Another limitation is that since the study was online, we did not have the privilege of observing participant’s full activity cycles. It is likely that remote sessions also lead to participants employing less think-aloud than when participating in person. We also constrained participation to people between the age of 35 and 65 who were either on multiple prescription medications or played the role of caregivers to others on multiple medications. While this allowed us to capture insights for that specific population, these insights do not necessarily generalize to other populations. Our calendar designs were also suited for relatively large screens such as laptops and tablets and were not evaluated on mobile devices. Finally, the study was only limited to tasks that relate to reading calendar entries. In the future, tasks such as adding and modifying medication entries should be included.

Another limitation is that since the study was online, we did not have the privilege of observing participant’s full activity cycles. It is likely that remote sessions also lead to participants employing less think-aloud than when participating in person. We also constrained participation to people between the age of 35 and 65 who were either on multiple prescription medications or played the role of caregivers to others on multiple medications. While this allowed us to capture insights for that specific population, these insights do not necessarily generalize to other populations. Our calendar designs were suited for relatively large screens such as laptops and tablets and were not evaluated on mobile devices. Given the focus of our study on medication entries, we opted for assigning the same color to all non-medication calendar entries. However, events in real-life calendars are often of several colors. The added colors likely increase visual complexity and visual clutter that must be considered in future studies. Finally, the study was only limited to tasks that relate to reading calendar entries. In the future, tasks such as adding and modifying medication entries should be included.


Other notable modiﬁcations include: (a) gated residual instead of a sum residual in the ResNet block (Figure 4), for improved training stability; (b) more ﬂexible observation model based on a discretized mixture of logistics instead of a discretized logistic distribution, for improved performance and (c) mixed-precision (Micikevicius et al., 2018), which reduced memory requirements thus allowing training with larger batches. Without SDNs, we refer to this architecture as IAF-VAE+.

Other notable include Figure 4) training stability b more ﬂexible based on a mixture of logistics instead of a discretized logistic distribution, for improved performance; and (c) mixed-precision (Micikevicius et al., 2018), which reduced memory requirements thus allowing training with larger batches. Without SDNs, we refer to this architecture as IAF-VAE+.


Table 1 reports the results. We have the following observations. First, the ASR of E VAS is significantly higher than ResNet18 and the other two random arches. For instance, on CIFAR10, E VAS is 21.8%, 28.3%, and 34.5% more effective than ResNet18 and random arches, respectively. Second, E VAS has the highest ASR across all the datasets. Recall that we use the same arch throughout different datasets. This indicates that the attack vulnerability probably resides at the arch level and is agnostic to concrete datasets. Third, all the arches show higher ASR on simpler datasets such as CIFAR10. This may be explained by that more complex datasets ( e.g. , more classes, higher resolution) imply more intricate manifold structures, which may interfere with arch-level backdoors.

Table 1 reports the results. We have the following observations. First, the ASR of E VAS is significantly higher than ResNet18 and the other two random arches. For instance, on CIFAR10, E VAS is 21.8%, 28.3%, and 34.5% more effective than ResNet18 and random arches, respectively. Second, E VAS has the highest ASR across all the datasets. Recall that we use the same arch throughout different datasets. This indicates that the attack vulnerability probably resides at the arch level and is insensitive to concrete datasets, which corroborates with prior work on NAS: one performant arch found on one dataset often transfers across different datasets (Liu et al., 2019). This may be explained as follows. An arch α essentially defines a function family F α , while a trained model f α ( · ; θ ) is an instance in F α , thereby carrying the characteristics of F α ( e.g. , effective to extract important features or exploitable by a trigger generator). Third, all the arches show higher ASR on simpler datasets such as CIFAR10. This may be explained by that more complex datasets ( e.g. , more classes, higher resolution) imply more intricate manifold structures, which may interfere with arch-level backdoors.


These fusion modules enable information ﬂow from one branch to another. This network architecture employs intermediate fusion, according the categorization of multi-modal fusion strategy in the deep learning literature (Ngiam et al., 2011; Atrey et al., 2010; Baltrušaitis et al., 2018). It has demonstrated competitive performance against multi-modal DNNs with late fusion in many tasks (Perez et al., 2018; Joze et al., 2020; Anderson et al., 2018; Wang et al., 2020b).

These fusion modules enable information ﬂow from one branch to another. According to categorization of multi-modal fusion strategies in the deep learning literature, this is intermediate fusion (Ngiam et al., 2011; Atrey et al., 2010; Baltrušaitis et al., 2018). It has demonstrated competitive performance against multi-modal DNNs with late fusion in many tasks (Perez et al., 2018; Joze et al., 2020; Anderson et al., 2018; Wang et al., 2020b).

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In this work, we introduced SuperWeight Ensembles, a method for parameter sharing in heterogeneous ensembles. SuperWeight Ensembles outperform existing works on the anytime prediction task by leveraging gradient information for effective parameter sharing. We find our automatic sharing improves single member performance by 2% compared to the baselines. SuperWeight Ensembles also match performance of efficient ensembles in the low-parameter regime, while adding flexibility to adjust parameters which prior work doesn’t have. When we add parameters, we outperform even standard ensembles on ImageNet with just 50% of the parameters. We believe that SuperWeight Ensembles represent a promising step forward in parameter-efficiency. Future work will include more deeply exploring architecture diversity; Gontijo-Lopes et al. (2021) show that model architecture heterogeneity can be a key contributor to ensemble diversity on challenging tasks.

We introduce SuperWeight Ensembles, a method for parameter sharing in heterogeneous ensembles. SuperWeight Ensembles outperform existing anytime prediction work by leveraging gradient information for parameter sharing. Our automatic sharing improves single member performance by 2% compared to the baselines. SuperWeight Ensembles also match performance of efficient ensembles in the low-parameter regime, compared to prior work. When we add parameters, we outperform even deep ensembles on ImageNet with 50% of the parameters. We believe that SuperWeight Ensembles are a promising step forward in parameter-efficiency. Future work will include more deeply exploring architecture diversity; Gontijo-Lopes et al. (2021) show that model architecture heterogeneity can be a key contributor to ensemble diversity on challenging tasks.

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Datasets. Missing counterfactual makes it infeasible to evaluate the ground-truth PEHE over observational benchmarks. Following Liuyi et al. ; Uri et al. (2017), experiments are conducted on two semi-synthetic benchmarks. Specifically, the IHDP benchmark aims to estimate the effect of specialist home visits on infant’s future cognitive scores, with 747 observations and 25 covariates; the ACIC dataset comes from the collaborative perinatal project (Niswander & Gordon, 1972), with 4802 observations and 58 covariates.

Datasets. Missing counterfactuals impedes the evaluation of PEHE with observational benchmarks. Following Liuyi et al. ; Shalit et al. (2017), experiments are conducted on two semi-synthetic benchmarks. Specifically, the IHDP benchmark aims to estimate the effect of specialist home visits on infants’ potential cognitive scores, with 747 observations and 25 covariates; the ACIC dataset comes from the collaborative perinatal project, with 4802 observations and 58 covariates.

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With a generative classifier, our GMMSeg handles anomaly segmentation naturally, without neither external datasets of outliers, nor additional image resynthesis models. It also greatly differs from most uncertainty estimation-based methods that are post-processing techniques adjusting the prediction scoresofsoftmax-basedsegmentationnetworks[17,18,41–43].Themostrelevantonesaremaybeafew densityestimation-basedmodels[56,108,109],whichdirectlymeasurethelikelihoodofsamplesw.r.t. the data distribution. However, they are either limited to pre-trained representation [56] or specialized for anomaly detection with simple data [108,109]. To our best knowledge, this is the first time to report promising results on both closed-set and open-world large-scale settings, through a single model instance without any change of network architecture as well as training and inference protocols.

With a generative classifier, our GMMSeg handles anomaly segmentation naturally, without neither external datasets of outliers, nor additional image resynthesis models. It also greatly differs from most uncertainty estimation-based methods that are post-processing techniques adjusting the prediction scoresofsoftmax-basedsegmentationnetworks[17,18,41–43].Themostrelevantonesaremaybeafew densityestimation-basedmodels[56,109,110],whichdirectlymeasurethelikelihoodofsamplesw.r.t. the data distribution. However, they are either limited to pre-trained representation [56] or specialized for anomaly detection with simple data [109,110]. To our best knowledge, this is the first time to report promising results on both closed-set and open-world large-scale settings, through a single model instance without any change of network architecture as well as training and inference protocols.


Rotamers with d torsional angles can be viewed as points on the ddimensional torus. The D -dimensional torus  is the product of D circles S 1 , i.e. T D = S 1 ×· · ·× S 1 . Therefore, we can model the distribution onT D by modeling the joint distribution of D variables on S 1 . In specific, we adopt the coupling layer technique to model the joint distribution (Dinh et al., 2016). On each coupling layer, we update one dimension using the bijective for S 1 , keeping the other D − 1 dimensions fixed and using them along with the hidden representation of the amino acid as the condition to parameterize the bijective (Figure 2B):

Rotamers with D torsional angles can be viewed as points on the Ddimensional torus which is the product of D circles S 1 , i.e. T D = S 1 × · · · × S 1 . To model the distribution on T D , we adopt the coupling layer technique to model the joint distribution (Dinh et al., 2016). Each coupling layer updates one dimension using the bijective for S 1 , keeping the other D − 1 dimensions fixed, and uses the D − 1 dimensions along with the hidden representation of the residue as the conditioner to parameterize the bijective (Figure 2B):

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Zero-shot recognition of novel concepts. To master a novel hierarchical concept and directly use it for classiﬁcation and detection only given its relational graph structure, we need a way to compose the In this paper, we use i to index different concepts, j to index relations, and n to index example images. previous concept and relation energy based models. Here we introduce the hierarchical composition rule, using an English letter “F” as an illustrative example 5 .

Zero-shot recognition of novel concepts. To master a novel hierarchical concept and directly use it for classiﬁcation and detection only given its relational graph structure, we need a way to compose the previous concept and relation energy based models. Here we introduce the hierarchical composition rule, using an English letter “F” as an illustrative example 6 .

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Human Evaluation of KG Explanations We conduct a user study to measure  KG-augmented models’ ability to give plausible explanations, using the original KG or RLRR perturbed KG. For both KGs, we sample 30 questions from the CSQA and OBQA test sets which were correctly answered by MHGRN. For each question, we retrieve the top-scoring path for each answer choice via MHGRN’s path decoder attention. We then ask three human subjects to rate each path for readability and usability, with ratings aggregated via majority voting. Readability (Read) is whether the path makes sense, usability (Use) is whether the path is relevant to the given question-answer pair, and both are measured on a [0 , 1] scale. We obtain a Fleiss’ κ of 0 . 1891 , indicating slight agreement between raters. To illustrate, we provide examples of explanation paths and their consensus ratings.

Human Evaluation of KG Explanations We conduct a user study to measure the plausibility of KG-augmented models’ path-based explanations. For both the original KG and RL-RR perturbed KG, we sample 30 questions from the CSQA and OBQA test sets which were correctly answered by MHGRN. For each question, we retrieve the top-scoring path for each answer choice via MHGRN’s path decoder attention. We then ask three human subjects to rate each path for readability and usability, with ratings aggregated via majority voting. Readability (Read) is whether the path makes sense, usability (Use) is whether the path is relevant to the given question-answer pair, and both are measured on a [0 , 1] scale. We obtain a Fleiss’ κ of 0 . 1891 , indicating slight agreement between raters. To illustrate, we provide examples of explanation paths and their consensus ratings.

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Feature afﬁnity. We consider propagating the labels from an annotated set to an unlabeled set by nearest neighbor search in the featurespace. We assume that  semantic  clustersemerge during training with sparse supervision, reinforced by aforementioned pixel-to-segment relationships. By propagating labels in the feature space, we reinforce the learning of semantic clusters.

Feature afﬁnity. Our goal is to learn a pixel-wise feature that indicates semantic segmentation. It is thus reasonable to assume that pixels and segments of the same semantics form a cluster in the feature space, and we reinforce such clusters with a featural smoothness prior: We ﬁnd nearest neighbours in the feature space and propagate labels accordingly.

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After deriving the theoretical relation between learning rate, batch size, gradient covariance and properties of the minima, such as loss and width, we experimentally verify that the controllable noise n c = η/S controls the width and height of the minima towards which SGD converges. We also show the impact of the controllable noise on the memorization phenomenon. We discussed the limitations of the theory and in what situations it breaks down, exempliﬁed by when the learning rate gets too large. We also experimentally verify that η and S are exchangeable as long as the controllable noise η/S remains the same. This is true for both cyclic and constant schedules. In the cyclic case, we experimentally show hints that cyclical learning rates oscillate between sharp/deep and wide/shallow minima as long as the stage of increased noise is long enough to allow for mixing.

Further, we experimentally verify that the controllable noise n c = η/S determines the width and height of the minima towards which SGD converges. We also show the impact of this controllable noise on the memorization phenomenon. We discussed the limitations of the theory and in what situations it breaks down, exempliﬁed by when the learning rate gets too large. We also experimentally verify that η and S can vary in linear proportion as long as the controllable noise η/S remains the same. In addition, our experiments suggest that cyclical learning rates oscillate between sharp/deep and wide/shallow minima as long as the stage of increased noise is long enough to allow for mixing.

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Finally, we trained SoftHebb on a more difﬁcult dataset, namely Fashion-MNIST (Xiao et al., 2017) which contains grey-scale images of clothing products. A supervised MLP of the same size achieved a test accuracy of (90 . 55 ± 0 . 04)% on this task. The SoftHebb model achieved an accuracy of (87 . 46 ± 0 . 05)% , whereas a hard WTA also reached a similar accuracy of (87 . 49 ± 0 . We did not test the speed advantages of SoftHebb on this dataset. SoftHebb’s generative interpolations (Fig. 3B) are reconﬁrmed  as are its robustness to attacks, whereas, with very small adversarial perturbations, the MLP drops to an accuracy lower than the SoftHebb model (dashed lines in Fig. 2).

Finally, we performed preliminary tests on two more difﬁcult datasets, namely Fashion-MNIST (Xiao et al., 2017), which contains grey-scale images of clothing products, and CIFAR-10 (Krizhevsky et al., 2009), which contains RGB images of animals and vehicles. We did not tune the Hebbian networks’ hyper-parameters extensively, so accuracies on these tasks are not deﬁnitive but do give a good indication. On F-MNIST, the SoftHebb model achieved a top accuracy of 87 . 46% whereas a hard WTA reached a similar accuracy of 87 . A supervised MLP of the same size achieved a test accuracy of 90 . SoftHebb’s generative interpolations (Fig. 3B) are reconﬁrmed on the F-MNIST dataset, as are its robustness to attacks, whereas, with very small adversarial perturbations, the MLP drops to an accuracy lower than the SoftHebb model (dashed lines in Fig. 2). On CIFAR-10’s preliminary results, the hard WTA and SoftHebb achieved an accuracy of 49 . 78% and 50 . 27% respectively.


The notch(a black area where no drawing is performed) located in  the center of the top edge on theMacBook Pro (2021) display (Figure 1). Although the cursor can enter the notch area, it is  partially or entirelyhidden by the notch when the cursor entersto the notch.

A notch is used to position the web camera in a display for increasing the usable area of the display. For example, the MacBook Pro (2021) has a notch at the center of the top edge on the display (Figure 1). It is the black area on the display that cannot be used; however, the cursor can enter this area and is hidden partially or entirely when the cursor enters the notch.


Learning Geometry with Mirror Maps: Common to most DP model training algorithms, including DP-SGD, DP-FTRL (Kairouz et al., 2021b), and our algorithm, is a DP estimator of the gradient of the loss ∇ θ L ( θ t ; D priv ) = (cid:80) d ∈ D priv ∇ θ (cid:96) ( θ t ; d ) generated by the private data set D priv at a given model state θ t ∈ R p . This DP estimator essentially adds isotropic Gaussian noise N (0 , σ 2 I p ) to ∇ θ L ( θ t ; D priv ) , where σ depends on the privacy parameters ( ε, δ ) and the maximum allowable value of (cid:107)∇ θ (cid:96) ( θ t ; d ) (cid:107) 2 (a.k.a. the clipping norm (Abadi et al., 2016)). 1 It is well known that for most learning tasks, the set of gradients vectors in L ( θ t ; D priv ) are seldom isotropic (Gur-Ari et al., 2018; Agarwal et al., 2019). Hence, it is natural to wonder if the Gaussian noise in the DP estimator can be made to respect the geometry of the gradients. Prior works (Zhou et al., 2020; Asi et al., 2021; Kairouz et al., 2021a) have used public data ( D pub ) to explicitly learn this geometry, mostly in the form of preconditioner matrices (Duchi et al., 2011) to be multiplied to the estimated noisy gradients. In this paper, we take an implicit approach towards respecting this geometry, by using the loss L ( θ ; D pub ) generated by the public data as the mirror map in classical mirror descent. As a ﬁrst order approximation (formalized in Section 4), one can view it as doing DP-SGD on L ( θ ; D priv ) while using L ( θ ; D pub ) as a regularizer. This approach has the following advantages: (i) The information of the geometry is “free”, i.e., one does not need to learn the preconditioner explicitly from the public data, (ii) Unlike prior works (Zhou et al., 2020; Kairouz et al., 2021a), one does not need to assume that the gradients of L ( θ ; D priv ) lie in a ﬁxed rank subspace, (iii) The achieved excess population risk guarantees have better dependence on n pub = | D pub | compared to prior results (Asi et al., 2021), and (iv) Because the geometry is implicitly deﬁned, the implementation does not need to maintain an additional data structure for the preconditioner, and hence is much easier to implement. Empirically, under our best-effort comparison, our baseline algorithm improves over the state of the art (Asi et al., 2021). We note that differentially private mirror descent has been considered before by Talwar et al. (2014) and Wang et al. Their results are not directly comparable to ours because (i) they do not have access to in-distribution public data (ii) as shown in Bassily et al. (2014), without public data, it is impossible to achieve the dimension independent bounds we achieve (iii) in our experiments we solve unconstrained optimization problems, but those works choose the mirror map based on the constraint set rather than the data set. We note that the utility bounds we prove in this paper also apply to a public data-assisted variant of the accelerated mirror descent algorithm considered in Wang et al.

Learning Geometry with Mirror Maps: Common to most DP model training algorithms, including DP-SGD, DPFTRL [24], and our algorithm, is a DP estimator of the gradient of the loss ∇ θ L ( θ t ; D priv ) = (cid:80) d ∈ D priv ∇ θ (cid:96) ( θ t ; d ) generated by the private data set D priv at a given model state θ t ∈ R p . This DP estimator essentially adds isotropic Gaussian noise N (0 , σ 2 I p ) to ∇ θ L ( θ t ; D priv ) , where σ depends on the privacy parameters ( ε, δ ) and the maximum allowable value of (cid:107)∇ θ (cid:96) ( θ t ; d ) (cid:107) 2 (a.k.a. the clipping norm [1]). 1 It is well known that for most learning tasks, the set of gradients vectors in L ( θ t ; D priv ) are seldom isotropic [2, 20]. Hence, it is natural to wonder if the Gaussian noise in the DP estimator can be made to respect the geometry of the gradients. Prior works [4, 23, 39] have used public data ( D pub ) to explicitly learn this geometry, mostly in the form of preconditioner matrices [15] to be multiplied to the estimated noisy gradients. In this paper, we take an implicit approach towards respecting this geometry, by using the loss L ( θ ; D pub ) generated by the public data as the mirror map in classical mirror descent. As a ﬁrst order approximation (formalized in Section 4), one can view it as doing DP-SGD on L ( θ ; D priv ) while using L ( θ ; D pub ) as a regularizer. This approach has the following advantages: (i) The information of the geometry is “free”, i.e., one does not need to learn the preconditioner explicitly from the public data, (ii) Unlike prior works [23, 39], one does not need to assume that the gradients of L ( θ ; D priv ) lie in a ﬁxed rank subspace, (iii) The achieved excess population risk guarantees have better dependence on n pub = | D pub | compared to prior results [4], and (iv) Because the geometry is implicitly deﬁned, the implementation does not need to maintain an additional data structure for the preconditioner, and hence is much easier to implement. Empirically, under our best-effort comparison, our baseline algorithm improves over the state of the art [4]. We note that differentially private mirror descent has been considered before by [36] and [38]. Their results are not directly comparable to ours because (i) they do not have access to in-distribution public data (ii) as shown in [6], without public data, it is impossible to achieve the dimension independent bounds we achieve (iii) in our experiments we solve unconstrained optimization problems, but those works choose the mirror map based on the constraint set rather than the data set. We note that the utility bounds we prove in this paper also apply to a public data-assisted variant of the accelerated mirror descent algorithm considered in [38].


In this paper, we give generalization error bounds of deep ReLU networks for a Besov space and its variant with mixed smoothness , which includes the H ¨ older space, the Sobolev space, and the function class with total variation as special cases 1 . By doing so, (i) we show that deep learning achieves the minimax optimal rate on the Besov space and notably it outperforms any linear estimator such as the kernel ridge regression, and (ii) we show that deep learning can avoid the curse of dimensionality on the mixed smooth Besov space and achieves the minimax optimal rate. As related work, Mhaskar & Micchelli (1992); Mhaskar (1993); Chui et al. (1994); Mhaskar (1996); Pinkus (1999) also developed an approximation error analysis which essentially leads to analyses for Besov spaces. However, the ReLU activation is basically excluded and comprehensive analyses for the Besov space have not been given. Consequently, it has not been clear whether ReLU neural networks can outperform another representative methods such as kernel methods. As a summary, the contribution of this paper is listed as follows:

In this paper, we give generalization error bounds of deep ReLU networks for a Besov space and its variant with mixed smoothness , which includes the H ¨ older space, the Sobolev space, and the function class with total variation as special cases. By doing so, (i) we show that deep learning achieves the minimax optimal rate on the Besov space and notably it outperforms any linear estimator such as the kernel ridge regression, and (ii) we show that deep learning can avoid the curse of dimensionality on the mixed smooth Besov space and achieves the minimax optimal rate. As related work, Mhaskar & Micchelli (1992); Mhaskar (1993); Chui et al. (1994); Mhaskar (1996); Pinkus (1999) also developed an approximation error analysis which essentially leads to analyses for Besov spaces. However, the ReLU activation is basically excluded and comprehensive analyses for the Besov space have not been given. As a summary, the contribution of this paper is listed as follows:

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Linear Unit ( ELU ) (Clevert et al., 2015), the Gaussian Error Linear Unit ( GELU ) (Hendrycks & Gimpel, 2016) and the Swish function ( Swish ) (Ramachandran et al., 2017). The details of each activation function can refer to Appendix A. We also compare PIAC with standard activation functions with the layer-wise adaptive slopes (Jagtap et al., 2020a). PIAC setups. We employ the PIAC in a layer-wise manner by default. We set the candidate function set F as { sin , tanh , GELU , Swish , Softplus } . The learnable parameters { α i } Ni =1 are initialized as zeros and optimized jointly with the weights and biases of neural networks. The scaling factors { β i } Ni =1 are initialized as ones and can be fixed or learnable.

Linear Unit ( GELU ) and the Swish function ( Swish ). We compare PIAC to other adaptive activation functions which could provide higher-order derivatives, including SLAF Goyal et al. PAU Molina et al. and ACON Ma et al. The details of each activation function can refer to Appendix A and Appendix B. We also compare PIAC with standard activation functions with the layer-wise adaptive slopes (Jagtap et al., 2020a). We employ the PIAC in a layer-wise manner by default. We set the candidate function set F as { sin , tanh , GELU , Swish , Softplus } . The learnable parameters { α i } Ni =1 are initialized as zeros and optimized jointly with the weights and biases of PINNs. The scaling factors { β i } Ni =1 are initialized as ones and can be fixed or learnable.


Correlation Between Estimated Entropies and B-factors B-factor is an experimental measurement of conformational flexibility. We collect the average b-factor of sidechain atoms of amino acids in the test split of the PDB-REDO dataset. Simultaneously, we use the RDE to estimate the conformational entropy for each amino acid in the testset. The average Pearson correlation coefficient between these two quantities is 0.4637 and the average Spearman coefficient is 0.4282. Detailed results are presented in Table 10 in the appendix. In summary, this supports that  the entropy estimated by the RDE is correlated to experimentally determined conformational flexibility.

Correlation Between Estimated Entropy and B-factors B-factor is an experimental measurement that quantifies the conformational flexibility. We calculate the average b-factor of sidechain atoms of residues in the test split of the PDB-REDO dataset. Then, we estimate the conformational entropy for each residue in the test split using the RDE. The average Pearson correlation coefficient between these two quantities is 0.4637, and the average Spearman coefficient is 0.4282. Detailed results are presented in Table 8 in the appendix. In summary, this indicates that there is a correlation between the entropy estimated by the RDE and experimentally determined conformational flexibility measured by B-factor.


The irregular structure of meshes leads to a variety of approaches to deﬁne convolutions. Closely related to our method are graph based methods which are often based on variations of graph convolutional networks Kipf & Welling (2017); Defferrard et al. GCNs have been applied on spherical meshes Perraudin et al. and cortical surfaces Cucurull et al. ; Zhao et al. Verma et al. (2018) augment GCNs with anisotropic kernels which are dynamically computed via an attention mechanism over graph neighbours.

The irregular structure of meshes leads to a variety of approaches to deﬁne convolutions. Closely related to our method are graph based methods which are often based on variations of graph convolutional networks (Kipf & Welling, 2017; Defferrard et al., 2016). GCNs have been applied on spherical meshes (Perraudin et al., 2019) and cortical surfaces (Cucurull et al., 2018; Zhao et al., 2019a). Verma et al. (2018) augment GCNs with anisotropic kernels which are dynamically computed via an attention mechanism over graph neighbours.

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Ensembles, where multiple neural networks are trained individually and their predictions are averaged, have been shown to be widely successful for improving both the accuracy and predictive uncertainty of single neural networks. However, an ensemble’s cost for both training and testing increases linearly with the number of networks.

Ensembles, where multiple neural networks are trained individually and their predictions are averaged, have been shown to be widely successful for improving both the accuracy and predictive uncertainty of single neural networks. However, an ensemble’s cost for both training and testing increases linearly with the number of networks, which quickly becomes untenable.


We instead study the setting where all instances come with instance-speciﬁc features; this is a natural and practical assumption [27, 30] that encompasses numerical representations of the instance itself— e.g. bits representing a query or a graph—or other information such as weather or day of the week. These are passed to functions—e.g. linear predictors, neural nets, or trees—whose parameters can be learned from data. We study linear predictors, which are often amenable to similar analyses as above since the composition of a convex and afﬁne function is convex. For example, it is straightforward to extend the matching results to bound the regret and sample complexity of learning a linear predictor of duals. Page migration is more challenging because the outputs must lie in the simplex, which we solve by restricting to matrices with columns in the simplex, i.e. rectangular stochastic matrices. Both sets of results are shown in Appendix C. Notably, for page migration our guarantees cover the auto-regressive setting where the server probabilities are determined by a ﬁxed linear transform of past states.

We instead study the setting where all instances come with instance-speciﬁc features, a natural and practical assumption [27, 30] that encompasses numerical representations of the instance itself— e.g. bits representing a query or a graph—or other information such as weather or day of the week. These are passed to functions—e.g. linear predictors, neural nets, or trees—whose parameters can be learned from data. We study linear predictors, which are often amenable to similar analyses as above since the composition of a convex and afﬁne function is convex. For example, it is straightforward to extend the matching results to learning linear predictors of duals. OPM is more challenging because the outputs must lie in the simplex, which can be solved by learning rectangular stochastic matrices. Both sets of results are shown in Appendix C. Notably, for page migration our guarantees cover the auto-regressive setting where the server probabilities are determined by a ﬁxed linear transform of past states.

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Figure 6 shows a schematic of the task. In experiment 1, we strictly set the starting position of the trial. In experiment 2, we did not set the starting position of the trial as a condition. The starting area was a rectangle, and the trials started by simply clicking an area once. Except for this change, the task was the same as  in experiment 1.

Figure 6 shows a schematic of the task. We did not set the starting position of the trial as a condition, unlike that in Experiment 1. The starting area was a rectangle, and the trials started by simply clicking an area once. Except for this change, the task was the same as that in Experiment 1.

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In multiset-equivariant tasks, we want the output multiset to be ordered the same as the input multiset: the “ﬁrst” element in the input corresponds to the “ﬁrst” in the output, but the overall order is still irrelevant. We can make iDSPN multiset-equivariant to the input – not just to its initialization Y 0 – by concatenating the input set to the set being optimized. In general, when wealready know parts of the desired Y (e.g. certain elements or the values in certain dimensions), we can help the model by keeping those parts ﬁxed and not optimizing them in the forward optimization. We use this in

When we already know parts of the desired Y (e.g. speciﬁc elements in the set or the values along certain dimensions), we can help the model by keeping those parts ﬁxed and not optimizing them in Equation 7. For example, in Section 4.1 we know that the ﬁrst few dimensions in the output multiset should be the exactly same as in the input multiset. If we ﬁx these dimensions during optimization, the model only needs to learn the remaining dimensions that we do not already know. We use this in

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Problem 1 deﬁnes a feature matching problem similar to many inverse reinforcement learning(IRL) [10] and IOC [11] formulations. We design a set of features f as well as a risk feature f φ ξthat capture thedriving and risk management preferences of the demonstrator. From the demonstration data we then learn the risk metric parameters ξ and the weights that combine the feature values. This combination gives us a cost model. Solving for a trajectory that optimizes this costmodel results in driving behaviors similar to the demonstrator (in the context of our deﬁned featuresand risk models). Existing IOC approaches are able to imitate standard driving behaviors, wherethe probability P θ is assumed to be the exponential distribution. As such, the generated trajectories from the cost model are exponentially more preferred by the agent. In our case, the additionof the risk measure allows us to match drivers better under risky situations. While none of theseapproaches exactly mimic the demonstrator, they capture the driving style of the demonstrator andallow generalization to new risky scenarios.

Problem 1 deﬁnes a feature matching problem similar to many inverse reinforcement learning(IRL) [10] and IOC [11] formulations. We design a set of features f to capture driving preferencesin non-risky situations, but also include a risk-based feature f φ ξ capturing the risk managementpreferences seen in demonstrations. From the demonstration data we then learn the risk metric parameters ξ and the weights that combine the feature values. This combination gives us a cost modelthat, when solved, yields driving behaviors similar to the demonstrator (in the context of our deﬁnedfeatures and risk models). While existing IOC approaches are able to imitate standard driving behaviors such that the generated trajectories from the cost model are exponentially more preferred bythe agent (under maximum-entropy formulations, e.g. [11]). In our case, the additional risk feature provides the capacity to generalize our model better under risky situations without hindering theability to replicate the driving style of human demonstrators in normal scenarios.

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As we know that the performance of hand-designed LR schedule is very sensitive to the initial learning rates. To avoid carefully tuning the initial learning rates, we learn the LR schedule from an interval. We set γ = 1 for image tasks, and γ = 20 for text tasks  to eliminate the inﬂuence of loss magnitude between two  tasks.

As we know that the performance of hand-designed LR schedules and HPO methods is very sensitive to the initial LR. To avoid carefully tuning the initial LR, we learn the LR schedules from an interval [0 , γ ] , and now the initial LR is determined by the output of the MLR-SNet. We set γ = 1 for image tasks, and γ = 40 for text tasks in all our experiments to eliminate the inﬂuence of loss magnitude between two different tasks.


This is processed by a graph attention network (GAT) (Veliˇckovi´c et al., 2017) which learns action relations. In a GAT, the attention weights are expected to be high for closely interdependent actions. The resultant nodes are pooled together to form a compact relational summary of the  action space. This summaryvector and each action’s relational representation is used to compute an action’s utility, which is used as a Q-value oras a probability logit for policy gradient methods (Figure 2).

A graph attention network (GAT) (Veliˇckovi´c et al., 2017) processes the action graph and learns action relations. The attention weights in the GAT would be high for closely related actions such as a nail and a hammer in Figure 1. A utility network uses the GAT’s resulting relational action representations, the state, and a pooled together action set summary to compute each action’s Qvalue or probability logit for policy gradient methods(Figure 2).

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To be speciﬁc, we assign a semantic label for each unlabeled segment by ﬁnding its nearest labeled segment in the feature space. We denote this new labeled set (together with the original labeled set) as ˆ C  , where ˆ C + denotes all the same-category segments other than segment s in ˆ C and pixel i belongs to the segment s . Such a formulation is based on three assumptions: 1) The size of the original labeled set is large enough to cover the feature space, 2) the labeled segments are distributed uniformly in the feature space, and 3) the embedding already encodes certain semantic information. Therefore, we only apply this relationship to propagate the keypoint annotations in the DensePose dataset, where each body part  is annotated by a point.

Speciﬁcally, we assign a semantic label to each unlabeled segment by ﬁnding its nearest labeled segment in the feature space. We denote this expanded labeled set by ˆ C . For pixel i , we deﬁne its positive (negative) segment set ˆ C + ( ˆ C − ) according to whether a segment has the same label as i . Our feature afﬁnity relationship works best when: 1) the original labeled set is large enough to cover the feature space, 2) the labeled segments are distributed uniformly in the feature space, and 3) the pixel-wise already encodes certain semantic information. We thus only apply to DensePose keypoint annotations in our experiments, where each body part in the training image is annotated by a point.

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A critical problem in the literature on post hoc explanations is the lack of a common foundational goal among methods. For example, some methods are motivated by function approximation, some use game theoretic notions such as Shapley-Aumann values, and some are ad hoc, driven by the goal of obtaining clean visualizations. Such fragmentation of goals  not only prevents a coherent conceptual understanding of post hoc explainability but also causes the practical challenge of not knowing which method to use when.

A critical problem in the ﬁeld of post hoc explainability is the lack of a common foundational goal among methods. For example, some methods are motivated by function approximation, some by game theoretic notions, and some by obtaining clean visualizations. This fragmentation of goals causes not only an inconsistent conceptual understanding of explanations but also the practical challenge of not knowing which method to use when.

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There are a lot of measure benchmarking studies considering node classiﬁcation and clustering for both generated graphs and  datasets (Fouss et al., 2012; Sommer et al., 2016; 2017; Avrachenkov et al., 2017; Ivashkin & Chebotarev, 2016; Guex et al., 2018; 2019; Aynulin, 2019a;b; Courtain et al., 2020; Leleux et al., 2020), etc. Although a large number of experimental results, theoretical results  still look unattainable. One of the most important theoretical results for graph measures is a work  Luxburg et al. (2010), where problems of  Commute Time  on big graphs were shown theoretically, and a substantiated amendment was proposed to correct the problem. The paper shows how difﬁcult such proves. Besides difﬁcult proves, there is still no complex empirical understanding of what effects need to be proven. Our empirical work has two main advantages from  previous ones. Firstly, we consider the vast amount of graph measures, which for the ﬁrst time gives the full picture. Secondly, unlike these studies concluding with a global leaderboard, we are looking for the leading measures for each set of  LFR parameters.

There are a lot of measure benchmarking studies considering node classiﬁcation and clustering for both generated graphs and real-world datasets (Fouss et al., 2012; Sommer et al., 2016; 2017; Avrachenkov et al., 2017; Ivashkin & Chebotarev, 2016; Guex et al., 2018; 2019; Aynulin, 2019a;b; Courtain et al., 2020; Leleux et al., 2020), etc. Despite a large number of experimental results, theoretical results are still a matter of the future. One of the most interesting theoretical results on graph measures is the work by Luxburg et al. (2010), where some unattractive features of the Commute Time distance on large graphs were explained theoretically, and a reasonable amendment was proposed to ﬁx the problem. Beyond the complexity of such proofs, there is still very little empirical understanding of what effects need to be proven. Our empirical work has two main differences from the previous ones. First, we consider a large number of graph measures, which for the ﬁrst time gives a fairly complete picture. Second, unlike these studies concluding with a global leaderboard, we are looking for the leading measures for each set of the LFR parameters.

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Limitations and future work. The main downside of SDN remains the computation time. However, we suspect that a more optimized implementation could substantially improve  runtime performance. Another possible direction is to explore the applicability of SDN  to other generative models such as generative adversarial networks (Goodfellow et al., 2014).

Limitations and future work. The main downside of SDN remains the computation time. However, we suspect that a more optimized implementation could substantially improve the runtime performance of SDN. In the future, it would be beneﬁcial to explore the applicability of SDN in other settings, e.g. to apply it to other generative models such as generative adversarial networks (Goodfellow et al., 2014) or to any other image processing task such as image super-resolution or image segmentation.


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+} BS & DBS stand for beam search and its variant diverse beam search (Vijayakumar et al., 2016). In our conﬁguration, we use Hamming distance as the diversity function and set the diversity strength to 1.5, following Vijayakumar et al.

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+} I 5 like 6 this 7 show play musical performance p = 0. play musical amazing …… …… 0.2 0. p = 0. BS & DBS stand for beam search and its variant diverse beam search (Vijayakumar et al., 2018). In our conﬁguration, we use Hamming distance as the diversity function and set the diversity strength to 1.5, following Vijayakumar et al.


However, the implementation of Model-X requires the generation of the so called ’knockoffs’, which has limited existing methods. The goal of our paper is to ﬁll in the gap by proposing a model-free method for generating knockoffs with relaxed assumptions for the data, leveraging the power of the recent development in deep generative models. Speciﬁcally, we propose a framework to generate knockoffs based on latent variable Z which captures the correlation structure of X . This provides a great generalization from the existing knockoff generation methods for hidden Markov model (HMM) and mixture Gaussian to (but not limited to) Hidden Markov Random Field. The contribution of this paper is in two folds: ﬁrst, we tackles the problem of generating Model-X knockoffsfor any data set by proposing and justifying a framework based on latent variable Z ; second, the proposed framework is a novel application for variational auto encoder and its variants. Compared with other deep learning architecture to generate knockoff, the proposed framework has the advantage in less computational complexity and thus easier to be  implemented by domain scientists.

However, the implementation of Model-X requires the generation of the so called ’knockoffs’, which has limited existing methods. The goal of our paper is to ﬁll in the gap by proposing a model-free method for generating knockoffs with relaxed assumptions for the data, leveraging the power of the recent development in deep generative models. Speciﬁcally, we propose a framework to generate knockoffs based on latent variable Z which captures the correlation structure of X . This provides a great generalization from the existing knockoff generation methods for hidden Markov model (HMM) and mixture Gaussian to (but not limited to) Hidden Markov Random Field. The contribution of this paper is it propose a framework to tackle the problem of generating Model-X knockoffs based on latent variable Z . We provide theoretical justiﬁcation of the approach and also demonstrate the state of art variational auto encoder (VAE) can achieve promising results in the task of FDR controlled variable selection. This paper is among the ﬁrst works for the new application of representative learning in generating model-X knockoffs. And it is a nature generalization of the simple latent variable models based methods in the literature and can be easily implemented with variational auto-encoder. Our discussion and preliminary results for goodness-of-ﬁt also shed lights on future improvement of VAE for generating knockoffs.

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MCDP-EMD and MCDP-MMD are also obtained similarly. Note that they share the same E and P as the sample mean is the same. The D of q ∗ x is expected to be the best uncertainty measure that OPU can approach theoretically (when the amortization loss is zero, see Sec. 2.4). The baselines for SGLD are obtained in a similar way. Students use categorical entropy (for BDK) or D (for DPN) and P of the output distributions.

MCDP-EMD and MCDP-MMD are also obtained similarly. Note that they share the same E and P as the sample mean is the same. The D of q ∗ x is expected to be the best uncertainty measure that OPU can approach theoretically (when the amortization loss is zero, see Sec. 2.4). For fairness, the same set of posterior particles is used for obtaining D. The baselines for SGLD are obtained in a similar way. Students use categorical entropy (for BDK) or D (for DPN) and P of the output distributions.


Hyperparameter tuning: For our PHN method we select hyperparameters based on the HV computed on a validation set. Selecting hyperparameters for the baselines is non-trivial as there is no clear criteria that is reasonable in terms of runtime; In order to select hyperparameters based on HV, each approach needs to be trained multiple times on all rays. We therefore select hyperparameters based on a single ray, and apply those for all rays. Our selection criterion is as follow: we collect all models trained using all hyperparameter conﬁgurations, and ﬁlter out the dominated solutions. Finally, we select the combination of hyperparametrs with the highest uniformity.

Hyperparameter tuning: For PHN, we select hyperparameters based on the HV computed on a validation set. Selecting hyperparameters for the baselines is challenging because there are no clear criteria that can be computed quickly with many rays. Selecting HPs based on HV requires to train each baseline multiple times on all rays. Therefore, we select hyperparameters based on a single ray and apply the selected hyperparameters to all rays. Speciﬁcally, we collect all models trained using all hyperparameter conﬁgurations, and remove dominated solutions. Finally, we select the combination of HPs with the highest uniformity.

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We modify the grid world environment (Chevalier-Boisvert et al., 2018) where an agent navigates a 2D maze with two lava rivers to reach a goal. The agent always has access to 4 direction movements and a turn-left skill (Figure 5). However, two  actions  out of 4 special skills: turn-right, step-forward, dig-orange-lava and dig-pink-lava , are randomly sampled for the agent in every task instance. The agent can walk over lava for one timestep without dying, but it can remove the lava by using the matching dig-lava skill. Thus, when available, dig-lava skills can be used to build shortcut paths to the goal on the other end, and thus receive a higher reward. We use PPO to train all methods.

We modify the grid world environment (Chevalier-Boisvert et al., 2018) where an agent navigates a 2D maze with two lava rivers to reach a goal. The agent always has access to four direction movements and a turn-left skill (Figure 5). There are two additional actions randomly sampled out of four special skills: turn-right, step-forward, dig-orange-lava and dig-pink-lava . If the agent enters the lava, it will die unless it uses the matching dig-lava skill to remove the lava in the immediately next timestep. Thus, when available, dig-lava skills can be used to create shortcut paths to the goal and receive a higher reward. We use PPO for all experiments in this environment.

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To mitigate the burden of data labeling, we aim at improving data efﬁciency for both classiﬁcation and regression setups in deep learning. However, the current focus is on classiﬁcation problems while rare attention has been paid to deep regression, which usually requires more human effort to labeling. Further, due to the intrinsic difference between categorical and continuous label space, the common intuitions for classiﬁcation, e.g. cluster assumptions or pseudo labeling strategies, cannot be naturally adapted into deep regression. To this end, we ﬁrst delved into the existing data-efﬁcient methods in deep learning and found that they either encourage invariance to data stochasticity ( e.g. , consistency regularization under different augmentations) or model stochasticity ( e.g. , difference penalty for predictions of models with different dropout). To take the power of both worlds, we propose a novel χ -model by simultaneously encouraging the invariance to data stochasticity and model stochasticity. Further, the χ -model plays a minimax game between the feature extractor and task-speciﬁc heads to further enhance the invariance to model stochasticity. Extensive experiments verify the superiority of the χ -model among various tasks, from a single-value prediction task of age estimation to a dense-value prediction task of keypoint localization, a 2D synthetic and a 3D realistic dataset, as well as a multi-category object recognition task.

To mitigate the burden of data labeling, we aim at improving data efﬁciency for both classiﬁcation and regression setups in deep learning. However, the current focus is on classiﬁcation problems while rare attention has been paid to deep regression, which usually requires more human effort to labeling. Further, due to the intrinsic difference between categorical and continuous label space, the common intuitions for classiﬁcation, e.g. cluster assumptions or pseudo labeling strategies, cannot be naturally adapted into deep regression. To this end, we ﬁrst delved into the existing data-efﬁcient methods in deep learning and found that they either encourage invariance to data stochasticity ( e.g. , consistency regularization under different augmentations) or model stochasticity ( e.g. , difference penalty for predictions of models with different dropout). To take the power of both worlds, we propose a novel χ -model by simultaneously encouraging the invariance to data stochasticity and model stochasticity. Extensive experiments verify the superiority of the χ -model among various tasks, from a single-value prediction task of age estimation to a dense-value prediction task of keypoint localization, a 2D synthetic and a 3D realistic dataset, as well as a multi-category object recognition task.

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Very Deep Network. To apply SRP to  very deep networks, like over 400 Conv layers in RCAN, we revise RCAN(Zhang et al., 2018b) by removing all  channel attention modules. We set the channel number in the revised RCAN  as 96 and then prune it to 64. For × 2, we reduce the number of parameters from 34.5M to 15.3M and name the compressed model as SRPN.

Deep Networks. To apply SRP to the very deep network RCAN (Zhang et al., 2018b), a representative top-performing deep SR network with over 400 Conv layers, we revise RCAN by removing all the channel attention modules (Zhang et al., 2018b). The channel number in the revised RCAN is chosen as 96 and then pruned to 64. For × 2 scale, we reduce the number of parameters from 34.5M to 15.3M and dub the compressed model as SRPN.

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These models use an inference network that takes as input the full video frame sequences to predict the locations of 2D object bounding boxes, as well as frame-to-frame displacements, in order to minimize  view prediction error in 2D. We were not able to produce meaningful results from their inference networks. The success of Hsieh et al. (2018) may partially depend on carefully selected priors for  2D object bounding box location and object size parameters that match the moving MNIST dataset statistics  used in the paper, as suggested by the publicly available code. We do not assume knowledgeor existence ofsuch object location or size priors for our CARLA data.

These models consume a video as input, and predict the locations of 2D object bounding boxes, as well as frame-to-frame displacements, in order to minimize a view regression error. We were not able to produce meaningful results from these models. The success of Hsieh et al. (2018) may partially depend on carefully selected priors for the 2D bounding box locations and sizes, to match the statistics of the “Moving MNIST” dataset used in that work (as suggested in the ofﬁcial code). For our CARLA experiments, we do not assume knowledge of priors for location or size.

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We also compare with Outlier Exposure (Hendrycks et al., 2019) (OE), an approach that relies on the real outlier data for model regularization. For a fair comparison, we train the object detector on PASCAL-VOC using the same architecture ResNet-50, and use the OE objective for the classiﬁcation branch. The real outliers for OE training are sampled from the OpenImages dataset (Kuznetsova et al., 2020). We perform careful deduplication to ensure there is no overlap between the outlier training data and PASCAL-VOC. Our method achieves OOD detection performance on COCO (AUROC: 88.44%) that favorably matches OE (AUROC: 89.41%), and does not impose strong data assumption.

We also compare with Outlier Exposure (Hendrycks et al., 2019) (OE). OE serves as a strong baseline since it relies on the real outlier data. We train the object detector on PASCAL-VOC using the same architecture ResNet-50, and use the OE objective for the classification branch. The real outliers for OE training are sampled from the OpenImages dataset (Kuznetsova et al., 2020). We perform careful deduplication to ensure there is no overlap between the outlier training data and PASCAL-VOC. Our method achieves OOD detection performance on COCO (AUROC: 88.66%) that favorably matches OE (AUROC: 90.18%), and does not require external data.


When domain labels d are available (not the case in latent domain learning) one strategy  is to modulate networks by constraining the underlying generating function of residual networks (He et al., 2016) Φ( x ) = x + f ( x ) to allow at most a linear change V d per each domain from some pretrained mapping Φ 0 (with f 0 in every layer), whereby Φ( x ) − Φ 0 ( x ) = V d x . Note the slight abuse of notation here in letting x denote a feature map with channels C . Rearranging this yields:

When domain labels d are available (not the case in latent domain learning) one strategy established by Rebufﬁ et al. (2017) is to modulate networks by constraining the layerwise transformation of residual networks (He et al., 2016) Φ( x ) = x + f ( x ) to allow at most a linear change V d per each domain from some pretrained mapping Φ 0 (with f 0 in every layer), whereby Φ( x ) − Φ 0 ( x ) = V d x . Note the slight abuse of notation here in letting x denote a feature map with channels C . Rearranging this yields:


SegSort (Hwang et al., 2019) is an end-to-end image segmentation model that generates a pixel-wise embedding and a consequent over-segmentation. It assumes that each segment has an independent normal distribution on a hypersphere. Spherical K-Means clustering (Banerjee et al., 2005) is used for segmenting an image and learning the discriminative feature clustering jointly. It is worth noting that, such an assumption indicates homogeneous representations within each segment. To learn the embedding, SegSort formulates a maximum likelihood loss thatmaximizes the  discrimination between segments. In addition, soft neighborhood assignment (Goldberger et al., 2005) is incorporated to enforce grouping of semantically similar segments. During inference, the segment labels are predicted by K-Nearest Neighbor retrievals.

SegSort (Hwang et al., 2019) is an end-to-end segmentation model that generates a pixel-wise feature map and a resulting segmentation. Assuming independent normal distributions for individual segments, SegSort seeks a maximum likelihood estimation of the feature mapping, so that the feature induced partitioning in the image and clustering across images provide maximum discrimination among segments. During inference, the segment label is predicted by K-Nearest Neighbor retrievals.

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Specially, we randomly sample 20% of nodes V s in data ﬂow, mask direct edges connecting these sampled nodes by add an inﬁnitely negative value in the mask matrix, and then predict these masked edges E mask . Taking the variable x 11 in Figure 2 for an example, we ﬁrst mask edges (cid:104) x 7 , x 11 (cid:105) and (cid:104) x 9 , x 11 (cid:105) in the graph and then let the model to predict these edges. Formally, the pre-training objective of the task is calculated as Equation 5, where E c = V s × V ∪ V × V s is a set of candidates for edge prediction, δ ( e ij ∈ E ) is 1 if (cid:104) v i , v j (cid:105) ∈ E otherwise 0, and the probability p e ij of existing an edge from i -th to j -th node is calculated by dot product following a sigmoid function using representations of two nodes from GraphCodeBERT. loss EdgePred

Specially, we randomly sample 20% of nodes V s in data ﬂow, mask direct edges connecting these sampled nodes by add an inﬁnitely negative value in the mask matrix, and then predict these masked edges E mask . Taking the variable x 11 in Figure 2 for an example, we ﬁrst mask edges (cid:104) x 7 , x 11 (cid:105) and (cid:104) x 9 , x 11 (cid:105) in the graph and then let the model to predict these edges. Formally, the pre-training objective of the task is calculated as Equation 7, where E c = V s × V ∪ V × V s is a set of candidates for edge prediction, δ ( e ij ∈ E ) is 1 if (cid:104) v i , v j (cid:105) ∈ E otherwise 0, and the probability p e ij of existing an edge from i -th to j -th node is calculated by dot product following a sigmoid function using representations of two nodes from GraphCodeBERT. To balance positive-negative ratio of examples, we sample negative and positive samples with the same number for E c . loss EdgePred


This is equivalent to Temperature Scaling (Hinton et al., 2015), a mechanism that also maintains the probabilistic interpretation of the output. It can also be implemented by a normalized layer of exponential neurons, and are compatible with our theoretical derivations and the optimization by the plasticity rule of Eq. 8. Moreover, we show in the Appendix that soft WTA models can be constructed by rectiﬁed linear units (ReLU) or in general by neurons with any non-negative monotonically increasing activation function, and their weights are also optimized by the same plasticity rule.

This is equivalent to Temperature Scaling (Hinton et al., 2015), a mechanism that also maintains the probabilistic interpretation of the output. It can also be implemented by a normalized layer of exponential neurons, and are compatible with our theoretical derivations and the optimization by the plasticity rule of Eq. 8. This allows us to integrate the hard WTA into the SoftHebb framework, as a special case with an inﬁnite base. Therefore, the hard WTA, if used with the plasticity rule that we derived, is expected to have some similar properties to a soft WTA implementation. Moreover, we show in the Appendix that soft WTA models can be constructed by rectiﬁed linear units (ReLU) or in general by neurons with any non-negative monotonically increasing activation function, and their weights are also optimized by the same plasticity rule.


Human eye gaze is a crucial non-verbal cue used in a wide variety of applications, such as gazecontingent rendering in virtual reality [1, 2, 3, 4, 5], gaze-based interaction [6, 7], gaze-assisted collaboration [8, 9], as well as eye movement-based task recognition [10, 11]. Given the importance of eye gaze, many researchers have focused on the problem of gaze estimation [12, 13, 14, 15], i.e. estimating gaze position or direction from eye images. However, the collection of the large amounts of eye images required to train deep learning models, or the exchange of such data across networks, can pose significant privacy risks. In addition, the heterogeneous data distribution across different users in real-world settings (in-the-wild settings) can significantly hinder the training process of gaze estimation methods [16, 17]. Therefore, preserving privacy and maintaining high performance for heterogeneous data become two main challenges for gaze estimation.

Human eye gaze is a crucial non-verbal cue used in a wide variety of applications, such as gaze-contingent rendering in virtual reality Hu et al. (2019, 2020b, 2021) ; Hu (2020); Hu et al. (2020a) , gaze-based interaction Mardanbegi et al. ; Piumsomboon et al. (2017), gaze-assisted collaboration Higuch et al. ; Zhang et al. (2017c), as well as eye movementbased task recognition Hu et al. ; Coutrot et al. Given the importance of eye gaze, many researchers have focused on the problem of gaze estimation Baluja and Pomerleau (1993); Liang et al. ; Choi et al. ; Lu et al. (2014), i.e. estimating gaze position or direction from eye images. However, the collection of the large amounts of eye images required to train deep learning models, or the exchange of such data across networks, can pose significant privacy risks. In addition, the heterogeneous data distribution across different users in real-world settings (in-the-wild settings) can significantly hinder the training process of gaze estimation methods Zhang et al. Therefore, preserving privacy and maintaining high performance for heterogeneous data become two main challenges for gaze estimation.


VSG VSG VSG Ƹ𝑟 3 ො𝑎ො𝑎ො𝑎ො𝑣ො𝑣ො𝑣Ƹ𝑠Ƹ𝑠(b) While training, the world model is learned with collected experience, the policy is improved on trajectories unrolled using the world model and new episodes are collected by deploying the policy in the environment. An initial set of episodes are collected using a random policy. As training progresses, new episodes are collected using the latest policy to further improve the world model.

VSG VSG VSG Ƹ𝑟 3 ො𝑎ො𝑎ො𝑎ො𝑣ො𝑣ො𝑣Ƹ𝑠Ƹ𝑠(b) trajectories unrolled using the world model and new episodes are collected by deploying the policy in the environment. An initial set of episodes are collected using a random policy. As training progresses, new episodes are collected using the latest policy to further improve the world model.


The code for PEPPP and experiments is in the anonymous GitHub repository at https://github. com/barterer/lp . We use QPyTorch (Zhang et al., 2019) to simulate low-precision formats on standard hardware. The 87 datasets and 99 low-precision conﬁgurations in experiments are listed in Appendix A. The datasets consist of natural and medical images from various domains. Apart from

The code for PEPPP and experiments is in the GitHub repository at https://github.com/ chengrunyang/peppp . We use QPyTorch (Zhang et al., 2019) to simulate low-precision formats on standard hardware. The 87 datasets and 99 low-precision conﬁgurations in experiments are listed in Appendix A. The datasets consist of natural and medical images from various domains. Apart from


In contrast to the regular case in Proposition 2.1, the optimal plans γ [-
-] j transporting P 0 to P j need not be unique. However, the projection for ﬁxed γ
 j is unique.

The optimal plans γ [-
-]0 j transporting P 0 to P j need not be unique if P j lies outside the cut locus of P 0 , i.e., when there is more than one optimal way to transport P 0 onto P j . However, the projection for ﬁxed γ
 j is always unique by virtue of the linear regression.


P T =
 ψ ( r ) are the representation distributions under treated and untreated groups, respectively, induced by the map r = ψ ( x ) . The discrepancycould be minimized by updating the representation map ψ with gradient-based optimizers, as it is differentiable with respect to ψ (Flamary et al., 2021).

P T =
 ψ ( r ) are the distributions of representations in treated and untreated groups, respectively, induced by the mapping r = ψ ( x ) . The discrepancy is differentiable with respect to ψ (Flamary et al., 2021), thus can be minimized by updating the representation mapping ψ with gradient-based optimizers.

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The core idea of hypernetworks is to use one network to predict parameters for another network (Ha et al., 2017; Knyazev et al., 2021). The original goal of hypernetwork  is to decrease the number of training parameters (Ha et al., 2017), by training a hypernetwork with a smaller size to generate the parameters of another network with a larger size. Because of its promising performance, hypernetwork has been gradually applied to various tasks (Krueger et al., 2017; Zhang et al., 2019; von

The original goal of hypernetwork proposed in (Ha et al., 2017) is to decrease the number of training parameters , by training a hypernetwork with a smaller size to generate the parameters of another network with a larger size on a fixed dataset. Because of its promising performance, hypernetwork has been gradually applied to various tasks (Krueger et al., 2017; Zhang et al., 2019; von


Similar to standard rSLDS, the dynamics are conditionally linear given a leaf node z t . It is intuitive to expect that  nearby regions in  latent space have similar dynamics. In the context of the tree-structured stick breaking partitions that share a common parent should have similar dynamics. We explicitly model this by enforcing a hierarchical tree-structured prior on the dynamics.

Similar to standard rSLDS, the dynamics are conditionally linear given a leaf node z t . A priori, it is natural to expect that locally linear dynamics of nearby regions in the latent space are similar. Thus, in the context of tree-structured stick breaking, we impose that partitions that share a common parent should have similar dynamics. We explicitly model this by enforcing a hierarchical prior on the dynamics that respects the tree structure.

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Figure 5 reports the performance metrics for CIFAR10 and CIFAR100 under corruption. Tabulated results can be found in Section G in the Appendix. The general trend of these results is that the function-space prior frequently provides gains in OOD UQ with only a small decrease in (uncorrupted) test performance. Moreover, the shared function-space prior resulted in remarkable consistency across models, compared to the variety seen in weight-space priors. Due to the predictive prior, the ECE was signiﬁcantly higher for corruption 0, but this underﬁtting provided superior ECE when OOD. This suggests the function-space prior should be ﬁne-tuned (i.e. empirical Bayes) if superior ECE is desired in-distribution. For CIFAR100, the higher prior regularization due to higher dimensionality (see Section B.2), resulted in reduced beneﬁt over weight-space models compared to CIFAR10, with improved performance only evident at stronger corruptions. Figure 7 summarizes the performance difference due to the function-space prior for log-likelihood.

Figure 5 reports the performance metrics for CIFAR10 and CIFAR100 under corruption. Tabulated results can be found in Section G in the Appendix. The general trend of these results is that the function-space prior frequently provides gains in OOD UQ with only a small decrease in (uncorrupted) test performance. This trade-off between accuracy and robustness has been observed and discussed in the adversarial robustness setting (Tsipras et al., 2019; Yang et al., 2020), and remains an open problem if and how both qualities can be achieved in practice. Moreover, the shared function-space prior resulted in remarkable consistency across models, compared to the variety seen in weight-space priors. Due to the predictive prior, the ECE was signiﬁcantly higher for corruption 0, but this underﬁtting provided superior ECE when OOD. This suggests the function-space prior should be ﬁne-tuned (i.e. empirical Bayes) if superior ECE is desired in-distribution. For CIFAR100, the higher prior regularization due to higher dimensionality (see Section B.2), resulted in reduced beneﬁt over weight-space models compared to CIFAR10, with improved performance only evident at stronger corruptions. Figure 7 summarizes the LLH difference due to the function-space prior.


Le et al., 2019) can enjoy polynomial sample complexity under appropriate coverage assumptions, but the guarantee relies on the strong Bellman-completeness assumption on the function class; marginalized importance sampling (MIS) methods, which have gained significant attention recently (Liu et al., 2018; Xie et al., 2019; Uehara et al., 2020; Nachum et al., 2019a), use two function classes to simultaneously approximate the value and the density-ratio (or weight) function and optimize minimax objectives. Notably, it is the only family of methods known to produce accurate return estimates with a polynomial sample complexity, when the function classes only satisfy the relatively weak realizability assumptions (i.e., they contain the true value and weight functions).

Le et al., 2019) can enjoy polynomial sample complexity under appropriate coverage assumptions, but the guarantee requires the function class to satisfy the strong Bellmancompleteness assumption, i.e. closure under the Bellman operator (Chen & Jiang, 2019; Xie et al., 2021). Marginalized importance sampling (MIS) methods, which have gained significant attention recently (Liu et al., 2018; Xie et al., 2019; Uehara et al., 2020; Nachum et al., 2019a), use two function classes to simultaneously approximate the value and the density-ratio (or weight) function and optimize minimax objectives. Notably, it is the only family of methods known to produce accurate return estimates with a polynomial sample complexity, when the function classes only satisfy the relatively weak realizability assumptions (i.e., they contain the true value and weight functions).


Similar to Fung et al. , we ﬁnd that this is not only faster to run and easier to implement (no need to solve a linear system involving H ), but also leads to better results than standard implicit differentiation. Hence, we apply this simpliﬁcation in all of our experiments.

Other motivations for this approximation are discussed in Fung et al. (2022), where they call this Jacobian-free backpropagation. Matching their results, we ﬁnd that this is not only faster to run and easier to implement (no need to solve a linear system involving H ), but also leads to better results than standard implicit differentiation in preliminary experiments. Hence, we apply this approximation to differentiate Equation 7 with respect to z and θ in all of our experiments.


We investigate this empirically as follows: we start by selecting 2 . 5% random data points and train a model on this subset. At line 7 of Alg. 1, we enrich the unlabelled pool U by 10% with datapoints from another dataset and select the 200 data point with the highest score given by uncertaintysampling. If the selection would be random, then 10% of the selected data points would come from the out-of-domain dataset.

We investigate this empirically as follows: we start by selecting 2 . 5% random data points and train a model on this subset. At line 7 of Alg. 1, we enrich the unlabelled pool U by 10% with data points from another dataset. We perform three runs, where in each run we inject random data points from only one of the three other datasets. Afterwards the 200 data point with the highest score given by uncertainty sampling are selected. If the selection would be random, then 10% of the selected data points would come from the out-of-domain dataset.


We characterize the expressiveness of the matrices used in our method. In particular, we prove that block butterﬂy retains the expressiveness of butterﬂy, and that ﬂat butterﬂy can accurately approximate the residual form of butterﬂy. Moreover, ﬂat block butterﬂy + low-rank (an instance of sparse + low-rank) is more expressive than sparse or low-rank matrices alone. Finally, we analyze the training convergence and generalization of networks with sparse weights. All proofs are in the Appendix.

We characterize the expressiveness of the matrices used in our method. In particular, we prove that block butterfly retains the expressiveness of butterfly, and that flat butterfly can accurately approximate the residual form of butterfly. Moreover, flat block butterfly + low-rank (an instance of sparse + low-rank) is more expressive than sparse or low-rank matrices alone. Finally, we analyze the training convergence and generalization of networks with sparse weights. All proofs are in the Appendix.


MNIST dataset with 1000 training images. All external regularization schemes except learning rate decay and batch normalization have been turned off. We perform the following experiments : 1 ) Full-batch gradient descent with β =
 (i.e., GD) for various learning rate h and the best test accuracy is noted (in Figure 2) to be 95 . Full-batch gradient descent with momentum (GD+M) performed for various β with a fixed step-size h − 0 . 1 and the best test-accuracy is noted (in Figure 3) to be 96 . Our observation is that the best performance of GD (across all learning rates) is worse than the best performance of (GD+M) (across all β ’s). (Cohen et al., 2021) showed that gradient descent (GD) has an overwhelming tendency to increase the sharpness 2 till it reaches 2 h , called “the edge of stability”. And for (GD+M), the sharpness can reach up to 2(1+ β ) h , hence allowing it to enter a sharper region before becoming unstable. As greater allowable sharpness for (GD+M) than that of (GD) may suggest a higher test accuracy for (GD), this is not what we observe from the above experiment. We think the  implicit regularization  for (GD+M) plays a part in it. We believe IGR for momentum outweighs the sharpness effect in achieving better test accuracy.

MNIST dataset with 1000 training images. All external regularization schemes except learning rate decay and batch normalization have been turned off. We perform the following experiments : 1 ) Full-batch gradient descent with β =
 (i.e., GD) for various learning rate h and the best test accuracy is noted (in Figure 2) to be 95 . Full-batch gradient descent with momentum (GD+M) performed for various β with a fixed step-size h = 0 . 1 and the best test-accuracy is noted (in Figure 3) to be 96 . Our observation is that the best performance of GD (across all learning rates) is worse than the best performance of (GD+M) (across all β ’s). This observation failed to be explained by the known theory of edge of stability 3 but can be well-explained by our implicit regularization theory for (GD+M) as adding momentum increases the strength of the IGR.

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This deﬁnition of µ ( θ ; · ) is inspired by discussion on the effective update of parameters in the literature (Van Laarhoven, 2017; Zhang et al., 2019; Brock et al., 2021). Previous studies found that when normalization techniques are applied, such as batch normalization (Ioffe & Szegedy, 2015),  the update of the direction of  θ ,  i.e., θ / || θ || 22 , reﬂects how much the update on θ changes the model f in order to ﬁt the batch of samples.

This deﬁnition of µ ( θ ; i ) is inspired by discussion on the effective update of parameters in the literature (Van Laarhoven, 2017; Zhang et al., 2019; Brock et al., 2021; Hoffer et al., 2018). When normalization techniques, such as batch normalization (Ioffe & Szegedy, 2015), are applied to the DNNs, the key property of the weight vector, θ , is its direction, i.e., θ / || θ || 22 . Thus, we measure the update on θ using the norm of its gradient normalized by its norm.

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Having shown that networks do indeed lose some notion of target-ﬁtting capacity, we now turn our attention to the interaction between capacity and performance. To study this phenomenon further, we will use a computationally cheaper measure of network capacity which we call the effective dimension. An approximation of the rank of a feature embedding, it is both more computationally efﬁcient – as it does not require training a network for several thousand steps – and requires fewer hyperparameter design choices than the previous capacity metric. While this does not take into account the ability of the network to update early layers, Appendix B shows that it correlates reasonably well with target-ﬁtting capacity while also being a cheap, low-variance estimator against which to compare an agent’s relatively noisy performance.

Having shown that networks do indeed lose some notion of target-ﬁtting capacity, we now turn our attention to the interaction between capacity and performance. To study this phenomenon further, we will use a computationally cheaper measure of network capacity which we call the effective dimension. An approximation of the rank of a feature embedding, it is both more computationally efﬁcient – as it does not require training a network for several thousand steps – and requires fewer hyperparameter design choices than the previous capacity metric. This notion of capacity also measures more directly the ability of the network to distinguish input observations and so provides a better measure of how network updates will generalize to other states. This notion of state similarity is particularly relevant to sparse-reward environments, where policy improvement depends on the agent’s ability to distinguish a handful of rewarding states from the vast non-rewarding majority.


Prioritized Experience Replay (Schaul et al., 2015b) ( PER ) improves the performance of DQN (Mnih et al., 2015) by biasing sampling in favor of experiences that cause large temporal-difference (TD) errors. TD errors may signal rare events that would convey useful information to the learner, especially if the rewards are rare. Alternatively ER can be used to train transition models in planning-based RL (Pan et al., 2018), or to train off-policy learners on auxiliary tasks (Schaul et al., 2015a; Jaderberg et al., 2017) helping to shape the network features. Finally, when rewards are extremely rare, exploration may be guided by training the agent to reach rarely-visited states (Andrychowicz et al., 2017). de Bruin et al. (2015) proposes a modiﬁcation to ER that increases the diversity of behaviors contained in the RM, which is the opposite of what ReF-ER achieves. The ideas proposed by de Bruin et al.

Prioritized Experience Replay (Schaul et al., 2015b) ( PER ) improves the performance of DQN (Mnih et al., 2015) by biasing sampling in favor of experiences that cause large temporal-difference (TD) errors. TD errors may signal rare events that would convey useful information to the learner. ER can be used to train transition models in planning-based RL (Pan et al., 2018), or to train off-policy learners on auxiliary tasks (Schaul et al., 2015a; Jaderberg et al., 2017) helping to shape the network features. When rewards are very sparse, RL agents can be trained to repeat previous outcomes (Andrychowicz et al., 2017) or to reproduce successful states or episodes (Oh et al., 2018; Goyal et al., 2018). de Bruin et al. (2015) proposes a modiﬁcation to ER that increases the diversity of behaviors contained in the RM, which is the opposite of what ReF-ER achieves. The ideas proposed by de Bruin et al.


Our contribution is twofold. First, we propose the design of calen- dars that support medication prescriptions. The calendars allow for the scheduling of medication  alongside other everyday activities and provide a way for rendering and resolving scheduling conflicts when they are raised by unsafe drug interactions due to violation of admin- istration constraints specified in the prescription. Second, we present the results of a qualitative study with twelve participants interact- ing with the calendar designs. Results indicate that calendars can be designed to support the integration of medication prescriptions and that people would generally be favorable to using a calendar to manage their medications, if the design does not deviate from familiar calendars. Results also show that conflicts arising from unsafe rescheduling can be rendered within the calendars. These results inform five additional design goals that an integrated calendar should address: the use of familiar design (DG2), avoiding clutter (DG3), allowing for personalization (DG3), supporting personal reflection (DG5), and highlighting for user attention (DG6).

Our contribution is twofold. First, we identify and discuss consid- erations for the design of calendars that support medication prescriptions. Such calendars allow for the scheduling of medication actions alongside other everyday activities and provide a way for rendering and resolving conflicts when they are raised by unsafe schedules (i.e., schedules that violate constraints specified in the prescriptions). Second, we present the results of a qualitative study with twelve participants interacting with alternative calendar designs. Results indicate the potential benefit of equipping electronic calendars that already in use by many patients with additional functionality to support the scheduling of medication prescriptions. Users are generally in favour of using such an integrated approach that leverages their familiarity with existing tools. Results also show that it is feasible to design calendars that effectively communicate unsafe medication schedules. These results inform five additional design goals that an integrated calendar should address: the use of familiar design (DG2), avoiding clutter (DG3), allowing for personalization (DG3), support- ing personal reflection (DG5), and highlighting for user attention (DG6).

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Gaze Estimation Methods Gaze estimation methods can be generally categorised as either modelbased or appearance-based [24]. Model-based methods employ features detected from eye images to estimate gaze direction and can hardly obtain good performance in real-world settings because accurate eye feature detection relies on high-resolution images and homogeneous illumination [24, 25]. In contrast, appearance-based approaches directly regress gaze direction from eye images [12, 13, 14, 15] and can handle low-resolution images and different gaze ranges. However, appearance-based methods require a larger number of training eye images than model-based approaches to cover the significant variability in eye appearance [24], which poses serious privacy risks since eye images contain ample personal information, such as gender [26], identity [27], and personality traits [28]. Moreover, the heterogeneous data distribution across different users in real-world settings, which is caused by many factors including the differences in gaze range, head pose, illumination condition, and personal appearance [24, 17], can significantly hinder the training process of appearance-based gaze estimation methods.

Gaze estimation methods can be generally categorised as either model-based or appearance-based Zhang et al. (2017b). Model-based methods employ features detected from eye images to estimate gaze direction and can hardly obtain good performance in real-world settings because accurate eye feature detection relies on high-resolution images and homogeneous illumination Zhang et al. (2017b, 2019). In contrast, appearance-based approaches directly regress gaze direction from eye images Baluja and Pomerleau (1993); Liang et al. ; Choi et al. ; Lu et al. and can handle low-resolution images and different gaze ranges. However, appearance-based methods require a larger number of training eye images than model-based approaches to cover the significant variability in eye appearance Zhang et al. (2017b), which poses serious privacy risks since eye images contain ample personal information, such as gender Sammaknejad et al. (2017), identity Cantoni et al. and personality traits Hoppe et al. Moreover, the heterogeneous data distribution across different users in real-world settings, which is caused by many factors including the differences in gaze range, head pose, illumination condition, and personal appearance Zhang et al. (2017b, 2018), can significantly hinder the training process of appearance-based gaze estimation methods. Recent works Li et al. ; Bozkir et al.


Specifically, as shown in Figure 2 (b), existing methods (Yao et al., 2019) just assume unconfoundedness in Assumption A.1  to circumvent the UCE issue. Specifically, given two units r i ∈ P T = 1 ψ ( r ) and r j ∈ P T = 0 ψ ( r ) , the optimal transport in Definition 3.2 directly calculates the unit-wise distance as D ij ∶= ( r i − r j ) 2 . Given Assumption A.1 holds, this approach mitigates the treatment selection bias since it blocks the backdoor path X → T  by balancing the distribution of the observed covariates in a latent space. However, Assumption A.1 is usually violated in real scenarios, which invalidates existing methods as the backdoor path X ′ → T is not blocked.

CFR (Shalit et al., 2017), the unconfoundedness assumption A.1 (see Appendix A) is often taken to circumvent the UCE issue (Ma et al., 2022). Given two units r i ∈ P T = 1 ψ ( r ) and r j ∈ P T = 0 ψ ( r ) , for instance, optimal transport in Definition 3.2 calculates the unit-wise distance as D ij ∶= ∥ r i − r j ∥ 2 . If Assumption A.1 holds, this approach mitigates the treatment selection bias since it blocks the backdoor path X → T in Figure 3(a) by balancing the confounders across groups in a latent space. However, Assumption A.1 is usually violated in practice as per Figure 3(b), which hinder existing methods including OT from handling treatment selection bias since the backdoor path X ′ → T is not blocked.

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Meta learning, also referred to as “learning to learn”, usually learns a prior model from multiple tasksso that the learned model is able to quickly adapt to unseen tasks. Meta learning has been successfullyapplied to few-shot learning and multi-task learning [2,12,20,25,37]. While there are many exciting meta learning methods today, in this paper, we will study a representative meta learning settingwhere the goal is to learn a shared initial model that can quickly adapt to task-speciﬁc models. This adaptation may take an explicit form such as the output of one gradient descent step, which is referred to as the model agnostic meta learning (MAML) method [20]. Alternatively, the adaptation step may take an implicit form such as the solution of another optimization problem, which is referred to as the implicit MAML (iMAML) method [33]. Since both MAML and iMAML will solve a bileveloptimization problem, we term them the gradient-based meta learning thereafter. In many cases,overparameterized models are used as the initial models in  meta learning for quick adaptation. Forexample, ResNets-based MAML models typically have around 6 million parameters, but are trainedon 1-3 million meta-training data [11]. Training such initial models is often difﬁcult in meta learningbecause the number of training data is much smaller than the dimension of the model parameter.

Meta learning, also referred to as “learning to learn”, usually learns a prior model from multipletasks so that the learned model is able to quickly adapt to unseen tasks. Meta learning has beensuccessfully applied to few-shot learning and multi-task learning [2, 11, 19, 24, 35]. While there are many exciting meta learning methods today, in this paper, we will study a representative metalearning setting where the goal is to learn a shared initial model that can quickly adapt to task-speciﬁc models. This adaptation may take an explicit form such as the output of one gradient descent step, which is referred to as the model agnostic meta learning (MAML) method [19]. Alternatively, the adaptation step may take an implicit form such as the solution of another optimization problem, which is referred to as the implicit MAML (iMAML) method [31]. Since both MAML and iMAML will solve a nested optimization problem, we term them the nested meta learning thereafter. In manycases, overparameterized models are used as the initial models in nested meta learning for quickadaptation. However, training such initial models is often difﬁcult in meta learning because thenumber of training data is much smaller than the dimension of the model parameter.

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Proteins rarely act alone but usually interact with other proteins to orchestrate biological processes (Alberts & Miake-Lye, 1992; Kastritis & Bonvin, 2013). For example, antibodies, a kind of immune system protein, recognize pathogens by binding to proteins on their surface and elicit immune responses by interacting with the receptor protein of immune cells (Lu et al., 2018). Since proteinprotein interactions determine a large number of  biological functions, developing methods to modulate protein-protein interactions is critical. A typical way to modulate protein-protein interactions is to mutate amino acids on the interface — some mutations improve the strength of binding while others weaken and even disrupt the interaction (Gram et al., 1992; Barderas et al., 2008). Biologists  choose either to increase or decrease the binding strength depending on  specific goals. For example, if one would like to enhance the effect of a neutralizing antibody against a virus, it is usually necessary to increase the binding strength between the antibody and the viral protein. However, as the combinatorial space of amino acid mutations is large,  it is not always feasible or affordable to conduct wet-lab assays to test all the viable mutations. Therefore, computational approaches are needed to guide the identification of desirable mutations via predicting the mutational effect on binding strength measured by the change in binding free energy ( ∆∆ G ).

Proteins rarely act alone and usually interact with other proteins to perform a diverse range of biological functions (Alberts & Miake-Lye, 1992; Kastritis & Bonvin, 2013). For example, antibodies, a type of immune system protein, recognize and bind to proteins on pathogens’ surfaces, eliciting immune responses by interacting with the receptor protein of immune cells (Lu et al., 2018). Given the importance of protein-protein interactions in many biological processes, developing methods to modulate these interactions is critical. A common strategy to modulate protein-protein interactions is to mutate amino acids on the interface: some mutations enhance the strength of binding, while others weaken or even disrupt the interaction (Gram et al., 1992; Barderas et al., 2008). Biologists may choose to increase or decrease binding strength depending on their specific goals. For example, enhancing the effect of a neutralizing antibody against a virus usually requires increasing the binding strength between the antibody and the viral protein. However, the combinatorial space of amino acid mutations is large, so it is not always feasible or affordable to conduct wet-lab assays to test all viable mutations. Therefore, computational approaches are needed to guide the identification of desirable mutations by predicting their mutational effects on binding strength, typically measured by the change in binding free energy ( ∆∆ G ).

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