Inspired by BatchNorm, there has been an explosion of normalization layers in deep learning. Recent works have identified a multitude of beneficial properties in BatchNorm to explain its success. However, given the pursuit of alternative normalization layers, these properties need to be generalized so that any given layer's success/failure can be accurately predicted. In this work, we take a first step towards this goal by extending known properties of BatchNorm in randomly initialized deep neural networks (DNNs) to several recently proposed normalization layers. Our primary findings follow: (i) similar to BatchNorm, activations-based normalization layers can prevent exponential growth of activations in ResNets, but parametric techniques require explicit remedies; (ii) use of GroupNorm can ensure an informative forward propagation, with different samples being assigned dissimilar activations, but increasing group size results in increasingly indistinguishable activations for different samples, explaining slow convergence speed in models with LayerNorm; and (iii) small group sizes result in large gradient norm in earlier layers, hence explaining training instability issues in Instance Normalization and illustrating a speed-stability tradeoff in GroupNorm. Overall, our analysis reveals a unified set of mechanisms that underpin the success of normalization methods in deep learning, providing us with a compass to systematically explore the vast design space of DNN normalization layers.

Inspired by BatchNorm, there has been an explosion of normalization layers in deep learning. Recent works have identified a multitude of beneficial properties in BatchNorm to explain its success. However, given the pursuit of alternative normalization layers, these properties need to be generalized so that any given layer's success/failure can be accurately predicted. In this work, we take a first step towards this goal by extending known properties of BatchNorm in randomly initialized deep neural networks (DNNs) to several recently proposed normalization layers. Our primary findings follow: (i) similar to BatchNorm, activations-based normalization layers can prevent exponential growth of activations in ResNets, but parametric techniques require explicit remedies; (ii) use of GroupNorm can ensure an informative forward propagation, with different samples being assigned dissimilar activations, but increasing group size results in increasingly indistinguishable activations for different samples, explaining slow convergence speed in models with LayerNorm; and (iii) small group sizes result in large gradient norm in earlier layers, hence explaining training instability issues in Instance Normalization and illustrating a speed-stability tradeoff in GroupNorm. Overall, our analysis reveals a unified set of mechanisms that underpin the success of normalization methods in deep learning, providing us with a compass to systematically explore the vast design space of DNN normalization layers.

For any > 0, the -covering number of the Euclidean ball Bd(R):= {x 2Rd: kxk2 R} with radius R> 0 in the Euclidean metric is upper bounded by (1+2R/)d. Let F0 F 1 ... FT be a filtration and let X1,X2,...,XT be real random variables such that Xt is Ft-measurable, E[Xt|Ft 1]=0, |Xt| balmost surely, and PT t=1 E[X2t |Ft 1] V for some fixed V> 0and b> 0. Then for any 2(0,1), we have with probability at least 1, For any linear MDP satisfying Definition 3.1, we must have that k (s,a)k2 1/ p d for all s and a, and k,hk2 1/ p d for all and h. By Definition 3.1, we know that Ph( |s,a)= h (s,a),µh()i forms a valid probability distribution, and that k R S |dµh(s)|k2 p d. This yields the first equality. Repeating this calculation h 1more times yields the final equality. Lemma A.8. Fix some h and i (s,a)| 1, and kvk2 p d. Proof. By the linear MDP structure (see Proposition 2.3 of Jin et al. (2020)), for any j, Q j (s,a)= h (s,a),w j i = h (s,a), ji+ Z We first consider the case where u = h for some h which is a valid reward satisfying Definition 3.1. Assume that the reward in our MDP is set such that for h0 6= h, h0 =0 .

While much progress has been made in understanding the minimax sample complexity of reinforcement learning (RL)--the complexity of learning on the "worstcase" instance--such measures of complexity often do not capture the true difficulty of learning. In practice, on an "easy" instance, we might hope to achieve a complexity far better than that achievable on the worst-case instance. In this work we seek to understand the "instance-dependent" complexity of learning near-optimal policies (PACRL) in the setting of RL with linear function approximation. We propose an algorithm, PEDEL, which achieves a fine-grained instance-dependent measure of complexity, the first of its kind in the RL with function approximation setting, thereby capturing the difficulty of learning on each particular problem instance. Through an explicit example, we show that PEDEL yields provable gains over low-regret, minimax-optimal algorithms and that such algorithms are unable to hit the instance-optimal rate. Our approach relies on a novel online experiment design-based procedure which focuses the exploration budget on the "directions" most relevant to learning a near-optimal policy, and may be of independent interest.

Multi-view anchor graph clustering selects representative anchors to avoid full pair-wise similarities and therefore reduce the complexity of graph methods. Although widely applied in large-scale applications, existing approaches do not pay sufficient attention to establishing correct correspondences between the anchor sets across views. To be specific, anchor graphs obtained from different views are not aligned column-wisely. Such an Anchor-Unaligned Problem (AUP) would cause inaccurate graph fusion and degrade the clustering performance. Under multi-view scenarios, generating correct correspondences could be extremely difficult since anchors are not consistent in feature dimensions.

A.1 Network Architecture Here, we describe the architecture of the eVAE presented in Figure 1 of the main paper, in more detail. Event Context Network: We adapt the architecture proposed in [21] for the event context network, but without the feature transformation preprocessing steps. In our implementation, we use three Conv1d layers of 64, 128 and 1024 channels each followed by BatchNorm and a ReLU activation. At the end of the ECN, we add the temporal features (see Appendix A.2) to the N 1024 feature tensor, and execute the max operation to result in a context vector. The sizes of the intermediate features and the context feature are hyperparameters that can be varied based on the application, data complexity etc. Encoder: The encoder for the VAE is composed of two layers, of sizes 1024 and 256 respectively, resulting in two output vectors of 1 8 each, corresponding to the mean and standard deviation for the latent space vector.