The task we want to address is the inference of the n A = n S − 1 unknown, ancestral sequences (Joy et al., 2016). We show that our probabilistic model, called Draupnir, is about on par with or better than the accuracy of established ASR methods for a standard experimentallyderived data set (Alieva et al., 2008; Randall et al., 2016) and several simulated data sets.

The task we want to address is the inference of the n A ≤ n S − 1 unknown, ancestral sequences (Joy et al., 2016). We show that our probabilistic model, called Draupnir, is about on par with or better than the accuracy of established ASR methods for a standard experimentallyderived data set (Alieva et al., 2008; Randall et al., 2016) and several simulated data sets. In addition, we show that Draupnir is capable of capturing coevolution among sequence positions, unlike conventional ASR methods.


Meanwhile, a feature transformation can be applied to increase ﬁlter strength (Li et al., 2019b). Recap theinput of GDN in Section 3.2.2 is the smoothed representations H (cid:48) and the recovered structure A (cid:48) , the inverse version of GCN can be written as:

Following GCN (Kipf & Welling, 2017), a feature transformation is applied to increase ﬁlter strength. Recap the GDN in Section 3.2, the inverse version of GCN can be written as:

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MIMO’s experiments build onthe Uncertainty Baselines framework. This framework allows us to benchmark the performance and to compare against high-quality, well-optimized implementations of baseline methods (see framework for further baselines than ones highlighted here). We looked at three model/dataset combinations: ResNet28-10/CIFAR10, ResNet28-10/CIFAR100, and ResNet50/ImageNet. MIMO’s code will be open-sourced.

We described and analyzed MIMO. In this section, we compare MIMO on benchmarks building on Uncertainty Baselines. This framework allows us to benchmark the performance and to compare against high-quality, well-optimized implementations of baseline methods (see framework for further baselines than ones highlighted here). We looked at three model/dataset combinations: ResNet28-10/CIFAR10, ResNet28-10/CIFAR100, and ResNet50/ImageNet. MIMO’s code is open-sourced.

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Dataset Every set has a ﬁxed size with n elements of dimensionality d . As previously mentioned, every element is sampled i.i.d. from N ( 0 , I ) . We assign 64,000 samples to the training set and 6, samples to the test set. Changing the random seed also changes the dataset. As loss and evaluation metric, we use Hungarian matching with  Huber loss  as pairwise loss. We always use the results for a model after the ﬁnal epoch without any early-stopping (we did not observe overﬁtting) or selection based on best loss.

Dataset. Every set has a ﬁxed size with n elements of dimensionality d . Every element is sampled i.i.d. from N ( 0 , I ) . We assign 64,000 samples to the training set and 6,400 samples to the test set. Changing the random seed also changes the dataset. As loss and evaluation metric, we use Hungarian matching with the Huber loss (Huber, 1964) (quadratic for distances below 1, linear for distances above 1) as pairwise loss. We always use the results for a model after the ﬁnal epoch without any early-stopping (we did not observe overﬁtting) or selection based on best loss.


One remaining issue, however, is the ignorance of the cognitive effort required of the human for understanding an explanation. In previous work, the human is expected to understand any explanation providedbefore the task execution, regardless of how much information is present. In this work, we argue that explanations, especially complex ones, should be provided in an online fashion, which intertwines the communication of explanations with plan execution. In such a manner, an online explanation requires less cognitive effort at any speciﬁc point of time. The challenge here, however, is that the different parts of an explanation are dependent on each other, which must be taken into account when generating online explanations. The online explanation generation process spreads out the information to be communicated while ensuring that they do not introduce cognitive gaps so that the different parts of the information are perceived in a smooth fashion.

One remaining issue, however, is the ignorance of the mental workload required of the human for understanding an explanation. In most earlier work on explanation generation, the human is expected to understand any explanation provided regardless of how much information is present and no discussion has been provided on the process for presenting the information. In this work, we argue that explanations, especially complex ones, should be provided in an online fashion, which intertwines the communication of explanations with plan execution. In such a manner, an online explanation requires less mental workload at any speciﬁc point of time. One of the main challenges here, however, is that the different parts of an explanation could be dependent on each other, which must be taken into account when generating online explanations. The online explanation generation process spreads out the information to be communicated while ensuring that they do not introduce cognitive dissonance so that the different parts of the information are perceived in a smooth fashion.


Even without exact duplicates in the input, very similar elements can still cause problems for set-equivariant models. For example, in order to learn the push apart function, it is necessary for a model to make a decision of which element is larger. This decision is fundamentally discontinuous. Since most models used in deep learning cannot represent discontinuous jumps, near the discontinuity (when the  elements are similar) the modeling error will likely be high.

Even without exact duplicates in the input, very similar elements can still cause problems for set-equivariant models. For example, in order to learn the push apart function, it is necessary for a model to decide which element is larger. This decision is fundamentally discontinuous, but most models in deep learning cannot represent discontinuous jumps. A continuous model must approximate the discontinuity, which means that close to the discontinuity (where elements are similar) the modeling error will likely be high. This is a general problem when trying to map similar elements in the input (e.g. due to random initialization of Y 0 ) to dissimilar elements in the output.


Visual Comparisons. We further provide visual comparisons ( × 4) in Fig.4 for challenging cases. We can observe that most of the compared methods cannot recover structural details with proper directions and suffer from blurring artifacts. In  contrast, our SRPN-L can better alleviate the blurring artifacts and recover more structures. These visual comparisons are consistent with the quantitative results, demonstrating the  superiority of our SRP method.

Visual Comparisons. The visual comparisons at × 4 scale are shown in Fig. It is easy to see that most of the comparison algorithms can barely recover the lost texture details in the correct direction; moreover, they suffer from blurring artifacts. In stark contrast, our SRPN-Lite effectively alleviates the blurring artifacts and recovers sharper structural ﬁne textures in the right way, justifying the performance superiority of our SRP approach over other methods.

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Architecture. Inspired by Carion et al. (2020), GPE employs  a ConvNeXt (Liu et al., 2022) pretrained on ImageNet (Russakovsky et al., 2015)  to extract spatial features of input video frames.

Architecture. Inspired by Carion et al. (2020), GPE employs the combination of convolution- and attention-based models to extract features. Concretely, a ConvNeXt (Liu et al., 2022) pre-trained on ImageNet (Russakovsky et al., 2015) is used to extract spatial features of input video frames.


We have developed a projection method between sets of probability measures supported on R d based on the tangential structure of the 2-Wasserstein space. Our method seeks to best approximate some  target distribution that is potentially multivariate, using some chosen set of control distributions. We provide an implementation which gives unique, interpretable weights in a setting of regular probability measures. For general probability measures, we construct our projection by ﬁrst creating a regular tangent space through applying barycentric projection to optimal transport plans. Our application to evaluating the ﬁrst- and second-order effects of Medicaid expansion in Montana via an extension of the synthetic controls estimator (Abadie & Gardeazabal, 2003, Abadie et al., 2010) demonstrates the method’s efﬁciency and the necessity to have a method that is applicable for general proabbility measures. The approach still works without restricting optimal weights to be in the unit simplex, which would allow for extrapolation beyond the convex hull of the control units, providing a notion of tangential regression. It can also be extended to a continuum of measures, using established consistency results of barycenters (e.g. Le Gouic & Loubes, 2017).

We have developed a projection method between sets of probability measures supported on R d based on the tangent cone structure of the 2-Wasserstein space. Our method seeks to best approximate some general target measure using some chosen set of control measures. In particular, it provides a global (and in most cases unique) optimal solution. Our application to evaluating the ﬁrst- and second-order effects of Medicaid expansion in Montana via an extension of the synthetic controls estimator (Abadie & Gardeazabal, 2003, Abadie et al., 2010) demonstrates the method’s utility in allowing for a method that is applicable for general probability measures. The method still works without restricting optimal weights to be in the unit simplex, which would allow for extrapolation beyond the convex hull of the control units, providing a notion of tangential regression. It can also be extended to a continuum of measures, using established consistency results of barycenters (e.g. Le Gouic & Loubes, 2017).

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However, recent work has questioned the efficacy of current OOD methods and reported sobering experimental results under rigorous examination [4, 14]. Nagarajan et al. [12] analyzed OOD failure modes, and found that spurious correlations induce two kinds of skews in the data: geometric and statistical skew. Geometric skew occurs when there is an imbalance between groups of types of datapoints, and leads to misclassification when the balance of groups changes. This understanding hasmotivated to simply remove data  from the training data to balance between groups of data points [2].

However, recent work has questioned the efficacy of current OOD methods and reported sobering experimental results under rigorous examination [3, 10]. Nagarajan et al. [8] analyzed OOD failure modes, and found that spurious correlations induce two kinds of skews in the data: geometric and statistical skew. Geometric skew occurs when there is an imbalance between groups of types of data points (such as data points from different environments) which induces a spurious correlation, and leads to misclassification when the balance of groups changes. This understanding has motivated simply removing data points from the training data to balance between groups of data points [2].


By applying the implicit function theorem on Equation 8, we obtain the Jacobians ∂ Y ∗ ( z , θ ) /∂ z and ∂ Y ∗ ( z , θ ) /∂ θ , which is all we need in order for iDSPN to ﬁt into an autodiff framework. Appendix B contains the full details of how this works.

The implicit function theorem allows us to differentiate Equation 8 and obtain the Jacobians ∂ Y ∗ ( z , θ ) /∂ z and ∂ Y ∗ ( z , θ ) /∂ θ , which is all we need in order for iDSPN to ﬁt into an autodiff framework. Appendix B contains the full details on how this works.

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We introduce a proactive multi-camera collaboration framework based on multi-agent reinforcement learning (MARL) for real-time distributive adjustments of multi-camera formation for 3D HPE in a human crowd. In our approach, multiple camera agents perform seamless collaboration for successful reconstructions of 3D human poses. Additionally, it is a decentralized framework which offers flexibility over the size of formation  and eliminates dependency to a control hierarchy or a centralized entity. In regards of the first challenge, we argue the utmost importance of the model’s capability to anticipate human motions and future states of the environment. To model these properties into the state representations, we incorporate World Dynamics Learning into the training of our model, i.e ., learning with five auxiliary tasks to predict  target’s position, pedestrians’ positions, self state, teammates’ states and team reward.

In in work, we introduce a proactive multi-camera collaboration framework based on multi-agent reinforcement learning (MARL) for real-time distributive adjustments of multi-camera formation for 3D HPE in a human crowd. In our approach, multiple camera agents perform seamless collaboration for successful reconstructions of 3D human poses. Additionally, it is a decentralized framework that offers flexibility over the formation size and eliminates dependency on a control hierarchy or a centralized entity. For the first challenge, we argue that the model’s ability to predict human movements and environmental changes is crucial. Thus, we incorporate World Dynamics Learning (WDL) to train a state representation with these properties, i.e ., learning with five auxiliary tasks to predict the target’s position, pedestrians’ positions, self state, teammates’ states, and team reward.

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We observe that after training, TR 2 -GPT2 exhibits an understanding of an optimal strategy to determine when to rotate or not. As shown in Fig. 4, whenever the agent is in a chamber which permits rotation, the agent attends to positions between the next chamber or next next chamber, all of which are indicative of the orientation of the upcoming corner. Attending to these locations enables the agent to successfully bridge the high- to low domain gap in Couch Moving. A video of full trajectory attention analysis can be found in the supplementary materials and our project page .

We observe that after training, TR 2 -GPT2 exhibits an understanding of an optimal strategy to determine when to rotate or not. As shown in Fig. 4, whenever the agent is in a chamber which permits rotation, the agent attends to positions between the next chamber or next next chamber, all of which are indicative of the orientation of the upcoming corner. Attending to these locations enables the agent to successfully bridge the high- to low domain gap in Couch Moving. Moreover, the agent learns to pay attention mostly to locations up ahead and learns that the past parts are uninformative, despite being given the full abstract trajectory to process at each timestep. A video of full trajectory attention analysis can be found in the supplementary materials and our project page .


In this paper, we first empirically show that winning tickets are actually more vulnerable to label noisesetting compared to the subnetwork created with the large learning rate; that is, the generalizationability of winning tickets is degraded due to the learning rate constraint. To explain this, we nextapply the PAC-Bayesian theory to LTH  and show that it can explain the relationship between LTHand generalization behavior. We use the PAC-Bayes bound for a spike-and-slab distribution toanalyze winning tickets, which is based on our experimental findings that reducing the expectedsharpness restricted to an unpruned parameter space and adding the regularization of distance fromthe initial weights can enhance the test performance of winning tickets. Finally, we revisit existingalgorithms such as IMP, continuous sparsification [34] from the point of view of the PAC-Bayesbound optimization. This consideration gives an interpretation of these methods as an approximationof bound optimization.

In this paper, we first empirically show that winning tickets are actually more vulnerable to label noisesetting compared to the subnetwork created with the large learning rate; that is, the generalizationability of winning tickets is degraded due to the learning rate constraint. In this connection, we thenfocus on the two concepts flatness and the distance from the initial weights of the winning tickets. Wenext apply the PAC-Bayesian theory to LTH on the basis of the flatness motivation and show that itcan explain the relationship between LTH and generalization behavior. We use the PAC-Bayes boundfor a spike-and-slab distribution to analyze winning tickets, which is based on our experimentalfindings that reducing the expected sharpness restricted to an unpruned parameter space and addingthe regularization of distance from the initial weights can enhance the test performance of winningtickets. Finally, we revisit existing algorithms such as IMP, continuous sparsification [34] from thepoint of view of the PAC-Bayes bound optimization. This consideration gives an interpretation ofthese methods as an approximation of bound optimization.


CIFAR-10 [20] is a classical image classification dataset with 10 categories. Each category has 5000images for training and 1000 images for testing. Existing methods [5, 21, 34] evaluate CIFAR-10 inthe separate case, where one class is viewed as normal samples, and others serve as anomalies. Incontrast, we propose the unified case, which is detailed in Sec. 4.4.

CIFAR-10 [22] is a classical image classification dataset with 10 categories. Existing methods [6, 23,36 ] evaluate CIFAR-10 mainly in the one-versus-many setting, where one class is viewed as normalsamples, and others serve as anomalies. Semantic AD [1, 9] proposes a many-versus-one setting,treating one class as anomalous and the remaining classes as normal. Different from both, we proposea unified case ( many-versus-many setting), which is detailed in Sec. 4.4. Metrics .


MIS task. Another limitation is that applying DIMES to a broader ranges of NP-complete problemsthat variables can take multiple values, such as Mixed Integer Programming (MIP), is not trivial andneeds further understanding of the nature of the problems. Checklist For all authors...

CO solvers. There do exist problems beyond this assumption, e.g., Mixed Integer Programming (MIP), where variables can take multiple integer values instead of binary values.


COCO-Soft with adversarial constraints: Adversarial constraints are more difﬁcult to satisfy buthave been considered in the literature [24, 20, 16, 6, 22]. For COCO-Soft with adversarial constraints,the authors in [24] developed an online mirrored descent type algorithm that achieves O p? T q regretand O p T 3 { 4 q violation. Later, [6, 16] generalized the baseline in [24] and still achieve O p ? T q regret and O p T 3 { 4 q violation. With Slater’s condition, [20] presents an online gradient descent algorithmbased on the drift-plus-penalty method [19], which achieves O p? T q regret and O p? T q violation and[22] extended it to an online optimization with sub-modular losses. Note the key improvement inthese work is the Lyapunov drift technique that can provide a reﬁned bound on virtual queues (ordual variables) with Slater’s condition, which thus achieves a smaller soft constraint violation. Itremains open that whether O p T 3 { 4 q violation can be reduced with adversarial constraints (soft orhard) while keeping O p? T q regret without Slater’s condition.

COCO-Soft with adversarial constraints: Adversarial constraints are more difﬁcult to satisfy buthave been considered in the literature [22, 18, 14, 5, 20]. For COCO-Soft with adversarial constraints,the authors in [22] developed an online mirrored descent type algorithm that achieves O p? T q regretand O p T 3 { 4 q violation. Later, [5, 14] generalized the baseline in [22] and still achieve O p ? T q regret and O p T 3 { 4 q violation. With Slater’s condition, [18] presents an online gradient descent algorithmbased on the drift-plus-penalty method [17], which achieves O p? T q regret and O p? T q violationand [20] extended it to an online optimization with sub-modular losses. It remains open that whether O p T 3 { 4 q violation can be reduced with adversarial constraints (soft or hard) while keeping O p? T qregret without Slater’ condition.

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Training ﬂow-based encoders The next challenge is to design a training procedure for our newly proposed architecture. The main issue is that the statistical distance is not differentiable (as classiﬁer µ ∗ is binary) so wewant to replace it with a differentiable proxy. Pinsker’s inequality guarantees that the statistical distance between Zand Z 1 can be bounded using symmetrized KL divergence between Z 0 and Z 1 (see Appendix A. for more detailed proof). We show a high-level description of our training procedure in Algorithm 1. In each step, we sample a batch of x 0 and xfrom the respective distributions and encode them to the representations z 0 and z 1 . We then estimate a symmetrized KL-divergence between distributions Z 0 and Z 1 , denoted as L 0 + L 1 , and combine it with a classiﬁcation loss L clf using tradeoff parameter γ , and run a gradient step to minimize the joint loss. While we use a convex scalarization scheme to obtain the joint loss in Algorithm 1, our approach is independent of the concrete multi-objective optimization objective. We will demonstrate the compatibility with other scalarization schemes in Appendix C.

Training ﬂow-based encoders The next challenge is to design a training procedure for our newly proposed architecture. The main issue is that the statistical distance is not differentiable (as classiﬁer µ ∗ is binary) so we replace it with a differentiable proxy based on symmetrized KL divergence, shown in Lemma 5.3 (proof is shown in ?? ). We show a high-level description of our training procedure in Algorithm 1. In each step, we sample a batch of x 0 and x 1 from the respective distributions and encode them to the representations zand z 1 . We then estimate a symmetrized KL divergence between distributions Z 0 and Z 1 , denoted as L 0 + L 1 , and combine it with a classiﬁcation loss L clf using tradeoff parameter γ , and run a gradient step to minimize the joint loss. While we use a convex scalarization scheme to obtain the joint loss in Algorithm 1, our approach is independent of the concrete multi-objective optimization objective. We will demonstrate the compatibility with other scalarization schemes in Appendix C.

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Representation-based methods mitigate the treatment selection bias and enhance the overall performance. In particular, CFR-WASS reaches an out-of-sample PEHE of 3.207 on ACIC, significantly outperforming most statistical methods. It also gets an AUUC of 0.715 on IHDP, exceeding all other baselines. However,  MSE and UCE issues hinder the mitigation of treatment selection bias by these methods. The proposed ESCFR achieves significant improvement over most metrics compared with various state-of-the-art baselines 3 . Combined with aforementioned comparisons, we attribute its superiority to our design of the  RMPR and PFOR regularizers, making it robust to MSE and UCE.

Representation-based methods mitigate the treatment selection bias and enhance overall performance. In particular, CFR-WASS reaches an out-of-sample PEHE of 3.207 on ACIC, significantly outperforming most statistical methods. However, the MSE and UCE issues impede these methods from solving the treatment selection bias. The proposed ESCFR achieves significant improvement over most metrics compared with various prevalent baselines 4 . Combined with the comparisons above, we attribute its superiority to the proposed RMPR and PFOR regularizers, which makes it robust to MSE and UCE.

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For the dog images in Figure 2, although they are quite different at the pixel level, they contain similar semantic meanings. Meanwhile, they  have a small augmented distance. Thus, the semantic distance can be partially characterized by the proposed augmented distance. Based on the augmented distance, we now introduce the ( σ, δ ) -augmentation to measure the concentration of augmented data.

For the dog images in Figure 2 as an example, even though their pixel-level differences are significant, their semantic meanings are similar. Meanwhile, they also have a small augmented distance. Thus, the proposed augmented distance can partially capture the semantic distance. Based on the augmented distance, we now introduce the ( σ, δ ) -augmentation to measure the concentration of augmented data.

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Figure 13(b) shows the result on both train and test actions. AGILE-Tuned, With pre-summrizer , and No twin-GAT showed the similar performance which is better than No target-q-change . The difference between AGILE-Tuned and No target-q-change shows thatsince the cascaded networkconsiders the intermediate list constructed in decision-making, in the computation of the target qvalues the agent also needs to refer to the  q-value of the next position in the same list instead of another list from the future time-step.

Figure 13(b) shows the result on both train and test actions. AGILE-Tuned, With pre-summrizer , and No twin-GAT showed the similar performance which is better than No target-q-change . The difference between AGILE-Tuned and No target-q-change is that that the cascaded network in AGILETuned uses the target q-value from the intermediate list constructed. This is a more accurate target q-value as compared to the target q-value from another list from a future time-step.

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Sequence-based models do not  predict ∆∆ G accurately for protein-protein binding in accordance with the discussion in Section 2.2. Figure 3 plots the distribution of per-complex correlation coefficients. Please refer to Section B of the appendix for more results and discussion.

Sequence-based models do not accurately predict ∆∆ G for protein-protein binding, as discussed in Section 2.2. Figure 3 shows the distribution of per-complex correlation coefficients. Please refer to Section B of the appendix for more results and discussion.

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• We formulate  active  3D human pose estimation in a human crowd problem as a Dec-POMDP, and proposed a novel multi-camera ( n ≥ 3) collaboration framework. • We  propose CTCR to improve credit assignment  in multi-camera collaboration and demonstrate notable improvements in reconstruction accuracy compared to both passive and active baselines.• We introduce five auxiliary tasks called to help the model learn environment dynamics, further enhancing the model’s ability to handle highly dynamic scenes. • We contribute high-fidelity environments built for simulating realistic- looking human crowds with authentic behaviors, along with visualization software for frame-by-frame video analysis.

• We formulate the active multi-camera 3D human pose estimation problem as a Dec-POMDP, and proposed a novel multi-camera ( n ≥ 3) collaboration framework based on MARL. • We introduce five auxiliary tasks to help the model learn the dynamics of the environment, further enhancing the model’s ability to handle highly dynamic scenes. • We propose CTCR to address the credit assignment problem in MARL and demonstrate notable improvements in reconstruction accuracy compared to both passive and active baselines. • We contribute high-fidelity environments built for simulating realistic-looking human crowds with authentic behaviors, along with visualization software for frame-by-frame video analysis.

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Attention in the policy network are as follows, d model = 64 , n head = 6 , d v = 32 , d k = 32 . The maximum number of search epochs N is limited to 300 (including NFS, DIFER and F ETCH ), and the  number of sampling also parallelized workers per round, is W = 24 . The maximum feature order K is set by K = 3 . Other methods are limited to run for 5 hours respectively, which is the average running time of F ETCH . All methods take their default parameters wherever possible.

Attention in the policy network are as follows, d model = 64 , n head = 6 , d v = 32 , d k = 32 . The maximum number of search epochs N is limited to 300 (including DIFER and F ETCH ). Due to the requirements of NFS in their paper, we set N to 100 epochs for it. The number of sampling also parallelized workers per round, is W = 24 . The maximum feature order K is set by K = 3 . Other methods are limited to run for 5 hours respectively, which is the average running time of F ETCH . All methods take their default parameters wherever possible.


Jung et al. (2021) proposed a standardized max-logit approach for detecting outliers in semantic segmentation, which is a post-hoc approach. Zhao et al. ; Grcic et al. (2021) trained a generative model and synthesize outliers in the pixel space, which cannot be applied to object detection where a scene consists of both known and unknown objects. Their regularization terms are based on entropy maximization, which is different from VOS .

Jung et al. (2021) proposed to detect outliers for semantic segmentation task. Jung et al. Grcic et al. (2021) trained a generative model and synthesize outliers in the pixel space, which cannot be applied to object detection where a scene consists of both known and unknown objects. The regularization is based on entropy maximization, which is different from VOS .

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By default, we sample a random initial set Y 0 ∼ N ( 0 , I / 10) to start the optimization with. Similar to DSPN, we can also use a learned initial set Y 0 to start closer to a solution. Unlike DSPN however, implicit differentiation treats the optimizer of Equation 7 as a black box, so there is no gradient signal for Y 0 . We therefore need to include a regularizer in Equation 6 to give us a gradient for Y 0 , for example by adding the regularizer from Rajeswaran et al. (2019): λ  || Y − Y
 || 2 . Wewill use this when comparing iDSPN to  DSPN performance in Subsection 4.2.

By default, we sample a random initial set Y 0 ∼ N ( 0 , I / 10) to start the optimization with. Similar to DSPN, we can also use a learned initial set Y 0 to start closer to a solution. However, implicit differentiation treats the optimizer of Equation 7 as a black box, so there is no gradient signal for Y 0 . We therefore need to include a regularizer in Equation 6 to give us a gradient for Y 0 , for example by adding the regularizer from Rajeswaran et al. λ 2 || Y − Y
 || 2 . We use this only in Section 4.2 to make the forward passes of iDSPN and DSPN the same so that we can compare them fairly.

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Shape Classification. We fine-tune Point-M2AE on two shape classification datasets: the widely adopted ModelNet40 [43] and the challenging ScanObjectNN [37]. We follow Point-BERT to usethe voting strategy [25] for fair comparison on ModelNet40, which tests the model for several timeswith different point cloud augmentation and ensembles the predictions. To handle the noisy spatial structures, we increase k of k -NN into {32, 16, 16} for ScanObjectNN to encode local patterns with larger receptive fields. As reported in Table 2, Point-M2AE achieves 94.0% accuracy on ModelNet40with 1024 points per sample, which surpasses Point-BERT fine-tuned with 1024 points by +0.8% and 8192 points by +0.2%. For ScanObjectNN in Table 3, our Point-M2AE outperforms the secondbest Point-BERT by a significant margin, +3.79%, +0.69% and +3.36%, respectively for the three splits, indicating our great advantages under complex circumstances by multi-scale encoding. As

Shape Classification. We fine-tune Point-M2AE on two shape classification datasets: the widely adopted ModelNet40 [53] and the challenging ScanObjectNN [45]. For local spatial attention layers, we set the ball queries’ radii of 3-scale point clouds as {0.32, 0.64, 1.28}. We follow Point-BERT to use the voting strategy [29] for fair comparison on ModelNet40. To handle the noisy spatial structures, we increase k of k -NN into {32, 16, 16} for ScanObjectNN to encode local patterns with larger receptive fields. As reported in Table 2, Point-M2AE achieves 94.0% accuracy on ModelNet with 1024 points per sample, which surpasses Point-BERT fine-tuned with 1024 points by +0.8% and 8192 points by +0.2%. For ScanObjectNN in Table 3, our Point-M2AE outperforms the secondbest Point-BERT by a significant margin, +3.79%, +0.69% and +3.36%, respectively for the three splits, indicating our great advantages under complex circumstances by multi-scale encoding. As


This section proposes the ESCFR approach  to tackle the treatment selection bias. It is built upon the stochastic optimal transport framework for distribution discrepancy minimization across treatment groups. Based on the framework, we  propose a relaxed mass preserving regularizer to address the sampling effect, and a proximal factual outcome regularizer to handle the unobserved confounders.

In this section, we present the proposed Entire Space CounterFactual Regression (ESCFR) approach based on optimal transport to tackle the treatment selection bias. We first illustrate the stochastic optimal transport framework for distribution discrepancy minimization across treatment groups. Based on the framework, we then propose a relaxed mass-preserving regularizer to address the sampling effect, and a proximal factual outcome regularizer to handle the unobserved confounders.


• Full dense : original large and dense model, with N parameters; • Thin dense : original model with less wide layers, such that it had (1 − s ) N parameters; • Static sparse : original model reparameterized to sparsity s , then trained with connectivity ﬁxed; • Compressed sparse : state-of-the-art compression of the original model by iterative pruning and retraining the original model to target sparsity s (see Appendix A for details of implementation).

• Full dense : original large and dense model, with N parameters; • Thin dense : original model with less wide layers, such that it had (1 − s ) N parameters; • Static sparse : original model reparameterized to sparsity s , then trained with connectivity ﬁxed; • Compressed sparse : state-of-the-art compression of the original model by iterative pruning and retraining the original model to target sparsity s (Zhu & Gupta, 2017); • DeepR : sparse model trained by using Deep Rewiring (Bellec et al., 2017); • SET : sparse model trained by using Sparse Evolutionary Training (Mocanu et al., 2018). Note that compressed sparse is a compression method that starts training with a dense model, whereas DeepR and SET , like ours, are dynamic reparameterization techniques that maintain sparsity throughout training. See Appendix A for hyperparameters used in the experiments.


The easiest way to ensure m n = m n +1 , such that the acceptance probability remains nonzero, is to set the time step h = 0 . This, however, makes the sampler useless. If we take the limit h → 0 instead, the Euler-Maruyama scheme gets arbitrarily close to the true SDE trajectory. Crucially however, per theorem 1, the acceptance probability remains 0 for any h > 0 . Thus, it is impossible to use the acceptance probability to monitor the discretisation error. The Euler-Maruyama scheme cannot satisfy detailed balance. Note that, so far, we have not considered stochastic gradients: this result is valid forany choice of ∇ θ U ( θ n ) .

The easiest way to ensure m n = m n +1 , such that the acceptance probability remains nonzero, is to set the time step h = 0 . This, however, makes the sampler useless. If we take the limit h → 0 instead, the Euler-Maruyama scheme gets arbitrarily close to the true SDE trajectory. Crucially however, per theorem 1, the acceptance probability remains 0 for any h > 0 . Thus, it is impossible to use the acceptance probability to monitor the discretisation error. The Euler-Maruyama scheme cannot satisfy detailed balance. So far, we have not considered stochastic gradients, but this result includes them: it still holds if we substitute an arbitrary g n for ∇ θ U ( θ n ) .

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Prescriptions come with constraints associated with drug dosage units, frequency and  indications. Constraints can occur within the same medication and between different medications. An example of within-medication constraints is found in the the prescription Take 600 mg of Ibuprofen three times a day as needed with food , that has three constraints: 1) that 600mg should be taken at a given time; 2) that the maximum number of intakes per day is three; and 3) that the drug must be taken with food. But medications are often more complex. Consider for example the following prescription:

Prescriptions come with constraints, e.g., drug dosage, adminis- tration frequency and other indications. Constraints may relate to a single medication or a set of different medications. An example of within-medication constraints is found in the the prescription Take 600 mg of Ibuprofen three times a day as needed with food , that has three constraints: 1) that 600mg should be taken at a given time; 2) that the maximum number of intakes per day is three; and 3) that the drug must be taken with food. But medications are often more complex. Consider for example the following prescription:

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The dice-enterprise produces unbiased ancestor variables. For details on efﬁciency and correctness, please refer to Section 3.1. Our proposed VSMC-PRC bound is constructed through a marginal likelihood estimator obtained by combining the SMC sampler with a PRC step and dice-enterprise . The variance of estimators obtained through SMC-PRC particle ﬁlter is usually low (Peters et al., 2012). Therefore, we expect VSMC-PRC to be a tighter bound compared to the standard SMC based bounds used in recent works (Maddison et al., 2017; Naesseth et al., 2017; Le et al., 2017).

The dice-enterprise produces unbiased ancestor variables. Note that we can easily control the efﬁciency of the proposed dice-enterprise through hyper-parameter M (similar to Eq. 2) in contrast to existing Bernoulli factory algorithms (Flegal et al., 2012; Schmon et al., 2019). For details on efﬁciency and correctness, please refer to Section 3.1 and Section 3.3. Our proposed VSMC-PRC bound is constructed through a marginal likelihood estimator obtained by combining the SMC sampler with a PRC step and dice-enterprise . The variance of estimators obtained through SMC-PRC particle ﬁlter is usually low (Peters et al., 2012). Therefore, we expect VSMC-PRC to be a tighter bound compared to the standard SMC based bounds used in recent works (Maddison et al., 2017; Naesseth et al., 2017; Le et al., 2017).


Attention mechanism in transformer is the key component which models relations between feature representations. In this section, we visualize the self-attentions mechanism of our Entroformer, focusing on a few points in the image. The attention map is rescaled for a better view. In Figure 8, we visualize self-attention heads from different points These attention heads exhibit different behaviours that seem related to structure, semantic and color respectively. These visualizations show how the Entroformer ﬁnds related context to support its distribution prediction for the current latent. This allows the Entroformer to capture richer dependencies of the latents and achieve better compression performance. In Figure 9, we visualize self-attentions heads separately from one point.

Attention mechanism in transformer is the key component which models relations between feature representations. In this section, we visualize the self-attentions mechanism of our Entroformer, focusing on a few points in the image. In Figure 8, the attention maps from different points exhibit different behaviours that seem related to structure, semantic and color respectively. It show how the Entroformer ﬁnds related context to support its distribution prediction for the current latent. In Figure 9, we visualize self-attentions heads separately from one point.

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SuperWeight Clustering approach described in Section 3.3. We provide two baselines: a manually defined heuristic, depth binning, which as used for our initialSuperWeight Clusters in Section 3. and prior work that proposed to clusters coefficients α in Eq. (1) to group layers (Plummer et al., 2022). Our results show that clustering based on the coefficients performs on par with the heuristic baseline. In contrast, our gradient analysis approach (Section 3.3) takes into account the direction of change rather than just the current value of the coefficients. As a result we obtain a 2% gain on individual models and a small boost to ensembling performance with our approach (Table 3). We show a visualization of SuperWeight cluster assignment in Appendix G.

SuperWeight Clustering approach described in Section 3.3. We provide four baselines: Shared Coefficients , which learns SuperWeight Clusters, but shares coefficients between all layers ( i.e ., removing Section 3.2.2); Single SuperWeight Cluster , which allows layers to have their own coefficients, but does not learn clusters ( i.e ., removing Section 3.3); Depth-binning , a heuristic used for our initial SuperWeight Clusters in Section 3.3; and Coefficient Clustering (Plummer et al., 2022), which clusters coefficients α in Eq. (1) to group layers. Our results show that our approach outperforms all of our baselines. Notably, we find we that Coefficient Clustering performs in par or worse than other baselines. In contrast, our gradient analysis approach (Section 3.3) takes into account the direction of change rather than just the current value of the coefficients. Thus, we obtain a 2% gain on individual models and a small boost to ensembling performance with our approach (Table 3). We show a visualization of SuperWeight cluster assignment in Appendix C.6.


Methods for predicting mutational effects for single proteins are either structure-based or sequencebased (evolution-based). Structure-based approaches can also be divided into biophysical methods, statistical methods, and deep learning-based methods. They aim at predicting thermal stability or fitness of the protein rather than  binding free energy between proteins (Schymkowitz et al., 2005; Park et al., 2016; Alford et al., 2017). The mutational effects of single proteins can also be predicted using only sequences via mining its evolutionary history. Traditionally, this is done by performing statistics on multiple sequence alignments (MSAs), which are constructed by searching from largescale sequence databases (Hopf et al., 2017; Riesselman et al., 2018; Rao et al., 2021; Luo et al., 2021; Frazer et al., 2021; Notin et al., 2022). Recent studies show that protein language models (PLMs)trained on large sequence databases are capable of evaluating mutations without MSAs (Meier et al., 2021).

The prediction of mutational effects for single proteins can be achieved using either structure-based or sequence-based (evolution-based) approaches. Structure-based methods can be categorized into biophysical, statistical, and deep learning-based methods, which aim to predict the thermal stability or fitness of the protein rather than the binding free energy between proteins (Schymkowitz et al., 2005; Park et al., 2016; Alford et al., 2017; Lei et al., 2023). Sequence-based methods rely on the mining of evolutionary history, done by performing statistics on multiple sequence alignments (MSAs) constructed from large-scale sequence databases (Hopf et al., 2017; Riesselman et al., 2018; Rao et al., 2021; Luo et al., 2021; Frazer et al., 2021), or leveraging protein language models (PLMs) (Meier et al., 2021; Notin et al., 2022).

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Window based critical event prediction: Intervention is a binary decision. At each time-step adecision must be made to intervene or not. This is often modeled directly as a binary classiﬁcation problem where we look for event occurrence within a speciﬁed future window. In window basedintervention (WBI), time-steps where an event is present within a lookahead period are labeled aspositive class examples and the rest are labeled as negative examples. A classiﬁer may then be learntto predict the probability of the positive class. An optimal threshold on this probability is then tunedover a validation set that measures costs of triggered interventions directly to choose an optimal triggerthreshold. WBI policies are standard practice in predictive maintenance[33, 2, 22, 2, 38, 40, 13].

Window based critical event prediction: Intervention is a binary decision. At each time-step adecision must be made to intervene or not. This is often modeled directly as a binary classiﬁcation analysis approach. We may treat unobserved critical events as censored. See Section 4.1 for detailsa stopping time in probability theory is random a variable τ such that 1 ( τ = n ) is a function of X n . So we can determine if τ = j or not by only considering X j [29]. problem where we look for event occurrence within a speciﬁed future window. In window basedintervention (WBI), time-steps where an event is present within a lookahead period are labeled aspositive class examples and the rest are labeled as negative examples. A classiﬁer may then be learntto predict the probability of the positive class. An optimal threshold on this probability is then tunedover a validation set that measures costs of triggered interventions directly to choose an optimal triggerthreshold. WBI policies are standard practice in predictive maintenance[33, 2, 22, 2, 38, 40, 13].


Conflict overlays were easily identifiable on all the designs. The user-preferred way to represent such conflicts is to use  indicators for the position of medication entries that are involved in the conflict. The connectors  for conflicting pairs should avoid thick solid lines, that create clutter. Instead, thin or dotted lines should be employed. While lines are effective in connecting conflicting entries, employing line style to indicate the nature of the conflict is contrariwise. Dif- ferent line styles may appear similar at a distance and hence fail in communicating the intended message. Line style should be carefully employed to imply an action.

Conflict overlays were easily identifiable on all the designs. Par- ticipants preferred the use of indicators for the position of medication entries that are involved in the conflict. The connectors (lines) for conflicting entries should use thin or dotted lines rather than thick solid lines. Participants found that different line styles may appear similar at a distance and hence fail in communicating the nature of a conflict.

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Relative position encoding is necessary for transformer-based networks because fine-grained positioninformation may be lost in high-level features with the deepening of the network. To make better useof position information to facilitate multi-scale feature learning in our case, we adopt a scale-awareadaptive relative position encoding strategy inspired by [30, 46, 52], which can generate the positionalbias dynamically with scales for different head groups.

The 3D point cloud feature generally contains the original coordinates information, which voxelswill inherit. However, the fine-grained location information may be blurred with the deepening ofthe network, so the relative position encoding is necessary. Furthermore, since MsSVT can extractmulti-scale features, the position-coding should differ at different scales, even for the same relativeposition. In light of these, we adopt an adaptive, scale-aware relative position encoding strategyinspired by [25, 38, 44].


Graph Attention Network : The action graph G is input to a GAT. Since the graph is fullyconnected, we choose an attention-based graph architecture which can learn to focus on the most relevant actions in the given set. A similar insight was employed by Zambaldi et al. (2018) where the entities inferred from the visual observation are assumed to form a fully connected graph. To enable propagation of sufﬁcient relational information between actions, we use two graph attention layers with an ELU activation in between (Clevert et al., 2015). We found a residual connection after the second GAT layer was crucial in experiments.

Graph Attention Network : The action graph G is input to a GAT. Since the graph is fullyconnected, we choose an attention-based graph network that can learn to focus on the most relevant actions in the available action set. A similar insight was employed by Zambaldi et al. (2018) where the entities inferred from the visual observation are assumed to form a fully connected graph. To enable propagation of sufﬁcient relational information between actions, we use two graph attention layers with an ELU activation in between (Clevert et al., 2015). We found a residual connection after the second GAT layer was crucial in experiments, while multi-headed attention did not help.


Before we deﬁne the diffusion distance, we brieﬂy introduce the intuition behind it: two nodes are considered similar when they diffuse in a similar way through the graph, and therefore when they inﬂuence the other nodes in a similar manner (Fouss et al., 2012). In other words, two nodes are close if they are in the same cluster which has a consistent local structure. More precisely, the diffusion distance at time K between nodes i and j is deﬁned as follows:

Two nodes are considered similar when they are diffused in a similar way through the graph, and therefore when they inﬂuence the other nodes in a similar manner (Fouss et al., 2012). In other words, two nodes are close if they are in the same cluster which has a consistent local structure. More precisely, the diffusion distance at time K between nodes i and j is deﬁned as follows:

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Proof. Let A psoc be a real-world adversary that semi-honestly corrupts T out of N servers at the beginning of the protocol Π train . We now present the steps of the ideal-world adversary (simulator) S f for A psoc . Note that, in the semi-honest setting S f already posses the input of A psoc and the final output shares of b val . S f acts on behalf of N − T honest servers, sets their shares as random values in Z 2 ℓ and simulates each step of Π train protocol to the corrupt servers as follows:

Proof. Given the training framework securely realizes each of the building block used in protocol Π train , we now argue Π train securely realizes functionality F pTrain . Let A psoc be a real-world adversary that semi-honestly corrupts T out of N servers at the beginning of the protocol Π train . We now present the steps of the ideal-world adversary (simulator) S f for A psoc . Note that, in the semi-honest setting S f already posses the input of A psoc and the final output shares of b val . S f acts on behalf of N − T honest servers, sets their shares as random values in Z 2 ℓ and simulates each step of Π train to the corrupt servers as follows:


Data and Evaluation. We use DIV2K dataset (Timofte et al., 2017)  as training data, following most recent works (Timofte et al., 2017; Lim et al., 2017; Zhang et al., 2018a; Haris et al., 2018). For testing, we use ﬁve standard benchmark datasets: Set5 (Bevilacqua et al., 2012), Set14 (Zeyde et al., 2010), B100 (Martin et al., 2001), Urban100 (Huang et al., 2015), and Manga109 (Matsui et al., 2017). The SR results are evaluated with PSNR and SSIM (Wang et al., 2004) on  Y channel of transformed YCbCr space. We also provide model size and FLOPs (a.k.a. Mult-Adds) comparisons.

Data and Evaluation. We use DIV2K dataset (Timofte et al., 2017) and Flickr2K Lim et al. (2017) as training data, following most recent works (Timofte et al., 2017; Lim et al., 2017; Zhang et al., 2018a; Haris et al., 2018). For testing, we use ﬁve standard benchmark datasets: Set5 (Bevilacqua et al., 2012), Set14 (Zeyde et al., 2010), B100 (Martin et al., 2001), Urban100 (Huang et al., 2015), and Manga109 (Matsui et al., 2017). The SR results are evaluated with PSNR and SSIM (Wang et al., 2004) on the Y channel in YCbCr space.


We have presented an analytical method to predict the precision required for partial sum accumulation in the three GEMM functions in deep learning training. Our results prove that our method is able to accurately pinpoint the minimum precision needed for the convergence of benchmark networks to the full-precision baseline. While we have demonstrated the applicability of our work to computer vision datasets, in principle, the theoretical concepts are application agnostic. On the practical side, this analysis is a useful tool for hardware designers implementing reduced-precision FPUs, who in the past have resorted to computationally prohibitive brute-force emulations. We believe this work addresses a critical missing link on the path to truly low-precision ﬂoating-point hardware for DNN training.

We have presented an analytical method to predict the precision required for partial sum accumulation in the three GEMM functions in deep learning training. Our results prove that our method is able to accurately pinpoint the minimum precision needed for the convergence of benchmark networks to the full-precision baseline. Our theoretical concepts are application agnostic, and an interesting extension would be to consider recurrent architectures such as LSTMs. In particular, training via backpropagation in time could make the GRAD accumulation very large depending on the number of past time-steps used. In such a case, our analysis is of great relevance to training precision optimization. On the practical side, this analysis is a useful tool for hardware designers implementing reduced-precision FPUs, who in the past have resorted to computationally prohibitive brute-force emulations. We believe this work addresses a critical missing link on the path to truly low-precision ﬂoating-point hardware for DNN training.


What makes OOD detection particularly challenging? To explain this, modern neural networks are commonly optimized on thein-distribution (ID) data only, and lack explicit knowledge of unknowns during training time. The resulting decision boundary, despite being useful on ID tasks such as classiﬁcation, can undesirably cover OOD data. We illustrate this in Figure 1. The ID data (gray) consists of three class-conditional Gaussians, on which a three-way softmax classiﬁer is trained. The resulting classiﬁer is overconﬁdent for regions far away from the ID data (see the red shade in Figure 1(b)). When such models are directly employed, the decision boundary can be ill-fated for OOD detection. Ideally, a model should learn a more compact decision boundary that produces low uncertainty for the ID data, with high OOD uncertainty elsewhere ( e.g. , Figure 1(c)). However, achieving this goal is non-trivial due to the lack of supervision signal of unknowns. It is hard to comprehensively anticipate unknown data in advance under a large space of OOD uncertainty. This prompts thefollowing question: How can we enable unknown-aware deep neural networks without explicit knowledge of the unknowns in advance?

The vulnerability to OOD inputs arises due to the lack explicit knowledge of unknowns during training time. In particular, neural networks are typically optimized only on the in-distribution (ID) data. The resulting decision boundary, despite being useful on ID tasks such as classification, can be ill-fated for OOD detection. We illustrate this in Figure 1. The ID data (gray) consists of three class-conditional Gaussians, on which a three-way softmax classifier is trained. The resulting classifier is overconfident for regions far away from the ID data (see the red shade in Figure 1(b)), causing trouble for OOD detection. Ideally, a model should learn a more compact decision boundary that produces low uncertainty for the ID data, with high OOD uncertainty elsewhere ( e.g. , Figure 1(c)). However, achieving this goal is non-trivial due to the lack of supervision signal of unknowns. This motivates the question: Can we synthesize virtual outliers for effective model regularization?

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In general, Theorem 3.1 does not provide a straightforward path to analyzing the decay of the NTK power series coefﬁcients for depths greater than two. This is  due to the difﬁculty in analyzing F ( p, k, ¯ α l − 1 ) , which we recall is the sum of all ordered products of k elements of ¯ α l − 1 whose indices sum to p , deﬁned in (4). However, in the setting where the squares of the Hermite coefﬁcients, and therefore the series ( α p, 2 ) ∞ p =0 , decay at an exponential rate, this quantity can be characterized and therefore an analysis of the impact of depth conducted. Although admittedly limited in scope, we highlight that this setting is relevant for the study of Gaussian activation functions and radial basis function (RBF) networks, which to the best of our knowledge have not been analyzed in prior works. Under an additional  assumption that the activation function has zero bias, which helps simplify the analysis, the following lemma precisely describes the evolution of the coefﬁcients of the related Gaussian Process kernel. We leave  relaxing this zero bias assumption, which we expect to not be materially different from the non-zero bias setting, as well as only enforcing exponential decay asymptotically and exploring other decay patterns, to future work.

In general, Theorem 3.1 does not provide a straightforward path to analyzing the decay of the NTK power series coefﬁcients for depths greater than two. This is at least in part due to the difﬁculty of analyzing F ( p, k, ¯ α l − 1 ) , which recall is the sum of all ordered products of k elements of ¯ α l −whose indices sum to p , deﬁned in (4). However, in the setting where the squares of the Hermite coefﬁcients, and therefore the series ( α p, 2 ) ∞ p =0 , decay at an exponential rate, this quantity can be characterized and therefore an analysis, at least to a certain degree, of the impact of depth conducted. Although admittedly limited in scope, we highlight that this setting is relevant for the study of Gaussian activation functions and radial basis function (RBF) networks, which to the best of our knowledge have not been analyzed in prior works. We will also make the additional simplifying assumption that the activation function has zero bias, unfortunately this further reduces the applicability of the following results to any activation function commonly used in practice. We leave the study of relaxing this zero bias assumption, perhaps only enforcing exponential decay asymptotically, as well as a proper exploration of other decay patterns, to future work. The following lemma precisely describes, in the speciﬁc setting considered here, the evolution of the coefﬁcients of the Gaussian Process kernel with depth.

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• Majority voting of the labeling functions, where ties are broken by choosing the most common class as speciﬁed by the labling functions. • BERT [23] is a transformer-based language model that has shown exceptional performance on a wide-range of NLP tasks. We train BERT with the majority vote labels from our LFs.• MSWS [16] is a denoising method for multi-source weak supervision that co-trains cotrainsa rule denoiser with a neural classiﬁer to learn optimal weightings for rules and labelunmatched samples.

• Majority voting of the labeling functions, where ties are broken by choosing the most common class as speciﬁed by the labeling functions. • BERT [23] is a transformer-based language model that has shown exceptional performance on a wide-range of NLP tasks. We train BERT with the majority vote labels from our LFs. • Epoxy [26] is a recent weak supervision paradigm that uses combined pretrained embeddings with anchored weakly-labeled examples to enable interactive model training. • MSWS [16] is a denoising method for multi-source weak supervision that co-trains a rule denoiser with a neural classiﬁer to learn optimal weightings for rules and label unmatched samples. • Fully Supervised We train a BERT model with a fully connected layer to provide baseline performance for a model trained with ground-truth labels.


The property of exclusive multiset-equivariance motivates us to study DSPN in greater detail. There are of course other factors that determine the quality of a set predictor, so we begin by pointing out some remaining problems of DSPN.

Despite being exclusively multiset-equivariant, DSPN is outperformed by the set-equivariant Slot Attention (Locatello et al., 2020). We begin by highlighting some issues in DSPN that might be the cause of this, then propose approximate implicit differentiation as a solution.


For all experiments we use an image size of 128 × 128 and a batch size of 12 to 16 depending on memory usage. We use the RMSProp (Tieleman & Hinton. (2012)) optimizer with a learning rate of 1 × 10 − 5 for the foreground module and Adam (Kingma & Ba (2014)) optimizer with a learning rate of 1 × 10 − 3 for the background module except for Figure 5, for which we use a learning rate of 1 × 10 − 4 as SPAIR to ensure fair comparison. We use gradient clipping with a maximum norm of 1.0. For Atari games, we ﬁnd it beneﬁcial to set α to be ﬁxed for the ﬁrst several thousand steps, and vary the actual value and number of steps for different games. This allows both the foreground as well as the background module to learn in the early stage of training.

For all experiments we use an image size of 128 × 128 and a batch size of 12 to 16 depending on memory usage. For the foreground module, we use the RMSProp (Tieleman & Hinton. (2012)) optimizer with a learning rate of 1 × 10 − 5 except for Figure 5, for which we use a learning rate of 1 × 10 − 4 as SPAIR to ensure fair comparison. For the background module, we use the Adam (Kingma & Ba (2014)) optimizer with a learning rate of 1 × 10 − 3 . We use gradient clipping with a maximum norm of 1.0. For Atari games, we ﬁnd it beneﬁcial to set α to be ﬁxed for the ﬁrst several thousand steps, and vary the actual value and number of steps for different games. This allows both the foreground as well as the background module to learn in the early stage of training.

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One block comprised a random ordering of 2 ( A ) × 2 ( W ) × 3 ( I ) × = 12 conditions. Participants first practiced 48 trials (4 blocks) to become familiar with the movement of Figure 9. Then,  participants performed 20 blocks (240 trails) for data collection. Each participant took approximately 10 min for this experiment. We instructed participants the same instruction as in  experiment 2.

One set comprised a random ordering of 2 ( A ) × 2 ( W ) × 3 ( I ) =conditions. Participants first practiced 48 trials (4 sets) to become fa- miliar with the movement shown in Figure 9. Then, the participants performed 20 sets (240 trials) for data collection. Each participant took approximately 10 min to complete this experiment, and the same instruction as in Experiment 2 were provided. No participant performed a clutching action in this experiment as well.


Calibrated modality utilization We train multi-modal DNNs as described in §5.2, using each training algorithm. We set the imbalance  parameter α to 0.01 and the re-balancing window size Q to 5 for ModelNet40, and α to 0.1 and Q to 5 for NVGesture.

Calibrated modality utilization We train multi-modal DNNs as described in §5.2, using the guided, the random, and the conventional training algorithm (referred to as vanilla ). For ModelNet40, we set the imbalance tolerance parameter α to 0.01 and the re-balancing window size Q to 5. For NVGesture, we use α of 0.1 and Q of 5.

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Macro-intent labeling function. We extract weak macro-intent labels ˆ g kt for each player k as done in (Zheng et al., 2016). We segment the left half-court into a 10 × 9 grid of 5 ft × 5 ft boxes. The weak macro-intent ˆ g kt at time t is a 1-hot encoding of dimension 90 of the next box in which player k is stationary (speed (cid:107) x kt +1 − x kt (cid:107) 2 below a threshold). The shared global macro-intent g t is the concatenation of individual macro-intents. Figure 4 shows the distribution ofextracted macro-intents for each player and pseudocode can be found in appendix C.

Macro-intent labeling function. We extract weak macro-intent labels ˆ g kt for each player k as done in (Zheng et al., 2016). We segment the left half-court into a 10 × 9 grid of 5 ft × 5 ft boxes. The weak macro-intent ˆ g kt at time t is a 1-hot encoding of dimension 90 of the next box in which player k is stationary (speed (cid:107) x kt +1 − x kt (cid:107) 2 below a threshold). The shared global macro-intent g t is the concatenation of individual macro-intents. Figure 4 shows the distribution of macro-intents for each player. We refer to this labeling function as LF-stationary (pseudocode in appendix D).


To resolve the aforementioned problem, researchers and practitioners focused on the use of the model-parallel technique to partition large models over multiple accelerators/workers; this allows to further scale up model parameters signiﬁcantly. However, in conventional model parallelism, computing resources are severely underutilized as only one accelerator can work at a time. Recently, some studies proposed the pipeline-parallel technique to accelerate conventional model parallelism.

To resolve the aforementioned problem, researchers and practitioners focused on the use of the model-parallel technique to partition large models over multiple accelerators/workers; this allows to further scale up model parameters signiﬁcantly. However, the conventional model parallelism, which includes inter-layer model parallelism and intra-layer model parallelism, either suffers from low resource utilization or high communication overhead Narayanan et al. Recently, some studies proposed the pipeline-parallel technique to accelerate conventional model-parallel training.


C1) Focusing on the relatively tractable linear models, we derive the  excess risk forthe minimum- norm solution to overparameterized gradient-based meta learning including MAML andiMAML. Speciﬁcally, the excess risk upper bound adopts the following form Cross-task variance + Per-task variance + Bias where the cross-task variance quantiﬁes the error caused by ﬁnite task number and thevariation of the ground truth task speciﬁc parameter, which is a unique term compared to

C1) We derive the upper bound of the excess risk for overparameterized nested meta learningincluding MAML and iMAML, with their minimum norm solution. Speciﬁcally, the excess risk upper bound adopts the following formCross-task variance + Per-task variance + Bias where the cross-task variance quantiﬁes the error caused by ﬁnite task number and thevariation of the ground truth task speciﬁc parameter, which is a unique term compared to

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Figure 3 is an example of how to plug numbers into Eq. 6 to compute CTCR for each of the three cameras. There is a breakdown of Eq. 6 just below it. But just in case if it seems too obscure, here is a more intuitive description of Eq. The CTCR is incentivized by the Shapley Value, so the main idea is that the overall optimality needs to also account for the optimality of every possible sub-formation. In the context of an active HPE task, for a camera agent to receive the highest CTCR possible, its current position and view must be optimal both in terms of its current formation and any sub-formation possible.

Figure 3 is an example of using Eq. 6 to compute CTCR for each of the three cameras. The CTCR is incentivized by the Shapley Value. The main idea is that the overall optimality needs to also account for the optimality of every possible sub-formation. For a camera agent to receive the highest CTCR possible, its current position and view must be optimal both in terms of its current formation and any sub-formation possible.

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While neuronal spiking occurs on a temporal scale of milliseconds, the behavior spans the timescales from milliseconds to hours and even days (Mathis et al., 2018). As a result, the recorded neuronal and behavioral variables may operate at different timescales and exhibit different statistics.

While neuronal spiking occurs on a temporal scale of milliseconds, the behavior spans the timescales As a result, the recorded neuronal and behavioral variables may operate at different timescales and exhibit different statistics.


Euclidean norm, and the plasticity rule we derived updated its weights and biases in an unsupervised manner. We used K = 2000 neurons. First we trained the network for 100 epochs, i.e. randomly ordered presentations of the 60000 training digits. In our validation testing we found that softmax with a base of 1000 (see Section 2.7) performed best. The learning rate η of Eq. 8 decreased linearly from 0.03 to 0 throughout training. Each training experiment we will describe was repeated ﬁve times with varying random initializations and input order. We will report the mean and standard deviation of accuracies. Inference of the input labels by the WTA network of 2000 neurons was performed in two different ways. The ﬁrst approach is single-layer, where after training the network we assigned a label to each of the 2000 neurons, in a standard approach that is used in unsupervised clustering. Namely, for each neuron, we found the label of the training set that makes it win the WTA competition most often. In this single-layer approach, this is the only time when labels were used, and at no point were weights updated using labels. The second approach was two-layer and based on supervised training of a perceptron classiﬁer on top of the WTA layer. The classiﬁer layer was trained with the Adam optimizer and cross-entropy loss for 60 epochs, while the previously-trained WTA parameters were frozen. SoftHebb achieved an accuracy of (96 . 18 ± 0 . 06)% and (96 . 94 ± 0 . 02)% in its 1- and 2-layer form respectively. To conﬁrm the strength of the soft WTA approach combined with training the priors through biases, which makes the network Bayesian, we also trained the weights of anetwork with a hard-WTAsetup, i.e. where the strongest-activated neuron’s output y k is 1, and the other neurons are suppressed to 0, for each input. We found that an initial learning rate of 0.05 was best for the hardWTA network. The SoftHebb model  outperformed the hard WTA (Fig. 1A).

Euclidean norm, and the plasticity rule we derived updated its weights and biases in an unsupervised manner. We used K = 2000 neurons. First we trained the network for 100 epochs, i.e. randomly ordered presentations of the 60000 training digits. Each training experiment we will describe was repeated ﬁve times with varying random initializations and input order. We will report the mean and standard deviation of accuracies. Inference of the input labels by the WTA network ofneurons was performed in two different ways. The ﬁrst approach is single-layer, where after training the network we assigned a label to each of the 2000 neurons, in a standard approach that is used in unsupervised clustering. Namely, for each neuron, we found the label of the training set that makes it win the WTA competition most often. In this single-layer approach, this is the only time when labels were used, and at no point were weights updated using labels. The second approach was two-layer and based on supervised training of a perceptron classiﬁer on top of the WTA layer. The classiﬁer layer was trained with the Adam optimizer and cross-entropy loss for 100 epochs, while the previously-trained WTA parameters were frozen. SoftHebb achieved an accuracy of (96 . 31 ± 0 . 06)% and (97 . 80 ± 0 . 02)% in its 1- and 2-layer form respectively. To test the strengths of the soft-WTA approach combined with training the priors through biases, which makes the network Bayesian, we also trained the weights of a hard-WTA network. The SoftHebb model slightly outperformed the hard WTA (Fig.

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Nevertheless, these pre-training strategies do not  capture well the foundation of protein-protein interactions. More recently, the success of protein language models in has drawn interest in adopting the mask-predict(BERT) paradigm to protein 3D structures (Wang et al., 2018; Shroff et al., 2020; Jing et al., 2020; Yang et al., 2022; Zhang et al., 2022; Hsu et al., 2022). These methods partially mask amino acid types on a given protein backbone and use neural networks to recover the masked information. It has been reported that the difference in the probability of amino acid types before and after mutation show correlation to the change in binding free energy (Yang et al., 2022). Hence, they can serve as unsupervised predictors of the mutational effects on binding.

However, most pre-training tasks are not designed to capture the foundation of protein-protein interactions. Unsupervised models adopt the mask-predict paradigm to protein 3D structures, partially masking amino acid types on a given protein backbone, and recovering the masked information using neural networks (Wang et al., 2018; Shroff et al., 2020; Jing et al., 2020; Yang et al., 2022; Hsu et al., 2022). These models can serve as unsupervised predictors of the mutational effects on binding, as the difference in the probability of amino acid types before and after mutation correlates mildly to the change in binding free energy.

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Visualization of Pruning Process . To ﬁguratively understand how SRP works, in Fig. 3 we plot the mean L 1 -norm of ﬁlters in two layers of EDSR baseline during SRP training. The ﬁlters are split into two groups, pruned and kept. As seen, the mean L 1 -norm of the pruned ﬁlters goes down gradually because the penalty grows stronger and stronger, driving them towards zero. Interestingly, note the L 1 -norms of the kept ﬁlters arise themselves (we do not have any regularization term to encourage them to grow larger). It means the network learns to recover  itself , akin to the compensation effect  in human brain (Duffau et al., 2003). We provide more visualization results in the appendix.

Visualization of Pruning Process . In Fig. 3, we visualize the pruning process by plotting the mean L 1 -norm of ﬁlters in two layers of EDSR baseline during SRP training. The ﬁlters are split into two groups, kept ﬁlters and pruned ﬁlters. As is shown in the ﬁgure, the mean L 1 -norm of the pruned ﬁlters goes down gradually because the penalty grows stronger and stronger, driving them towards zero. Interestingly, note the L 1 -norms of the kept ﬁlters arise themselves. Recall that there is no explicit regularization to promote them to grow. In other words, the network learns to recover by itself , akin to the compensation effect found in human brain (Duffau et al., 2003).

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Although the theoretical understanding of the predictive capacity of high-dimensional ML models continues to advance rapidly, a parallel rigorous theory for UQ is comparatively lagging. The prominent heuristic in modern ML that larger models will typically perform better has become almost axiomatic. However, it is only more recently that this heuristic has become represented in statistical theory  (Bartlett et al., 2020; Wang et al., 2021; Derezinski et al., 2020b). Typically, these arguments involve applications of random matrix theory (Edelman & Rao, 2005; Paul & Aue, 2014), most notably the Marchenko-Pastur law, concerning limits of spectral distributions under large data/large dimension regimes.

Although the theoretical understanding of the predictive capacity of high-dimensional ML models continues to advance rapidly, a parallel rigorous theory for UQ is comparatively lagging. The prominent heuristic in modern ML that larger models will typically perform better has become almost axiomatic. However, it is only more recently that this heuristic has become represented in the theory through the characterisation of benign overfitting (Bartlett et al., 2020). In particular, the double descent curve extends the bias-variance tradeoff curve to account for improving performance with higher model complexity (Belkin et al., 2019; Wang et al., 2021; Derezinski et al., 2020b) (see Figure 1(right)). Typically, these arguments involve applications of random matrix theory (Edelman & Rao, 2005; Paul & Aue, 2014), notably the Marchenko-Pastur law, concerning limits of spectral distributions under large data/large dimension regimes.


In the previous section, we saw that reward hacking often leads to phase transitions in agent behaviour. Furthermore, in applications like trafﬁc routing or COVID response, the true reward may be observed only sporadically or not at all. Blindly optimizing the proxy in these cases can lead to catastrophic failure.

In the previous section, we saw that reward hacking often leads to phase transitions in agent behaviour. Furthermore, in applications like trafﬁc routing or COVID response, the true reward may be observed only sporadically or not at all. Blindly optimizing the proxy in these cases can lead to catastrophic failure (Zhuang & Hadﬁeld-Menell, 2020; Taylor, 2016).


We denote O + as the set of segments among images with overlapping object categories. For example, if pixel i is labeled as sofa and another image also contains sofa, all the segments from that image are included in O + ; otherwise they are considered negative segments  O − . This semantic context relationship does not require localization annotations or cues yet enhances the global and higher-level regularization on the pixel embedding.

Let O + denote the set of segments in images with overlapping categories. For example, if pixel i is labeled as sofa and another image also contains sofa , all the segments from that image are included in O + ; otherwise they are considered negative segments in O − . This semantic context relationship does not require localized annotations yet imposes regularization on pixel feature learning.

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Result Analysis. We empirically evaluate this method on CIFAR-10 with ResNet-18 backbone (details in Appendix B.1). From Figure 2a, we can see that as we adopt a larger memory bank with a large aggregation step s , the linearprobing accuracycontinues to grow larger and larger. Although the overall accuracy is still relatively low, we can observe a clear benefit of multi-stage aggregation for preventing feature collapse, which previously has onlybeen possible via asymmetric architectural designs. The success of applying multi-stage aggregation  to contrastive learning not only suggests a new approach for negative-free contrastive learning, but also helps verify our established connection between contrastive learning and MP-GNNs. In Section D, we also show that theproposed multi-stage aggeregation can also bring clear benefits  on benchmark datasets.

Result Analysis. From Figure 2a, we can see that when directly applying the multi-stage alignment objective, a larger number of aggregation epochs could indeed improve linear accuracy and avoid full feature collapse. Although the overall accuracy is still relatively low, we can observe a clear benefit of multi-stage aggregation (better than random guess when s ≥ 2 ), which was only possible using asymmetric modules, e.g., SimSiam’s predictor (Chen & He, 2021). Inspired by this observation, we further combine multi-stage aggregation with SimSiam to further alleviate feature collapse. As shown in Table 1a, the multi-stage mechanism indeed brings cosistent and significant improvements over SimSiam on all three datasets, which clearly demonstrates the benefits of multi-stage aggregation on improving existing non-contrastive methods by further alleviating their feature collapse issues.


One way to check for any correspondences is to simply group the validation samples into M equalmass bins (henceforth called validation-bins) and compare the confidence with the training samples that fall into the same validation-bins. But first we  clarify a few notations that are of interest.

One way to check for any correspondences is to simply group the validation samples into M equalmass bins (henceforth called validation-bins) and compare the confidence with the training samples that fall into the respective validation-bin boundaries. Before proceeding further, we first clarify a few notations of interest.

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Rotated MNIST and Permuted MNIST. Since DPGrad is designed  for linear regression, we provide two variants of DPGrad — one is a modiﬁcation suitable for multi-class classiﬁcation, the other is a modiﬁcation suitable for non-linear featurizers. Detailed numbers and ﬁgures can be found in Appendix E. In brief, both algorithms alleviate catastrophic forgetting and perform much better than vanilla SGD. Furthermore, the performance of both is much more stable than OGD and the accuracy remains at a high level across tasks.

Rotated MNIST and Permuted MNIST. Since DPGrad is designed speciﬁcally for linear regression, we provide two variants of DPGrad (without provable guarantees on their performance, of course)— one is a modiﬁcation suitable for multi-class classiﬁcation, the other is a modiﬁcation suitable for non-linear featurizers. Detailed numbers and ﬁgures can be found in Appendix E. In brief, both algorithms alleviate catastrophic forgetting and perform much better than vanilla SGD. Furthermore, the performance of both is much more stable than OGD and the accuracy remains at a high level across tasks.


To visualize the advantage of our approach, considerthe semi-circle domain in Figure 1, adapted from [3]:a 2-dimensional agent must navigate to a goal, lo cated somewhere on the semi-circle. A task therefore corresponds to the goal location, and the task distribution is uniform on the 1-dimensional semi-circle.

To visualize the advantage of our approach, consider the HalfCircle domain in Figure 1, adapted 3]: a 2- dimensional agent must navigate to a goal, lo cated somewhere on the half-circle. A task therefore corresponds to the goal location, and the task distribution is uniform on the 1-dimensional half- circle.

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In Table 1, we evaluate BiLAW using two relatively small networks on two datasets: MNIST (LeCun & Cortes, 2010) and Fashion MNIST (Xiao et al., 2017). Tiny-CNN is a convolutional network withconvolutional and 2 dense layers. FC1 corresponds to a single hidden layer feedforward network with 1024 hidden units. The details of the architectures are given in the Appendix. We consider robustness with respect to (cid:96) ∞ distance. We use two criteria: clean test accuracy (TA) and robust test accuracy (PGD) for a given threshold (cid:15) . Robust test accuracy is computed using Projected Gradient Descent (PGD) (Madry et al., 2018) with 20 iterations. In all testcases, our method matches the performance of GAIRAT and out-performs the other methods. However, we note the overall distribution of both clean and robust accuracy is tight. We note a potential drawback of reweighting algorithms: the MNIST and Fashion-MNIST datasets contain a non-trivial number of misclassiﬁed samples which can inﬂuence performance (Müller & Markert, 2019). For algorithms which perform weighted training, possible large weights on outliers or mislabeled examples may inﬂuence classiﬁcation performance. We will investigate this in the context of adversarial training in future work

In Table 1, we evaluate BiLAW using two relatively small networks on two datasets: MNIST (LeCun & Cortes, 2010) and Fashion MNIST (Xiao et al., 2017). Tiny-CNN is a convolutional network with 2 convolutional and 2 dense layers. FC1 corresponds to a single hidden layer feedforward network with 1024 hidden units. The details of the architectures are given in the Appendix. We consider robustness with respect to (cid:96) ∞ distance. We use three criteria: clean test accuracy (clean), robust test accuracy (PGD) for a given threshold (cid:15) and AutoAttack (AA). Robust test accuracy is computed using Projected Gradient Descent (PGD) (Madry et al., 2018) with 20 iterations. In all testcases, our method matches the performance of GAIRAT and out-performs the other methods for clean and PGD accuracy and we out-perform all reweighting methods on AA accuracy. However, we note the overall distribution of both clean and robust accuracy is tight. We note a potential drawback of reweighting algorithms: the MNIST and F-MNIST datasets contain a non-trivial number of misclassiﬁed samples which can inﬂuence performance (Müller & Markert, 2019). For algorithms which perform weighted training, possible large weights on outliers or mislabeled examples may inﬂuence classiﬁcation performance. We will investigate this in the context of adversarial training in future work


The addition of medication entries in the calendar introduces new visual elements. Medication entries also communicate more details (D1 - D5) than the standard title, time, and location of standard calendar events. Therefore, a design that integrates prescription visualization to a calendar should satisfy the following prescription- related usability requirement:

The addition of medication entries in the calendar introduces new visual elements. Medication entries communicate more details (D1 - D5) than the standard title, time, and location of regular calendar entries [66–68]. Integrating prescriptions in a generalpurpose calendar involves integrating Personal Health Information (PHI) and related activities into calendars [69–71]. Therefore, a design that integrates prescription visualization to a calendar should satisfy the following prescription-related usability requirement:


We are particularly interested in understanding why capacity loss occurs. Two possible causes are immediate: the effect of bootstrapping , and the effect of sequential training . The effect of bootstrapping on capacity has been studied in other contexts (Mobahi et al., 2020; Kumar et al., 2021). We aim to isolate the effect of sequential prediction tasks on capacity loss. To minimize the potential for confounding factors to inﬂuence our results, we construct a toy  prediction problem on the MNIST data set. We ﬁrst consider labels computed by a randomly initialized neural network f ✓ : we transform input-label pairs ( x, y ) from the canonical MNIST dataset to ( x, f ✓ ( x )) , where f ✓ ( x ) is the network output. To generate a new task, we simply reinitialize the network; our evaluations consist of 10 such iterations. We further consider a ‘sparse-reward’ version of MNIST: for each of 10 iterations i , we use the label ˆ y i = [ y < i ] . This mimics sparse-reward environments where the agent initially obtains no reward in  the environment, then gradually improves its policy and thus increases its prediction targets over the course of training.

We are particularly interested in understanding why capacity loss occurs. Two possible causes are immediate: the effect of bootstrapping , and the effect of sequential training . The effect of bootstrapping on capacity has been studied in other contexts (Mobahi et al., 2020; Kumar et al., 2021). We aim to isolate the effect of sequential prediction tasks on capacity loss. To minimize the potential for confounding factors to inﬂuence our results, we construct our toy iterative prediction problems on the MNIST data set, which consists of images of handwritten digits and corresponding labels, and manually construct a sequence of targets which the network must ﬁt over the course of training. We ﬁrst consider labels computed by a randomly initialized neural network f ✓ : we transform input-label pairs ( x, y ) from the canonical MNIST dataset to ( x, f ✓ ( x )) , where f ✓ ( x ) is the network output. To generate a new task, we simply reinitialize the network; our evaluations consist of 10 iterations of label-generation followed by a training period during which we run a gradient-based optimizer on the network starting from the parameters we obtained in the previous iteration. We further consider a ‘sparse-reward’ version of MNIST: for each of 10 iterations i , we use the label ˆ y i = [ y < i ] , where y is the true label of the image. For example, at the ﬁrst iteration, all images are assigned label zero. At the second iteration, the images of the digit zero are assigned label one, while all other inputs retain the zero label. This continues until all inputs are assigned label one at the ﬁnal iteration. We follow the same training procedure in both cases: optimizing the network for a ﬁxed number of steps on one set of labels, then generating new labels and running the optimization algorithm again from the parameters obtained in the previous phase.


Note that we use a ResNet18 image encoder, while Slot Attention uses a simpler 4-layer convolutional neural network with 5 × 5 kernels. This difference is not so easily compared: we compress the image into a 512d vector while they operate on a feature map. Their ﬁnal feature map for CLEVR has 32x spatial positions with 64d each, so it can be argued that their latent space is 65536-dimensional. It is natural that a tighter bottleneck requires more processing. ResNet18 also applies several strided convolutions early, so the amount of processing between it and the Slot Attention image encoder are not too dissimilar. This is reﬂected by the fact that the time taken to process a sample is similar between ResNet18 + iDSPN and CNN + Slot Attention.

Note that we use a ResNet18 image encoder, while Slot Attention uses a simpler 4-layer convolutional neural network with 5 × 5 kernels. This difference is not so easily compared: we compress the image into a 512d vector while they operate on a feature map. Their ﬁnal feature map for CLEVR has 32x spatial positions with 64d each, so it can be argued that their latent space is 65536-dimensional. It is natural that a tighter bottleneck requires more processing. ResNet18 also applies several strided convolutions early, so the amount of processing between it and the Slot Attention image encoder are not too dissimilar. This is reﬂected by the fact that the time taken to process a sample is similar between ResNet18 + iDSPN and CNN + Slot Attention, even though we use a smaller batch size and could gain a speed-up from better parallelization for higher batch sizes. A small difference between the default Slot Attention setup and our setup is that they normalize the 3d coordinates to be within [0, 1] while we use an interval of [-1, 1]. In Table 3, Slot Attention* uses an interval of [-1, 1] (the same as iDSPN) while Slot Attention † uses the interval of [-3, 3] (the default coordinate range for this dataset). This increases the weight on the coordinates versus the other attributes, so it improves AP for strict thresholds but trades off classiﬁcation performance at a threshold of inﬁnity.


We propose an approach for learning non-parametric spatio-temporal drift and diffusion functions of stochastic differential equation (SDE) systems such that the resulting simulated state distributions match data. The experiment on a real world data set shows that our model can better ﬁt complex dynamics than the spatial counterpart. An interesting future research direction is the study of various vector ﬁeld kernels, such as divergence-free, curl-free or spectral kernels [12]. The model could be extended to have an observation model, e.g., GPLVM or deep neural network, rather than PCA.

We propose an approach for learning non-parametric spatio-temporal drift and diffusion functions of stochastic differential equation (SDE) systems such that the resulting simulated state distributions match data. The experiment on a real world data set shows that our model can better ﬁt complex dynamics than the spatial counterpart. This increase in model capacity, however, results in larger data set requirements and makes the model more vulnerable to overﬁtting, which could be better accounted for using e.g. variational inference. An interesting future research direction is the study of various vector ﬁeld kernels, such as divergence-free, curl-free or spectral kernels [12]. The model could be extended to have an observation model, e.g., GPLVM or deep neural network, rather than PCA.


Our considerations are focused on invariances of typical combinatorial problems under speciﬁc transformations of the cost vector. These transformations usually manifest as projections or normalizations , e.g. as an immediate consequence of the linearity of the objective, the combinatorial solver is agnostic to normalization of the cost vector. Such invariances, if unattended, can hinder fast convergence when used in combination with adaptive optimizers, or can result in divergence and

Our considerations are focused on invariances of typical combinatorial problems under speciﬁc transformations of the cost vector. These transformations usually manifest as projections or normalizations , e.g. as an immediate consequence of the linearity of the objective, the combinatorial solver is agnostic to normalization of the cost vector. Such invariances, if unattended, can hinder fast convergence due to the noise of spurious irrelevant updates, or can result in divergence and


In our prior work (Chakraborti et al. 2017), we encapsulate such inconsistencies as model differences , while considering the discrepancies between the human and its own model when generating explanations. An explanation then becomes a request to the human to adjust the model differences in his mind so that the robot’s behavior would make sense in the updated model, which captures the human’s expectation of the robot. The general decision-making process of an agent in the presence of such model differences is termed model reconciliation (Chakraborti et al. 2017; Zhang et al. 2017).

To address this challenge, the agent should consider the discrepancies between the human and its own model while generating explanations. In our prior work [7], we encapsulate such inconsistencies as model differences . An explanation then becomes a request to the human to adjust the model differences in his mind so that the robot’s behavior would make sense in the updated model, which is used to produce the human’s expectation of the robot. The general decision-making process of an agent in the presence of such model differences is termed model reconciliation [7], [8].

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DDIM sampler) and for higher number of function evaluations performance is better than the DDIM sampler (beneﬁt coming from the Probability Flow Momentum Sampler). That said, there is no spot in which the Predictor-Correct performs better than both DDIM and the Probability Flow Momentum Sampler. We encourage future research in identifying  better pairs of Predictor-Correctors that might outperform both the Predictor and the Corrector in some regime.

DDIM sampler) and for higher number of function evaluations performance is better than the DDIM sampler (beneﬁt coming from the Probability Flow Momentum Sampler). There is a spot, at NFEs, where the Predictor-Corrector sampler is better than both the Predictor and the Corrector. We encourage future research in identifying even better pairs of Predictor-Corrector samplers.

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