Neural Tangent Kernel (NTK) is widely used to analyze overparametrized neural networks due to the famous result by Jacot et al. (2018): in the infinite-width limit, the NTK is deterministic and constant during training. However, this result cannot explain the behavior of deep networks, since it generally does not hold if depth and width tend to infinity simultaneously. In this paper, we study the NTK of fully-connected ReLU networks with depth comparable to width. We prove that the NTK properties depend significantly on the depth-to-width ratio and the distribution of parameters at initialization. In fact, our results indicate the importance of the three phases in the hyperparameter space identified in Poole et al. (2016): ordered, chaotic and the edge of chaos (EOC). We derive exact expressions for the NTK dispersion in the infinite-depth-and-width limit in all three phases and conclude that the NTK variability grows exponentially with depth at the EOC and in the chaotic phase but not in the ordered phase. We also show that the NTK of deep networks may stay constant during training only in the ordered phase and discuss how the structure of the NTK matrix changes during training.

Group equivariant convolutional neural networks (G-CNNs) are generalizations of convolutional neural networks (CNNs) which excel in a wide range of technical applications by explicitly encoding symmetries, such as rotations and permutations, in their architectures. Although the success of G-CNNs is driven by their explicit symmetry bias, a recent line of work has proposed that the implicit bias of training algorithms on particular architectures is key to understanding generalization for overparameterized neural nets. In this context, we show that L-layer full-width linear G-CNNs trained via gradient descent for binary classification converge to solutions with low-rank Fourier matrix coefficients, regularized by the 2/L-Schatten matrix norm. Our work strictly generalizes previous analysis on the implicit bias of linear CNNs to linear G-CNNs over all finite groups, including the challenging setting of non-commutative groups (such as permutations), as well as band-limited G-CNNs over infinite groups. We validate our theorems via experiments on a variety of groups, and empirically explore more realistic nonlinear networks, which locally capture similar regularization patterns. Finally, we provide intuitive interpretations of our Fourier space implicit regularization results in real space via uncertainty principles.

The use of sparse neural networks has seen rapid growth in recent years, particularly in computer vision. Their appeal stems largely from the reduced number of parameters required to train and store, as well as in an increase in learning efficiency. Somewhat surprisingly, there have been very few efforts exploring their use in Deep Reinforcement Learning (DRL). In this work we perform a systematic investigation into applying a number of existing sparse training techniques on a variety of DRL agents and environments. Our results corroborate the findings from sparse training in the computer vision domain –sparse networks perform better than dense networks for the same parameter count– in the DRL domain. We provide detailed analyses on how the various components in DRL are affected by the use of sparse networks and conclude by suggesting promising avenues for improving the effectiveness of sparse training methods, as well as for advancing their use in DRL.

Permutation invariant neural networks are a promising tool for predictive modeling of set data. We show, however, that existing architectures struggle to perform well when they are deep. In this work, we mathematically and empirically analyze normalization layers and residual connections in the context of deep permutation invariant neural networks. We develop set norm, a normalization tailored for sets, and introduce the ``clean path principle'' for equivariant residual connections alongside a novel benefit of such connections, the reduction of information loss. Based on our analysis, we propose Deep Sets++ and Set Transformer++, deep models that reach comparable or better performance than their original counterparts on a diverse suite of tasks. We additionally introduce Flow-RBC, a new single-cell dataset and real-world application of permutation invariant prediction. We open-source our data and code here: https://github.com/rajesh-lab/deeppermutationinvariant.

We present a conceptual framework, \emph{datamodeling}, for analyzing the behavior of a model class in terms of the training data. For any fixed ``target'' example $x$, training set $S$, and learning algorithm, a {\em datamodel} is a parameterized function $2^S \to \mathbb{R}$ that for any subset of $S' \subset S$---using only information about which examples of $S$ are contained in $S'$---predicts the outcome of training a model on $S'$ and evaluating on $x$. Despite the complexity of the underlying process being approximated (e.g. end-to-end training and evaluation of deep neural networks), we show that even simple {\em linear} datamodels successfully predict model outputs. We then demonstrate that datamodels give rise to a variety of applications, such as:accurately predicting the effect of dataset counterfactuals; identifying brittle predictions; finding semantically similar examples; quantifying train-test leakage; and embedding data into a well-behaved and feature-rich representation space.

Adversarial training (AT) is a widely recognized defense mechanism to gain the robustness of deep neural networks against adversarial attacks. It is built on min-max optimization (MMO), where the minimizer (i.e., defender) seeks a robust model to minimize the worst-case training loss in the presence of adversarial examples crafted by the maximizer (i.e., attacker). However, the conventional MMO method makes AT hard to scale. Thus, Fast-AT and other recent algorithms attempt to simplify MMO by replacing its maximization step with the single gradient sign-based attack generation step. Although easy to implement, FAST-AT lacks theoretical guarantees, and its empirical performance is unsatisfactory due to the issue of robust catastrophic overfitting when training with strong adversaries. In this paper, we advance Fast-AT from the fresh perspective of bi-level optimization (BLO). We first show that the commonly-used Fast-AT is equivalent to using a stochastic gradient algorithm to solve a linearized BLO problem involving a sign operation. However, the discrete nature of the sign operation makes it difficult to understand the algorithm performance. Inspired by BLO, we design and analyze a new set of robust training algorithms termed Fast Bi-level AT (Fast-BAT), which effectively defends sign-based projected gradient descent (PGD) attacks without using any gradient sign method or explicit robust regularization. In practice, we show that our method yields substantial robustness improvements over multiple baselines across multiple models and datasets.

Deep metric learning aims to learn distance metrics that measure similarities and dissimilarities between samples. The existing approaches typically focus on designing different hard sample mining or distance margin strategies and then minimize a pair/triplet-based or proxy-based loss over the training data. However, this can lead the model to recklessly learn all the correlated distances found in training data including the spurious distance (e.g., background differences) that is not the distance of interest and can harm the generalization of the learned metric. To address this issue, we study metric learning from a causality perspective and accordingly propose deep causal metric learning (DCML) that pursues the true causality of the distance between samples. DCML is achieved through explicitly learning environment-invariant attention and task-invariant embedding based on causal inference. Extensive experiments on several benchmark datasets demonstrate the superiority of DCML over the existing methods.

Data poisoning causes misclassification of test time target examples, by injecting maliciously crafted samples in the training data. Existing defenses are often effective only against a specific type of targeted attack, significantly degrade the generalization performance, or are prohibitive for standard deep learning pipelines. In this work, we propose an efficient defense mechanism that significantly reduces the success rate of various data poisoning attacks, and provides theoretical guarantees for the performance of the model. Targeted attacks work by adding bounded perturbations to a randomly selected subset of training data to match the targets’ gradient or representation. We show that: (i) under bounded perturbations, only a number of poisons can be optimized to have a gradient that is close enough to that of the target and make the attack successful; (ii) such effective poisons move away from their original class and get isolated in the gradient space; (iii) dropping examples in low-density gradient regions during training can successfully eliminate the effective poisons, and guarantees similar training dynamics to that of training on full data. Our extensive experiments show that our method significantly decreases the success rate of state-of-the-art targeted attacks, including Gradient Matching and Bullseye Polytope, and easily scales to large datasets.

Incorporating symmetries can lead to highly data-efficient and generalizable models by defining equivalence classes of data samples related by transformations. However, characterizing how transformations act on input data is often difficult, limiting the applicability of equivariant models. We propose learning symmetric embedding networks (SENs) that encode an input space (e.g. images), where we do not know the effect of transformations (e.g. rotations), to a feature space that transforms in a known manner under these operations. This network can be trained end-to-end with an equivariant task network to learn an explicitly symmetric representation. We validate this approach in the context of equivariant transition models with 3 distinct forms of symmetry. Our experiments demonstrate that SENs facilitate the application of equivariant networks to data with complex symmetry representations. Moreover, doing so can yield improvements in accuracy and generalization relative to both fully-equivariant and non-equivariant baselines.

The federated learning (FL) framework enables edge clients to collaboratively learn a shared inference model while keeping privacy of training data on clients. Recently, many heuristics efforts have been made to generalize centralized adaptive optimization methods, such as SGDM, Adam, AdaGrad, etc., to federated settings for improving convergence and accuracy. However, there is still a paucity of theoretical principles on where to and how to design and utilize adaptive optimization methods in federated settings. This work aims to develop novel adaptive optimization methods for FL from the perspective of dynamics of ordinary differential equations (ODEs). First, an analytic framework is established to build a connection between federated optimization methods and decompositions of ODEs of corresponding centralized optimizers. Second, based on this analytic framework, a momentum decoupling adaptive optimization method, FedDA, is developed to fully utilize the global momentum on each local iteration and accelerate the training convergence. Last but not least, full batch gradients are utilized to mimic centralized optimization in the end of the training process to ensure the convergence and overcome the possible inconsistency caused by adaptive optimization methods.

Byzantine resilience emerged as a prominent topic within the distributed machine learning community.Essentially, the goal is to enhance distributed optimization algorithms, such as distributed SGD, in a way that guarantees convergence despite the presence of some misbehaving (a.k.a., {\em Byzantine}) workers. Although a myriad of techniques addressing the problem have been proposed, the field arguably rests on fragile foundations. These techniques are hard to prove correct and rely on assumptions that are (a) quite unrealistic, i.e., often violated in practice, and (b) heterogeneous, i.e., making it difficult to compare approaches. We present \emph{RESAM (RESilient Averaging of Momentums)}, a unified framework that makes it simple to establish optimal Byzantine resilience, relying only on standard machine learning assumptions. Our framework is mainly composed of two operators: \emph{resilient averaging} at the server and \emph{distributed momentum} at the workers. We prove a general theorem stating the convergence of distributed SGD under RESAM. Interestingly, demonstrating and comparing the convergence of many existing techniques become direct corollaries of our theorem, without resorting to stringent assumptions. We also present an empirical evaluation of the practical relevance of RESAM.

The teacher-student (TS) framework, training a (student) network by utilizing an auxiliary superior (teacher) network, has been adopted as a popular training paradigm in many machine learning schemes, since the seminal work---Knowledge distillation (KD) for model compression and transfer learning. Many recent self-supervised learning (SSL) schemes also adopt the TS framework, where teacher networks are maintained as the moving average of student networks, called the momentum networks. This paper presents TSPipe, a pipelined approach to accelerate the training process of any TS frameworks including KD and SSL. Under the observation that the teacher network does not need a backward pass, our main idea is to schedule the computation of the teacher and student network separately, and fully utilize the GPU during training by interleaving the computations of the two networks and relaxing their dependencies. In case the teacher network requires a momentum update, we use delayed parameter updates only on the teacher network to attain high model accuracy. Compared to existing pipeline parallelism schemes, which sacrifice either training throughput or model accuracy, TSPipe provides better performance trade-offs, achieving up to 12.15x higher throughput.

Federated learning allows clients to collaboratively learn statistical models while keeping their data local. Federated learning was originally used to train a unique global model to be served to all clients, but this approach might be sub-optimal when clients' local data distributions are heterogeneous. In order to tackle this limitation, recent personalized federated learning methods train a separate model for each client while still leveraging the knowledge available at other clients. In this work, we exploit the ability of deep neural networks to extract high quality vectorial representations (embeddings) from non-tabular data, e.g., images and text, to propose a personalization mechanism based on local memorization. Personalization is obtained by interpolating a collectively trained global model with a local $k$-nearest neighbors (kNN) model based on the shared representation provided by the global model. We provide generalization bounds for the proposed approach in the case of binary classification, and we show on a suite of federated datasets that this approach achieves significantly higher accuracy and fairness than state-of-the-art methods.

We justify the fast equilibrium conjecture on stochastic gradient descent from (Li et al. 2020) under the assumptions that critical points are non-degenerate and the stochastic noise is a standard Gaussian. In this case, we prove an SGD with constant effective learning rate consists of three stages: descent, diffusion and tunneling, and explicitly identify temporary equilibrium states in the normalized parameter space that can be observed within practical training time. This interprets the gap between the mixing time in the fast equilibrium conjecture and the previously known upper bound. While our assumptions do not represent typical implementations of SGD of neural networks in practice, this is the first description of the three-stage mechanism in any case. The main finding in this mechanism is that a temporary equilibrium of local nature is quickly achieved after polynomial time (in term of the reciprocal of the intrinsic learning rate) and then stabilizes within observable time scales; and that the temporary equilibrium is in general different from the global Gibbs equilibrium, which will only appear after an exponentially long period beyond typical training limits. Our experiments support that this mechanism may extend to the general case.

Recently, Optimization-Derived Learning (ODL) has attracted attention from learning and vision areas, which designs learning models from the perspective of optimization. However, previous ODL approaches regard the training and hyper-training procedures as two separated stages, meaning that the hyper-training variables have to be fixed during the training process, and thus it is also impossible to simultaneously obtain the convergence of training and hyper-training variables. In this work, we design a Generalized Krasnoselskii-Mann (GKM) scheme based on fixed-point iterations as our fundamental ODL module, which unifies existing ODL methods as special cases. Under the GKM scheme, a Bilevel Meta Optimization (BMO) algorithmic framework is constructed to solve the optimal training and hyper-training variables together. We rigorously prove the essential joint convergence of the fixed-point iteration for training and the process of optimizing hyper-parameters for hyper-training, both on the approximation quality, and on the stationary analysis. Experiments demonstrate the efficiency of BMO with competitive performance on sparse coding and real-world applications such as image deconvolution and rain streak removal.