Moreover, we replace the original constraint function g t p¨q with ˆ g ` t ´ 1 p¨q and the dual variables λ twith Q p t ´ 1 q such that Q p t ´ 1 q ˆ g ` t ´ 1 p x q is a rectiﬁed approximator of λ t g t p x q . We also added thethe regularization (or smooth) term α t } x ´ x t } 2 that helps the stability of the algorithm. Note thisdesign is related to penalty-based proximal optimization where we aim to minimize an approximatedf t p x q w.r.t. proximal operator on the “old” rectiﬁed function ˆ g ` t ´ 1 p x q . In the penalty update of Q p t q , we ﬁrst rectify the original constraint function g t p¨q with ˆ g ` t p¨q andadd it to Q p t ´ 1 q such that penalty increases when violation occurs in each round. Further werectify Q p t q with a round-dependent constant η t to impose a “minimum” penalty price. The designof rectiﬁed penalty update induces conservative decisions in the decision-making step to minimizeconstraint violation. Note that it is different with the traditional primal-dual algorithm that doesnot rectify the constraint violation and impose a minimum penalty price, and when the price (dualvariable) is zero, the algorithm can take very aggressive decisions that lead to large hard violation. This is not a problem when primal-dual algorithm is used as a numerical method for solving aconstrained optimization problem, but leads to overly aggressive decisions and large violation whenapplying it to COCO. For a similar reason, RECOO rectiﬁes the amount of violation g t p x q in thedecision-making step so the violation does not become negative to prevent overly optimistic decisions. We will see that the rectiﬁers in both decision-making and penalty update leads toan upper boundof “regret + violation” as a whole (Lemma 1) such that we can establish regret and (hard) constraintviolation directly. These techniques are different from the primal-dual optimization that quantiﬁesthe constraint violation indirectly by bounding the dual variables/virtual queues. Finally, we comment that our algorithm only needs to solve an “almost” unconstrained optimizationproblem ( X is usually a simple set like the box constraints). Therefore, the gradient-based methodsare sufﬁcient to ﬁnd the minimizer or we might even ﬁnd its close-form with the inverse operationof the function by taking “gradient “ zero”.

Moreover, we replace the original constraint function g t p¨q with ˆ g ` t ´ 1 p¨q and the dual variables λ twith Q p t ´ 1 q such that Q p t ´ 1 q ˆ g ` t ´ 1 p x q is a rectiﬁed approximator of λ t g t p x q . We also added thethe regularization (or smooth) term α t } x ´ x t } 2 that helps the stability of the algorithm. In the penalty update of Q p t q , we ﬁrst rectify the original constraint function g t p¨q with ˆ g ` t p¨q and addit to Q p t ´ 1 q such that penalty increases when violation occurs in each round. Further we rectify Q p t qwith a round-dependent constant η t to impose a “minimum” penalty price. The design of rectiﬁedpenalty update induces conservative decisions in the decision-making step to minimize constraintviolation. Note that it is different with the traditional primal-dual algorithm that does not rectifythe constraint violation and impose a minimum penalty price, and when the price (dual variable)is zero, the algorithm can take very aggressive decisions that lead to large hard violation. This isnot a problem when primal-dual algorithm is used as a numerical method for solving a constrainedoptimization problem, but leads to overly aggressive decisions and large violation when applying it to COCO. For a similar reason, RECOO rectiﬁes the amount of violation g t p x q in the decision-makingstep so the violation does not become negative to prevent overly optimistic decisions. We will see thatthe rectiﬁers in both decision-making and penalty update lead to a small cumulative hard constraintviolation ř Tt “ 1 g ` t p x t q .

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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] (b) Did you mention the license of the assets? [No] (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? [Yes] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] 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]

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] (b) Did you mention the license of the assets? [No] (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? [Yes] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] 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]


The core of this proof is based on substituting a neural network for the inner product between the branch net and  trunk net. The neural network with a low complexity could approximate the inner product with better performance since the inner product is an infinitely differentiable function. The DeepONet showed defects as it could be replaced with a neural network with E ( u ) and y as input. A large number of basis p in DeepONet increases the number of parameters of the trunk net, which can be considered a target network in HyperDeepONet. Models such as Shift-DeepONet and flexDeepONet could achieve the desired accuracy with a small number of basis. Still, there was a trade-off in which the first hidden layer of the target network required numerous units. There was no restriction on the dimension of the last hidden layer in the target network for NOMAD, which uses a fully nonlinear reconstruction. However, the first hidden layer of the target network  had to be wide enough, increasing the number of parameters. Details can be found in Appendix C.

The core of this proof is showing that the inner product between the branch net and the trunk net could be replaced with a neural network that has a low complexity (Lemma 1). Therefore, the entire structure of DeepONet could be replaced with a neural network that receives [ E ( u ) , y ] ∈ R d y + m as input. It gives the lower bound for the number of parameters in DeepONet based on Theorem 1. The proof can be found in Appendix C.1. The analogous results holds for variant models of DeepONet. Models such as Shift-DeepONet and flexDeepONet could achieve the desired accuracy with a small number of basis. Still, there was a trade-off in which the first hidden layer of the target network required numerous units. There was no restriction on the dimension of the last hidden layer in the target network for NOMAD, which uses a fully nonlinear reconstruction. However, the first hidden layer of the target network also had to be wide enough, increasing the number of parameters. Details can be found in Appendix C.2.

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TransBoost’s performance is compared to the standard inductive (fully supervised) performance. Forinstance, using 20% of the training set and 25% of the test set, we obtained the best top-1 accuracygain of +3.58% in the transductive setting while the performance in the inductive setting degraded by-1.21%. We note that the performance in the inductive setting (highlighted in blue) is evaluated usingthe test instances that were not used at training time.

Table 4 presents 16 experiments of TransBoost’s procedure performed on all combinations of the instance, % of training set and 25% of the test set, we obtained the best top gain of +3.58% in the transductive setting while the performance in the inductive setting degraded by We note that the performance in the inductive setting (highlighted in blue) is evaluated usingthe test instances that were not used at training time.


Four participants answered that they intentionally used edges (Figure 5 (iv)) in all conditions. However, they were affected by the cursor hidden by the notch, causing them to lose sight of the cursor. All participants answered that when the cursor was hidden by the notch, they tried to find the cursor by moving the mouse vigorously.

Four participants answered that they intentionally used edges (Figure 5 (iv)) in all conditions. However, the notch hid the cursor, which caused them to lose sight of the cursor. All participants answered that they attempted to find the cursor by moving the mouse vigorously when the cursor was hidden by the notch.

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To make this point more concrete, we implement push apart from the main text using the Jacobian of sorting. We use sort : R n → R n to denote sorting of a multiset with scalar elements inascending order. This is extended to FSPool with higher-dimensional elements by independently sorting each dimension across the multiset. We begin by deﬁning the function g ([ a, b ]) = sort ([ a, b ]) · [ − 1 , 1] . This multiplies the smaller value with − 1 and the larger with 1 (ties broken arbitrarily), then adds them together. Taking the gradient of g with respect to the input set means following the permutation that the sort applied backwards (similar to how max pooling is differentiated), so the − 1 gets propagated to the smaller element andto the larger one. In other words, ∇ [ a,b ] g = [ − 1 , 1] if a ≤ b (alternatively a < b ), else [1 , − 1] . Now we can deﬁne push apart ([ a, b ]) = [ a, b ] + ∇ [ a,b ] g to obtain exactly the same function as in the main text.

To make this point more concrete, we implement push apart from the main text using the Jacobian of sorting. We omit the transposes in the following for brevity. We begin by deﬁning the function g ([ a, b ]) = sort ([ a, b ]) · [ − 1 , 1] . This multiplies the smaller value with − 1 and the larger with(ties broken arbitrarily), then adds them together. Taking the gradient of g with respect to the input set means following the permutation that the sort applied in reverse ( S (cid:62) X ), so the − 1 gets propagated to the smaller element and 1 to the larger one. In other words, ∇ [ a,b ] g = [ − 1 , 1] if a ≤ b (alternatively a < b ), else [1 , − 1] . Now we can deﬁne push apart ([ a, b ]) = [ a, b ] + ∇ [ a,b ] g to obtain exactly the same function as in the main text.


Because typical targets on GUIs are rectangular, target height ( H ) also affects the movement time [3,8,14,20,21,27]. Accot and Zhai [1] proposed a model for a bivariate (2D) pointing tasks that takes H . Zhang et al. [28] proposed to balance the effects of W and H (Eq. 2).

Target height ( H ) also affects the movement time because typical targets on grafical user interfaces (GUIs) are rectangular [3,8,14,20, 22,30]. Accot and Zhai [1] proposed a model for a bivariate (2D) pointing tasks that considers H . Further, Zhang et al. [31] proposed to balancing the effects of W and H .

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According to the greedy learner hypothesis, it is the  speed at which a multi-modal DNN learns from each modality that leads to  imbalance in conditional utilization rate. If we appropriately intervene in the learning process to adjust these speeds, we may be able to prevent the hurtful imbalance across input modalities. We characterize such learning speed by introducing a metric, called conditional learning speed . We empirically show that it is a reasonable proxy for conditional utilization rate. We introduce a training algorithm, balanced multi-modal learning which accelerates the model to learn  from either modality alternatively according to their conditional learning speeds. We show that it produces models that learn to use all modalities appropriately and achieve stronger generalization on three multi-modal datasets: Colored MNIST dataset (Kim et al., 2019), ModelNet40 dataset of 3D objects (Su et al., 2015) and NVGesture Dataset (Molchanov et al., 2015).

According to the greedy learner hypothesis, it is the diverged speed at which a multi-modal DNN learns from different modalities that leads to an imbalance in conditional utilization rate. If we intervene in the training process to adjust these speeds, we may be able to prevent the hurtful imbalance across input modalities. We analyze the learning dynamics of model components and propose a metric named by conditional learning speed using the gradient norm and weight norm of models’ parameters. It measures the relative learning speed at which the model learns from one modality against the other modality. We empirically show that it is a reasonable proxy for conditional utilization rate. We introduce a training algorithm, balanced multi-modal learning which guides the model to learn intentionally from one of the modalities according to the conditional learning speeds. We show that models trained with this algorithm learn to use all modalities appropriately and achieve stronger generalization on three multi-modal datasets: Colored MNIST dataset (Kim et al., 2019), ModelNet40 dataset of 3D objects (Su et al., 2015) and NVGesture Dataset (Molchanov et al., 2015).


Although the mouse cursor can enter the notch area at the top  of the MacBook Pro (2021) display, it is partially or entirely hidden by the notch. Avoiding the notch or moving the cursor carefully around the notch can increase the movement time. In this study, we conducted a series of experiments to evaluate the effect of the notch on the movement of the mouse cursor. Experiment 1 showed that the notch increased the pointing movement time in specific situations. Experiment 2 showed that it is better to avoid the notch than enter the notch in the  current specification of the notch. Experiment 3 showed that changing the notch to an area where the cursor cannot enter is an effective way of pointing at the target more rapidly and accurately, if the target is adjacent to the notch. Consequently, the outer edge of the notch stops the cursor, resulting in faster and more accurate target-pointing. Therefore, the notch should be an area where the cursor cannot enter .

The notch on the top edge of the MacBook Pro (2021) display hides the mouse cursor even though the cursor can move under this area. Avoiding the notch or moving the cursor carefully around the notch can increase the movement time. In this study, we perform three experiments to evaluate the effect of the notch on the movement of the mouse cursor. In Experiment 1, we showed that the notch increases the pointing movement time under specific scenarios. In Experiment 2, we showed that it is better to avoid the notch instead of moving the cursor under the notch given its current specification. Finally, in Experiment 3, we showed that changing the notch to an area where the cursor cannot enter is an effective approach that allows the user to point at the target more rapidly and accurately if the target is adjacent to the notch. This is because the outer edge of the notch stops the cursor, and this results in faster and more accurate target pointing. Thus, the notch should be an area where the cursor cannot enter .

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When dealing with conflicts, arrows are effective in communi- cating the suggested conflict resolution action. End-of-line arrows can be used to indicate that the medication entries which have been scheduled too close together should be taken apart and vice-versa. The calendar should also have support for indication that a given entry is optional. Suchsupport targets entries such as prescription medication that should be taken as needed and non-prescription medication that is bought over the counter. These design decisions are influenced by everyday activities that users are familiar with.

When dealing with conflicts, arrows are effective in communi- cating the suggested conflict resolution action. End-of-line arrows can be used to indicate that medication entries that have been sched- uled too close together should be taken apart and vice-versa. The calendar should also have support for indication that a given entry is optional. Such entries would be used for medications that should be administered “as needed” and non-prescription medications that are sold “over the counter”. These design decisions are influenced by everyday activities that users are familiar with.

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The goal is to reconstruct the input with a permutation-invariant latent vector as bottleneck. Varying the set size n and dimensionality of set elements d allows us to control the difﬁculty of the task. Note that while this appears like a toy task, it is also likely harder than many real-world datasets with similar n and d . This is because the i.i.d. nature of the elements means that there is no global structure for the model to rely on; the only way to solve this task is to memorize every element exactly, which is especially difﬁcult when we want the latent space to be permutation-invariant.

The goal is to reconstruct the input with a permutation-invariant latent vector as bottleneck. Varying the set size n and dimensionality of set elements d allows us to control the difﬁculty of the task. Note that while this appears like a toy task, it is also likely harder than many real-world datasets with similar n and d . This is because the independent nature of the elements means that there is no global structure for the model to rely on; the only way to solve this task is to memorize every element exactly, which is especially difﬁcult when we want the latent space to be permutation-invariant. This task is also challenging because it indirectly requires a version of push apart : similar elements in the initialization Y 0 might need to be moved far away from each other in the output.


Patrick et al. proposed the Mouse Ether technique on finding out that  when using multiple displays with different resolutions, a user loses the cursor because of unnatural cursor movement between displays [5]. The results showed that the technique improved performance by up to 28% by preventing unnatural warping when the cursor was moved between displays.

Patrick et al. found out that a user loses the cursor when using multiple displays with different resolutions based on an unnatural cursor movement between displays, and proposed a Mouse Ether technique [5]. The proposed technique improved performance by up to 28% by preventing unnatural warping when the cursor was moved between displays.

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To leverage the natural sparsity of point clouds andfurther improve efficiency, we sparsely implementall our window center searching, window gathering, and balanced window sampling into CUDAoperations. These operations are mainly based on a hash map that establishes the mapping fromcoordinate space to voxel index [23]. Taking the window gathering operation as an example, wequery each possible position w.r.t. the given center within the window, and retrieve the correspondingfeatures if the position is a valid key in the pre-built hash map. More details are available in the supplementary materials.

We implement all our window center searching,window gathering, and balanced window samplingsparsely in cuda operations to leverage the natural sparsity of point clouds and improve efficiency. These operations are mainly based on a hash mapwhich establishes the mapping from coordinatespace to voxel index as in [20]. For example, forwindow gathering, we query each possible positionwrt the given center within the window and retrievethe feature if the position is a valid key in our prebuilt hash map. More details are available in the supplementary materials.

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The surprising result of our experimental evaluation is that, for runtimes of 1h, the black box optimizers are hardly ever able to improve upon Naive AutoML. In fact, the experiments even show that the “Ex-def” baseline itself is already quite strong in this time frame. This is not a contradiction to the results in Thornton et al. (2013), where a timeout of 30 hours was used. While this observations calls for more exhaustive experiments (with longer runtimes), it is already evident that simple baselines are much stronger than they were supposed to be until now.

The surprising result of our experimental evaluation is that, for runtimes of 1h, the black box optimizers are hardly ever able to improve upon Naive AutoML. In fact, the experiments even show that the “Ex-def” baseline itself is already quite strong in this time frame. This is not a contradiction to the results in (Thornton et al., 2013), where a timeout of 30 hours was used. While this observations calls for more exhaustive experiments, it is already evident that simple baselines are stronger than perhaps believed.

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It is known that the edge target (placing the target adjacent to the edge of the screen) can reduce the movement time [3,9,10,24,25]. Pointing at a target in the center of the screen requires the cursor to stop just inside the target. When pointing an edge target, the cursor stops at the edge. As a result, the pointing  can be completed by moving a cursor horizontally relative to the edge at which the target is adjacent. Additionally, the target adjacent to the corner of the screen can be pointed  fast simply by hitting the corner with the cursor [24].

An edge target (target adjacent to the edge of the screen) can reduce the movement time [3,9,10,27,28]. Pointing at a target at the center of the screen requires the cursor to stop inside the target. The cursor stops at the edge when pointing at an edge target. Thus, the pointing task can be completed by moving a cursor horizontally relative to the edge to which the target is adjacent. Further, a target adjacent to the corner of the screen can be pointed at fast simply by hitting the corner with the cursor [27].

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Given the probability density P ( θ ) , we are now interested in deriving the probability of ending at a given minimum, θ A , which we will denote by lowercase p A = ˜ p A C , where C is a normalization constant  (the unnormalized probability ˜ p A is all we are interested in when estimating the relative probability of ﬁnishing in a given minimum compared to another one). This probability is derived in Appendix D, and given in the following theorem, which is the core result of our theory. ) algorithm in the Bayesian learning setting (Sato & Nakagawa, 2014). While in this setting it was found that the stochastic noise can be ignored, we arrive at a different conclusion.

Given the probability density P ( θ ) , we are now interested in deriving the probability of ending at a given minimum, θ A , which we will denote by lowercase p A = ˜ p A C , where C is a normalization constant which is the same for every mimnima (the unnormalized probability ˜ p A is all we are interested in when estimating the relative probability of ﬁnishing in a given minimum compared to another one). This probability is derived in Appendix D, and given in the following theorem, which is the core result of our theory. context of the stochastic gradient langevin dynamics (SGLD) algorithm in the Bayesian learning setting (Sato & Nakagawa, 2014). While in this setting it was found that the stochastic noise can be ignored, we arrive at a different conclusion.


We instructed the participants to (1) point the target as quickly and accurately as possible after clicking the starting position, (2) avoid any clutching action (floating the mouse in the middle of an operation) during the trail, and (3) check the presented conditions before starting the trial. Clutching action decreases the model fit of Fitts’ law [7]. We  instructed  participants to avoid  clutching action as possible in order to restrict the effect for  model  fits to the experimental conditions. Note that any A was a distance that the cursor could be moved without  clutching action.

We instructed the participants to (1) point the target as quickly and accurately as possible after clicking the starting position, (2) avoid any clutching action (floating the mouse in the middle of an operation) during the trial, and (3) check the presented conditions before starting the trial. The clutching action decreases the model fit of Fitts’ law [7]. We also instructed the participants to avoid the clutching action to restrict the effect for the model that fits the experimental conditions. A was set the distance where the cursor could be moved without a clutching action; no participant performed the clutching action during the trial.


Active learning is a powerful technique to get the best out of an annotation budget. It consists of integrating the current model itself in the selection of which data points should be annotated next, which are selected based on an heuristic that is supposed to maximize the improvement of the model.

Active learning is a powerful technique to get the best out of an annotation budget for machine learning services in general, and Natural Language Processing ones in particular. It consists of integrating the current model itself in the selection of which data points should be annotated next, which are selected based on an heuristic that is supposed to maximize the improvement of the model.


Estimating individual treatment effects from observational data is very challenging due to the existence of treatment selection bias. Most existing representation-based methods mitigate this issue by aligning distributions of different treatment groups in the representation space. However, they still suffer from two critical problems: (1) Mini-batch Sampling Effects (MSE), where the alignment easily fails due to the outcome imbalance or outliers in the batch; (2) Unobserved Confounder Effects (UCE), where the unobserved confounders damage the correct alignment.

Estimating individual treatment effects from observational data is highly challenging due to the existence of treatment selection bias. Most prevalent approaches mitigate this issue by aligning distributions of different treatment groups in the representation space. However, there are two critical problems circumvented: (1) mini-batch sampling effects (MSE), where the alignment easily fails due to the outcome imbalance or outliers at a mini-batch level; (2) unobserved confounder effects (UCE), where the unobserved confounders damage the correct alignment.

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The learning problem can be arbitrarily difficult, especially when f ∗ ( z ) is close to 1 2 , in which case itwill be difficult to determine the true value of r ( z ) . To give a problem-dependent bound, we assume the following bounded noise assumption. In the literature on statistical learning, this assumption is also referred to as Massart noise [Massart and Nédélec, 2006, Giné and Koltchinskii, 2006, Hanneke et al., 2014]. Our framework can also work under the low noise assumption - due to space limit, wedefer the discussion to Appendix F.3.

The learning problem can be arbitrarily difficult, especially when f ∗ ( z ) is close to 1 2 , in which caseit will be difficult to determine the true value of r ( z ) . However, the marginal case is rare in manyreal-world problems, and the learning goal is not that difficult to identify for human beings. To give a problem-dependent bound, we assume the following bounded noise assumption. In the literature on statistical learning, this assumption is also referred to as Massart noise [Massart and Nédélec, 2006, Giné and Koltchinskii, 2006, Hanneke et al., 2014].


SDE model. Figures 2c-dclearly illustrate that time-variant drift function leads to reduced prediction error, which hints that the dynamics underlying a golf swing motion are learned better. We also see that the error consistently decreases as the number of inducing points M is increased, and reaches the minimum at M = 80 .

SDE model. Figures 2c-d illustrate that time-variant drift function can also reduce prediction error, at least for golf swing trajectories that span approximately the same part of the state space. We also see that the error consistently decreases as the number of inducing points M is increased, and reaches the minimum at M = 80 .

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One can fine-tune the global model on the local data [41], perform MAML-based personalizedapproaches [37, 38], or achieve the personalization by local batch normalization layers [11]. Our proposed method is perpendicular to the above studies and potentially can be combined with them for further improvement.

One can fine-tune the global model on the local data [37], perform MAML-based personalizedapproaches [38, 39], or achieve the personalization by local batch normalization layers [11]. Thereare also many other emerging explorations dealing with FL data heterogeneity, such as heterogeneousoptimization [40, 41, 42], robust aggregation [15], etc. Our proposed method is perpendicular to the above studies and potentially can be combined with them for further improvement.


General classes of network formation methods include: 1) exponential random graph models (ERGMs) Lusher et al. [2013]Pattison and Wasserman [1999], meta-networks, and meta-matrices Carley and Hill [2001]Krackhardt and Carley [1998] for multilayer social networks. 2) block modeling Guimerà and Sales-Pardo [2009] 3) geographic or characteristic based approaches, Boucherand Mourifié [2012]Leung [2014]; 4) link formation techniques Christakis et al. [2010]Bramoulléet al. [2012], and 5) subgraph model-based approaches (SUGMs) Chandrasekhar and Jackson [2016].

General classes of network formation methods include: 1) exponential random graph models (ERGMs) [Lusher et al., 2013][Pattison and Wasserman, 1999], meta-networks, and meta-matrices [Carley and Hill, 2001][Krackhardt and Carley, 1998] for multilayer social networks. 2) block modeling [Guimerà and Sales-Pardo, 2009] 3) geographic or characteristic based approaches, [Boucher and Mourifié, 2012][Leung, 2014]; 4) link formation techniques [Christakis et al., 2010][Bramoullé et al., 2012], and 5) subgraph model-based approaches (SUGMs) [Chandrasekhar and Jackson, 2016].


This is done by applying a matrix ρ ( g q → p ) ∈ R C out × C in to the coefﬁcients of the feature at q , in order to obtain the coefﬁcients of the feature vector transported to p , which can be used for the convolution at p . The transporter depends on the geometric type  of the feature, denoted by ρ . Details of how the tangent space is deﬁned, how to compute the map to the tangent space, angles θ pq , and the parallel transporter are given in Appendix A.

This is done by applying a matrix ρ ( g q → p ) ∈ R C out × C in to the coefﬁcients of the feature at q , in order to obtain the coefﬁcients of the feature vector transported to p , which can be used for the convolution at p . The transporter depends on the geometric type (group representation) of the feature, denoted by ρ and described in more detail below. Details of how the tangent space is deﬁned, how to compute the map to the tangent space, angles θ pq , and the parallel transporter are given in Appendix A.


To avoid the significant variance in the IHDP benchmark, we conduct ablation study  on the ACIC benchmark, to evaluate the effectiveness of ESCFR’s components and validate our claims in Section 3. In Table 2, ESCFR firstly augments TARNet with stochastic optimal transport in Section 3.1, which effectively reduces the out-of-sample PEHE from 3.254 to 3.207. Afterwards, it  mitigates the MSE issue with RMPR in Section 3.2 and the UCE issue with PFOR in Section 3.3, reducing the out-ofsample PEHE to 2.768 and 2.633, respectively. Finally, ESCFR combines the RMPR and PFOR in a unified framework in Section 3.4 and further reduces the out-of-sample PEHE to 2.316.

To verify the effectiveness of individual components, an ablation study is conducted on the ACIC benchmark in Table 2. Specifically, ESCFR first augments TARNet with stochastic optimal transport in Section 3.1, which effectively reduces the out-of-sample PEHE from 3.254 to 3.207. Then, it  mitigates the MSE issue with RMPR in Section 3.2 and the UCE issue with PFOR in Section 3.3, reducing the out-of-sample PEHE to 2.768 and 2.633, respectively. Finally, ESCFR combines the RMPR and PFOR in a unified framework in Section 3.4, reducing the out-of-sample PEHE to 2.316.

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We refer to the training steps at which we perform forward and backward passes normally as regular steps . We introduce re-balancing steps at which we force the model to update only one of the unimodal branches  in order to accelerate learning from the associated input modality. See Appendix A. for the full explanation of the re-balancing step.

We refer to the training steps at which we perform forward and backward passes normally as regular steps . We introduce re-balancing steps at which we update one of the uni-modal branches intentionally in order to accelerate the model to learn from the corresponding modality. See Appendix A.2 for the full explanation of the re-balancing step.

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We compare against four baseline methods: PLOT, NeuralUCB, Greedy (no exploration), and decayed ϵ -greedy method. For ϵ -greedy, we follow (Kveton et al., 2019) and use a decayed schedule, dropping to 0.1% exploration by T=2500. In addition we evaluate the performance of the Standalone Adversarial classifier as an ablation study.

We compare against four baseline methods: PLOT, NeuralUCB, Greedy (no exploration), and decayed ϵ -greedy method. For ϵ -greedy, we follow (Kveton et al., 2019) and use a decayed schedule, dropping to 0.1% exploration by T=2500. We combine PLOT with ϵ -greedy selection of pseudolabel candidates as in Pacchiano et al. In addition we evaluate the performance of the standalone adversarially de-biased classifier as an ablation study.


Deep CNN is ﬁrstly utilized for image SR in SRCNN (Dong et al., 2014) and  continuously shows promising SR performance. There are only three convolutional (Conv) layers in SRCNN, hindering its performance. Kim et al . increased the network depth in VDSR (Kim et al., 2016a) with residual learning and obtained notable improvements over SRCNN. Lim et al . (Lim et al., 2017) built a much deeper network EDSR by using simpliﬁed residual blocks. Zhang et al . (Zhang et al., 2018b) proposed RCAN, which is one of the deepest networks in SR. With increased network size, very deep networks, like EDSR (Lim et al., 2017) and RCAN (Zhang et al., 2018b), have achieved remarkable SR performance. However, they also suffer from heavy model parameters, number of operations, and inference time. Therefore, it is  infeasible to directly deploy them on resource-limited platformswithout neural processing units or off-chip memory (Lee et al., 2020).

Deep CNN for image SR is pioneered by SRCNN (Dong et al., 2014) and has continuously shown promising SR performance. There are only three convolutional (Conv) layers in SRCNN, constraining its expressivity. Kim et al . increased the network depth in VDSR (Kim et al., 2016a) with residual learning and obtained notable improvements over SRCNN. Lim et al . (Lim et al., 2017) built a much deeper network EDSR by using simpliﬁed residual blocks. Zhang et al . (Zhang et al., 2018b) proposed RCAN, which is one of the deepest networks in SR at present. Empowered by increased network size, deep SR models like EDSR (Lim et al., 2017) and RCAN (Zhang et al., 2018b) have seen remarkable SR performance. However, as a cost, the large model size brings about problems such as excessive memory footprint, slow inference speed. It is thereby infeasible to directly deploy them on resource-constrained platforms (Lee et al., 2020).

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Our handcrafted backdoor attacks directly modify a pre-trained model’s parameters to introduce malicious functionality. Because our attack does not require training, knowledge of or access to the training data is unnecessary. More importantly, handcrafted attacks have more degrees of freedom in optimizing a model’s behaviors for malicious purposes. Our handcrafted attack works by injecting a decision path between the trigger that appears in the input neurons and the output of the neural network, so that the models exhibit different behaviors in the presence of the trigger.

Our handcrafted backdoor attacks directly modify a pre-trained model’s parameters to introduce malicious functionality. Because our attack does not require training, knowledge of or access to the training data is unnecessary. More importantly, handcrafted attacks have more degrees of freedom in optimizing a model’s behaviors for malicious purposes. Our handcrafted attack works by injecting 36th Conference on Neural Information Processing Systems (NeurIPS 2022). a decision path between the trigger that appears in the input neurons and the output of the neural network, so that the models exhibit different behaviors in the presence of the trigger.


We further derive the upper bound of PEHE in the stochastic batch form as in Theorem 3.1 based on Uri et al. (2017), which demonstrates that the PEHE can be optimized by iteratively minimizing the factual outcome estimation error and the optimal transport discrepancy at a mini-batch level .

To further investigate the effectiveness of this shortcut, Theorem 3.1 demonstrates that PEHE can be optimized by iteratively minimizing the factual outcome estimation error and the mini-batch group discrepancy (6). The proof of the theorem can be found in Appendix A.3.

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To validate the model and inference procedure, we used the neural spike train data recorded from the primary visual cortex of an anesthetized macaque monkey collected by Graf et al. The dataset is composed of short trials where the monkey viewed periodic temporal pattern of motions of 72 orientations, each repeated 50 times. Previous state space modeling of the dataset showed that for each orientation of the drifting grating stimulus, the neural response oscillates over time, but in a stimulus dependent geometry  (Zhao & Park, 2017b). We used 25 trials each from a subset of 4 stimulus orientations grouped in two (140 and 150 degrees vs 230 and 240 degrees). Each trial contained 140 neurons, and their spike trains were binarized with a 10 ms window. We truncated the onset and offset neural responses, resulting in 111 time bins per trial.

To validate the model and inference procedure, we used the neural spike train data recorded from the primary visual cortex of an anesthetized macaque monkey collected by Graf et al. The dataset is composed of short trials where the monkey viewed periodic temporal pattern of motions of 72 orientations, each repeated 50 times. Dimensionality reduction of the dataset showed that for each orientation of the drifting grating stimulus, the neural response oscillates over time, but in a stimulus dependent geometry captured in 3-dimensions (Zhao & Park, 2017). We used 50 trials each from a subset of 4 stimulus orientations grouped in two (140 and 150 degrees vs. 230 anddegrees) where each trial contained 140 neurons. Out of the 140 neurons, we selected 63 well-tuned neurons. The spike trains were binarized with a 10 ms window for Bernoulli observation model and we truncated the onset and offset neural responses, resulting in 111 time bins per trial.


To evaluate the proposed spatial-wise and channel-wise sparse-complementary (SW-SC and CWSC) convolutional kernels, we applied them onto on state-of-the-art network architectures for the image classiﬁcation task, including ResNet (He et al. (2016a;b)) and DenseNet (Huang et al. for the CIFAR-10/100 (Krizhevsky & Hinton(2009)) and ImageNet-1K (Russakovsky et al. (2015)) datasets. For all experiments, we replace all 3 × 3 and 1 × 1 kernels by SW-SC and CW-SC convolutional kernels, respectively. We add a sufﬁx − sc to refer to our models with either SW-SC,

To evaluate the proposed spatial-wise and channel-wise sparse-complementary (SW-SC and CWSC) convolutional kernels, we applied them onto on state-of-the-art network architectures for the image for the CIFAR-10/ Krizhevsky Hinton and ImageNet-1K ) datasets. For all experiments, we replace all 3 × 3 and 1 × 1 kernels by SW-SC and CW-SC convolutional kernels, respectively. We add a sufﬁx − sc to refer to our models with either SW-SC,

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In this section, we would like to clarify the complexity of the DeepONet required for the approximation A and reconstruction R based on the theory in Galanti & Wolf (2020). Furthermore,using the results on the upper bound for the complexity of hypernetwork Galanti & Wolf (2020), we will show that the HyperDeepONet entails a relatively lower complexity than the DeepONet.

In this section, we would like to clarify the complexity of the DeepONet required for the approximation A and reconstruction R based on the theory in Galanti & Wolf (2020). Furthermore, we will show that the HyperDeepONet entails a relatively lower complexity than the DeepONet using the results on the upper bound for the complexity of hypernetwork (Galanti & Wolf, 2020).

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Previous work has shown GFlowNets are useful in settings with multi-modal posteriors. This is of particular interest to us where many admissible structures can explain the observed data equally well. Next, we  discuss a toy system in which has many modes in section 5, then present our GFlowNet-based solution in section 4.

Previous work has shown GFlowNets are useful in settings with multi-modal posteriors. This is of particular interest to us where many admissible structures can explain the observed data equally well. Next, we present our GFlowNet-based solution in section 4, then discuss a toy system which has many modes in section 5.

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We would like to highlight that, sequence-based(evolution-based) methodsfor single proteins are not suitable for  protein-protein interactions due to the lacking of evolutionary information in most cases — protein-protein interactions involve two or more chains. The chains might belong to different species (e.g. host and virus). Sometimes inter-chain co-evolution simply does not happen (e.g. immune response where effective antibodies are produced to induce rapid clearance of pathogens, leaving no time for the pathogen to evolve). These make it  infeasible to predict mutational effects via mining sequence databases with existing powerful tools such as multiple sequence alignments, protein language models, etc. Hence, currently, effective ways for predicting mutational effects on binding are based on structures rather than sequences alone.

However, it is important to note that sequence-based methods are not suitable for predicting mutational effects on general protein-protein interactions due to the lack of evolutionary information in many cases. Protein-protein interactions typically involve two or more chains, which may belong to different species or may not experience inter-chain co-evolution. As such, it is infeasible to predict mutational effects via mining sequence databases using existing powerful tools such as MSAs or PLMs. Thus, effective ways for predicting mutational effects on protein-protein interaction rely on structure-based approaches rather than sequences alone.

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Previous works on unsupervised learning for CO have studied max-cut Yao et al. and TSP problems Hudson et al. (2021), while these works depend on carefully selected problem-specific objectives. Some works have investigated satisfaction problems Amizadeh et al. ; Toenshoff et al. Applying these approaches to general CO problems requires problem reductions. The works most relevant to us are Karalias & Loukas (2020), Wang et al. and Schuetz et al. Karalias & Loukas (2020) propose an unsupervised learning framework EGN for general CO problems based on the Erd˝os’s probabilistic method, which bonds the quality of the final solutions with probability. Wang et al. (2022) generalize EGN and prove that if the CO objective can be relaxed into an entry-wise concave form, a solution of good quality can be deterministically achieved. This further inspires the design of proxy objectives for the CO problems that may not have closed-form objectives, such as those in circuit design. Schuetz et al. (2022) have recently extended

Previous works on unsupervised learning for CO have studied max-cut (Yao et al., 2019) and TSP problems (Hudson et al., 2021), while these works depend on carefully selected problem-specific objectives. Some works have investigated satisfaction problems (Amizadeh et al., 2018; Toenshoff et al., 2019). Applying these approaches to general CO problems requires problem reductions. The works most relevant to us are (Karalias & Loukas, 2020), (Wang et al., 2022) and (Schuetz et al., 2022). Karalias & Loukas (2020) propose an unsupervised learning framework EGN for general CO problems based on the Erd˝os’ probabilistic method, which bonds the quality of the final solutions with probability. Wang et al. (2022) generalize EGN and prove that if the CO objective can be relaxed into an entry-wise concave form, a solution of good quality can be deterministically achieved. This further inspires the design of proxy objectives for CO problems that may not have closed-form objectives, such as those in circuit design. Schuetz et al. (2022) have recently extended


Deep Learning Approaches. The use of recurrent networks is a natural approach to dealing with the problem of human motion. LSTM and RNN networks are trained to generate output sequentially, each output conditioned on the previous elements in the sequence. These networks have shown much success in Natural Language Processing for generating text Sutskever et al. (2011), hand written characters Graves (2013); Gregor et al. (2015), and even captioning images Vinyals et al.

Deep Learning Approaches. The use of recurrent networks is a natural approach to dealing with the problem of human motion. LSTM and RNN networks are trained to generate output sequentially, each output conditioned on the previous elements in the sequence. These networks have shown much success in Natural Language Processing for generating text (Sutskever et al., 2011), hand written characters (Graves, 2013; Gregor et al., 2015), and even captioning images (Vinyals et al., 2014).


RDE-Linear achieves performance comparable  to Rosetta and outperforms some unsupervised representation learning baselines. Though it does not outperform most of the baselines over the whole SKEMPI2 dataset, we find that its performance is much better when considering only single-point mutations (Table 6 in the appendix). The reason might be  that simple linear models cannot capture well the non-linear relationship dominating multi-point mutations. Anyway, RDE-Linear shows that using simple statistics of the estimated rotamer density alone can predict ∆∆ G , laying the foundation for the more accurate RDE-Network.

RDE-Linear achieves comparable performance to Rosetta and outperforms some unsupervised learning baselines. While it does not surpass most baseline methods over the entire SKEMPI dataset, we observe that its performance is better when considering only single-point mutations (Table 6 in the appendix). This might be attributed to the fact that simple linear models cannot capture well the non-linear relationship dominating multi-point mutations. Nevertheless, RDE-Linear demonstrates that using the basic statistics of the estimated rotamer density alone can predict ∆∆ G , which lays the foundation for the more accurate RDE-Network.

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• The strongest beneﬁt of employing digital circuits (in the form of ASICs or FPGAs) to implement XOR gates is that all XOR gates can be performed simultaneously (unlike GPUs or CPUs where each core needs to simulate only a few XOR gates). Thus, all XOR operations of our proposed decoder are completed within just one clock cycle. •  (with shift registers) would increase the latency. Throughput, however, maintains to be the same regardless of  through pipelining technique which is a basic hardware design principle. • verall, the design complexity (in terms of area overhead and latency) of XOR-gate decoders is extremely low (note that one XOR gate consumes only 6 transistors). • XOR-gate decoders would work as memory decompressors (located in-between memory and computation logic). In the view of computational units that receive outputs of an XOR-gate decoder, then, the amount of memory is simply reduced while regular memory access patterns are not disturbed. • In the case of , the overall design cost would be 2 additional clock cycles for the latency and a few hundred XOR gates (which would be approximately equivalent to a few thousand transistors). Overall, we can provide full memory bandwidth (based on regular memory access patterns through ﬁxed-to-ﬁxed sparsity formats) while the overall hardware design cost is only marginal. • While designing DNN inference accelerators is gaining increasing attention, our work can provide a new research direction.

• The strongest beneﬁt of employing digital circuits (in the form of ASICs or FPGAs) to implement XOR gates is that all XOR gates can be performed simultaneously (unlike GPUs or CPUs where each core needs to simulate only a few XOR gates). Thus, all XOR operations of our proposed decoder are completed within just one clock cycle. • N s (with shift registers) would increase the latency. Throughput, however, maintains to be the same regardless of N s through pipelining technique which is a basic hardware design principle. • Overall, the design complexity (in terms of area overhead and latency) of XOR-gate decoders is extremely low (note that one XOR gate consumes only 6 transistors). • XOR-gate decoders would work as memory decompressors (located in-between memory and computation logic). In the view of computational units that receive outputs of an XOR-gate decoder, then, the amount of memory is simply reduced while regular memory access patterns are not disturbed. • Given N s , an XOR-gate decoder requires N s additional clock cycles for the latency.. • Since M matrix has the size of N out × N in and an element of M is randomly ﬁlled withor 1 , the number of XOR gates is ( N out · N in / 2) . Thus, the total number of transistors to design an XOR-gate decoder is (3 · N out · N in ) . • Overall, we can provide full memory bandwidth (based on regular memory access patterns through ﬁxed-to-ﬁxed sparsity formats) while the overall hardware design cost is only marginal. • While designing DNN inference accelerators is gaining increasing attention, our work can provide a new research direction.


The most closely related works to ours are Bonneel et al. and Werenski et al. The former develops a regression approach in barycentric coordinates with applications in computer graphics as well as color and shape transport problems. Their method  requires solving a computationally costly bilevel optimization problem, which does not need to achieve a global solution. The latter works on a tangential structure like we do, but is based on “Karcher means” (Karcher, 2014, Zemel & Panaretos, 2019). This implies that their method works between absolutely continuous measures with densities that are bounded away from zero, with the target measure lying in the convex hull of the control measures.

The most closely related works to ours are Bonneel et al. (2016), Mérigot et al. (2020), and Werenski et al. The ﬁrst develops a regression approach in barycentric coordinates with applications in computer graphics as well as color and shape transport problems. Their method is deﬁned directly on W 2 and requires solving a computationally costly bilevel optimization problem, which does not necessarily yield global solutions. The second introduces a linearization of the 2 -Wasserstein space by lifting it to a L 2 -space anchored at measure that is absolutely continuous with respect to Lebesgue measure. This approach relies on the existence of optimal transport maps between this absolutely continuous “anchor” distribution and other distributions and hence only deﬁnes tangent spaces at absolutely continuous measures. The third works on a tangential structure based on “Karcher means” (Karcher, 2014, Zemel & Panaretos, 2019), which is more restrictive still. This implies that their method requires all involved measures to be absolutely continuous measures with densities that are bounded away from zero, with the target measure lying in the convex hull of the control measures.


Second, it makes the model not translation-equivariant because a unique positional encoding vector is added to every one patch. The translation equivalence plays an important role in classiﬁcation because we hope the networks’ responses changes accordingly as the object moves in the image. One may note that the ﬁrst issue can be remedied by removing the positional encodings since except for the positional encodings, all other components ( e.g ., MHSA and FFN) of the vision transformer can directly be applied to longer sequences. However, this solution severely deteriorates the performance model has no way to extract the order without the positional encodings. The experiment results on

Second, it makes the model not translation-equivariant because a unique positional encoding vector is added to every one patch. The translation equivalence plays an important role in classiﬁcation because we hope the networks’ responses changes accordingly as the object moves in the image. can directly be applied to longer sequences. However, this solution severely deteriorates the performance. This is understandable because the order of the input sequence is an important clue and the model has no way to extract the order without the positional encodings. The experiment results on


In this paper, we reveal the high sensitivity of PINNs to the choice of activation functions and relate it to various characteristics of the underlying PDE system. To avoid the inefficient manual selection of activation functions when solving different PDEs, we sought to learn specialized activation functions automatically for PINNs. The proposed physics-informed activation function is presented as learnable combinations of a set of candidate functions, whose coefficients can be adapted to the governing PDEs. Intuitively, we show that PIAC makes the neural tangent kernel of PINNs learnable, which partially explains the performance improvement of PIAC. Extensive experiments on a series of challenging benchmarks demonstrate the effectiveness and generalization ability of PIAC.

In this paper, we reveal the high sensitivity of PINNs to the choice of activation functions and relate it to various characteristics of the underlying PDE system. We sought to learn specialized activation functions automatically for PINNs to avoid the inefficient manual selection of activation functions and to alleviate the optimization difficulty of PINNs. The proposed physics-informed activation function is presented as learnable combinations of a set of candidate functions, whose coefficients can be adapted to the governing PDEs. Intuitively, we show that PIAC makes the neural tangent kernel of PINNs learnable, which partially explains the performance improvement of PIAC. Extensive experiments on a series of challenging benchmarks demonstrate the effectiveness and generalization ability of PIAC.


Assuming a bound  N ≥ 2 on the number of days and B > 0 on the buy price we can show that U t is bounded and Lipschitz w.r.t. λ . We can thus run randomized exponentiated gradient  over the product set [ N ] × { δ/ 2 , . . , 1 − δ/ 2 }on the functions U t ( · , x t ) for some δ s.t.
 /δ ∈ Z ≥ 2 to get a bound on the expected regret (proof in A.3).

Assuming a bound of N ≥ 2 on the number of days and B > 0 on the buy price implies that U t is bounded and Lipschitz w.r.t. λ . We can thus run exponentiated gradient on the functions U t to learn a categorical distribution over the product set [ N ] × { δ/ 2 , . . , 1 − δ/ 2 } for some δ s.t.
 /δ ∈ Z ≥ 2 . This yields the following bound on the expected regret (proof in A.3).


In the literature, there is a line of research on estimation and inference of the heterogeneous treatment effects (HTE) (Athey & Imbens, 2016; Taddy et al., 2016; Wager & Athey, 2018; Yu et al., 2020). In particular, Yu et al. (2020) proposed an online test for QTE. We remark that QTE and QTE are related yet fundamentally different hypotheses. There are cases where both the alternative hypothesis in Yu et al. and the null hypothesis in our paper hold. Consequently, applying their test will fail in our setting.

In the literature, there is a line of research on estimation and inference of the heterogeneous treatment effects (HTE) (Athey & Imbens, 2016; Taddy et al., 2016; Wager & Athey, 2018; Yu et al., 2020). In particular, Yu et al. (2020) proposed an online test for HTE. We remark that HTE and QTE are related yet fundamentally different hypotheses. There are cases where HTE exists whereas QTE does not. See Figure 1 for an illustration. Consequently, applying their test will fail in our setting.


Topology aware segmentation Multiple studies have described the importance of topologically correct segmentations in various computer vision applications. Clough et al. (2020) used the concept of Betti numbers as a prior for the segmentation network. A key publication by Hu et al. (2019) introduced the Wasserstein loss as a variation of the Wasserstein distance to improve image segmentation with neural networks. The method computes persistence diagrams for the ground truth and the prediction and matches the points  in the persistence diagrams  of dimension-1 by minimizing the squared distance of matched points. However, this matching algorithm has a fundamental limitation, in that it cannot guarantee that the 0-dim and 1-dim structures matched by the algorithm are spatially related in any sense, see Figure 2. Put succinctly, thematched cyclesand connected components are matched irrespective of the location within the image,  frequently leading to situations where the resulting loss will emphasize the wrong features and penalize the correct features. See Figure 2 for two schematic examples and Figures 1,11 for the result of the Wasserstein matching method on the CREMI dataset. Moreover, the method only considers 1-dimensional structures in their implementation and restricts to superlevel filtrations. Persistent homology and distance-based barcode matching have been usedin different settings, for example, by Abousamra et al. (2021) for crowd localization, by Wang et al. (2022) for gland segmentation, and by Waibel et al. (2022) for reconstructing 3D cell shapes from 2D images. An alternative approach to persistent homology is employed by Hu & Chen (2021), whose method penalizes different locations of critical points using the warping error metric Jain et al. (2010). Otherfrequently used methods in topology-aware segmentation focus on overlap-based segmentation, where the overlapped structures are not the entire volume but certain topologically relevant structures. For example, theskeleta of curvilinear structures play this role in the clDice score Shit et al. (2021), which is calculated as the harmonic mean of the overlap of the prediction skeleton with the label volume and the label skeleton with the prediction volume.

Topology aware segmentation Multiple works have highlighted the importance of topologically correct segmentations in various computer vision applications. Persistent homology is a popular tool from algebraic topology to address this issue. A key publication by Hu et al. (2019) introduced the Wasserstein loss as a variation of the Wasserstein distance to improve image segmentation. They match points of dimension 1 in the persistence diagrams – an alternative to barcodes as descriptor of persistent homolgy – of ground truth and prediction by minimizing the squared distance of matched points. However, this matching has a fundamental limitation, in that it cannot guarantee that the matched structures are spatially related in any sense (see Fig. 1 and App. A). Put succinctly, the cycles are matched irrespective of the location within the image, which frequently has an adverse impact during training (see App. F). Clough et al. (2020) follows a similar approach and train without knowing the explicit ground truth segmentation, but only the Betti numbers it ought to have. Persistent homology has also been used by Abousamra et al. (2021) for crowd localization and by Waibel et al. (2022) for reconstructing 3D cell shapes from 2D images. Other methods incorporate pixel-overlaps of topologically relevant structures. For example, the clDice score, introduced by Shit et al. (2021), computes the harmonic mean of the overlap of the predicted skeleton with the ground truth volume and vice versa. Hu & Chen (2021) and Jain et al.


We test a ResNet-18 [5] on each environment we generate of ∼ 3600 images, and optimize it usingthe Empirical Risk Minimization (ERM) objective [16]. For the sake ofcompleteness, we also testwith group DRO [15], a popular OOD objective that accounts for the environment label e attached toeach data point ( x , y, e ) , and present its results in Appendix A. To test for each DGP construction (Figure 2 a), b) and c)), we  construct four experiments within eachregime. In each experiment, we train on three environments and test on one held-out environment. The distributions are detailed in Table 1.

We conduct tests in which we aim to evaluate the impact of statistical and geometric skews on the performance of a vision classifier. To achieve this, we use a ResNet-18 [4] as our base model architecture and construct four generalization tasks for each skew regime given in Figure 2. Further, we compare the performance of ERM against gDRO: while ERM shuffles all the data in its training domain, gDRO assigns an environment label e to each environment and optimizes the worst-case loss. Results are shown in Table 1, and experiments were performed using NVIDIA 3090 GPUs.


Figure 2 shows the results. We can observe that in both experiments N. + emp + opt successfullyexploits the fast spectral decay of Gaussian kernel and signiﬁcantly outperforms other methods. Also,even without using the knowledge of any expectations, N. + emp (and Thinning ) show a decentconvergence rate comparable to Herding or iid Bayes , which actually use the additional information.

Figure 2 shows the results. We can observe that in both experiments N. + emp + opt successfullyexploits the fast spectral decay of Gaussian kernel and significantly outperforms other methods. Also,even without using the knowledge of any expectations, N. + emp shows a decent convergence ratecomparable to Herding or iid Bayes .

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Simulation-based Methods. Simulation-based techniques are able to produce physically plausible animations Levine & Popovi´c (2012); Clegg et al. ; Hämäläinen et al. ; Ha et al. (2012), including realistic balancing, motion on terrain of various heights, and recovery from falling. These techniques tend to consider physical constraints on the human skeleton while optimizing a motion objective. For example, in the work of Levine et al. Levine & Popovi´c (2012), one task they employ is moving the skeleton in a certain direction without falling over. Similarly, in Ha et al. Ha et al. (2012), given initial fall conditions, they seek to minimize joint stress due to landing impact while ensuring a desired landing pose. Though the output is plausible and realistic, they require speciﬁc objectives and constraints for each individual task. It is not feasible to use such explicit objectives or constraints for stylized motion. The correct arch of the back and ﬂourishes of the limbs for a dance cannot be described a priori and must instead be inferred from the data.

Simulation-based Methods. Simulation-based techniques are able to produce physically plausible animations (Levine & Popovi´c, 2012; Clegg et al., 2015; Hämäläinen et al., 2015; Ha et al., 2012), including realistic balancing, motion on terrain of various heights, and recovery from falling. These techniques tend to consider physical constraints on the human skeleton while optimizing a motion objective. For example, in the work of Levine et al. (Levine & Popovi´c, 2012), one task they employ is moving the skeleton in a certain direction without falling over. Similarly, in Ha et al. (Ha et al., 2012), given initial fall conditions, they seek to minimize joint stress due to landing impact while ensuring a desired landing pose. Though the output is plausible and realistic, they require speciﬁc objectives and constraints for each individual task. It is not feasible to use such explicit objectives or constraints for stylized motion. The correct arch of the back and ﬂourishes of the limbs for a dance cannot be described a priori and must instead be inferred from the data.


In this area, leading approaches include NSGA-III (Deb & Jain, 2013) and MOEA/D (Zhang & Li, 2007). However, these gradient free methods scale poorly with the number of parameters and are not suitable for training large scale neural networks. Sener & Koltun (2018) proposed a gradient based MOO algorithm for MTL, based on MDGA (Désidéri, 2012), suitable for training large scale neural networks. Other recent works include (Lin et al., 2019; Mahapatra & Rajan, 2020) detailed in Section 2. Another recent approach that tries to get a more complete view of the Pareto front is Ma et al. This method extents a given Pareto optimal solution in its local neighborhood. In a concurrent work, Lin et al. propose extending Lin et al. for approximating the entire Pareto front, by utilizing hypernetworks. The proposed method is conceptually similar to our approach. Similarly, Dosovitskiy & Djolonga (2019) proposed learning a single model conditioning on the objective weight vector. The method uses feature-wise linear modulation Perez et al. and dynamically weighted LS loss criterion.

In this area, leading approaches include NSGA-III (Deb & Jain, 2013) and MOEA/D (Zhang & Li, However, these gradient-free methods scale poorly with the number of parameters and are not suitable for training large-scale neural networks. Sener & Koltun (2018) proposed a gradient-based MOO algorithm for MTL, based on MDGA (Désidéri, 2012), suitable for training large-scale neural networks. Other recent works include (Lin et al., 2019; Mahapatra & Rajan, 2020) detailed in Section 2. Another recent approach that aims at a more complete view of the Pareto front is Ma et al. They extend a given Pareto optimal solution in its local neighborhood. In a concurrent work, Lin et al. (2020) extends Lin et al. for approximating the entire Pareto front. They train a single hypernetwork by constantly changing the reference directions in the PMTL objective. The proposed method is conceptually similar to our approach, but since it builds on PMTL, it may not produce an exact mapping between the preference and the corresponding solution. Similarly, Dosovitskiy & Djolonga (2019) proposed learning a single model conditioning on the objective weight vector. The method uses feature-wise linear modulation Perez et al. and dynamically weighted LS loss criterion.


For the first category of experiments, we averaged our results over 400 random trials. An array of 5 different ResNet architectures were trained in this category: ResNet10, 18, 34, 50, and 101, following a simple pipeline in the Pytorch library example 1 . It is worth noting that we deliberately did not engineer strong features for this category of experiments. The mini-ImageNet dataset was used for these experiments, and 16-way classification was performed on both the validation and novel classes. For this purpose, 16 out of 20 novel classes were chosen at random once and fixed for all the evaluations. We split the validation and novel classes into 90% training and 10% held-out for accuracy evaluation. The imbalanced data settings were chosen to either have 7.5 or 15 average number of shots per class. For the second category of experiments, we used the WideResNet-28pre-trained feature backbones from Mangla et al. ; Yang et al These experiments were conducted on three benchmarks, and each hyper-parameter configuration was averaged over 10, random trials. Additional details and hyper-parameters are covered in the subsequent sections.

For the first category of experiments, we averaged our results over 400 random trials. An array of 5 different ResNet architectures were trained in this category: ResNet10, 18, 34, 50, and 101, following a simple pipeline in the Pytorch library example 1 . It is worth noting that we deliberately did not engineer strong features for this category of experiments. The mini-ImageNet dataset was used for these experiments, and 16-way classification was performed on both the validation and novel classes. For this purpose, 16 out of 20 novel classes were chosen at random once and fixed for all the evaluations. We split the validation and novel classes into 90% training and 10% held-out for accuracy evaluation. The imbalanced data settings were chosen to either have 7.5 or 15 average number of shots per class. For the second category of experiments, we used the WideResNet-28pre-trained feature backbones from Mangla et al. ; Yang et al. These experiments were conducted on three benchmarks, and each hyper-parameter configuration was averaged over 10, random trials. The backbone parameters and extracted features for all datasets are publicly available in the Illinois Data Bank repository (Saleh et al., 2022). Additional details and hyper-parameters are covered in the subsequent sections.


Snuba [14] generates candidate rules using a collection of weak learner primitives (e.g. decision stumps, k-nearest neighbors) and then synthesizes and prunes this set of rules to generate ﬁnal rules for labeling. ReGAL combines elements  this framework to allow iterative communication betweendownstream classiﬁers and rule selectors to allow each to mutually enhance the other. This allows ReGAL to uniquely offer the ease of model-generated LFs while still gaining the nuanced insight ofhuman annotator input.

Snuba [14] generates candidate rules using a collection of weak learner primitives (e.g. decision stumps, k-nearest neighbors) and then synthesizes and prunes this set of rules to generate ﬁnal rules for labeling. ReGAL combines elements of this framework to allow iterative communication between downstream classiﬁers and rule selectors to allow each to mutually enhance the other. ReGAL differs from these by soliciting user feedback on rules, thus enabling it complement model-generated LFs with annotator guidance.

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Recommender system  is a natural application of the problem of varying action space where items to recommend change often. Speciﬁcally, action interdependence is  apparent in the case of listwise actions. For example, a list of diverse ads is more likely to get a user click than ads from the same domain (Zhou et al., 2010). In this work, we experiment with the listwise metric of Complementary Product Recommendation (CPR) (Hao et al., 2020).

Recommender system (RecSys) is a natural application of varying action space RL — for instance, news articles or videos to recommend are updated daily. Action interdependence is distinctly apparent in the case of listwise actions. A recommended list of diverse videos is more likely to get a user click than videos about the same thing (Zhou et al., 2010). In this work, we experiment with the listwise metric of Complementary Product Recommendation (CPR) (Hao et al., 2020).


Now, the question is: given a full model f , is it possible to ﬁnd a more efﬁcient reparameterization f ψ that, trained de novo , can generalize comparably well? The success of various model compression techniques suggests that such reparameterizations might exist for most well-known models, and a recent study (Frankle & Carbin, 2018) made successful post hoc identiﬁcations of sparse reparameterized small networks with precisely such properties. Nevertheless, all attempts of training small networks de novo have yielded results signiﬁcantly underperforming those compressed from the original models (Zhu & Gupta, 2017). This has led to a commonly held belief that gross overparameterization is necessary for effective training, and it is further hypothesized that this necessity is due to the haphazard occurrences of good reparameterizations and their sensitivity to initialization (Frankle & Carbin, 2018). To date, a method capable of training a  sparse network de novo to match the performance of a dense model compressed to the same size remains out of reach. In this study we present a dynamic sparse reparameterization technique that successfully overcame this challenge.

Now, the question is: given a full model f , is it possible to ﬁnd a more efﬁcient reparameterization f ψ that, trained de novo , can generalize comparably well? The success of various model compression techniques suggests that such reparameterizations might exist for most well-known models, and a recent study (Frankle & Carbin, 2018) made successful post hoc identiﬁcations of sparse reparameterized small networks with precisely such properties. Nevertheless, attempts at training small networks de novo typically yield results signiﬁcantly underperforming networks obtained by compressing larger models (Zhu & Gupta, 2017). This has led to a commonly held belief that gross overparameterization is necessary for effective training. Here we argue that this is not true by presenting a dynamic sparse reparameterization technique able to train sparse models de novo without the need to compress a large model, a desirable feature for training on memory- and power-constrained devices.

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During the translation/transformation, some domain-speciﬁc attributes are changed, such as the colors, texture, and semantics of certain image regions (see e.g. the examples in Fig 1). Although there is no supervised information for these changes, certain consistency during the transformation is desirable. Inspired by graph-based semi-supervised learning (Zhu et al., 2003; Zhu, 2006), we introduce smoothness terms to unpaired image-to-image translation (Zhu et al., 2017a) by providing a stronger regularization for the translation/transformation between the source and target domains, aiming to exploit the “manifold structure” of the source and target domains. For a pair of similar samples (two different locations in an image; one can think of them as two patches although the receptive ﬁelds of CNN are quite large), we add a smoothness term to minimize a weighted distance of the corresponding locations in the target image. Note that two spatially distant samples might be neighbors in the feature space. We name our algorithm HarmonicGAN as it behaves harmonically along with the circularity and adversarial constraints to learn a pair of dual translations between the source and the target domains. Metrics deﬁned on two alternative features are adopted: (1) a low-level soft RGB histograms; and (2) CNN (VGG) features with pre-trained semantics.

During the translation/transformation, some domain-speciﬁc attributes are changed, such as the colors, texture, and semantics of certain image regions. Although there is no supervised information for these changes, certain consistency during the transformation is desirable, meaning that for image contents similar in the source space should also be similar in the target space. Inspired by graph-based semi-supervised learning (Zhu et al., 2003; Zhu, 2006), we introduce smoothness terms to unpaired image-to-image translation (Zhu et al., 2017a) by providing a stronger regularization for the translation/transformation between the source and target domains, aiming to exploit the “manifold structure” of the source and target domains. For a pair of similar samples (two different locations in an image; one can think of them as two patches although the receptive ﬁelds of CNN are quite large), we add the smoothness term to minimize a weighted distance of the corresponding locations in the target image. Note that two spatially distant samples might be neighbors in the feature space. We name our algorithm HarmonicGAN as it behaves harmonically along with the circularity and adversarial constraints to learn a pair of dual translations between the source and target domains, as shown in Fig. Distance metrics deﬁned on two alternative features are adopted: (1) a low-level soft RGB histograms; and (2) CNN (VGG) features with pre-trained semantics.


The experimental results are summarized in Table 5, where we take CIFAR- 10 as ID data and CIFAR- 100 as OOD data. Here, the common watermarklearning setup (common) can already lead to improved performance in near-OOD detection compared to the cases without watermarking (w/o watermark). Moreover, watermarking with shiftingaugmentations (perm and rotate) can further boost the detection power of the models, leading to atmost 8 . 60 and 4 . 70 improvements in FPR95 for the softmax and the free energy scoring.

The near OOD experiments are summarized in Table 5, where we take CIFAR- 10 as ID data and CIFAR- 100 as OOD data. Here, the common learning setup (common) already leads to improved performance compared to the cases without watermarking (w/o watermark). Moreover, watermarking with shifting augmentations (permute and rotate) can further boost the detection power of the models, leading to at most 8 . 60 and 4 . 70 improvements in FPR95 for the softmax and the free energy scoring.

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SuperWeights, which are linear combination of templates which get reused by multiple layers . These SuperWeights capture a single operation on the input features ( e.g ., edge or texture detectors), and are themselves generated via a weighted combination of one or more templates of trainable parameters held by Weight Templates. Thus, to generate the weights for a single layer, we must first construct SuperWeights from the trainable parameters held by Weight Templates (discussed in Section 3.1), and then concatenate together all SuperWeights used by the layer to create its final weights (process illustrated in center-right column of Figure 2).

SuperWeights, which are linear combination of templates and get reused by multiple layers and capture a single operation on the input features ( e.g ., edge or texture detectors). Then, to generate the weights for a single layer, we must first construct SuperWeights from the trainable parameters held by Weight Templates (discussed in Section 3.1), and then concatenate together all SuperWeights used by the layer to create its final weights (process illustrated in center-right column of Figure 2).

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Results for K = 20 are depicted in Figure 2b. All approaches update the Q-function with a learning rate of 10 − 3 on the same ﬁxed batch of 10 3 episodes with a percentage of 10% non-optimal transitions. For the multi-step approaches, we set rollout length n = 4 . Since there is no generalization among states in the tabular setting, we update the Shifted Q-function with a learning rate of 10 −and the Truncated Q-functions with a learning rate of 10 − 3 . Please note, however, that the Shifted Q-function bootstraps from the full Q-estimate.

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The datasets for training the rotamer density estimator are derived from PDB-REDO (Joosten et al., 2014),  a database containing refined X-ray structures in PDB. Structures with resolution worse than 3.5 ˚A are removed. All the protein chains are clustered by 50% sequence identity. This process leads to 38,413 chain clusters, and they are randomly divided into the training set, the validation set, and the test set by 95%/0.5%/4.5%. At training time, the data loader randomly selects a cluster and then randomly chooses a chain from the cluster. During training, the structure is cropped into a patch containing 128 amino acids. The patch is constructed by first choosing a seed amino acid, and then finding its 127 nearest neighbors according to C-beta distances. To emulate mutations,  the rotamers of 10% of amino acids in the patch are masked. Noise is added  to the rotamers of amino acids whose C-beta distance to the closest masked amino acid is less than 8 ˚A.

The dataset for training the rotamer density estimator is derived from PDB-REDO (Joosten et al., 2014), which is a database containing refined X-ray structures in PDB. The protein chains are clustered based on 50% sequence identity, leading to 38,413 chain clusters, which are randomly divided into the training, validation, and test sets by 95%/0.5%/4.5% respectively. During training, the data loader randomly selects a cluster and then randomly chooses a chain from the cluster to ensure balanced sampling. We crop structures into patches containing 128 residues by first choosing a seed residue, and then selecting its 127 nearest neighbors based on C-beta distances. To simulate mutations, we masked the rotamers of 10% of residues in the patch, and we added noise to the rotamers of residues whose C-beta distance to the closest masked residue was less than 8 ˚A.

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NVIDIA Dynamic Hand Gesture Dataset, or NVGesture in short (Molchanov et al., 2015), consists of 1,532 video clips (1,050 training and 482 test ones) of hand gestures in 25 classes. We sample 20% training samples as the validation set and use depth and RGB as  two modalities. We adopt the conﬁguration by Joze et al. (2020) for data preparation and use  I3D Carreira & Zisserman (2017) as uni-modal branches and MMTMs as fusion modules in the six ﬁnal inception modules.

NVIDIA Dynamic Hand Gesture Dataset (or NVGesture (Molchanov et al., 2015)), consists of 1, video clips (1,050 training and 482 test ones) of hand gestures in 25 classes. We sample 20% training examples as the validation set and use depth and RGB as the two modalities. We adopt the data preparation steps used in Joze et al. and use the I3D architecture (Carreira & Zisserman, 2017) as uni-modal branches and MMTMs as fusion modules in the six ﬁnal inception modules.

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An on-calendar prescription visualization should support tasks that relate to reading calendar entries (e.g., reading calendar events and accessing events’ information) and to managing calendar entries (e.g., adding, editing and deleting calendar entries). Because such a system requires adding visual information about  prescriptions on top of an existing calendar, the first step in the design process is to understand how to support tasks related to reading calendar entries - that include information about prescriptions and their potential conflicts. Below we introduce basic requirements for reading tasks related to reading  calendar entries. A design that integrates prescription visualization to a calendar must be presented on a calendar. Therefore, such a design should satisfy the following basic usability requirement:

An on-calendar prescription visualization should support tasks that relate to reading calendar entries (e.g., reading calendar events and accessing event information) and to managing calendar entries (e.g., adding, editing and deleting calendar entries). Representing medication prescriptions in general-purpose calen- dars requires adding visual information about these prescriptions on top of an existing planned schedule. This means that such a calendar must support tasks related to reading calendar entries (basic function of a calendar) as well as reading information about prescriptions and their potential conflicts. Tasks related to reading standard calendar entries have been dis- cussed in previous work [63–65]. These include tasks related to retrieving temporal features and reading event-related information such as date, time, location and purpose. From this, we derive the following basic usability requirement:


The usual approach to representing a  set of vectors in memory is to store it as a matrix, for which we need to choose some arbitrary order for the elements. To  avoid being affected by this arbitrary order, recent set prediction models make use of a property called permutation-equivariance (Zaheer et al., 2017): changing the order of the input elements should change the order of the output elements in the same way.

The usual approach to representing a multiset or set of vectors in memory is to store it as a list in which the elements are placed in an arbitrary order. To still treat the list as if it were a multiset or set, recent set prediction models avoid being affected by this arbitrary order by using list-to-list functions that are enforced to be permutation-equivariant (Zaheer et al., 2017): changing the order of the input elements should change the order of the output elements in the same way.


As demonstrated in §3.2, the imbalance in conditional utilization rates is a sign of the model exploiting the connection between the target and only one of the input modalities, ignoring crossmodal information. Conditional utilization rate was however can be measured once training was done, making it difﬁcult to use it in real-time during training. We instead derive a proxy metric, called conditional learning speed , that captures relative learning speed between modalities during training.

As demonstrated in §3.2, the imbalance in conditional utilization rates is a sign of the model exploiting the connection between the target and only one of the input modalities, ignoring crossmodal information. However, conditional utilization rates are measured after training is done, making it expensive to use them in real-time during training. We instead derive a proxy metric, called conditional learning speed , that captures relative learning speed between modalities during training.

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Theorem 1 indicates  we can not identify the disentangled model because we can not tell whether the latent explanatory factors are entangled by h or not from the marginal distribution. Compared to the parameter-space identiﬁability in nonlinear ICA (Khemakhem et al., 2020), identifying a disentangled model is more challenging because even if we can identify the right distribution, the function h may still entangle the latent explanatory factors. In fact, recent generative models in disentangled representationsalways learn the ground-truth distribution, which is an isotropic Gaussian, via regularization (Kumar et al., 2017; Locatello et al., 2020). Therefore, we formulate our identiﬁability in the function space:

Theorem 1 indicates that we can not identify the disentangled model because we can not tell whether the latent explanatory factors are entangled by h or not from the marginal distribution. Compared to the parameter-space identiﬁability in nonlinear ICA (Khemakhem et al., 2020), identifying a disentangled model is more challenging because even if we can identify the right distribution, the function h may still entangle the latent explanatory factors. Another difference is that the groundtruth distribution of the latent variables in disentangled representations is assumed to be isotropic Gaussian, which diminishes the importance of the parameter-space identiﬁability. Therefore, we formulate our identiﬁability in the function space:


• Support Vector Machines (SVMs): on top of texts represented via sparse, TF-IDF bag-ofwords (BoW) vectors. SVM is implemented using a linear kernel. • Convolutional Neural Networks (CNNs): the text is represented by a (cid:96) × d matrix, where (cid:96) is the sequence length (120) and d is the dimension of the Glove embeddings [Pennington et al., 2014]. For the convolution, we applied ﬁlter sizes of 3, 4, and 5, with 128 ﬁlters persize.• Bidirectional Long Short-Term Memory (BiLSTM): words are also represented as Glove embeddings, and the maximum sentence length is set such that it covers 90% of the documents [Lowell et al., 2019].

• Support Vector Machines (SVMs): on top of texts represented via sparse, TF-IDF bag-ofwords (BoW) vectors. SVM is implemented using a linear kernel. • Convolutional Neural Networks (CNNs): the text is represented by a (cid:96) × d matrix, where (cid:96) is the sequence length (120) and d is the dimension of the Glove embeddings [Pennington et al., 2014]. For the convolution, we applied ﬁlter sizes of 3, 4, and 5, with 128 ﬁlters per size. • Bidirectional Long Short-Term Memory (BiLSTM): words are also represented as Glove embeddings, and the maximum sentence length is set such that it covers 90% of the documents [Lowell et al., 2019].


Estimating individual treatment effect (ITE) with randomized controlled trials (RCTs) is a common practice in causal inference, which has been widely used in e-commerce (Betlei et al., 2021), education (Cordero et al., 2018) and health care (Schwab et al., 2020). For example,application managers would impose a marketing strategy on the randomly sampled users to see its potential benefit on the click-through rate; drug developers would conduct clinical A/B tests to evaluate the drug effects. Although RCT is the golden standard  for causal inference (Judea & Dana, 2018), it is always too expensive to conduct randomized experiments. Hence, observational data that can be acquired without intervention has been a tempting shortcut. For example, drug developers tend to assess drug effects with post-marketing monitoring reports instead of the clinical A/B trials. With the growing access to observational data, estimating ITE from observational data has attracted intense research interest.

Estimating individual treatment effect (ITE) with randomized controlled trials is a common practice in causal inference, which has been widely used in e-commerce (Betlei et al., 2021), education (Cordero et al., 2018), and health care (Schwab et al., 2020). For example, drug developers would conduct clinical A/B tests to evaluate the drug effects. Although randomized controlled trials are the gold standard (Pearl & Mackenzie, 2018) for causal inference, it is often prohibitively expensive to conduct such experiments. Hence, observational data that can be acquired without intervention has been a tempting shortcut. For example, drug developers tend to assess drug effects with post-marketing monitoring reports instead of clinical A/B trials. With the growing access to observational data, estimating ITE from observational data has attracted intense research interest.

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The trend we observe from Lemma 3.2 is that activation functions whose Hermite coefﬁcients decay quickly, such as ω σ , result in a faster decay of the NTK coefﬁcients. We remark that analyzing the rates of decay for l ≥ 3 is challenging due to the calculation of F ( p, k, ¯ α l − 1 ) (4). In Appendix B. we provide some preliminary results in this direction, illustrating how, in a  speciﬁc setting, depth can lead to slower coefﬁcient decay. Additionally, in Appendix B.4 we show that the zeroth coefﬁcient strictly increases with depth. Finally, we brieﬂy pause here to highlight the potential for using a truncation of (5) in order to perform efﬁcient numerical approximation of the inﬁnite width NTK.

The trend we observe from Lemma 3.2 is that activation functions whose Hermite coefﬁcients decay quickly, such as ω σ , result in a faster decay of the NTK coefﬁcients. We remark that analyzing the rates of decay for l ≥ 3 is challenging due to the calculation of F ( p, k, ¯ α l − 1 ) (4). In Appendix B. we provide preliminary results in this direction, upper bounding, in a very speciﬁc setting, the decay of the NTK coefﬁcients for depths l ≥ 2 . Finally, we brieﬂy pause here to highlight the potential for using a truncation of (5) in order to perform efﬁcient numerical approximation of the inﬁnite width NTK.

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According to Ethier et al. [58], a prescription is made of the follow- ing six building blocks which define the modalities of administration for a given drug: 1) drug prescription item (specifying actions re- lated to one or several drugs), 2) drug administration specification (specifying the drug product), 3) drug course specification (specify- ing duration, initiation, and termination), 4) drug dosage specification (specifying the dosage of a drug),5) drug dose administration specification (administration instructions), and6) drug dispensing specification (specifying the dispensing of a drug product). Kumar et al. [59] summarizes these building blocks as superscription (directive to take), inscription (name and dose), subscription (directions to the pharmacists), and signature (Instructions for Patient), and Fox [60] as drug name, drug dose, drug dose units, drug dose frequency, du- ration (comprising start and end date), and indication. Like others (e.g. [14,31]), we adopt the latter classification by Fox [60] because of the simplicity and directness of the naming convention.

According to Ethier et al. [58], a prescription is made of the differ- ent parts which define the modalities of administration for a given drug. The underlying building block is the drug prescription item. The drug prescription item comprises the drug administration specification, healthcare objective specification, and drug distribution specification. The drug administration specification is the part that contains information that pertains to drug administration. This in- formation includes the drug product specification (which indicates the name of the drug), drug dosage specification (which indicates the dosage, administration route, and dosing conditions), and drug course specification (which indicates the the starting condition and the duration of the prescription). Kumar et al. [59] summarizes these building blocks as superscription (directive to take), inscrip- tion (name and dose), subscription (directions to the pharmacists), and signature (instructions for Patient), and Fox [60] as drug name, drug dose, drug dose units, drug dose frequency, duration (comprising start and end date), and indication. Like others (e.g. [14,30]), we adopt the latter classification by Fox [60] because of the simplicity and directness of the naming convention.

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