activities over the course of training in tanh and ReLU networks respectively. ", + "bbox": [ + 176, + 209, + 825, + 239 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/8bbc07b14318e40e4fb277a40f930af47fb6bee0d59af7119d1a65cf2931ed0c.jpg", + "image_caption": [ + "Figure 16: Histogram of neural activities in a tanh network during training. The final three layers eventually saturate in the top and bottom bins corresponding to the saturation limits of the tanh activation function, explaining the compression observed in tanh. x-axis: training time in epochs. y-axis: Hidden activity bin values from lowest to highest. Colormap: density of hidden layer activities across all input examples. " + ], + "image_footnote": [], + "bbox": [ + 272, + 247, + 725, + 842 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/5f209aa29a04ff3253199dfe5ae51898ad1ec81c4d32a5a079b30809f462dc92.jpg", + "image_caption": [ + "Figure 17: Histogram of neural activities in a ReLU network during training. ReLU layers 1-5 have a roughly constant fraction of activities at zero, corresponding to instances where the ReLU is off; the nonzero activities disperse over the course of training without bound, yielding higher entropy distributions. The sigmoid output layer 6 converges to its saturation limits, and is the only layer that compresses during training (c.f. Fig. 1B). $\\mathbf { X }$ -axis: training time in epochs. y-axis: Hidden activity value. Colormap: density of hidden layer activities across all input examples. " + ], + "image_footnote": [], + "bbox": [ + 272, + 97, + 725, + 694 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "F INFORMATION PLANE DYNAMICS IN DEEPER LINEAR NETWORKS ", + "text_level": 1, + "bbox": [ + 173, + 845, + 740, + 861 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Supplementary Figure 18 shows information plane dynamics for a deep neural network with five hidden layers each containing 50 hidden units. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/cba1fe5de44777e480b97d883918063f6902a3728f471a1abd20879b524f24ab.jpg", + "image_caption": [ + "Figure 18: Information plane dynamics in a deep linear neural network. (A) Train and test error during learning. (B) Information plane dynamics. No compression is visible. " + ], + "image_footnote": [], + "bbox": [ + 178, + 104, + 818, + 270 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "G LINEAR MUTUAL INFORMATION CALCULATION ", + "text_level": 1, + "bbox": [ + 173, + 337, + 602, + 353 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "For the linear setting considered here, the mutual information between a hidden representation $T$ and the output $Y$ may be calculated using the relations ", + "bbox": [ + 173, + 367, + 823, + 397 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/c7bb894ca0901b7c568df05ec917564f6de8a14bb26aeff003d7a57a550511df.jpg", + "text": "$$\n\\begin{array} { c } { { { \\cal H } ( Y ) = \\displaystyle \\frac { N _ { o } } { 2 } \\log ( 2 \\pi e ) + \\displaystyle \\frac { 1 } { 2 } \\log | W _ { o } W _ { o } ^ { T } + \\sigma _ { o } ^ { 2 } I _ { N _ { o } } | , } } \\\\ { { { \\cal H } ( T ) = \\displaystyle \\frac { N _ { h } } { 2 } \\log ( 2 \\pi e ) + \\displaystyle \\frac { 1 } { 2 } \\log | \\bar { W } \\bar { W } ^ { T } + \\sigma _ { M I } ^ { 2 } I _ { N _ { h } } | , } } \\\\ { { { \\cal H } ( Y ; T ) = \\displaystyle \\frac { N _ { o } + N _ { h } } { 2 } \\log ( 2 \\pi e ) + \\displaystyle \\frac { 1 } { 2 } \\log | \\bar { W } \\bar { W } ^ { T } + \\sigma _ { M I } ^ { 2 } I _ { N _ { h } } \\quad \\quad \\bar { W } W _ { o } ^ { T } , } } \\\\ { { { \\cal I } ( Y ; T ) = { \\cal H } ( Y ) + { \\cal H } ( T ) - { \\cal H } ( Y ; T ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 228, + 402, + 767, + 520 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "H STOCHASTIC VS BATCH TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 536, + 495, + 553 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/547f68d75db31823ea92f3a340ffd0af1f2555a96902c610b5d4b54c25c651f4.jpg", + "image_caption": [ + "Figure 19 shows information plane dynamics for stochastic and batch gradient descent learning in a linear network. Randomness in the training process does not dramatically alter the information plane dynamics. ", + "Figure 19: Effect of stochastic training in linear networks. (A) Information plane dynamics for stochastic gradient descent in a linear network (same setting as Fig. 4). (B) Information plane dynamics for batch gradient descent. " + ], + "image_footnote": [], + "bbox": [ + 173, + 618, + 821, + 781 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "I GRADIENT SNR PHASE TRANSITION ", + "text_level": 1, + "bbox": [ + 174, + 863, + 508, + 880 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "The proposed mechanism of compression in Shwartz-Ziv & Tishby (2017) is noise arising from stochastic gradient descent training. The results in Section 4 of the main text show that compression still occurs under batch gradient descent learning, suggesting that in fact noise in the gradient updates is not the cause of compression. Here we investigate a related claim, namely that during training, networks switch between two phases. These phases are defined by the ratio of the mean of the gradient to the standard deviation of the gradient across training examples, called the gradient signal-to-noise ratio. In the first “drift” phase, the SNR is high, while in the second “diffusion” phase the SNR is low. Shwartz-Ziv & Tishby (2017) hypothesize that the drift phase corresponds to movement toward the minimum with no compression, while the diffusion phase corresponds to a constrained diffusion in weight configurations that attain the optimal loss, during which representations compress. However, two phases of gradient descent have been described more generally, sometimes known as the transient and stochastic phases or search and convergence phases (Murata, 1998; Chee & Toulis, 2017), suggesting that these phases might not be related specifically to compression behavior. ", + "bbox": [ + 174, + 895, + 825, + 924 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 257 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "In Fig. 20 we plot the gradient SNR over the course of training for the tanh and ReLU networks in the standard setting of Shwartz-Ziv & Tishby (2017). In particular, for each layer $l$ we calculate the mean and standard deviation as ", + "bbox": [ + 174, + 263, + 825, + 305 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/f293b8d1d9e4f8f91ea536c6fcee59a7fd3efa334024fbe6fd1f65b977d2185b.jpg", + "text": "$$\n\\begin{array} { r c l } { { m _ { l } } } & { { = } } & { { \\displaystyle \\left\\| \\left. \\frac { \\partial E } { \\partial W _ { l } } \\right. \\right\\| _ { F } } } \\\\ { { s _ { l } } } & { { = } } & { { \\displaystyle \\left\\| \\mathrm { S T D } \\left( \\frac { \\partial E } { \\partial W _ { l } } \\right) \\right\\| _ { F } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 403, + 386, + 591, + 459 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "where $\\langle \\cdot \\rangle$ denotes the mean and $S T D ( \\cdot )$ denotes the element-wise standard deviation across all training samples, and $\\left\\| \\cdot \\right\\| _ { F }$ denotes the Frobenius norm. The gradient SNR is then the ratio $m _ { l } / s _ { l }$ We additionally plot the norm of the weights $\\| W _ { l } \\| _ { F }$ over the course of training. ", + "bbox": [ + 174, + 540, + 825, + 583 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Both tanh and ReLU networks yield a similar qualitative pattern, with SNR undergoing a step-like transition to a lower value during training. Figures 9 and 10, fourth row, show similar plots for MNIST-trained networks. Again, SNR undergoes a transition from high to low over training. Hence the two phase nature of gradient descent appears to hold across the settings that we examine here. Crucially, this finding shows that the SNR transition is not related to the compression phenomenon because ReLU networks, which show the gradient SNR phase transition, do not compress. ", + "bbox": [ + 174, + 589, + 825, + 674 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Finally, to show the generality of the two-phase gradient SNR behavior and its independence from compression, we develop a minimal model of this phenomenon in a three neuron linear network. We consider the student-teacher setting of Fig. 3 but with $N _ { i } = N _ { h } = 1$ , such that the input and hidden layers have just a single neuron (as in the setting of Fig. 2). Here, with just a single hidden neuron, clearly there can be no compression so long as the first layer weight increases over the course of training. Figure 21AC shows that even in this simple setting, the SNR shows the phase transition but the weight norm increases over training. Hence again, the two phases of the gradient are present even though there is no compression. To intuitively understand the source of this behavior, note that the weights are initialized to be small and hence early in learning all must be increased in magnitude, yielding a consistent mean gradient. Once the network reaches the vicinity of the minimum, the mean weight change across all samples by definition goes to zero. The standard deviation remains finite, however, because on some specific examples error could be improved by increasing or decreasing the weights–even though across the whole dataset the mean error has been minimized. ", + "bbox": [ + 174, + 680, + 825, + 861 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Hence overall, our results show that a two-phase structure in the gradient SNR occurs in all settings we consider, even though compression occurs only in a subset. The gradient SNR behavior is therefore not causally related to compression dynamics, consistent with the view that saturating nonlinearities are the primary source of compression. ", + "bbox": [ + 176, + 867, + 823, + 924 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/fb8ca7128e6032058471eb13f01a0a5dba496e6d717ed509113be8ea4d25e22c.jpg", + "image_caption": [ + "Figure 20: Gradient SNR phase transition. (A) tanh networks trained in the standard setting of Shwartz-Ziv & Tishby (2017) show a phase transition in every layer. (B) ReLU networks also show a phase transition in every layer, despite exhibiting no compression. " + ], + "image_footnote": [], + "bbox": [ + 178, + 103, + 823, + 364 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/a9a3a5f252886534d662bb947459c62130ced799c330bd5e27586470c9beb20b.jpg", + "image_caption": [ + "Figure 21: Minimal model exhibiting gradient SNR phase transition. Here a three neuron linear network (architecture $1 - 1 - 1 )$ learns to approximate a teacher. Other parameters are teacher $S N R = 1$ , number of training samples $P = 1 0 0$ , learning rate .001. Left column: (A) The loss over training with SGD (minibatch size 1). (C) The resulting gradient SNR dynamics. Right column: (B) The loss over training with BGD. (D) The resulting gradient SNR dynamics averaging over all training samples (not minibatches, see text). 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In this work, we study the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 315, + 469, + 327 + ], + "spans": [ + { + "bbox": [ + 141, + 315, + 469, + 327 + ], + "score": 1.0, + "content": "information bottleneck (IB) theory of deep learning, which makes three specific", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 326, + 469, + 338 + ], + "spans": [ + { + "bbox": [ + 142, + 326, + 469, + 338 + ], + "score": 1.0, + "content": "claims: first, that deep networks undergo two distinct phases consisting of an", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 336, + 470, + 349 + ], + "spans": [ + { + "bbox": [ + 141, + 336, + 470, + 349 + ], + "score": 1.0, + "content": "initial fitting phase and a subsequent compression phase; second, that the compres-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 347, + 469, + 361 + ], + "spans": [ + { + "bbox": [ + 141, + 347, + 469, + 361 + ], + "score": 1.0, + "content": "sion phase is causally related to the excellent generalization performance of deep", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 359, + 469, + 371 + ], + "spans": [ + { + "bbox": [ + 141, + 359, + 469, + 371 + ], + "score": 1.0, + "content": "networks; and third, that the compression phase occurs due to the diffusion-like", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 370, + 469, + 382 + ], + "spans": [ + { + "bbox": [ + 141, + 370, + 469, + 382 + ], + "score": 1.0, + "content": "behavior of stochastic gradient descent. Here we show that none of these claims", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 381, + 469, + 392 + ], + "spans": [ + { + "bbox": [ + 141, + 381, + 469, + 392 + ], + "score": 1.0, + "content": "hold true in the general case. Through a combination of analytical results and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 392, + 469, + 404 + ], + "spans": [ + { + "bbox": [ + 141, + 392, + 469, + 404 + ], + "score": 1.0, + "content": "simulation, we demonstrate that the information plane trajectory is predominantly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 403, + 470, + 415 + ], + "spans": [ + { + "bbox": [ + 141, + 403, + 470, + 415 + ], + "score": 1.0, + "content": "a function of the neural nonlinearity employed: double-sided saturating nonlineari-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 414, + 469, + 425 + ], + "spans": [ + { + "bbox": [ + 141, + 414, + 469, + 425 + ], + "score": 1.0, + "content": "ties like tanh yield a compression phase as neural activations enter the saturation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 425, + 470, + 437 + ], + "spans": [ + { + "bbox": [ + 141, + 425, + 470, + 437 + ], + "score": 1.0, + "content": "regime, but linear activation functions and single-sided saturating nonlinearities", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 435, + 470, + 447 + ], + "spans": [ + { + "bbox": [ + 141, + 435, + 470, + 447 + ], + "score": 1.0, + "content": "like the widely used ReLU in fact do not. Moreover, we find that there is no evident", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 447, + 469, + 458 + ], + "spans": [ + { + "bbox": [ + 141, + 447, + 469, + 458 + ], + "score": 1.0, + "content": "causal connection between compression and generalization: networks that do not", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 458, + 470, + 469 + ], + "spans": [ + { + "bbox": [ + 141, + 458, + 470, + 469 + ], + "score": 1.0, + "content": "compress are still capable of generalization, and vice versa. Next, we show that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 468, + 469, + 482 + ], + "spans": [ + { + "bbox": [ + 141, + 468, + 469, + 482 + ], + "score": 1.0, + "content": "the compression phase, when it exists, does not arise from stochasticity in training", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 479, + 470, + 492 + ], + "spans": [ + { + "bbox": [ + 141, + 479, + 470, + 492 + ], + "score": 1.0, + "content": "by demonstrating that we can replicate the IB findings using full batch gradient", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 491, + 469, + 503 + ], + "spans": [ + { + "bbox": [ + 141, + 491, + 469, + 503 + ], + "score": 1.0, + "content": "descent rather than stochastic gradient descent. Finally, we show that when an", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 502, + 470, + 513 + ], + "spans": [ + { + "bbox": [ + 141, + 502, + 470, + 513 + ], + "score": 1.0, + "content": "input domain consists of a subset of task-relevant and task-irrelevant information,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 513, + 469, + 524 + ], + "spans": [ + { + "bbox": [ + 141, + 513, + 469, + 524 + ], + "score": 1.0, + "content": "hidden representations do compress the task-irrelevant information, although the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 523, + 470, + 536 + ], + "spans": [ + { + "bbox": [ + 141, + 523, + 470, + 536 + ], + "score": 1.0, + "content": "overall information about the input may monotonically increase with training time,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 534, + 470, + 547 + ], + "spans": [ + { + "bbox": [ + 141, + 534, + 470, + 547 + ], + "score": 1.0, + "content": "and that this compression happens concurrently with the fitting process rather than", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 546, + 307, + 558 + ], + "spans": [ + { + "bbox": [ + 142, + 546, + 307, + 558 + ], + "score": 1.0, + "content": "during a subsequent compression period.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 108, + 575, + 206, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 208, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 208, + 591 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "Deep neural networks (Schmidhuber, 2015; LeCun et al., 2015) are the tool of choice for real-world", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "tasks ranging from visual object recognition (Krizhevsky et al., 2012), to unsupervised learning", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "(Goodfellow et al., 2014; Lotter et al., 2016) and reinforcement learning (Silver et al., 2016). These", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 647 + ], + "score": 1.0, + "content": "practical successes have spawned many attempts to explain the performance of deep learning systems", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "(Kadmon & Sompolinsky, 2016), mostly in terms of the properties and dynamics of the optimization", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "problem in the space of weights (Saxe et al., 2014; Choromanska et al., 2015; Advani & Saxe, 2017),", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "or the classes of functions that can be efficiently represented by deep networks (Montufar et al.,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "2014; Poggio et al., 2017). This paper analyzes a recent inventive proposal to study the dynamics of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "learning through the lens of information theory (Tishby & Zaslavsky, 2015; Shwartz-Ziv & Tishby,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "2017). In this view, deep learning is a question of representation learning: each layer of a deep neural", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "network can be seen as a set of summary statistics which contain some but not all of the information", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "present in the input, while retaining as much information about the target output as possible. The", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 44.5 + } + ], + "page_idx": 0, + "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, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 79, + 380, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 381, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 381, + 97 + ], + "score": 1.0, + "content": "ON THE INFORMATION BOTTLENECK", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 98, + 323, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 98, + 323, + 117 + ], + "score": 1.0, + "content": "THEORY OF DEEP LEARNING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "list", + "bbox": [ + 112, + 134, + 491, + 168 + ], + "lines": [ + { + "bbox": [ + 111, + 134, + 380, + 148 + ], + "spans": [ + { + "bbox": [ + 111, + 134, + 380, + 148 + ], + "score": 1.0, + "content": "Andrew M. 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In this work, we study the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 315, + 469, + 327 + ], + "spans": [ + { + "bbox": [ + 141, + 315, + 469, + 327 + ], + "score": 1.0, + "content": "information bottleneck (IB) theory of deep learning, which makes three specific", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 326, + 469, + 338 + ], + "spans": [ + { + "bbox": [ + 142, + 326, + 469, + 338 + ], + "score": 1.0, + "content": "claims: first, that deep networks undergo two distinct phases consisting of an", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 336, + 470, + 349 + ], + "spans": [ + { + "bbox": [ + 141, + 336, + 470, + 349 + ], + "score": 1.0, + "content": "initial fitting phase and a subsequent compression phase; second, that the compres-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 347, + 469, + 361 + ], + "spans": [ + { + "bbox": [ + 141, + 347, + 469, + 361 + ], + "score": 1.0, + "content": "sion phase is causally related to the excellent generalization performance of deep", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 359, + 469, + 371 + ], + "spans": [ + { + "bbox": [ + 141, + 359, + 469, + 371 + ], + "score": 1.0, + "content": "networks; and third, that the compression phase occurs due to the diffusion-like", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 370, + 469, + 382 + ], + "spans": [ + { + "bbox": [ + 141, + 370, + 469, + 382 + ], + "score": 1.0, + "content": "behavior of stochastic gradient descent. Here we show that none of these claims", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 381, + 469, + 392 + ], + "spans": [ + { + "bbox": [ + 141, + 381, + 469, + 392 + ], + "score": 1.0, + "content": "hold true in the general case. Through a combination of analytical results and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 392, + 469, + 404 + ], + "spans": [ + { + "bbox": [ + 141, + 392, + 469, + 404 + ], + "score": 1.0, + "content": "simulation, we demonstrate that the information plane trajectory is predominantly", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 403, + 470, + 415 + ], + "spans": [ + { + "bbox": [ + 141, + 403, + 470, + 415 + ], + "score": 1.0, + "content": "a function of the neural nonlinearity employed: double-sided saturating nonlineari-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 414, + 469, + 425 + ], + "spans": [ + { + "bbox": [ + 141, + 414, + 469, + 425 + ], + "score": 1.0, + "content": "ties like tanh yield a compression phase as neural activations enter the saturation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 425, + 470, + 437 + ], + "spans": [ + { + "bbox": [ + 141, + 425, + 470, + 437 + ], + "score": 1.0, + "content": "regime, but linear activation functions and single-sided saturating nonlinearities", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 435, + 470, + 447 + ], + "spans": [ + { + "bbox": [ + 141, + 435, + 470, + 447 + ], + "score": 1.0, + "content": "like the widely used ReLU in fact do not. Moreover, we find that there is no evident", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 447, + 469, + 458 + ], + "spans": [ + { + "bbox": [ + 141, + 447, + 469, + 458 + ], + "score": 1.0, + "content": "causal connection between compression and generalization: networks that do not", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 458, + 470, + 469 + ], + "spans": [ + { + "bbox": [ + 141, + 458, + 470, + 469 + ], + "score": 1.0, + "content": "compress are still capable of generalization, and vice versa. Next, we show that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 468, + 469, + 482 + ], + "spans": [ + { + "bbox": [ + 141, + 468, + 469, + 482 + ], + "score": 1.0, + "content": "the compression phase, when it exists, does not arise from stochasticity in training", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 479, + 470, + 492 + ], + "spans": [ + { + "bbox": [ + 141, + 479, + 470, + 492 + ], + "score": 1.0, + "content": "by demonstrating that we can replicate the IB findings using full batch gradient", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 141, + 491, + 469, + 503 + ], + "spans": [ + { + "bbox": [ + 141, + 491, + 469, + 503 + ], + "score": 1.0, + "content": "descent rather than stochastic gradient descent. Finally, we show that when an", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 502, + 470, + 513 + ], + "spans": [ + { + "bbox": [ + 141, + 502, + 470, + 513 + ], + "score": 1.0, + "content": "input domain consists of a subset of task-relevant and task-irrelevant information,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 141, + 513, + 469, + 524 + ], + "spans": [ + { + "bbox": [ + 141, + 513, + 469, + 524 + ], + "score": 1.0, + "content": "hidden representations do compress the task-irrelevant information, although the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 523, + 470, + 536 + ], + "spans": [ + { + "bbox": [ + 141, + 523, + 470, + 536 + ], + "score": 1.0, + "content": "overall information about the input may monotonically increase with training time,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 534, + 470, + 547 + ], + "spans": [ + { + "bbox": [ + 141, + 534, + 470, + 547 + ], + "score": 1.0, + "content": "and that this compression happens concurrently with the fitting process rather than", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 142, + 546, + 307, + 558 + ], + "spans": [ + { + "bbox": [ + 142, + 546, + 307, + 558 + ], + "score": 1.0, + "content": "during a subsequent compression period.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 25.5, + "bbox_fs": [ + 141, + 294, + 470, + 558 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 575, + 206, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 208, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 208, + 591 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "Deep neural networks (Schmidhuber, 2015; LeCun et al., 2015) are the tool of choice for real-world", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "tasks ranging from visual object recognition (Krizhevsky et al., 2012), to unsupervised learning", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "(Goodfellow et al., 2014; Lotter et al., 2016) and reinforcement learning (Silver et al., 2016). These", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 647 + ], + "score": 1.0, + "content": "practical successes have spawned many attempts to explain the performance of deep learning systems", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "(Kadmon & Sompolinsky, 2016), mostly in terms of the properties and dynamics of the optimization", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "problem in the space of weights (Saxe et al., 2014; Choromanska et al., 2015; Advani & Saxe, 2017),", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "or the classes of functions that can be efficiently represented by deep networks (Montufar et al.,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "2014; Poggio et al., 2017). This paper analyzes a recent inventive proposal to study the dynamics of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "learning through the lens of information theory (Tishby & Zaslavsky, 2015; Shwartz-Ziv & Tishby,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "2017). In this view, deep learning is a question of representation learning: each layer of a deep neural", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "network can be seen as a set of summary statistics which contain some but not all of the information", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "present in the input, while retaining as much information about the target output as possible. The", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "amount of information in a hidden layer regarding the input and output can then be measured over the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "course of learning, yielding a picture of the optimization process in the information plane. Crucially,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "this method holds the promise to serve as a general analysis that can be used to compare different", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "architectures, using the common currency of mutual information. Moreover, the elegant information", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "bottleneck (IB) theory provides a fundamental bound on the amount of input compression and target", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "output information that any representation can achieve (Tishby et al., 1999). The IB bound thus", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "serves as a method-agnostic ideal to which different architectures and algorithms may be compared.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 600, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 159 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "amount of information in a hidden layer regarding the input and output can then be measured over the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "course of learning, yielding a picture of the optimization process in the information plane. Crucially,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "this method holds the promise to serve as a general analysis that can be used to compare different", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "architectures, using the common currency of mutual information. Moreover, the elegant information", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "bottleneck (IB) theory provides a fundamental bound on the amount of input compression and target", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "output information that any representation can achieve (Tishby et al., 1999). The IB bound thus", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "serves as a method-agnostic ideal to which different architectures and algorithms may be compared.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "A preliminary empirical exploration of these ideas in deep neural networks has yielded striking", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "findings (Shwartz-Ziv & Tishby, 2017). Most saliently, trajectories in the information plane appear to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "consist of two distinct phases: an initial “fitting” phase where mutual information between the hidden", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 209 + ], + "score": 1.0, + "content": "layers and both the input and output increases, and a subsequent “compression” phase where mutual", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "information between the hidden layers and the input decreases. It has been hypothesized that this", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "compression phase is responsible for the excellent generalization performance of deep networks, and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "further, that this compression phase occurs due to the random diffusion-like behavior of stochastic", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 176, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 176, + 254 + ], + "score": 1.0, + "content": "gradient descent.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 259, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "Here we study these phenomena using a combination of analytical methods and simulation. In Section", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "2, we show that the compression observed by Shwartz-Ziv & Tishby (2017) arises primarily due to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "the double-saturating tanh activation function used. Using simple models, we elucidate the effect of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "neural nonlinearity on the compression phase. Importantly, we demonstrate that the ReLU activation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "function, often the nonlinearity of choice in practice, does not exhibit a compression phase. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "discuss how this compression via nonlinearity is related to the assumption of binning or noise in the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 507, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 507, + 338 + ], + "score": 1.0, + "content": "hidden layer representation. To better understand the dynamics of learning in the information plane,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "in Section 3 we study deep linear networks in a tractable setting where the mutual information can be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "score": 1.0, + "content": "calculated exactly. We find that deep linear networks do not compress over the course of training for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "the setting we examine. Further, we show a dissociation between generalization and compression.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "In Section 4, we investigate whether stochasticity in the training process causes compression in the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "information plane. We train networks with full batch gradient descent, and compare the results to those", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 404 + ], + "score": 1.0, + "content": "obtained with stochastic gradient descent. We find comparable compression in both cases, indicating", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "that the stochasticity of SGD is not a primary factor in the observed compression phase. Moreover,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 426 + ], + "score": 1.0, + "content": "we show that the two phases of SGD occur even in networks that do not compress, demonstrating", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "that the phases are not causally related to compression. These results may seem difficult to reconcile", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "with the intuition that compression can be necessary to attain good performance: if some input", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 444, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 459 + ], + "score": 1.0, + "content": "channels primarily convey noise, good generalization requires excluding them. Therefore, in Section", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "5 we study a situation with explicitly task-relevant and task-irrelevant input dimensions. We show", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 465, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 481 + ], + "score": 1.0, + "content": "that the hidden-layer mutual information with the task-irrelevant subspace does indeed drop during", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "training, though the overall information with the input increases. However, instead of a secondary", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "compression phase, this task-irrelevant information is compressed at the same time that the task-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 514 + ], + "score": 1.0, + "content": "relevant information is boosted. Our results highlight the importance of noise assumptions in applying", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "information theoretic analyses to deep learning systems, and put in doubt the generality of the IB", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 523, + 483, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 483, + 534 + ], + "score": 1.0, + "content": "theory of deep learning as an explanation of generalization performance in deep architectures.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 551, + 366, + 564 + ], + "lines": [ + { + "bbox": [ + 104, + 550, + 367, + 566 + ], + "spans": [ + { + "bbox": [ + 104, + 550, + 367, + 566 + ], + "score": 1.0, + "content": "2 COMPRESSION AND NEURAL NONLINEARITIES", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "The starting point for our analysis is the observation that changing the activation function can", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "markedly change the trajectory of a network in the information plane. In Figure 1A, we show", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "our replication of the result reported by Shwartz-Ziv & Tishby (2017) for networks with the tanh", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "nonlinearity.1 This replication was performed with the code supplied by the authors of Shwartz-Ziv", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "& Tishby (2017), and closely follows the experimental setup described therein. Briefly, a neural", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "network with 7 fully connected hidden layers of width 12-10-7-5-4-3-2 is trained with stochastic", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "gradient descent to produce a binary classification from a 12-dimensional input. In our replication we", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "used 256 randomly selected samples per batch. The mutual information of the network layers with", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "respect to the input and output variables is calculated by binning the neuron’s tanh output activations", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "into 30 equal intervals between -1 and 1. Discretized values for each neuron in each layer are then", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "used to directly calculate the joint distributions, over the 4096 equally likely input patterns and true", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "output labels. In line with prior work (Shwartz-Ziv & Tishby, 2017), the dynamics in Fig. 1 show a", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 46.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 115, + 722, + 496, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 719, + 497, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 719, + 497, + 733 + ], + "score": 1.0, + "content": "1Code for our results is available at https://github.com/artemyk/ibsgd/tree/iclr2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 159 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 82, + 506, + 161 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "A preliminary empirical exploration of these ideas in deep neural networks has yielded striking", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "findings (Shwartz-Ziv & Tishby, 2017). Most saliently, trajectories in the information plane appear to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "consist of two distinct phases: an initial “fitting” phase where mutual information between the hidden", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 209 + ], + "score": 1.0, + "content": "layers and both the input and output increases, and a subsequent “compression” phase where mutual", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "information between the hidden layers and the input decreases. It has been hypothesized that this", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "compression phase is responsible for the excellent generalization performance of deep networks, and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "further, that this compression phase occurs due to the random diffusion-like behavior of stochastic", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 176, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 176, + 254 + ], + "score": 1.0, + "content": "gradient descent.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 165, + 505, + 254 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 259, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "Here we study these phenomena using a combination of analytical methods and simulation. In Section", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "2, we show that the compression observed by Shwartz-Ziv & Tishby (2017) arises primarily due to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "the double-saturating tanh activation function used. Using simple models, we elucidate the effect of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "neural nonlinearity on the compression phase. Importantly, we demonstrate that the ReLU activation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "function, often the nonlinearity of choice in practice, does not exhibit a compression phase. We", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "discuss how this compression via nonlinearity is related to the assumption of binning or noise in the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 507, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 507, + 338 + ], + "score": 1.0, + "content": "hidden layer representation. To better understand the dynamics of learning in the information plane,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "in Section 3 we study deep linear networks in a tractable setting where the mutual information can be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "score": 1.0, + "content": "calculated exactly. We find that deep linear networks do not compress over the course of training for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "the setting we examine. Further, we show a dissociation between generalization and compression.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "In Section 4, we investigate whether stochasticity in the training process causes compression in the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "information plane. We train networks with full batch gradient descent, and compare the results to those", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 404 + ], + "score": 1.0, + "content": "obtained with stochastic gradient descent. We find comparable compression in both cases, indicating", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 415 + ], + "score": 1.0, + "content": "that the stochasticity of SGD is not a primary factor in the observed compression phase. Moreover,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 426 + ], + "score": 1.0, + "content": "we show that the two phases of SGD occur even in networks that do not compress, demonstrating", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "that the phases are not causally related to compression. These results may seem difficult to reconcile", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "with the intuition that compression can be necessary to attain good performance: if some input", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 444, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 459 + ], + "score": 1.0, + "content": "channels primarily convey noise, good generalization requires excluding them. Therefore, in Section", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "5 we study a situation with explicitly task-relevant and task-irrelevant input dimensions. We show", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 465, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 481 + ], + "score": 1.0, + "content": "that the hidden-layer mutual information with the task-irrelevant subspace does indeed drop during", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "training, though the overall information with the input increases. However, instead of a secondary", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "compression phase, this task-irrelevant information is compressed at the same time that the task-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 514 + ], + "score": 1.0, + "content": "relevant information is boosted. Our results highlight the importance of noise assumptions in applying", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "information theoretic analyses to deep learning systems, and put in doubt the generality of the IB", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 523, + 483, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 483, + 534 + ], + "score": 1.0, + "content": "theory of deep learning as an explanation of generalization performance in deep architectures.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 259, + 507, + 534 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 551, + 366, + 564 + ], + "lines": [ + { + "bbox": [ + 104, + 550, + 367, + 566 + ], + "spans": [ + { + "bbox": [ + 104, + 550, + 367, + 566 + ], + "score": 1.0, + "content": "2 COMPRESSION AND NEURAL NONLINEARITIES", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "The starting point for our analysis is the observation that changing the activation function can", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "markedly change the trajectory of a network in the information plane. In Figure 1A, we show", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "our replication of the result reported by Shwartz-Ziv & Tishby (2017) for networks with the tanh", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "nonlinearity.1 This replication was performed with the code supplied by the authors of Shwartz-Ziv", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "& Tishby (2017), and closely follows the experimental setup described therein. Briefly, a neural", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "network with 7 fully connected hidden layers of width 12-10-7-5-4-3-2 is trained with stochastic", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "gradient descent to produce a binary classification from a 12-dimensional input. In our replication we", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "used 256 randomly selected samples per batch. The mutual information of the network layers with", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "respect to the input and output variables is calculated by binning the neuron’s tanh output activations", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "into 30 equal intervals between -1 and 1. Discretized values for each neuron in each layer are then", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "used to directly calculate the joint distributions, over the 4096 equally likely input patterns and true", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "output labels. In line with prior work (Shwartz-Ziv & Tishby, 2017), the dynamics in Fig. 1 show a", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "transition between an initial fitting phase, during which information about the input increases, and a", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 444, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 444, + 106 + ], + "score": 1.0, + "content": "subsequent compression phase, during which information about the input decreases.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 578, + 506, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 154, + 506, + 476 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 154, + 506, + 476 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 154, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 506, + 476 + ], + "score": 0.974, + "type": "image", + "image_path": "ae057461c9856b2cafbb8dc433b0cc3cacf699acd8de55f5ace722373b08a275.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 154, + 506, + 261.3333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 261.3333333333333, + 506, + 368.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 368.66666666666663, + 506, + 475.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 489, + 506, + 654 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "Figure 1: Information plane dynamics and neural nonlinearities. (A) Replication of Shwartz-Ziv &", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "Tishby (2017) for a network with tanh nonlinearities (except for the final classification layer which", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 264, + 525 + ], + "score": 1.0, + "content": "contains two sigmoidal neurons). The", + "type": "text" + }, + { + "bbox": [ + 265, + 513, + 271, + 522 + ], + "score": 0.38, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "-axis plots information between each layer and the input,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "while the y-axis plots information between each layer and the output. The color scale indicates", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "training time in epochs. 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The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 190 + ], + "score": 1.0, + "content": "mutual information with the input monotonically increases in all ReLU layers, with no apparent", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "compression phase. 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Fig. C-D show that, again, tanh networks compressed but ReLU", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 254 + ], + "score": 1.0, + "content": "networks did not. Appendix B shows that similar results also obtain with the popular nonparametric", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 113, + 263 + ], + "score": 0.26, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "-nearest-neighbor estimator of Kraskov et al. (2004), and for other neural nonlinearities. Thus, the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 436, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 436, + 276 + ], + "score": 1.0, + "content": "choice of nonlinearity substantively affects the dynamics in the information plane.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 109, + 506, + 276 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 280, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "To understand the impact of neural nonlinearity on the mutual information dynamics, we develop", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 304 + ], + "score": 1.0, + "content": "a minimal model that exhibits this phenomenon. 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We assume a scalar Gaussian input distribution", + "type": "text" + }, + { + "bbox": [ + 410, + 303, + 464, + 315 + ], + "score": 0.91, + "content": "X \\sim \\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 303, + 505, + 315 + ], + "score": 1.0, + "content": ", which is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 266, + 327 + ], + "score": 1.0, + "content": "fed through the scalar first layer weight", + "type": "text" + }, + { + "bbox": [ + 266, + 315, + 279, + 325 + ], + "score": 0.87, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 313, + 448, + 327 + ], + "score": 1.0, + "content": ", and passed through a neural nonlinearity", + "type": "text" + }, + { + "bbox": [ + 448, + 315, + 466, + 326 + ], + "score": 0.9, + "content": "f ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 313, + 506, + 327 + ], + "score": 1.0, + "content": ", yielding", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 199, + 338 + ], + "score": 1.0, + "content": "the hidden unit activity", + "type": "text" + }, + { + "bbox": [ + 199, + 325, + 253, + 337 + ], + "score": 0.93, + "content": "h = f ( w _ { 1 } X )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 325, + 506, + 338 + ], + "score": 1.0, + "content": ". To calculate the mutual information with the input, this hidden", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 345, + 349 + ], + "score": 1.0, + "content": "unit activity is then binned yielding the new discrete variable", + "type": "text" + }, + { + "bbox": [ + 345, + 336, + 393, + 348 + ], + "score": 0.92, + "content": "T = \\dot { \\mathbf { b i n } } ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 335, + 505, + 349 + ], + "score": 1.0, + "content": "(for instance, into 30 evenly", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 177, + 360 + ], + "score": 1.0, + "content": "spaced bins from", + "type": "text" + }, + { + "bbox": [ + 177, + 347, + 187, + 357 + ], + "score": 0.31, + "content": "^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "to 1 for the tanh nonlinearity). This binning process is depicted in Fig. 2B. In", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 278, + 370 + ], + "score": 1.0, + "content": "this simple setting, the mutual information", + "type": "text" + }, + { + "bbox": [ + 279, + 358, + 313, + 370 + ], + "score": 0.93, + "content": "I ( T ; X )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 357, + 478, + 370 + ], + "score": 1.0, + "content": "between the binned hidden layer activity", + "type": "text" + }, + { + "bbox": [ + 479, + 358, + 487, + 367 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 369, + 313, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 144, + 381 + ], + "score": 1.0, + "content": "the input", + "type": "text" + }, + { + "bbox": [ + 144, + 369, + 154, + 378 + ], + "score": 0.8, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 369, + 313, + 381 + ], + "score": 1.0, + "content": "can be calculated exactly. 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This probability can be calculated exactly for monotonic nonlinearities", + "type": "text" + }, + { + "bbox": [ + 488, + 487, + 505, + 499 + ], + "score": 0.87, + "content": "f ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 248, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 234, + 509 + ], + "score": 1.0, + "content": "using the cumulative density of", + "type": "text" + }, + { + "bbox": [ + 234, + 498, + 244, + 507 + ], + "score": 0.85, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 497, + 248, + 509 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 453, + 506, + 509 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 513, + 411, + 528 + ], + "lines": [ + { + "bbox": [ + 198, + 513, + 411, + 528 + ], + "spans": [ + { + "bbox": [ + 198, + 513, + 411, + 528 + ], + "score": 0.88, + "content": "p _ { i } = P ( X \\geq f ^ { - 1 } ( b _ { i } ) / w _ { 1 } \\mathrm { a n d } X < f ^ { - 1 } ( b _ { i + 1 } ) / w _ { 1 } ) ,", + "type": "interline_equation", + "image_path": "af66e5c0522c9c7604634cf8f308e61b27609419bed5b8c877a971a4a87b2e43.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 198, + 513, + 411, + 528 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 533, + 284, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 531, + 285, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 133, + 547 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 532, + 161, + 545 + ], + "score": 0.94, + "content": "f ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 531, + 263, + 547 + ], + "score": 1.0, + "content": "is the inverse function of", + "type": "text" + }, + { + "bbox": [ + 264, + 533, + 281, + 546 + ], + "score": 0.91, + "content": "f ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 531, + 285, + 547 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 531, + 285, + 547 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 324, + 563 + ], + "score": 1.0, + "content": "As shown in Fig. 2C-D, as a function of the weight", + "type": "text" + }, + { + "bbox": [ + 325, + 552, + 337, + 561 + ], + "score": 0.86, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 550, + 506, + 563 + ], + "score": 1.0, + "content": ", mutual information with the input first", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "increases and then decreases for the tanh nonlinearity, but always increases for the ReLU nonlinearity.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 223, + 585 + ], + "score": 1.0, + "content": "Intuitively, for small weights", + "type": "text" + }, + { + "bbox": [ + 223, + 573, + 254, + 583 + ], + "score": 0.9, + "content": "w _ { 1 } \\approx 0", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 572, + 506, + 585 + ], + "score": 1.0, + "content": ", neural activities lie near zero on the approximately linear part", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 239, + 595 + ], + "score": 1.0, + "content": "of the tanh function. 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This", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "is more or less a coin flip, containing mutual information with the input of approximately 1 bit.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 207, + 640 + ], + "score": 1.0, + "content": "Hence the distribution of", + "type": "text" + }, + { + "bbox": [ + 207, + 627, + 216, + 637 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "collapses to a much lower entropy distribution, yielding compression for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "large weight values. 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It therefore remains", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 617, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 504, + 628 + ], + "score": 1.0, + "content": "unclear whether robustness of a representation to this sort of noise in fact influences generalization", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 628, + 262, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 262, + 640 + ], + "score": 1.0, + "content": "performance in deep learning systems.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "Furthermore, the addition of noise means that different architectures may no longer be compared in a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "common currency of mutual information: the binning/noise structure is arbitrary, and architectures", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "that implement an identical input-output map can nevertheless have different robustness to noise added", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "in their internal representation. 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(A) A linear", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 365, + 303 + ], + "score": 1.0, + "content": "teacher network generates a dataset by passing Gaussian inputs", + "type": "text" + }, + { + "bbox": [ + 366, + 290, + 376, + 299 + ], + "score": 0.8, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 289, + 505, + 303 + ], + "score": 1.0, + "content": "through its weights and adding", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "noise. 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In particular, we exploit recent results on the generalization dynamics", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 559, + 507, + 573 + ], + "spans": [ + { + "bbox": [ + 104, + 559, + 507, + 573 + ], + "score": 1.0, + "content": "in simple linear networks trained in a student-teacher setup (Seung et al., 1992; Advani & Saxe,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 584 + ], + "score": 1.0, + "content": "2017). In a student-teacher setting, one “student” neural network learns to approximate the output", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 580, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 596 + ], + "score": 1.0, + "content": "of another “teacher” neural network. 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Following the formulation", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 378, + 679 + ], + "score": 1.0, + "content": "of (Advani & Saxe, 2017), we assume multivariate Gaussian inputs", + "type": "text" + }, + { + "bbox": [ + 378, + 664, + 454, + 677 + ], + "score": 0.94, + "content": "\\begin{array} { r } { X \\sim \\mathcal { N } ( 0 , \\frac { 1 } { N _ { i } } \\overline { { I } } _ { N _ { i } } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 664, + 506, + 679 + ], + "score": 1.0, + "content": "and a scalar", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 135, + 690 + ], + "score": 1.0, + "content": "output", + "type": "text" + }, + { + "bbox": [ + 136, + 677, + 145, + 687 + ], + "score": 0.67, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 676, + 403, + 690 + ], + "score": 1.0, + "content": ". The output is generated by the teacher network according to", + "type": "text" + }, + { + "bbox": [ + 404, + 677, + 473, + 688 + ], + "score": 0.91, + "content": "Y = \\dot { W } _ { 0 } X + \\epsilon _ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 676, + 506, + 690 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 168, + 700 + ], + "score": 0.93, + "content": "\\epsilon _ { o } \\stackrel { - } { \\sim } \\mathcal { N } ( 0 , \\sigma _ { o } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "represents aspects of the target function which cannot be represented by a neural", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "network (that is, the approximation error or bias in statistical learning theory), and the teacher weights", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 107, + 710, + 122, + 721 + ], + "score": 0.87, + "content": "W _ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 708, + 247, + 723 + ], + "score": 1.0, + "content": "are drawn independently from", + "type": "text" + }, + { + "bbox": [ + 248, + 710, + 287, + 721 + ], + "score": 0.88, + "content": "{ \\mathcal N } ( 0 , \\sigma _ { w } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 708, + 506, + 723 + ], + "score": 1.0, + "content": ". Here, the weights of the teacher define the rule to be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 718, + 507, + 735 + ], + "spans": [ + { + "bbox": [ + 104, + 718, + 238, + 735 + ], + "score": 1.0, + "content": "learned. The signal to noise ratio", + "type": "text" + }, + { + "bbox": [ + 239, + 721, + 300, + 732 + ], + "score": 0.91, + "content": "\\mathrm { S N R } = \\sigma _ { w } ^ { 2 } / \\sigma _ { o } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 718, + 507, + 735 + ], + "score": 1.0, + "content": "determines the strength of the rule linking inputs to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "outputs relative to the inevitable approximation error. We emphasize that the “noise” added to the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "teacher’s output is fundamentally different from the noise added for the purpose of calculating mutual", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 159, + 117 + ], + "score": 1.0, + "content": "information:", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 160, + 106, + 170, + 115 + ], + "score": 0.85, + "content": "\\epsilon _ { o }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 170, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "models the approximation error for the task–even the best possible neural network", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "may still make errors because the target function is not representable exactly as a neural network–and", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 456, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 456, + 138 + ], + "score": 1.0, + "content": "is part of the construction of the dataset, not part of the analysis of the student network.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 642, + 507, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "outputs relative to the inevitable approximation error. We emphasize that the “noise” added to the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "teacher’s output is fundamentally different from the noise added for the purpose of calculating mutual", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 159, + 117 + ], + "score": 1.0, + "content": "information:", + "type": "text" + }, + { + "bbox": [ + 160, + 106, + 170, + 115 + ], + "score": 0.85, + "content": "\\epsilon _ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "models the approximation error for the task–even the best possible neural network", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "may still make errors because the target function is not representable exactly as a neural network–and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 456, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 456, + 138 + ], + "score": 1.0, + "content": "is part of the construction of the dataset, not part of the analysis of the student network.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 255 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 276, + 155 + ], + "score": 1.0, + "content": "To train the student network, a dataset of", + "type": "text" + }, + { + "bbox": [ + 276, + 144, + 285, + 153 + ], + "score": 0.81, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 142, + 505, + 155 + ], + "score": 1.0, + "content": "examples is generated using the teacher. The student", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "score": 1.0, + "content": "network is then trained to minimize the mean squared error between its output and the target output", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "using standard (batch or stochastic) gradient descent on this dataset. 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While linear activation functions stop the network from computing", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "complex nonlinear functions of the input, deep linear networks nevertheless show complicated", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "nonlinear learning trajectories (Saxe et al., 2014), the optimization problem remains nonconvex (Baldi", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "score": 1.0, + "content": "& Hornik, 1989), and the generalization dynamics can exhibit substantial overtraining (Fukumizu,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 244, + 225, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 225, + 255 + ], + "score": 1.0, + "content": "1998; Advani & Saxe, 2017).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 260, + 504, + 282 + ], + "lines": [ + { + "bbox": [ + 106, + 260, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 274 + ], + "score": 1.0, + "content": "Importantly, because of the simplified setting considered here, the true generalization error is easily", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 271, + 157, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 157, + 283 + ], + "score": 1.0, + "content": "shown to be", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 280, + 372, + 296 + ], + "lines": [ + { + "bbox": [ + 239, + 280, + 372, + 296 + ], + "spans": [ + { + "bbox": [ + 239, + 280, + 372, + 296 + ], + "score": 0.93, + "content": "E _ { g } ( t ) = | | W _ { o } - W _ { t o t } ( t ) | | _ { F } ^ { 2 } + \\sigma _ { o } ^ { 2 }", + "type": "interline_equation", + "image_path": "321c56854718da12e7f1927283f3e5bb80e2ef3af1ea37f630feb0175fb7ef5f.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 239, + 280, + 372, + 296 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 507, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 134, + 311 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 298, + 166, + 309 + ], + "score": 0.92, + "content": "W _ { t o t } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 296, + 466, + 311 + ], + "score": 1.0, + "content": "is the overall linear map implemented by the network at training epoch", + "type": "text" + }, + { + "bbox": [ + 466, + 299, + 471, + 308 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 296, + 507, + 311 + ], + "score": 1.0, + "content": "(that is,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 309, + 234, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 230, + 321 + ], + "score": 0.87, + "content": "W _ { t o t } = W _ { D + 1 } W _ { D } \\cdot \\cdot \\cdot W _ { 2 } W _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 309, + 234, + 322 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 325, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "Furthermore, the mutual information with the input and output may be calculated exactly, because", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 382, + 349 + ], + "score": 1.0, + "content": "the distribution of the activity of any hidden layer is Gaussian. Let", + "type": "text" + }, + { + "bbox": [ + 383, + 337, + 391, + 347 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "be the activity of a specific", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 192, + 360 + ], + "score": 1.0, + "content": "hidden layer, and let", + "type": "text" + }, + { + "bbox": [ + 192, + 347, + 204, + 358 + ], + "score": 0.85, + "content": "\\bar { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 347, + 472, + 360 + ], + "score": 1.0, + "content": "be the linear map from the input to this activity (that is, for layer", + "type": "text" + }, + { + "bbox": [ + 473, + 348, + 477, + 358 + ], + "score": 0.56, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 347, + 480, + 360 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 481, + 347, + 505, + 358 + ], + "score": 0.83, + "content": "\\bar { W } =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 107, + 359, + 165, + 370 + ], + "score": 0.9, + "content": "W _ { l } \\cdots W _ { 2 } W _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 358, + 196, + 371 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 196, + 358, + 238, + 369 + ], + "score": 0.91, + "content": "T = { \\bar { W } } X", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 358, + 350, + 371 + ], + "score": 1.0, + "content": ", the mutual information of", + "type": "text" + }, + { + "bbox": [ + 351, + 359, + 361, + 369 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 358, + 379, + 371 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 379, + 359, + 388, + 369 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "calculated using differential", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 370, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 506, + 382 + ], + "score": 1.0, + "content": "entropy is infinite. For the purpose of calculating the mutual information, therefore, we assume that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 321, + 393 + ], + "score": 1.0, + "content": "Gaussian noise is added to the hidden layer activity,", + "type": "text" + }, + { + "bbox": [ + 322, + 380, + 393, + 392 + ], + "score": 0.92, + "content": "T = \\bar { W } X + \\epsilon _ { M I }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 379, + 506, + 393 + ], + "score": 1.0, + "content": ", with mean 0 and variance", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 390, + 507, + 407 + ], + "spans": [ + { + "bbox": [ + 107, + 391, + 153, + 404 + ], + "score": 0.92, + "content": "\\sigma _ { M I } ^ { 2 } = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 390, + 507, + 407 + ], + "score": 1.0, + "content": ". This allows the analysis to apply to networks of any size, including overcomplete layers,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "score": 1.0, + "content": "but as before we emphasize that we do not add this noise either during training or testing. With these", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 413, + 334, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 160, + 426 + ], + "score": 1.0, + "content": "assumptions,", + "type": "text" + }, + { + "bbox": [ + 161, + 414, + 169, + 423 + ], + "score": 0.84, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 413, + 187, + 426 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 414, + 198, + 423 + ], + "score": 0.85, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 413, + 334, + 426 + ], + "score": 1.0, + "content": "are jointly Gaussian and we have", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 430, + 408, + 445 + ], + "lines": [ + { + "bbox": [ + 203, + 430, + 408, + 445 + ], + "spans": [ + { + "bbox": [ + 203, + 430, + 408, + 445 + ], + "score": 0.9, + "content": "I ( T ; X ) = \\log \\lvert \\bar { W } \\bar { W } ^ { T } + \\sigma _ { M I } ^ { 2 } I _ { N _ { h } } \\rvert - \\log \\lvert \\sigma _ { M I } ^ { 2 } I _ { N _ { h } } \\rvert", + "type": "interline_equation", + "image_path": "986b2b1744c46424503da00a715308fcc981ab1a8272cebe74c3deedeb7eb198.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 203, + 430, + 408, + 445 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 450, + 504, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 132, + 463 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 451, + 142, + 462 + ], + "score": 0.84, + "content": "\\left| \\cdot \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 449, + 474, + 463 + ], + "score": 1.0, + "content": "denotes the determinant of a matrix. Finally the mutual information with the output", + "type": "text" + }, + { + "bbox": [ + 474, + 451, + 483, + 460 + ], + "score": 0.82, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 449, + 505, + 463 + ], + "score": 1.0, + "content": ", also", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 461, + 436, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 436, + 474 + ], + "score": 1.0, + "content": "jointly Gaussian, can be calculated similarly (see Eqns. (22)-(25) of Appendix G).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "Fig. 3 shows example training and test dynamics over the course of learning in panel C, and the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "information plane dynamics in panel D. Here the network has an input layer of 100 units, 1 hidden", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "layer of 100 units each and one output unit. The network was trained with batch gradient descent on a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "dataset of 100 examples drawn from the teacher with signal to noise ratio of 1.0. The linear network", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "behaves qualitatively like the ReLU network, and does not exhibit compression. Nevertheless, it", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "score": 1.0, + "content": "learns a map that generalizes well on this task and shows minimal overtraining. Hence, in the setting", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 545, + 481, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 481, + 558 + ], + "score": 1.0, + "content": "we study here, generalization performance can be acceptable without any compression phase.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "The results in (Advani & Saxe (2017)) show that, for the case of linear networks, overtraining is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "worst when the number of inputs matches the number of training samples, and is reduced by making", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "the number of samples smaller or larger. Fig. 4 shows learning dynamics with the number of samples", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "matched to the size of the network. Here overfitting is substantial, and again no compression is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "seen in the information plane. Comparing to the result in Fig. 3D, both networks exhibit similar", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "information dynamics with respect to the input (no compression), but yield different generalization", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 628, + 162, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 162, + 639 + ], + "score": 1.0, + "content": "performance.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "Hence, in this linear analysis of a generic setting, there do not appear to be additional mechanisms that", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 656, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 667 + ], + "score": 1.0, + "content": "cause compression over the course of learning; and generalization behavior can be widely different", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 677 + ], + "score": 1.0, + "content": "for networks with the same dynamics of information compression regarding the input. We note that,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "in the setting considered here, all input dimensions have the same variance, and the weights of the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "teacher are drawn independently. Because of this, there are no special directions in the input, and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "each subspace of the input contains as much information as any other. It is possible that, in real", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "world tasks, higher variance inputs are also the most likely to be relevant to the task (here, have large", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 387, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 387, + 733 + ], + "score": 1.0, + "content": "weights in the teacher). We have not investigated this possibility here.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 49.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "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": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 82, + 505, + 138 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 255 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 276, + 155 + ], + "score": 1.0, + "content": "To train the student network, a dataset of", + "type": "text" + }, + { + "bbox": [ + 276, + 144, + 285, + 153 + ], + "score": 0.81, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 142, + 505, + 155 + ], + "score": 1.0, + "content": "examples is generated using the teacher. The student", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 166 + ], + "score": 1.0, + "content": "network is then trained to minimize the mean squared error between its output and the target output", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "using standard (batch or stochastic) gradient descent on this dataset. 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While linear activation functions stop the network from computing", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 211, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 211, + 505, + 224 + ], + "score": 1.0, + "content": "complex nonlinear functions of the input, deep linear networks nevertheless show complicated", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "nonlinear learning trajectories (Saxe et al., 2014), the optimization problem remains nonconvex (Baldi", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 246 + ], + "score": 1.0, + "content": "& Hornik, 1989), and the generalization dynamics can exhibit substantial overtraining (Fukumizu,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 244, + 225, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 225, + 255 + ], + "score": 1.0, + "content": "1998; Advani & Saxe, 2017).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 142, + 506, + 255 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 260, + 504, + 282 + ], + "lines": [ + { + "bbox": [ + 106, + 260, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 274 + ], + "score": 1.0, + "content": "Importantly, because of the simplified setting considered here, the true generalization error is easily", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 271, + 157, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 157, + 283 + ], + "score": 1.0, + "content": "shown to be", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 260, + 505, + 283 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 280, + 372, + 296 + ], + "lines": [ + { + "bbox": [ + 239, + 280, + 372, + 296 + ], + "spans": [ + { + "bbox": [ + 239, + 280, + 372, + 296 + ], + "score": 0.93, + "content": "E _ { g } ( t ) = | | W _ { o } - W _ { t o t } ( t ) | | _ { F } ^ { 2 } + \\sigma _ { o } ^ { 2 }", + "type": "interline_equation", + "image_path": "321c56854718da12e7f1927283f3e5bb80e2ef3af1ea37f630feb0175fb7ef5f.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 239, + 280, + 372, + 296 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 507, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 134, + 311 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 298, + 166, + 309 + ], + "score": 0.92, + "content": "W _ { t o t } ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 296, + 466, + 311 + ], + "score": 1.0, + "content": "is the overall linear map implemented by the network at training epoch", + "type": "text" + }, + { + "bbox": [ + 466, + 299, + 471, + 308 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 296, + 507, + 311 + ], + "score": 1.0, + "content": "(that is,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 309, + 234, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 230, + 321 + ], + "score": 0.87, + "content": "W _ { t o t } = W _ { D + 1 } W _ { D } \\cdot \\cdot \\cdot W _ { 2 } W _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 309, + 234, + 322 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 296, + 507, + 322 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 325, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "Furthermore, the mutual information with the input and output may be calculated exactly, because", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 382, + 349 + ], + "score": 1.0, + "content": "the distribution of the activity of any hidden layer is Gaussian. Let", + "type": "text" + }, + { + "bbox": [ + 383, + 337, + 391, + 347 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "be the activity of a specific", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 192, + 360 + ], + "score": 1.0, + "content": "hidden layer, and let", + "type": "text" + }, + { + "bbox": [ + 192, + 347, + 204, + 358 + ], + "score": 0.85, + "content": "\\bar { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 347, + 472, + 360 + ], + "score": 1.0, + "content": "be the linear map from the input to this activity (that is, for layer", + "type": "text" + }, + { + "bbox": [ + 473, + 348, + 477, + 358 + ], + "score": 0.56, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 347, + 480, + 360 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 481, + 347, + 505, + 358 + ], + "score": 0.83, + "content": "\\bar { W } =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 107, + 359, + 165, + 370 + ], + "score": 0.9, + "content": "W _ { l } \\cdots W _ { 2 } W _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 358, + 196, + 371 + ], + "score": 1.0, + "content": ". 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For the purpose of calculating the mutual information, therefore, we assume that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 321, + 393 + ], + "score": 1.0, + "content": "Gaussian noise is added to the hidden layer activity,", + "type": "text" + }, + { + "bbox": [ + 322, + 380, + 393, + 392 + ], + "score": 0.92, + "content": "T = \\bar { W } X + \\epsilon _ { M I }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 379, + 506, + 393 + ], + "score": 1.0, + "content": ", with mean 0 and variance", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 390, + 507, + 407 + ], + "spans": [ + { + "bbox": [ + 107, + 391, + 153, + 404 + ], + "score": 0.92, + "content": "\\sigma _ { M I } ^ { 2 } = 1 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 390, + 507, + 407 + ], + "score": 1.0, + "content": ". This allows the analysis to apply to networks of any size, including overcomplete layers,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 416 + ], + "score": 1.0, + "content": "but as before we emphasize that we do not add this noise either during training or testing. With these", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 413, + 334, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 160, + 426 + ], + "score": 1.0, + "content": "assumptions,", + "type": "text" + }, + { + "bbox": [ + 161, + 414, + 169, + 423 + ], + "score": 0.84, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 413, + 187, + 426 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 414, + 198, + 423 + ], + "score": 0.85, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 413, + 334, + 426 + ], + "score": 1.0, + "content": "are jointly Gaussian and we have", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 325, + 507, + 426 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 430, + 408, + 445 + ], + "lines": [ + { + "bbox": [ + 203, + 430, + 408, + 445 + ], + "spans": [ + { + "bbox": [ + 203, + 430, + 408, + 445 + ], + "score": 0.9, + "content": "I ( T ; X ) = \\log \\lvert \\bar { W } \\bar { W } ^ { T } + \\sigma _ { M I } ^ { 2 } I _ { N _ { h } } \\rvert - \\log \\lvert \\sigma _ { M I } ^ { 2 } I _ { N _ { h } } \\rvert", + "type": "interline_equation", + "image_path": "986b2b1744c46424503da00a715308fcc981ab1a8272cebe74c3deedeb7eb198.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 203, + 430, + 408, + 445 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 450, + 504, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 132, + 463 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 451, + 142, + 462 + ], + "score": 0.84, + "content": "\\left| \\cdot \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 449, + 474, + 463 + ], + "score": 1.0, + "content": "denotes the determinant of a matrix. Finally the mutual information with the output", + "type": "text" + }, + { + "bbox": [ + 474, + 451, + 483, + 460 + ], + "score": 0.82, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 449, + 505, + 463 + ], + "score": 1.0, + "content": ", also", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 461, + 436, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 461, + 436, + 474 + ], + "score": 1.0, + "content": "jointly Gaussian, can be calculated similarly (see Eqns. (22)-(25) of Appendix G).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 449, + 505, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "Fig. 3 shows example training and test dynamics over the course of learning in panel C, and the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "information plane dynamics in panel D. Here the network has an input layer of 100 units, 1 hidden", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "layer of 100 units each and one output unit. The network was trained with batch gradient descent on a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "dataset of 100 examples drawn from the teacher with signal to noise ratio of 1.0. The linear network", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "behaves qualitatively like the ReLU network, and does not exhibit compression. Nevertheless, it", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "score": 1.0, + "content": "learns a map that generalizes well on this task and shows minimal overtraining. Hence, in the setting", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 545, + 481, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 481, + 558 + ], + "score": 1.0, + "content": "we study here, generalization performance can be acceptable without any compression phase.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 478, + 506, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "The results in (Advani & Saxe (2017)) show that, for the case of linear networks, overtraining is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "worst when the number of inputs matches the number of training samples, and is reduced by making", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "the number of samples smaller or larger. Fig. 4 shows learning dynamics with the number of samples", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "matched to the size of the network. Here overfitting is substantial, and again no compression is", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "seen in the information plane. Comparing to the result in Fig. 3D, both networks exhibit similar", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "information dynamics with respect to the input (no compression), but yield different generalization", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 628, + 162, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 162, + 639 + ], + "score": 1.0, + "content": "performance.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 560, + 506, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "Hence, in this linear analysis of a generic setting, there do not appear to be additional mechanisms that", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 656, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 667 + ], + "score": 1.0, + "content": "cause compression over the course of learning; and generalization behavior can be widely different", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 505, + 677 + ], + "score": 1.0, + "content": "for networks with the same dynamics of information compression regarding the input. We note that,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "in the setting considered here, all input dimensions have the same variance, and the weights of the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "teacher are drawn independently. Because of this, there are no special directions in the input, and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "each subspace of the input contains as much information as any other. It is possible that, in real", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "world tasks, higher variance inputs are also the most likely to be relevant to the task (here, have large", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 387, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 387, + 733 + ], + "score": 1.0, + "content": "weights in the teacher). We have not investigated this possibility here.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 644, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 167, + 80, + 447, + 271 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 167, + 80, + 447, + 271 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 167, + 80, + 447, + 271 + ], + "spans": [ + { + "bbox": [ + 167, + 80, + 447, + 271 + ], + "score": 0.976, + "type": "image", + "image_path": "22b7932d2a586116824d010622a5076fd367f6f9658cd0958fd6f2a3a8c1187a.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 167, + 80, + 447, + 143.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 167, + 143.66666666666666, + 447, + 207.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 167, + 207.33333333333331, + 447, + 271.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 285, + 506, + 351 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "Figure 4: Overtraining and information plane dynamics. (A) Average training and test mean square", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 483, + 308 + ], + "score": 1.0, + "content": "error for a deep linear network trained with SGD. Overtraining is substantial. Other parameters:", + "type": "text" + }, + { + "bbox": [ + 483, + 296, + 505, + 307 + ], + "score": 0.89, + "content": "N _ { i } =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 307, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 125, + 319 + ], + "score": 1.0, + "content": "100,", + "type": "text" + }, + { + "bbox": [ + 126, + 307, + 159, + 317 + ], + "score": 0.85, + "content": "\\mathrm { P } = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 307, + 262, + 319 + ], + "score": 1.0, + "content": ", Number of hidden units", + "type": "text" + }, + { + "bbox": [ + 262, + 307, + 288, + 317 + ], + "score": 0.84, + "content": "= 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 307, + 336, + 319 + ], + "score": 1.0, + "content": ", Batch size", + "type": "text" + }, + { + "bbox": [ + 336, + 307, + 352, + 317 + ], + "score": 0.79, + "content": "= 5", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 307, + 506, + 319 + ], + "score": 1.0, + "content": "(B) Information plane dynamics. No", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 317, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 506, + 331 + ], + "score": 1.0, + "content": "compression is observed, and information about the labels is lost during overtraining. (C) Average", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 329, + 507, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 198, + 341 + ], + "score": 1.0, + "content": "train and test accuracy", + "type": "text" + }, + { + "bbox": [ + 199, + 329, + 208, + 339 + ], + "score": 0.69, + "content": "\\%", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 329, + 470, + 341 + ], + "score": 1.0, + "content": "correct) for nonlinear tanh networks exhibiting modest overfitting", + "type": "text" + }, + { + "bbox": [ + 471, + 329, + 501, + 339 + ], + "score": 0.86, + "content": "N = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 329, + 507, + 341 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 339, + 443, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 443, + 352 + ], + "score": 1.0, + "content": "(D) Information plane dynamics. Overfitting occurs despite continued compression.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "image", + "bbox": [ + 110, + 361, + 504, + 449 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 361, + 504, + 449 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 110, + 361, + 504, + 449 + ], + "spans": [ + { + "bbox": [ + 110, + 361, + 504, + 449 + ], + "score": 0.965, + "type": "image", + "image_path": "6fb26702a9042ab8cf86aa3fd585a3fceb5522ab227a6c80568e9d5d17d27d7b.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 110, + 361, + 504, + 390.3333333333333 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 110, + 390.3333333333333, + 504, + 419.66666666666663 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 110, + 419.66666666666663, + 504, + 448.99999999999994 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 465, + 505, + 499 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "Figure 5: Stochastic training and the information plane. (A) tanh network trained with SGD. (B) tanh", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 488 + ], + "score": 1.0, + "content": "network trained with BGD. (C) ReLU network trained with SGD. (D) ReLU network trained with", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "BGD. Both random and non-random training procedures show similar information plane dynamics.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "To see whether similar behavior arises in nonlinear networks, we trained tanh networks in the same", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 530, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 232, + 543 + ], + "score": 1.0, + "content": "setting as Section 2, but with", + "type": "text" + }, + { + "bbox": [ + 232, + 531, + 252, + 541 + ], + "score": 0.86, + "content": "30 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 530, + 506, + 543 + ], + "score": 1.0, + "content": "of the data, which we found to lead to modest overtraining.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "Fig. 4C-D shows the resulting train, test, and information plane dynamics. Here the tanh networks", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 553, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 506, + 565 + ], + "score": 1.0, + "content": "show substantial compression, despite exhibiting overtraining. This establishes a dissociation between", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "behavior in the information plane and generalization dynamics: networks that compress may (Fig. 1A)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 575, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 506, + 587 + ], + "score": 1.0, + "content": "or may not (Fig. 4C-D) generalize well, and networks that do not compress may (Figs.1B, 3A-B) or", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 586, + 257, + 598 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 257, + 598 + ], + "score": 1.0, + "content": "may not (Fig. 4A-B) generalize well.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 613, + 419, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 419, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 419, + 627 + ], + "score": 1.0, + "content": "4 COMPRESSION IN BATCH GRADIENT DESCENT AND SGD", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "Next, we test a core theoretical claim of the information bottleneck theory of deep learning, namely", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "that randomness in stochastic gradient descent is responsible for the compression phase. 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Then, the stationary distribution over weights", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "for this process will have maximum entropy, again subject to the training error constraint. 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Since the entropy of the input,", + "type": "text" + }, + { + "bbox": [ + 292, + 170, + 319, + 182 + ], + "score": 0.9, + "content": "H ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 171, + 505, + 183 + ], + "score": 1.0, + "content": ", is fixed, the result of the diffusion dynamics", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 187, + 195 + ], + "score": 1.0, + "content": "will be to minimize", + "type": "text" + }, + { + "bbox": [ + 187, + 181, + 312, + 193 + ], + "score": 0.91, + "content": "I ( X ; T ) \\mathrel { \\mathop : } = H ( X ) - \\bar { H } ( X | \\dot { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 181, + 394, + 195 + ], + "score": 1.0, + "content": "for a given value of", + "type": "text" + }, + { + "bbox": [ + 395, + 182, + 428, + 193 + ], + "score": 0.91, + "content": "I ( T ; Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "reached at the end", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 179, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 179, + 205 + ], + "score": 1.0, + "content": "of the drift phase.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 209, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 506, + 221 + ], + "score": 1.0, + "content": "However, this explanation does not hold up to either theoretical or empirical investigation. Let", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "score": 1.0, + "content": "us assume that the diffusion phase does drive the distribution of weights to a maximum entropy", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 504, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 504, + 243 + ], + "score": 1.0, + "content": "distribution subject to a training error constraint. Note that this distribution reflects stochasticity", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "of weights across different training runs. There is no general reason that a given set of weights", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "sampled from this distribution (i.e., the weight parameters found in one particular training run)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 168, + 277 + ], + "score": 1.0, + "content": "will maximize", + "type": "text" + }, + { + "bbox": [ + 168, + 264, + 205, + 276 + ], + "score": 0.93, + "content": "H ( X | T )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 264, + 468, + 277 + ], + "score": 1.0, + "content": ", the entropy of inputs given hidden layer activity. In particular,", + "type": "text" + }, + { + "bbox": [ + 468, + 264, + 505, + 276 + ], + "score": 0.93, + "content": "H ( X | T )", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 288 + ], + "score": 1.0, + "content": "reflects (conditional) uncertainty about inputs drawn from the data-generating distribution, rather", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 414, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 414, + 298 + ], + "score": 1.0, + "content": "than uncertainty about any kind of distribution across different training runs.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "score": 1.0, + "content": "We also show empirically that the stochasticity of the SGD is not necessary for compression. To do", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 325 + ], + "score": 1.0, + "content": "so, we consider two distinct training procedures: offline stochastic gradient descent (SGD), which", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "learns from a fixed-size dataset, and updates weights by repeatedly sampling a single example from", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "the dataset and calculating the gradient of the error with respect to that single sample (the typical", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 507, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 507, + 359 + ], + "score": 1.0, + "content": "procedure used in practice); and batch gradient descent (BGD), which learns from a fixed-size dataset,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "and updates weights using the gradient of the total error across all examples. Batch gradient descent", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 369, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 380 + ], + "score": 1.0, + "content": "uses the full training dataset and, crucially, therefore has no randomness or diffusion-like behavior in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 154, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 154, + 393 + ], + "score": 1.0, + "content": "its updates.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "We trained tanh and ReLU networks with SGD and BGD and compare their information plane", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "score": 1.0, + "content": "dynamics in Fig. 5 (see Appendix H for a linear network). We find largely consistent information", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "dynamics in both instances, with robust compression in tanh networks for both methods. Thus", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "randomness in the training process does not appear to contribute substantially to compression of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "information about the input. This finding is consistent with the view presented in Section 2 that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 406, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 406, + 464 + ], + "score": 1.0, + "content": "compression arises predominantly from the double saturating nonlinearity.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "Finally, we look at the gradient signal-to-noise ratio (SNR) to analyze the relationship between", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "compression and the transition from high to low gradient SNR. Fig. 20 of Appendix I shows the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 489, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 504 + ], + "score": 1.0, + "content": "gradient SNR over training, which in all cases shows a phase transition during learning. Hence", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "the gradient SNR transition is a general phenomenon, but is not causally related to compression.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "Appendix I offers an extended discussion and shows gradient SNR transitions without compression", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 523, + 294, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 294, + 534 + ], + "score": 1.0, + "content": "on the MNIST dataset and for linear networks.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "title", + "bbox": [ + 107, + 558, + 356, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 358, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 358, + 574 + ], + "score": 1.0, + "content": "5 SIMULTANEOUS FITTING AND COMPRESSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "Our finding that generalization can occur without compression may seem difficult to reconcile with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 599, + 507, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 507, + 613 + ], + "score": 1.0, + "content": "the intuition that certain tasks involve suppressing irrelevant directions of the input. In the extreme,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "if certain inputs contribute nothing but noise, then good generalization requires ignoring them. To", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "study this, we consider a variant on the linear student-teacher setup of Section 3: we partition the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 130, + 645 + ], + "score": 1.0, + "content": "input", + "type": "text" + }, + { + "bbox": [ + 130, + 633, + 141, + 643 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 633, + 275, + 645 + ], + "score": 1.0, + "content": "into a set of task-relevant inputs", + "type": "text" + }, + { + "bbox": [ + 276, + 633, + 296, + 644 + ], + "score": 0.91, + "content": "X _ { r e l }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 633, + 436, + 645 + ], + "score": 1.0, + "content": "and a set of task-irrelevant inputs", + "type": "text" + }, + { + "bbox": [ + 436, + 633, + 463, + 644 + ], + "score": 0.93, + "content": "X _ { i r r e l }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 633, + 506, + 645 + ], + "score": 1.0, + "content": ", and alter", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "the teacher network so that the teacher’s weights to the task-irrelevant inputs are all zero. Hence the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 133, + 668 + ], + "score": 1.0, + "content": "inputs", + "type": "text" + }, + { + "bbox": [ + 133, + 655, + 160, + 666 + ], + "score": 0.91, + "content": "X _ { i r r e l }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 655, + 288, + 668 + ], + "score": 1.0, + "content": "contribute only noise, while the", + "type": "text" + }, + { + "bbox": [ + 288, + 655, + 308, + 666 + ], + "score": 0.91, + "content": "X _ { r e l }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "contain signal. 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While the overall", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "dynamics show no compression phase, the information specifically about the task-irrelevant subspace", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "does compress over the course of training. This compression process occurs at the same time as", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "the fitting to the task-relevant information. Thus, when a task requires ignoring some inputs, the", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 46 + } + ], + "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": "text", + "bbox": [ + 107, + 82, + 505, + 203 + ], + "lines": [], + "index": 5, + "bbox_fs": [ + 105, + 83, + 506, + 205 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 106, + 209, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 506, + 221 + ], + "score": 1.0, + "content": "However, this explanation does not hold up to either theoretical or empirical investigation. Let", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 234 + ], + "score": 1.0, + "content": "us assume that the diffusion phase does drive the distribution of weights to a maximum entropy", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 504, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 504, + 243 + ], + "score": 1.0, + "content": "distribution subject to a training error constraint. Note that this distribution reflects stochasticity", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "of weights across different training runs. There is no general reason that a given set of weights", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "sampled from this distribution (i.e., the weight parameters found in one particular training run)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 168, + 277 + ], + "score": 1.0, + "content": "will maximize", + "type": "text" + }, + { + "bbox": [ + 168, + 264, + 205, + 276 + ], + "score": 0.93, + "content": "H ( X | T )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 264, + 468, + 277 + ], + "score": 1.0, + "content": ", the entropy of inputs given hidden layer activity. In particular,", + "type": "text" + }, + { + "bbox": [ + 468, + 264, + 505, + 276 + ], + "score": 0.93, + "content": "H ( X | T )", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 288 + ], + "score": 1.0, + "content": "reflects (conditional) uncertainty about inputs drawn from the data-generating distribution, rather", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 414, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 414, + 298 + ], + "score": 1.0, + "content": "than uncertainty about any kind of distribution across different training runs.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 209, + 506, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 506, + 316 + ], + "score": 1.0, + "content": "We also show empirically that the stochasticity of the SGD is not necessary for compression. To do", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 314, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 325 + ], + "score": 1.0, + "content": "so, we consider two distinct training procedures: offline stochastic gradient descent (SGD), which", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "learns from a fixed-size dataset, and updates weights by repeatedly sampling a single example from", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "the dataset and calculating the gradient of the error with respect to that single sample (the typical", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 507, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 507, + 359 + ], + "score": 1.0, + "content": "procedure used in practice); and batch gradient descent (BGD), which learns from a fixed-size dataset,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "and updates weights using the gradient of the total error across all examples. Batch gradient descent", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 369, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 380 + ], + "score": 1.0, + "content": "uses the full training dataset and, crucially, therefore has no randomness or diffusion-like behavior in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 154, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 154, + 393 + ], + "score": 1.0, + "content": "its updates.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 302, + 507, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "We trained tanh and ReLU networks with SGD and BGD and compare their information plane", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 505, + 419 + ], + "score": 1.0, + "content": "dynamics in Fig. 5 (see Appendix H for a linear network). We find largely consistent information", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "dynamics in both instances, with robust compression in tanh networks for both methods. Thus", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "randomness in the training process does not appear to contribute substantially to compression of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "information about the input. This finding is consistent with the view presented in Section 2 that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 406, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 406, + 464 + ], + "score": 1.0, + "content": "compression arises predominantly from the double saturating nonlinearity.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 396, + 506, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "Finally, we look at the gradient signal-to-noise ratio (SNR) to analyze the relationship between", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "compression and the transition from high to low gradient SNR. Fig. 20 of Appendix I shows the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 489, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 504 + ], + "score": 1.0, + "content": "gradient SNR over training, which in all cases shows a phase transition during learning. Hence", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "the gradient SNR transition is a general phenomenon, but is not causally related to compression.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "Appendix I offers an extended discussion and shows gradient SNR transitions without compression", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 523, + 294, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 294, + 534 + ], + "score": 1.0, + "content": "on the MNIST dataset and for linear networks.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 467, + 506, + 534 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 558, + 356, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 358, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 358, + 574 + ], + "score": 1.0, + "content": "5 SIMULTANEOUS FITTING AND COMPRESSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 589, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "Our finding that generalization can occur without compression may seem difficult to reconcile with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 599, + 507, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 507, + 613 + ], + "score": 1.0, + "content": "the intuition that certain tasks involve suppressing irrelevant directions of the input. In the extreme,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "if certain inputs contribute nothing but noise, then good generalization requires ignoring them. To", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "study this, we consider a variant on the linear student-teacher setup of Section 3: we partition the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 130, + 645 + ], + "score": 1.0, + "content": "input", + "type": "text" + }, + { + "bbox": [ + 130, + 633, + 141, + 643 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 633, + 275, + 645 + ], + "score": 1.0, + "content": "into a set of task-relevant inputs", + "type": "text" + }, + { + "bbox": [ + 276, + 633, + 296, + 644 + ], + "score": 0.91, + "content": "X _ { r e l }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 633, + 436, + 645 + ], + "score": 1.0, + "content": "and a set of task-irrelevant inputs", + "type": "text" + }, + { + "bbox": [ + 436, + 633, + 463, + 644 + ], + "score": 0.93, + "content": "X _ { i r r e l }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 633, + 506, + 645 + ], + "score": 1.0, + "content": ", and alter", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "the teacher network so that the teacher’s weights to the task-irrelevant inputs are all zero. Hence the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 133, + 668 + ], + "score": 1.0, + "content": "inputs", + "type": "text" + }, + { + "bbox": [ + 133, + 655, + 160, + 666 + ], + "score": 0.91, + "content": "X _ { i r r e l }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 655, + 288, + 668 + ], + "score": 1.0, + "content": "contribute only noise, while the", + "type": "text" + }, + { + "bbox": [ + 288, + 655, + 308, + 666 + ], + "score": 0.91, + "content": "X _ { r e l }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "contain signal. We then calculate the information", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "plane dynamics for the whole layer, and for the task-relevant and task-irrelevant inputs separately.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "Fig. 6 shows information plane dynamics for a deep linear neural network trained using SGD (5", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "samples/batch) on a task with 30 task-relevant inputs and 70 task-irrelevant inputs. While the overall", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "dynamics show no compression phase, the information specifically about the task-irrelevant subspace", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "does compress over the course of training. This compression process occurs at the same time as", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "the fitting to the task-relevant information. Thus, when a task requires ignoring some inputs, the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "information with these inputs specifically will indeed be reduced; but overall mutual information", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 272, + 281, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 281, + 286 + ], + "score": 1.0, + "content": "with the input in general may still increase.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 46, + "bbox_fs": [ + 104, + 588, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 80, + 502, + 169 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 80, + 502, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 80, + 502, + 169 + ], + "spans": [ + { + "bbox": [ + 108, + 80, + 502, + 169 + ], + "score": 0.967, + "type": "image", + "image_path": "3f982bb2a95db7292dd21c09073b2ac2ac7ccf908216d9ad8ed4dcda065e419e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 80, + 502, + 109.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 109.66666666666667, + 502, + 139.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 139.33333333333334, + 502, + 169.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 183, + 505, + 238 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "score": 1.0, + "content": "Figure 6: Simultaneous fitting and compression. (A) For a task with a large task-irrelevant subspace", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 193, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 193, + 506, + 207 + ], + "score": 1.0, + "content": "in the input, a linear network shows no overall compression of information about the input. (B)", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "The information with the task-relevant subspace increases robustly over training. (C) However, the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 214, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 229 + ], + "score": 1.0, + "content": "information specifically about the task-irrelevant subspace does compress after initially growing as", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 227, + 198, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 198, + 239 + ], + "score": 1.0, + "content": "the network is trained.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 504, + 284 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "information with these inputs specifically will indeed be reduced; but overall mutual information", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 272, + 281, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 281, + 286 + ], + "score": 1.0, + "content": "with the input in general may still increase.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 107, + 302, + 190, + 315 + ], + "lines": [ + { + "bbox": [ + 105, + 301, + 192, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 192, + 317 + ], + "score": 1.0, + "content": "6 DISCUSSION", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 328, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 106, + 328, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "Our results suggest that compression dynamics in the information plane are not a general feature of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "deep networks, but are critically influenced by the nonlinearities employed by the network. Double-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "saturating nonlinearities lead to compression, if mutual information is estimated by binning activations", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "or by adding homoscedastic noise, while single-sided saturating nonlinearities like ReLUs do not", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "compress in general. 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(D) The soft-plus activation function, a smoothed version of the ReLU, exhibits", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 416, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 428 + ], + "score": 1.0, + "content": "no compression. Hence double-saturating nonlinearities exhibit the compression effect while single-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 428, + 235, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 235, + 439 + ], + "score": 1.0, + "content": "saturating nonlinearities do not.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + } + ], + "index": 17.25 + }, + { + "type": "text", + "bbox": [ + 106, + 443, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "Figure 8A-B shows the result of applying this MI estimation method on the dataset and network", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "architecture of Shwartz-Ziv & Tishby (2017), with MI estimated on the full dataset and averaged", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 466, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 478 + ], + "score": 1.0, + "content": "over 50 repetitions. Mutual information was estimated using data samples from the test set, and we", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 475, + 507, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 203, + 491 + ], + "score": 1.0, + "content": "took the noise variance", + "type": "text" + }, + { + "bbox": [ + 203, + 476, + 241, + 487 + ], + "score": 0.91, + "content": "\\sigma ^ { 2 } = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 475, + 507, + 491 + ], + "score": 1.0, + "content": ". These results look similar to the estimate derived from binning,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 487, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 501 + ], + "score": 1.0, + "content": "with compression in tanh networks but no compression in ReLU networks. Relative to the binning", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 499, + 414, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 414, + 510 + ], + "score": 1.0, + "content": "estimate, it appears that compression is less pronounced in the KDE method.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 444, + 507, + 510 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 515, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 527 + ], + "score": 1.0, + "content": "Figure 1C-D of the main text shows the results of this estimation technique applied to a neural network", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 526, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 137, + 539 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 137, + 527, + 279, + 537 + ], + "score": 0.88, + "content": "7 8 4 - 1 0 2 4 - 2 0 - 2 0 - 2 0 - 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 526, + 506, + 539 + ], + "score": 1.0, + "content": "on the MNIST handwritten digit classification dataset.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "The network was trained using SGD with minibatches of size 128. As before, mutual information", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 548, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 444, + 560 + ], + "score": 1.0, + "content": "was estimated using data samples from the test set, and we took the noise variance", + "type": "text" + }, + { + "bbox": [ + 444, + 548, + 482, + 559 + ], + "score": 0.91, + "content": "\\sigma ^ { 2 } = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 549, + 505, + 560 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 560, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 571 + ], + "score": 1.0, + "content": "smaller layer sizes in the top three hidden layers were selected to ensure the quality of the kernel", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 570, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 506, + 582 + ], + "score": 1.0, + "content": "density estimator given the amount of data in the test set, since the estimates are more accurate for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 594 + ], + "score": 1.0, + "content": "smaller-dimensional data. Because of computational expense, the MNIST results are from a single", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 592, + 159, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 159, + 606 + ], + "score": 1.0, + "content": "training run.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 516, + 506, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 609, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 621 + ], + "score": 1.0, + "content": "More detailed results for the MNIST dataset are provided in Figure 9 for the tanh activation function,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "and in Figure 10 for the ReLU activation function. In these figures, the first row shows the evolution", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 643 + ], + "score": 1.0, + "content": "of the cross entropy loss (on both training and testing data sets) during training. The second row", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 641, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 505, + 655 + ], + "score": 1.0, + "content": "shows the mutual information between input and the activity of different hidden layers, using the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "nonparametric KDE estimator described above. The blue region in the second row shows the range", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 665, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 506, + 676 + ], + "score": 1.0, + "content": "of possible MI values, ranging from the upper bound described above (Eq. 10) to the following lower", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 609, + 506, + 676 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 255, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 255, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 255, + 97 + ], + "score": 1.0, + "content": "bound (Kolchinsky & Tracey, 2017),", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 99, + 460, + 175 + ], + "lines": [ + { + "bbox": [ + 149, + 99, + 460, + 175 + ], + "spans": [ + { + "bbox": [ + 149, + 99, + 460, + 175 + ], + "score": 0.94, + "content": "\\begin{array} { r c l } { { I ( T ; Y ) } } & { { \\geq } } & { { \\displaystyle - \\frac { 1 } { P } \\sum _ { i } \\log \\frac { 1 } { P } \\sum _ { j } \\exp \\left( - \\frac { 1 } { 2 } \\frac { \\left\\| h _ { i } - h _ { j } \\right\\| _ { 2 } ^ { 2 } } { 4 \\sigma ^ { 2 } } \\right) } } \\\\ { { } } & { { } } & { { \\displaystyle - \\sum _ { l } ^ { L } p _ { l } \\left[ - \\frac { 1 } { P _ { l } } \\sum _ { i , Y _ { i } = l } \\log \\frac { 1 } { P _ { l } } \\sum _ { j , Y _ { j } = l } \\exp \\left( - \\frac { 1 } { 2 } \\frac { \\left\\| h _ { i } - h _ { j } \\right\\| _ { 2 } ^ { 2 } } { 4 \\sigma ^ { 2 } } \\right) \\right] . } } \\end{array}", + "type": "interline_equation", + "image_path": "2b4d855811c068dfd8a5a6373771f34f8fdd4071c1a630b953dc7e58a0914035.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 149, + 99, + 460, + 124.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 149, + 124.33333333333333, + 460, + 149.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 149, + 149.66666666666666, + 460, + 175.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "score": 1.0, + "content": "The third row shows the mutual information between input and activity of different hidden layers,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "estimated using the binning method (here, the activity of each neuron was discretized into bins of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 199, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 455, + 214 + ], + "score": 1.0, + "content": "size 0.5). For both the second and third rows, we also plot the entropy of the inputs,", + "type": "text" + }, + { + "bbox": [ + 456, + 201, + 482, + 213 + ], + "score": 0.91, + "content": "H ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 199, + 506, + 214 + ], + "score": 1.0, + "content": ", as a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 210, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 159, + 226 + ], + "score": 1.0, + "content": "dashed line.", + "type": "text" + }, + { + "bbox": [ + 159, + 212, + 186, + 223 + ], + "score": 0.92, + "content": "H ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 210, + 372, + 226 + ], + "score": 1.0, + "content": "is an upper bound on the mutual information", + "type": "text" + }, + { + "bbox": [ + 372, + 212, + 406, + 224 + ], + "score": 0.91, + "content": "I ( X ; T )", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 210, + 506, + 226 + ], + "score": 1.0, + "content": ", and is computed using", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 221, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 104, + 221, + 506, + 236 + ], + "score": 1.0, + "content": "the assumption of a uniform distribution over the 10,000 testing points in the MNIST dataset, giving", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 234, + 195, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 191, + 246 + ], + "score": 0.91, + "content": "H ( X ) = \\log _ { 2 } { 1 0 0 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 234, + 195, + 247 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 506, + 361 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "Finally, the fourth row visualizes the dynamics of the SGD updates during training. For each layer and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 261, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 231, + 275 + ], + "score": 1.0, + "content": "epoch, the green line shows the", + "type": "text" + }, + { + "bbox": [ + 232, + 262, + 242, + 272 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 261, + 505, + 275 + ], + "score": 1.0, + "content": "norm of the weights. We also compute the vector of mean updates", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "across SGD minibatches (this vector has one dimension for each weight parameter), as well as the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 284, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 419, + 295 + ], + "score": 1.0, + "content": "vector of the standard deviation of the updates across SGD minibatches. The", + "type": "text" + }, + { + "bbox": [ + 419, + 284, + 430, + 294 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 284, + 505, + 295 + ], + "score": 1.0, + "content": "norm of the mean", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 294, + 507, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 263, + 307 + ], + "score": 1.0, + "content": "update vector is shown in blue, and the", + "type": "text" + }, + { + "bbox": [ + 264, + 294, + 274, + 306 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 294, + 507, + 307 + ], + "score": 1.0, + "content": "norm of the standard deviation vector is shown in orange.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "The gradient SNR, computed as the ratio of the norm of the mean vector to the norm of the standard", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "deviation vector, is shown in red. For both the tanh and ReLU networks, the gradient SNR shows a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 328, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 506, + 339 + ], + "score": 1.0, + "content": "phase transition during training, and the norm of the weights in each layer increases. Importantly,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "this phase transition occurs despite a lack of compression in the ReLU network, indicating that noise", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 348, + 342, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 342, + 363 + ], + "score": 1.0, + "content": "in SGD updates does not yield compression in this setting.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 14.5 + } + ], + "page_idx": 14, + "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, + 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": [ + 107, + 82, + 255, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 255, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 255, + 97 + ], + "score": 1.0, + "content": "bound (Kolchinsky & Tracey, 2017),", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 255, + 97 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 99, + 460, + 175 + ], + "lines": [ + { + "bbox": [ + 149, + 99, + 460, + 175 + ], + "spans": [ + { + "bbox": [ + 149, + 99, + 460, + 175 + ], + "score": 0.94, + "content": "\\begin{array} { r c l } { { I ( T ; Y ) } } & { { \\geq } } & { { \\displaystyle - \\frac { 1 } { P } \\sum _ { i } \\log \\frac { 1 } { P } \\sum _ { j } \\exp \\left( - \\frac { 1 } { 2 } \\frac { \\left\\| h _ { i } - h _ { j } \\right\\| _ { 2 } ^ { 2 } } { 4 \\sigma ^ { 2 } } \\right) } } \\\\ { { } } & { { } } & { { \\displaystyle - \\sum _ { l } ^ { L } p _ { l } \\left[ - \\frac { 1 } { P _ { l } } \\sum _ { i , Y _ { i } = l } \\log \\frac { 1 } { P _ { l } } \\sum _ { j , Y _ { j } = l } \\exp \\left( - \\frac { 1 } { 2 } \\frac { \\left\\| h _ { i } - h _ { j } \\right\\| _ { 2 } ^ { 2 } } { 4 \\sigma ^ { 2 } } \\right) \\right] . } } \\end{array}", + "type": "interline_equation", + "image_path": "2b4d855811c068dfd8a5a6373771f34f8fdd4071c1a630b953dc7e58a0914035.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 149, + 99, + 460, + 124.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 149, + 124.33333333333333, + 460, + 149.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 149, + 149.66666666666666, + 460, + 175.0 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 178, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "score": 1.0, + "content": "The third row shows the mutual information between input and activity of different hidden layers,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 505, + 201 + ], + "score": 1.0, + "content": "estimated using the binning method (here, the activity of each neuron was discretized into bins of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 199, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 455, + 214 + ], + "score": 1.0, + "content": "size 0.5). 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Mutual information for the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 353, + 376 + ], + "score": 1.0, + "content": "simple three neuron model shown in Fig. 2 with bin edges", + "type": "text" + }, + { + "bbox": [ + 353, + 363, + 487, + 376 + ], + "score": 0.36, + "content": "b _ { i } \\in \\mathrm { t a n h } ( \\operatorname* { l i n s p a c e } ( - 5 0 , 5 0 , N ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 364, + 505, + 376 + ], + "score": 1.0, + "content": ". 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This noise assumption is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "not present in the actual neural networks either during training or testing, and is made solely for the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 468, + 294, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 294, + 480 + ], + "score": 1.0, + "content": "purpose of calculating the mutual information.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 335, + 497 + ], + "score": 1.0, + "content": "Another strategy is to partition the continuous variable", + "type": "text" + }, + { + "bbox": [ + 335, + 485, + 343, + 495 + ], + "score": 0.81, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 485, + 441, + 497 + ], + "score": 1.0, + "content": "into a discrete variable", + "type": "text" + }, + { + "bbox": [ + 442, + 485, + 450, + 495 + ], + "score": 0.77, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 485, + 505, + 497 + ], + "score": 1.0, + "content": ", for instance", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "by binning the values (the approach taken in Shwartz-Ziv & Tishby (2017)). 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Again, however, in practice the network does not", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 236, + 531 + ], + "score": 1.0, + "content": "operate on the binned variables", + "type": "text" + }, + { + "bbox": [ + 237, + 518, + 245, + 527 + ], + "score": 0.77, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 517, + 376, + 531 + ], + "score": 1.0, + "content": "but on the continuous variables", + "type": "text" + }, + { + "bbox": [ + 377, + 518, + 384, + 528 + ], + "score": 0.75, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 517, + 506, + 531 + ], + "score": 1.0, + "content": ", and the binning is solely for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 527, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 543 + ], + "score": 1.0, + "content": "the purpose of calculating the mutual information. 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The choice of binning strategy is an assumption analogous to choosing a type", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "of noise to add to the representation in the continuous case: because there is in fact no binning in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "the operation of the network, there is no clear choice for binning methodology. The strategy we use", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "in binning-based experiments reported here is the following: for bounded activations like the tanh", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "activation, we use evenly spaced bins between the minimum and maximum limits of the function. For", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "unbounded activations like ReLU, we first train the network completely; next identify the minimum", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "and maximum hidden activation over all units and all training epochs; and finally bin into equally", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "spaced bins between these minimum and maximum values. We note that this procedure places no", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "restriction on the magnitude that the unbounded activation function can take during training, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 648, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 506, + 663 + ], + "score": 1.0, + "content": "yields the same MI estimate as using infinite equally spaced bins (because bins for activities larger", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 659, + 311, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 311, + 673 + ], + "score": 1.0, + "content": "than the maximum are never seen during training).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "As an example of another binning strategy that can yield markedly different results, we consider", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "evenly spaced bins in a neuron’s net input, rather than its activity. That is, instead of evenly spaced", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "bins in the neural activity, we determine the bin edges by mapping a set of evenly spaced values", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "through the neural nonlinearity. For tanh, for instance, this spaces bins more tightly in the saturation", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 719, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 735 + ], + "score": 1.0, + "content": "region as compared to the linear region. Figure 13 shows the results of applying this binning strategy", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 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": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 128, + 95 + ], + "score": 1.0, + "content": "Now", + "type": "text" + }, + { + "bbox": [ + 128, + 82, + 195, + 95 + ], + "score": 0.92, + "content": "H ( h | X ) = - \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 82, + 279, + 95 + ], + "score": 1.0, + "content": "since given the input", + "type": "text" + }, + { + "bbox": [ + 279, + 83, + 290, + 93 + ], + "score": 0.75, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 82, + 369, + 95 + ], + "score": 1.0, + "content": ", the hidden activity", + "type": "text" + }, + { + "bbox": [ + 369, + 83, + 376, + 92 + ], + "score": 0.79, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "is distributed as a delta function", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 116, + 106 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 117, + 94, + 140, + 106 + ], + "score": 0.92, + "content": "f ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 92, + 505, + 106 + ], + "score": 1.0, + "content": ". 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Mutual information for the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 353, + 376 + ], + "score": 1.0, + "content": "simple three neuron model shown in Fig. 2 with bin edges", + "type": "text" + }, + { + "bbox": [ + 353, + 363, + 487, + 376 + ], + "score": 0.36, + "content": "b _ { i } \\in \\mathrm { t a n h } ( \\operatorname* { l i n s p a c e } ( - 5 0 , 5 0 , N ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 364, + 505, + 376 + ], + "score": 1.0, + "content": ". In", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 375, + 461, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 461, + 387 + ], + "score": 1.0, + "content": "contrast to linear binning, the mutual information continues to increase as weights grow.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + } + ], + "index": 13.75 + }, + { + "type": "text", + "bbox": [ + 106, + 401, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 402, + 504, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 435, + 414 + ], + "score": 1.0, + "content": "To yield a finite mutual information, some noise in the mapping is required such that", + "type": "text" + }, + { + "bbox": [ + 435, + 402, + 470, + 414 + ], + "score": 0.93, + "content": "H ( h | X )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 402, + 504, + 414 + ], + "score": 1.0, + "content": "remains", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "finite. A common choice (and one adopted here for the linear network, the nonparametric kernel", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 212, + 436 + ], + "score": 1.0, + "content": "density estimator, and the", + "type": "text" + }, + { + "bbox": [ + 213, + 424, + 219, + 434 + ], + "score": 0.55, + "content": "\\mathbf { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "-nearest neighbor estimator) is to analyze a new variable with additive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 132, + 447 + ], + "score": 1.0, + "content": "noise,", + "type": "text" + }, + { + "bbox": [ + 132, + 435, + 180, + 445 + ], + "score": 0.91, + "content": "T = h + Z", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 434, + 210, + 447 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 210, + 435, + 219, + 444 + ], + "score": 0.79, + "content": "Z", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 434, + 365, + 447 + ], + "score": 1.0, + "content": "is a random variable independent of", + "type": "text" + }, + { + "bbox": [ + 365, + 435, + 375, + 444 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 434, + 402, + 447 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 402, + 434, + 477, + 447 + ], + "score": 0.93, + "content": "H ( T | X ) = H ( Z )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "which", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 446, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 228, + 458 + ], + "score": 1.0, + "content": "allows the overall information", + "type": "text" + }, + { + "bbox": [ + 228, + 446, + 337, + 458 + ], + "score": 0.93, + "content": "I ( T ; X ) = H ( T ) - H ( Z )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 446, + 506, + 458 + ], + "score": 1.0, + "content": "to remain finite. This noise assumption is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "not present in the actual neural networks either during training or testing, and is made solely for the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 468, + 294, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 294, + 480 + ], + "score": 1.0, + "content": "purpose of calculating the mutual information.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 402, + 506, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 335, + 497 + ], + "score": 1.0, + "content": "Another strategy is to partition the continuous variable", + "type": "text" + }, + { + "bbox": [ + 335, + 485, + 343, + 495 + ], + "score": 0.81, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 485, + 441, + 497 + ], + "score": 1.0, + "content": "into a discrete variable", + "type": "text" + }, + { + "bbox": [ + 442, + 485, + 450, + 495 + ], + "score": 0.77, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 485, + 505, + 497 + ], + "score": 1.0, + "content": ", for instance", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "by binning the values (the approach taken in Shwartz-Ziv & Tishby (2017)). This allows use", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "of the discrete entropy, which remains finite. 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Moreover, there are many possible binning", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "strategies, which yield different discrete random variables, and different mutual information with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 564 + ], + "score": 1.0, + "content": "respect to the input. The choice of binning strategy is an assumption analogous to choosing a type", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "of noise to add to the representation in the continuous case: because there is in fact no binning in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "the operation of the network, there is no clear choice for binning methodology. The strategy we use", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "in binning-based experiments reported here is the following: for bounded activations like the tanh", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "activation, we use evenly spaced bins between the minimum and maximum limits of the function. For", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "unbounded activations like ReLU, we first train the network completely; next identify the minimum", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "and maximum hidden activation over all units and all training epochs; and finally bin into equally", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "spaced bins between these minimum and maximum values. We note that this procedure places no", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "restriction on the magnitude that the unbounded activation function can take during training, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 648, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 104, + 648, + 506, + 663 + ], + "score": 1.0, + "content": "yields the same MI estimate as using infinite equally spaced bins (because bins for activities larger", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 659, + 311, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 311, + 673 + ], + "score": 1.0, + "content": "than the maximum are never seen during training).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 485, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "As an example of another binning strategy that can yield markedly different results, we consider", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "evenly spaced bins in a neuron’s net input, rather than its activity. That is, instead of evenly spaced", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "bins in the neural activity, we determine the bin edges by mapping a set of evenly spaced values", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "through the neural nonlinearity. For tanh, for instance, this spaces bins more tightly in the saturation", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 719, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 735 + ], + "score": 1.0, + "content": "region as compared to the linear region. Figure 13 shows the results of applying this binning strategy", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "to the minimal three neuron model with tanh activations. This binning scheme captures more", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "information as the weights of the network grow larger. Figure 14 shows information plane dynamics", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "for this binning structure. The tanh network no longer exhibits compression. (We note that the", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 521, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 533 + ], + "score": 1.0, + "content": "broken DPI in this example is an artifact of performing binning only for analysis, as discussed below).", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 677, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 150, + 82, + 464, + 403 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 150, + 82, + 464, + 403 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 150, + 82, + 464, + 403 + ], + "spans": [ + { + "bbox": [ + 150, + 82, + 464, + 403 + ], + "score": 0.975, + "type": "image", + "image_path": "12c7d31d8a03d347bac1cbe449138df659dc492793257cc0c61d13e2608ea4fa.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 150, + 82, + 464, + 189.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 150, + 189.0, + 464, + 296.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 150, + 296.0, + 464, + 403.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 421, + 505, + 455 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Figure 14: Effect of binning strategy on information plane dynamics. Results for the same tanh", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 326, + 444 + ], + "score": 1.0, + "content": "network and training regime as 1A, but with bin edges", + "type": "text" + }, + { + "bbox": [ + 326, + 433, + 346, + 443 + ], + "score": 0.88, + "content": "b _ { i } \\in", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 432, + 402, + 444 + ], + "score": 1.0, + "content": "tanh(linspace", + "type": "text" + }, + { + "bbox": [ + 402, + 432, + 456, + 444 + ], + "score": 0.81, + "content": "( - 5 0 , 5 0 , N )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "). Measured", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 443, + 373, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 373, + 456 + ], + "score": 1.0, + "content": "with this binning structure, there is no compression in most layers.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "to the minimal three neuron model with tanh activations. This binning scheme captures more", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "information as the weights of the network grow larger. Figure 14 shows information plane dynamics", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "for this binning structure. The tanh network no longer exhibits compression. (We note that the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 521, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 533 + ], + "score": 1.0, + "content": "broken DPI in this example is an artifact of performing binning only for analysis, as discussed below).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "Any implementation of a neural network on digital hardware is ultimately of finite precision, and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "hence is a binned, discrete representation. However, it is a very high resolution binning compared", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 558, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 573 + ], + "score": 1.0, + "content": "to that used here or by Shwartz-Ziv & Tishby (2017): single precision would correspond to using", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 139, + 583 + ], + "score": 1.0, + "content": "roughly", + "type": "text" + }, + { + "bbox": [ + 140, + 570, + 154, + 581 + ], + "score": 0.87, + "content": "2 ^ { 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "bins to discretize each hidden unit’s activity, as compared to the 30-100 used here. If the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 297, + 594 + ], + "score": 1.0, + "content": "binning is fine-grained enough that each input", + "type": "text" + }, + { + "bbox": [ + 297, + 582, + 307, + 591 + ], + "score": 0.76, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 582, + 474, + 594 + ], + "score": 1.0, + "content": "yields a different binned activity pattern", + "type": "text" + }, + { + "bbox": [ + 474, + 582, + 481, + 591 + ], + "score": 0.74, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 582, + 505, + 594 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 107, + 592, + 174, + 604 + ], + "score": 0.93, + "content": "H ( h ) \\stackrel { - } { = } \\log ( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 592, + 203, + 605 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 204, + 593, + 213, + 602 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "is the number of examples in the dataset, and there will be little to no", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 603, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 616 + ], + "score": 1.0, + "content": "change in information during training. As an example, we show in Fig. 15 the result of binning at full", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 615, + 183, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 183, + 626 + ], + "score": 1.0, + "content": "machine precision.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 507, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 507, + 644 + ], + "score": 1.0, + "content": "Finally, we note two consequences of the assumption of noise/binning for the purposes of analysis.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "score": 1.0, + "content": "First, this means that the data processing inequality (DPI) does not apply to the noisy/binned mutual", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "information estimates. The DPI states that information can only be destroyed through successive", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 662, + 507, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 662, + 211, + 678 + ], + "score": 1.0, + "content": "transformations, that is, if", + "type": "text" + }, + { + "bbox": [ + 212, + 664, + 273, + 675 + ], + "score": 0.92, + "content": "X h _ { 1 } h _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 662, + 382, + 678 + ], + "score": 1.0, + "content": "form a Markov chain, then", + "type": "text" + }, + { + "bbox": [ + 382, + 664, + 470, + 676 + ], + "score": 0.93, + "content": "I ( X ; \\bar { h _ { 1 } } ) \\geq I ( \\bar { X ; } h _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 662, + 507, + 678 + ], + "score": 1.0, + "content": "(see, eg,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "Tishby & Zaslavsky (2015)). Because noise is added only for the purpose of analysis, however, this", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 686, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 506, + 698 + ], + "score": 1.0, + "content": "does not apply here. In particular, for the DPI to apply, the noise added at lower layers would have to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 698, + 506, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 708 + ], + "score": 1.0, + "content": "propagate through the network to higher layers. That is, if the transformation from hidden layer 1", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 708, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 187, + 720 + ], + "score": 1.0, + "content": "to hidden layer 2 is", + "type": "text" + }, + { + "bbox": [ + 188, + 708, + 237, + 720 + ], + "score": 0.93, + "content": "h _ { 2 } = f ( h _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 708, + 256, + 720 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 256, + 708, + 315, + 719 + ], + "score": 0.91, + "content": "T _ { 1 } = h _ { 1 } + Z _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 708, + 506, + 720 + ], + "score": 1.0, + "content": "is the hidden layer activity after adding noise,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 275, + 734 + ], + "score": 1.0, + "content": "then the DPI would hold for the variable", + "type": "text" + }, + { + "bbox": [ + 275, + 720, + 434, + 733 + ], + "score": 0.92, + "content": "\\tilde { T } _ { 2 } = f ( T _ { 1 } ) + Z _ { 2 } = f ( h _ { 1 } + Z _ { 1 } ) + Z _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 719, + 505, + 734 + ], + "score": 1.0, + "content": ", not the quantity", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22 + } + ], + "page_idx": 20, + "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, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 150, + 82, + 464, + 403 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 150, + 82, + 464, + 403 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 150, + 82, + 464, + 403 + ], + "spans": [ + { + "bbox": [ + 150, + 82, + 464, + 403 + ], + "score": 0.975, + "type": "image", + "image_path": "12c7d31d8a03d347bac1cbe449138df659dc492793257cc0c61d13e2608ea4fa.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 150, + 82, + 464, + 189.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 150, + 189.0, + 464, + 296.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 150, + 296.0, + 464, + 403.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 421, + 505, + 455 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "Figure 14: Effect of binning strategy on information plane dynamics. 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Measured", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 443, + 373, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 373, + 456 + ], + "score": 1.0, + "content": "with this binning structure, there is no compression in most layers.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 532 + ], + "lines": [], + "index": 7.5, + "bbox_fs": [ + 105, + 487, + 506, + 533 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 505, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "Any implementation of a neural network on digital hardware is ultimately of finite precision, and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "hence is a binned, discrete representation. However, it is a very high resolution binning compared", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 558, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 573 + ], + "score": 1.0, + "content": "to that used here or by Shwartz-Ziv & Tishby (2017): single precision would correspond to using", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 569, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 139, + 583 + ], + "score": 1.0, + "content": "roughly", + "type": "text" + }, + { + "bbox": [ + 140, + 570, + 154, + 581 + ], + "score": 0.87, + "content": "2 ^ { 3 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 569, + 506, + 583 + ], + "score": 1.0, + "content": "bins to discretize each hidden unit’s activity, as compared to the 30-100 used here. If the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 582, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 297, + 594 + ], + "score": 1.0, + "content": "binning is fine-grained enough that each input", + "type": "text" + }, + { + "bbox": [ + 297, + 582, + 307, + 591 + ], + "score": 0.76, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 582, + 474, + 594 + ], + "score": 1.0, + "content": "yields a different binned activity pattern", + "type": "text" + }, + { + "bbox": [ + 474, + 582, + 481, + 591 + ], + "score": 0.74, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 582, + 505, + 594 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 107, + 592, + 174, + 604 + ], + "score": 0.93, + "content": "H ( h ) \\stackrel { - } { = } \\log ( P )", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 592, + 203, + 605 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 204, + 593, + 213, + 602 + ], + "score": 0.82, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "is the number of examples in the dataset, and there will be little to no", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 603, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 506, + 616 + ], + "score": 1.0, + "content": "change in information during training. As an example, we show in Fig. 15 the result of binning at full", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 615, + 183, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 183, + 626 + ], + "score": 1.0, + "content": "machine precision.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 538, + 506, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 631, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 507, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 507, + 644 + ], + "score": 1.0, + "content": "Finally, we note two consequences of the assumption of noise/binning for the purposes of analysis.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "score": 1.0, + "content": "First, this means that the data processing inequality (DPI) does not apply to the noisy/binned mutual", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "information estimates. The DPI states that information can only be destroyed through successive", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 662, + 507, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 662, + 211, + 678 + ], + "score": 1.0, + "content": "transformations, that is, if", + "type": "text" + }, + { + "bbox": [ + 212, + 664, + 273, + 675 + ], + "score": 0.92, + "content": "X h _ { 1 } h _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 662, + 382, + 678 + ], + "score": 1.0, + "content": "form a Markov chain, then", + "type": "text" + }, + { + "bbox": [ + 382, + 664, + 470, + 676 + ], + "score": 0.93, + "content": "I ( X ; \\bar { h _ { 1 } } ) \\geq I ( \\bar { X ; } h _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 662, + 507, + 678 + ], + "score": 1.0, + "content": "(see, eg,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "Tishby & Zaslavsky (2015)). Because noise is added only for the purpose of analysis, however, this", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 686, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 506, + 698 + ], + "score": 1.0, + "content": "does not apply here. In particular, for the DPI to apply, the noise added at lower layers would have to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 698, + 506, + 708 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 708 + ], + "score": 1.0, + "content": "propagate through the network to higher layers. That is, if the transformation from hidden layer 1", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 708, + 506, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 187, + 720 + ], + "score": 1.0, + "content": "to hidden layer 2 is", + "type": "text" + }, + { + "bbox": [ + 188, + 708, + 237, + 720 + ], + "score": 0.93, + "content": "h _ { 2 } = f ( h _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 708, + 256, + 720 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 256, + 708, + 315, + 719 + ], + "score": 0.91, + "content": "T _ { 1 } = h _ { 1 } + Z _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 708, + 506, + 720 + ], + "score": 1.0, + "content": "is the hidden layer activity after adding noise,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 275, + 734 + ], + "score": 1.0, + "content": "then the DPI would hold for the variable", + "type": "text" + }, + { + "bbox": [ + 275, + 720, + 434, + 733 + ], + "score": 0.92, + "content": "\\tilde { T } _ { 2 } = f ( T _ { 1 } ) + Z _ { 2 } = f ( h _ { 1 } + Z _ { 1 } ) + Z _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 719, + 505, + 734 + ], + "score": 1.0, + "content": ", not the quantity", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 630, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 149, + 81, + 456, + 232 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 149, + 81, + 456, + 232 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 81, + 456, + 232 + ], + "spans": [ + { + "bbox": [ + 149, + 81, + 456, + 232 + ], + "score": 0.972, + "type": "image", + "image_path": "e2187148ec2ae4d72ade118f8c38e42693609c24edb83c7050bd489b4d6120ca.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 149, + 81, + 456, + 131.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 149, + 131.33333333333334, + 456, + 181.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 149, + 181.66666666666669, + 456, + 232.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 249, + 506, + 295 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "Figure 15: Effect of binning at full machine precision. (A) ReLU network. (B) tanh network.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 260, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 285, + 274 + ], + "score": 1.0, + "content": "Information in most layers stays pinned to", + "type": "text" + }, + { + "bbox": [ + 285, + 261, + 345, + 273 + ], + "score": 0.92, + "content": "\\log _ { 2 } ( P ) = 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 260, + 506, + 274 + ], + "score": 1.0, + "content": ". Compression is only observed in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "highest and smallest layers near the very end of training, when the saturation of tanh is strong enough", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 284, + 227, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 227, + 295 + ], + "score": 1.0, + "content": "to saturate machine precision.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 504, + 340 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 229, + 328 + ], + "score": 0.9, + "content": "T _ { 2 } = h _ { 2 } + Z _ { 2 } = f ( h _ { 1 } ) + Z _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 315, + 482, + 329 + ], + "score": 1.0, + "content": "used in the analysis. Said another way, the Markov chain for", + "type": "text" + }, + { + "bbox": [ + 483, + 317, + 494, + 328 + ], + "score": 0.87, + "content": "T _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 315, + 505, + 329 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 107, + 326, + 504, + 342 + ], + "spans": [ + { + "bbox": [ + 107, + 329, + 194, + 339 + ], + "score": 0.9, + "content": "X h _ { 1 } h _ { 2 } T _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 326, + 213, + 342 + ], + "score": 1.0, + "content": ", not", + "type": "text" + }, + { + "bbox": [ + 213, + 328, + 300, + 339 + ], + "score": 0.92, + "content": "X h _ { 1 } T _ { 1 } T _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 326, + 411, + 342 + ], + "score": 1.0, + "content": ", so the DPI states only that", + "type": "text" + }, + { + "bbox": [ + 411, + 328, + 499, + 340 + ], + "score": 0.93, + "content": "I ( X ; h _ { 1 } ) \\ge I ( X ; T _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 326, + 504, + 342 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "A second consequence of the noise assumption is the fact that the mutual information is no longer", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 360, + 369 + ], + "score": 1.0, + "content": "invariant to invertible transformations of the hidden activity", + "type": "text" + }, + { + "bbox": [ + 360, + 356, + 367, + 366 + ], + "score": 0.74, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 355, + 506, + 369 + ], + "score": 1.0, + "content": ". A potentially attractive feature", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 506, + 379 + ], + "score": 1.0, + "content": "of a theory based on mutual information is that it can allow for comparisons between different", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "architectures: mutual information is invariant to any invertible transformation of the variables, so two", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "hidden representations could be very different in detail but yield identical mutual information with", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 400, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 506, + 413 + ], + "score": 1.0, + "content": "respect to the input. However, once noise is added to a hidden representation, this is no longer the case:", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 156, + 424 + ], + "score": 1.0, + "content": "the variable", + "type": "text" + }, + { + "bbox": [ + 156, + 411, + 204, + 421 + ], + "score": 0.92, + "content": "T = h + Z", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 410, + 367, + 424 + ], + "score": 1.0, + "content": "is not invariant to reparametrizations of", + "type": "text" + }, + { + "bbox": [ + 367, + 411, + 374, + 421 + ], + "score": 0.79, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 410, + 506, + 424 + ], + "score": 1.0, + "content": ". As a simple example, consider", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 104, + 422, + 296, + 434 + ], + "score": 1.0, + "content": "a minimal linear network with scalar weights", + "type": "text" + }, + { + "bbox": [ + 297, + 423, + 309, + 433 + ], + "score": 0.86, + "content": "w _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 422, + 329, + 434 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 329, + 423, + 342, + 433 + ], + "score": 0.86, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 422, + 449, + 434 + ], + "score": 1.0, + "content": "that computes the output", + "type": "text" + }, + { + "bbox": [ + 449, + 422, + 503, + 433 + ], + "score": 0.92, + "content": "{ \\hat { y } } = w _ { 2 } w _ { 1 } X", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 422, + 506, + 434 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 431, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 195, + 446 + ], + "score": 1.0, + "content": "The hidden activity is", + "type": "text" + }, + { + "bbox": [ + 196, + 433, + 237, + 443 + ], + "score": 0.91, + "content": "h = w _ { 1 } X", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 431, + 492, + 446 + ], + "score": 1.0, + "content": ". Now consider the family of networks in which we scale down", + "type": "text" + }, + { + "bbox": [ + 492, + 435, + 504, + 444 + ], + "score": 0.83, + "content": "w _ { 1 }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 442, + 507, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 159, + 457 + ], + "score": 1.0, + "content": "and scale up", + "type": "text" + }, + { + "bbox": [ + 159, + 445, + 172, + 455 + ], + "score": 0.85, + "content": "w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 442, + 219, + 457 + ], + "score": 1.0, + "content": "by a factor", + "type": "text" + }, + { + "bbox": [ + 219, + 444, + 243, + 455 + ], + "score": 0.91, + "content": "c \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 442, + 395, + 457 + ], + "score": 1.0, + "content": ", that is, these networks have weights", + "type": "text" + }, + { + "bbox": [ + 395, + 443, + 442, + 455 + ], + "score": 0.89, + "content": "\\tilde { w } _ { 1 } = w _ { 1 } / c", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 442, + 460, + 457 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 461, + 444, + 502, + 455 + ], + "score": 0.91, + "content": "\\tilde { w } _ { 2 } = c w _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 442, + 507, + 457 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 278, + 468 + ], + "score": 1.0, + "content": "yielding the exact same input-output map", + "type": "text" + }, + { + "bbox": [ + 278, + 455, + 444, + 466 + ], + "score": 0.92, + "content": "\\hat { y } = \\tilde { w } _ { 2 } \\tilde { w } _ { 1 } X = c w _ { 2 } ( w _ { 1 } / c ) X = w _ { 2 } w _ { 1 } X", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 454, + 506, + 468 + ], + "score": 1.0, + "content": ". Because they", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 465, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 479 + ], + "score": 1.0, + "content": "compute the same function, they necessarily generalize identically. However after introducing the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 477, + 282, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 282, + 488 + ], + "score": 1.0, + "content": "noise assumption the mutual information is", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 492, + 404, + 509 + ], + "lines": [ + { + "bbox": [ + 206, + 492, + 404, + 509 + ], + "spans": [ + { + "bbox": [ + 206, + 492, + 404, + 509 + ], + "score": 0.89, + "content": "\\begin{array} { l l l } { { I ( T ; X ) } } & { { = } } & { { \\log \\left( w _ { 1 } ^ { 2 } / c ^ { 2 } + \\sigma _ { M I } ^ { 2 } \\right) - \\log \\left( \\sigma _ { M I } ^ { 2 } \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "0c9da91e389db59f31738e85823c42564aea9f459391b32abdf8ec6fba0aab0e.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 206, + 492, + 404, + 509 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 336, + 526 + ], + "score": 1.0, + "content": "where we have taken the setting in Section 3 in which", + "type": "text" + }, + { + "bbox": [ + 337, + 514, + 347, + 523 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 513, + 505, + 526 + ], + "score": 1.0, + "content": "is normal Gaussian, and independent", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 218, + 538 + ], + "score": 1.0, + "content": "Gaussian noise of variance", + "type": "text" + }, + { + "bbox": [ + 219, + 524, + 239, + 537 + ], + "score": 0.92, + "content": "\\sigma _ { M I } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "is added for the purpose of MI computation. Clearly, the mutual", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 286, + 547 + ], + "score": 1.0, + "content": "information is now dependent on the scaling", + "type": "text" + }, + { + "bbox": [ + 286, + 538, + 292, + 545 + ], + "score": 0.71, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "of the internal layer, even though this is an invertible", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 547, + 504, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 504, + 558 + ], + "score": 1.0, + "content": "linear transformation of the representation. Moreover, this shows that networks which generalize", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "identically can nevertheless have very different mutual information with respect to the input when it", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 568, + 204, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 204, + 581 + ], + "score": 1.0, + "content": "is measured in this way.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 107, + 597, + 297, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 299, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 299, + 613 + ], + "score": 1.0, + "content": "D WEIGHT NORMS OVER TRAINING", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "Our argument relating neural saturation to compression in mutual information relies on the notion", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 632, + 507, + 647 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 507, + 647 + ], + "score": 1.0, + "content": "that in typical training regimes, weights begin small and increase in size over the course of training.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "We note that this is a virtual necessity for a nonlinearity like tanh, which is linear around the origin:", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "when initialized with small weights, the activity of a tanh network will be in this linear regime and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "the network can only compute a linear function of its input. Hence a real world nonlinear task can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "score": 1.0, + "content": "only be learned by increasing the norm of the weights so as to engage the tanh nonlinearity on some", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "examples. This point can also be appreciated from norm-based capacity bounds on neural networks,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "which show that, for instance, the Rademacher complexity of a neural network with small weights", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "must be low (Bartlett & Mendelson, 2002; Neyshabur et al., 2015). 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(A) ReLU network. (B) tanh network.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 260, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 260, + 285, + 274 + ], + "score": 1.0, + "content": "Information in most layers stays pinned to", + "type": "text" + }, + { + "bbox": [ + 285, + 261, + 345, + 273 + ], + "score": 0.92, + "content": "\\log _ { 2 } ( P ) = 1 2", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 260, + 506, + 274 + ], + "score": 1.0, + "content": ". Compression is only observed in the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "highest and smallest layers near the very end of training, when the saturation of tanh is strong enough", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 284, + 227, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 227, + 295 + ], + "score": 1.0, + "content": "to saturate machine precision.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 504, + 340 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 229, + 328 + ], + "score": 0.9, + "content": "T _ { 2 } = h _ { 2 } + Z _ { 2 } = f ( h _ { 1 } ) + Z _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 315, + 482, + 329 + ], + "score": 1.0, + "content": "used in the analysis. 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Moreover, this shows that networks which generalize", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "identically can nevertheless have very different mutual information with respect to the input when it", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 568, + 204, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 204, + 581 + ], + "score": 1.0, + "content": "is measured in this way.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 513, + 506, + 581 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 597, + 297, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 299, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 299, + 613 + ], + "score": 1.0, + "content": "D WEIGHT NORMS OVER TRAINING", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "Our argument relating neural saturation to compression in mutual information relies on the notion", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 632, + 507, + 647 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 507, + 647 + ], + "score": 1.0, + "content": "that in typical training regimes, weights begin small and increase in size over the course of training.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "We note that this is a virtual necessity for a nonlinearity like tanh, which is linear around the origin:", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "when initialized with small weights, the activity of a tanh network will be in this linear regime and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "the network can only compute a linear function of its input. Hence a real world nonlinear task can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "score": 1.0, + "content": "only be learned by increasing the norm of the weights so as to engage the tanh nonlinearity on some", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "examples. This point can also be appreciated from norm-based capacity bounds on neural networks,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "which show that, for instance, the Rademacher complexity of a neural network with small weights", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "must be low (Bartlett & Mendelson, 2002; Neyshabur et al., 2015). 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(B) Information plane", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 659, + 254, + 670 + ], + "spans": [ + { + "bbox": [ + 107, + 659, + 254, + 670 + ], + "score": 1.0, + "content": "dynamics for batch gradient descent.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 684, + 311, + 697 + ], + "lines": [ + { + "bbox": [ + 104, + 683, + 312, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 683, + 312, + 699 + ], + "score": 1.0, + "content": "I GRADIENT SNR PHASE TRANSITION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 710, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 504, + 722 + ], + "score": 1.0, + "content": "The proposed mechanism of compression in Shwartz-Ziv & Tishby (2017) is noise arising from", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "stochastic gradient descent training. The results in Section 4 of the main text show that compression", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "still occurs under batch gradient descent learning, suggesting that in fact noise in the gradient updates", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "is not the cause of compression. Here we investigate a related claim, namely that during training,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "networks switch between two phases. These phases are defined by the ratio of the mean of the gradient", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "to the standard deviation of the gradient across training examples, called the gradient signal-to-noise", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "ratio. In the first “drift” phase, the SNR is high, while in the second “diffusion” phase the SNR", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "is low. Shwartz-Ziv & Tishby (2017) hypothesize that the drift phase corresponds to movement", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "toward the minimum with no compression, while the diffusion phase corresponds to a constrained", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 507, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 507, + 172 + ], + "score": 1.0, + "content": "diffusion in weight configurations that attain the optimal loss, during which representations compress.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "However, two phases of gradient descent have been described more generally, sometimes known as", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "the transient and stochastic phases or search and convergence phases (Murata, 1998; Chee & Toulis,", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 482, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 482, + 205 + ], + "score": 1.0, + "content": "2017), suggesting that these phases might not be related specifically to compression behavior.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 710, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "still occurs under batch gradient descent learning, suggesting that in fact noise in the gradient updates", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "is not the cause of compression. Here we investigate a related claim, namely that during training,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "networks switch between two phases. These phases are defined by the ratio of the mean of the gradient", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "to the standard deviation of the gradient across training examples, called the gradient signal-to-noise", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "ratio. In the first “drift” phase, the SNR is high, while in the second “diffusion” phase the SNR", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "is low. Shwartz-Ziv & Tishby (2017) hypothesize that the drift phase corresponds to movement", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "toward the minimum with no compression, while the diffusion phase corresponds to a constrained", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 507, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 507, + 172 + ], + "score": 1.0, + "content": "diffusion in weight configurations that attain the optimal loss, during which representations compress.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "However, two phases of gradient descent have been described more generally, sometimes known as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "the transient and stochastic phases or search and convergence phases (Murata, 1998; Chee & Toulis,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 482, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 482, + 205 + ], + "score": 1.0, + "content": "2017), suggesting that these phases might not be related specifically to compression behavior.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "In Fig. 20 we plot the gradient SNR over the course of training for the tanh and ReLU networks in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 432, + 232 + ], + "score": 1.0, + "content": "the standard setting of Shwartz-Ziv & Tishby (2017). In particular, for each layer", + "type": "text" + }, + { + "bbox": [ + 433, + 221, + 437, + 230 + ], + "score": 0.4, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "we calculate the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 235, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 235, + 243 + ], + "score": 1.0, + "content": "mean and standard deviation as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 306, + 362, + 364 + ], + "lines": [ + { + "bbox": [ + 247, + 306, + 362, + 364 + ], + "spans": [ + { + "bbox": [ + 247, + 306, + 362, + 364 + ], + "score": 0.92, + "content": "\\begin{array} { r c l } { { m _ { l } } } & { { = } } & { { \\displaystyle \\left\\| \\left. \\frac { \\partial E } { \\partial W _ { l } } \\right. \\right\\| _ { F } } } \\\\ { { s _ { l } } } & { { = } } & { { \\displaystyle \\left\\| \\mathrm { S T D } \\left( \\frac { \\partial E } { \\partial W _ { l } } \\right) \\right\\| _ { F } } } \\end{array}", + "type": "interline_equation", + "image_path": "f293b8d1d9e4f8f91ea536c6fcee59a7fd3efa334024fbe6fd1f65b977d2185b.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 247, + 306, + 362, + 335.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 247, + 335.0, + 362, + 364.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 428, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 134, + 442 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 429, + 146, + 441 + ], + "score": 0.89, + "content": "\\langle \\cdot \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 428, + 241, + 442 + ], + "score": 1.0, + "content": "denotes the mean and", + "type": "text" + }, + { + "bbox": [ + 241, + 429, + 275, + 441 + ], + "score": 0.9, + "content": "S T D ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "denotes the element-wise standard deviation across all", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 439, + 502, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 195, + 453 + ], + "score": 1.0, + "content": "training samples, and", + "type": "text" + }, + { + "bbox": [ + 196, + 439, + 216, + 452 + ], + "score": 0.91, + "content": "\\left\\| \\cdot \\right\\| _ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 439, + 477, + 453 + ], + "score": 1.0, + "content": "denotes the Frobenius norm. The gradient SNR is then the ratio", + "type": "text" + }, + { + "bbox": [ + 478, + 440, + 502, + 452 + ], + "score": 0.92, + "content": "m _ { l } / s _ { l }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 450, + 428, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 287, + 464 + ], + "score": 1.0, + "content": "We additionally plot the norm of the weights", + "type": "text" + }, + { + "bbox": [ + 287, + 450, + 317, + 463 + ], + "score": 0.92, + "content": "\\| W _ { l } \\| _ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 450, + 428, + 464 + ], + "score": 1.0, + "content": "over the course of training.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "Both tanh and ReLU networks yield a similar qualitative pattern, with SNR undergoing a step-like", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "score": 1.0, + "content": "transition to a lower value during training. Figures 9 and 10, fourth row, show similar plots for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "MNIST-trained networks. Again, SNR undergoes a transition from high to low over training. Hence", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "the two phase nature of gradient descent appears to hold across the settings that we examine here.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "Crucially, this finding shows that the SNR transition is not related to the compression phenomenon", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 522, + 468, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 468, + 535 + ], + "score": 1.0, + "content": "because ReLU networks, which show the gradient SNR phase transition, do not compress.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "Finally, to show the generality of the two-phase gradient SNR behavior and its independence from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "compression, we develop a minimal model of this phenomenon in a three neuron linear network. We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 321, + 574 + ], + "score": 1.0, + "content": "consider the student-teacher setting of Fig. 3 but with", + "type": "text" + }, + { + "bbox": [ + 322, + 561, + 379, + 572 + ], + "score": 0.92, + "content": "N _ { i } = N _ { h } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 561, + 506, + 574 + ], + "score": 1.0, + "content": ", such that the input and hidden", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "layers have just a single neuron (as in the setting of Fig. 2). Here, with just a single hidden neuron,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "clearly there can be no compression so long as the first layer weight increases over the course of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "training. Figure 21AC shows that even in this simple setting, the SNR shows the phase transition but", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "the weight norm increases over training. Hence again, the two phases of the gradient are present even", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 617, + 504, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 504, + 628 + ], + "score": 1.0, + "content": "though there is no compression. To intuitively understand the source of this behavior, note that the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 628, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 506, + 639 + ], + "score": 1.0, + "content": "weights are initialized to be small and hence early in learning all must be increased in magnitude,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "yielding a consistent mean gradient. Once the network reaches the vicinity of the minimum, the mean", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "weight change across all samples by definition goes to zero. The standard deviation remains finite,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 659, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 674 + ], + "score": 1.0, + "content": "however, because on some specific examples error could be improved by increasing or decreasing the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 671, + 438, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 438, + 684 + ], + "score": 1.0, + "content": "weights–even though across the whole dataset the mean error has been minimized.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Hence overall, our results show that a two-phase structure in the gradient SNR occurs in all settings we", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "consider, even though compression occurs only in a subset. The gradient SNR behavior is therefore", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "not causally related to compression dynamics, consistent with the view that saturating nonlinearities", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 721, + 263, + 732 + ], + "spans": [ + { + "bbox": [ + 107, + 721, + 263, + 732 + ], + "score": 1.0, + "content": "are the primary source of compression.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + } + ], + "page_idx": 25, + "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": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "26", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 204 + ], + "lines": [], + "index": 5, + "bbox_fs": [ + 105, + 83, + 507, + 205 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "In Fig. 20 we plot the gradient SNR over the course of training for the tanh and ReLU networks in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 432, + 232 + ], + "score": 1.0, + "content": "the standard setting of Shwartz-Ziv & Tishby (2017). 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The gradient SNR is then the ratio", + "type": "text" + }, + { + "bbox": [ + 478, + 440, + 502, + 452 + ], + "score": 0.92, + "content": "m _ { l } / s _ { l }", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 450, + 428, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 287, + 464 + ], + "score": 1.0, + "content": "We additionally plot the norm of the weights", + "type": "text" + }, + { + "bbox": [ + 287, + 450, + 317, + 463 + ], + "score": 0.92, + "content": "\\| W _ { l } \\| _ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 450, + 428, + 464 + ], + "score": 1.0, + "content": "over the course of training.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 428, + 505, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "Both tanh and ReLU networks yield a similar qualitative pattern, with SNR undergoing a step-like", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "score": 1.0, + "content": "transition to a lower value during training. Figures 9 and 10, fourth row, show similar plots for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "MNIST-trained networks. Again, SNR undergoes a transition from high to low over training. Hence", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "the two phase nature of gradient descent appears to hold across the settings that we examine here.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "Crucially, this finding shows that the SNR transition is not related to the compression phenomenon", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 522, + 468, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 468, + 535 + ], + "score": 1.0, + "content": "because ReLU networks, which show the gradient SNR phase transition, do not compress.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 467, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "Finally, to show the generality of the two-phase gradient SNR behavior and its independence from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "compression, we develop a minimal model of this phenomenon in a three neuron linear network. We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 321, + 574 + ], + "score": 1.0, + "content": "consider the student-teacher setting of Fig. 3 but with", + "type": "text" + }, + { + "bbox": [ + 322, + 561, + 379, + 572 + ], + "score": 0.92, + "content": "N _ { i } = N _ { h } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 561, + 506, + 574 + ], + "score": 1.0, + "content": ", such that the input and hidden", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "layers have just a single neuron (as in the setting of Fig. 2). Here, with just a single hidden neuron,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "clearly there can be no compression so long as the first layer weight increases over the course of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "training. Figure 21AC shows that even in this simple setting, the SNR shows the phase transition but", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "the weight norm increases over training. 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The standard deviation remains finite,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 659, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 674 + ], + "score": 1.0, + "content": "however, because on some specific examples error could be improved by increasing or decreasing the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 671, + 438, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 438, + 684 + ], + "score": 1.0, + "content": "weights–even though across the whole dataset the mean error has been minimized.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 540, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "Hence overall, our results show that a two-phase structure in the gradient SNR occurs in all settings we", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "consider, even though compression occurs only in a subset. 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(A) tanh networks trained in the standard setting of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "score": 1.0, + "content": "Shwartz-Ziv & Tishby (2017) show a phase transition in every layer. 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Here a three neuron linear", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 197, + 669 + ], + "score": 1.0, + "content": "network (architecture", + "type": "text" + }, + { + "bbox": [ + 198, + 656, + 243, + 667 + ], + "score": 0.86, + "content": "1 - 1 - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "learns to approximate a teacher. 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