In this paper, we study the data-dependent convergence and generalization behavior of gradient methods for neural networks with smooth activation. Our first result is a novel bound on the excess risk of deep networks trained by the logistic loss via an alogirthmic stability analysis. Compared to previous works, our results improve upon the shortcomings of the well-established Rademacher complexity-based bounds. Importantly, the bounds we derive in this paper are tighter, hold even for neural networks of small width, do not scale unfavorably with width, are algorithm-dependent, and consequently capture the role of initialization on the sample complexity of gradient descent for deep nets. Specialized to noiseless data separable with margin $\gamma$ by neural tangent kernel (NTK) features of a network of width $\Omega(poly(\log(n)))$, we show the test-error rate $e^{O(L)}/{\gamma^2 n}$, where $n$ is the training set size and $L$ denotes the number of hidden layers. This results in an improvement in the test loss bound compared to previous works while maintaining the poly-logarithmic width conditions. We further investigate excess risk bounds for deep nets trained with noisy data, establishing that under a polynomial condition on the network width, gradient descent can achieve the optimal excess risk. Finally, we show that a large step-size significantly improves upon the NTK regime's results in classifying the XOR distribution. In particular, we show for a one-hidden layer neural network of constant width $m$ with quadratic activation and standard Gaussian initialization that SGD with linear sample complexity and with a large step-size $\eta=m$ reaches the perfect test accuracy after only $\lceil\log(d)\rceil$ iterations, where $d$ is the data dimension.

Adversarial training using empirical risk minimization (ERM) is the state-of-the-art method for defense against adversarial attacks, that is, against small additive adversarial perturbations applied to test data leading to misclassification. Despite being successful in practice, understanding the generalization properties of adversarial training in classification remains widely open. In this article, we take the first step in this direction by precisely characterizing the robustness of adversarial training in binary linear classification. Specifically, we consider the high-dimensional regime where the model dimension grows with the size of the training set at a constant ratio. Our results provide exact asymptotics for both standard and adversarial test errors under general -norm bounded perturbations ( ) in both discriminative binary models and generative Gaussian-mixture models with correlated features. We use our sharp error formulae to explain how the adversarial and standard errors depend upon the over-parameterization ratio, the data model, and the attack budget. Finally, by comparing with the robust Bayes estimator, our sharp asymptotics allow us to study the fundamental limits of adversarial training.

We investigate the generalization and optimization properties of shallow neural-network classifiers trained by gradient descent in the interpolating regime. Specifically, in a realizable scenario where model weights can achieve arbitrarily small training error $\epsilon$ and their distance from initialization is $g(\epsilon)$, we demonstrate that gradient descent with $n$ training data achieves training error $O(g(1/T)^2\big/T)$ and generalization error $O(g(1/T)^2\big/n)$ at iteration $T$, provided there are at least $m=\Omega(g(1/T)^4)$ hidden neurons. We then show that our realizable setting encompasses a special case where data are separable by the model's neural tangent kernel. For this and logistic-loss minimization, we prove the training loss decays at a rate of $\tilde O(1/ T)$ given polylogarithmic number of neurons $m=\Omega(\log^4 (T))$. Moreover, with $m=\Omega(\log^{4} (n))$ neurons and $T\approx n$ iterations, we bound the test loss by $\tilde{O}(1/ n)$. Our results differ from existing generalization outcomes using the algorithmic-stability framework, which necessitate polynomial width and yield suboptimal generalization rates. Central to our analysis is the use of a new self-bounded weak-convexity property, which leads to a generalized local quasi-convexity property for sufficiently parameterized neural-network classifiers. Eventually, despite the objective's non-convexity, this leads to convergence and generalization-gap bounds that resemble those found in the convex setting of linear logistic regression.

Normalized gradient descent has shown substantial success in speeding up the convergence of exponentially-tailed loss functions (which includes exponential and logistic losses) on linear classifiers with separable data. In this paper, we go beyond linear models by studying normalized GD on two-layer neural nets. We prove for exponentially-tailed losses that using normalized GD leads to linear rate of convergence of the training loss to the global optimum. This is made possible by showing certain gradient self-boundedness conditions and a log-Lipschitzness property. We also study generalization of normalized GD for convex objectives via an algorithmic-stability analysis. In particular, we show that normalized GD does not overfit during training by establishing finite-time generalization bounds.

Decentralized learning offers privacy and communication efficiency when data are naturally distributed among agents communicating over an underlying graph. Motivated by overparameterized learning settings, in which models are trained to zero training loss, we study algorithmic and generalization properties of decentralized learning with gradient descent on separable data. Specifically, for decentralized gradient descent (DGD) and a variety of loss functions that asymptote to zero at infinity (including exponential and logistic losses), we derive novel finite-time generalization bounds. This complements a long line of recent work that studies the generalization performance and the implicit bias of gradient descent over separable data, but has thus far been limited to centralized learning scenarios. Notably, our generalization bounds approximately match in order their centralized counterparts. Critical behind this, and of independent interest, is establishing novel bounds on the training loss and the rate-of-consensus of DGD for a class of self-bounded losses. Finally, on the algorithmic front, we design improved gradient-based routines for decentralized learning with separable data and empirically demonstrate orders-of-magnitude of speed-up in terms of both training and generalization performance.

Adversarial training using empirical risk minimization is the state-of-the-art method for defense against adversarial attacks, that is against small additive adversarial perturbations applied to test data leading to misclassification. Despite being successful in practice, understanding generalization properties of adversarial training in classification remains widely open. In this paper, we take the first step in this direction by precisely characterizing the robustness of adversarial training in binary linear classification. Specifically, we consider the high-dimensional regime where the model dimension grows with the size of the training set at a constant ratio. Our results provide exact asymptotics for both standard and adversarial test errors under ℓ∞-norm bounded perturbations in a generative Gaussian-mixture model. We use our sharp error formulae to explain how the adversarial and standard errors depend upon the overparameterization ratio, the data model, and the attack budget. Finally, by comparing with the robust Bayes estimator, our sharp asymptotics allow us to study fundamental limits of adversarial training.

Despite the popularity of Empirical Risk Minimization (ERM) algorithms, a theory that explains their statistical properties in modern high-dimensional regimes is only recently emerging. We characterize for the first time the fundamental limits on the statistical accuracy of convex ridge-regularized ERM for inference in high-dimensional generalized linear models. For a stylized setting with Gaussian features and problem dimensions that grow large at a proportional rate, we start with sharp performance characterizations and then derive tight lower bounds on the estimation and prediction error. Our bounds provably hold over a wide class of loss functions, and, for any value of the regularization parameter and of the sampling ratio. Our precise analysis has several attributes. First, it leads to a recipe for optimally tuning the loss function and the regularization parameter. Second, it allows to precisely quantify the sub-optimality of popular heuristic choices, such as optimally-tuned least-squares. Third, we use the bounds to precisely assess the merits of ridge-regularization as a function of the sampling ratio. Our bounds are expressed in terms of the Fisher Information of random variables that are simple functions of the data distribution, thus making ties to corresponding bounds in classical statistics.

We study convex empirical risk minimization for high-dimensional inference in binary models. Our first result sharply predicts the statistical performance of such estimators in the linear asymptotic regime under isotropic Gaussian features. Importantly, the predictions hold for a wide class of convex loss functions, which we exploit in order to prove a bound on the best achievable performance among them. Notably, we show that the proposed bound is tight for popular binary models (such as Signed, Logistic or Probit), by constructing appropriate loss functions that achieve it. More interestingly, for binary linear classification under the Logistic and Probit models, we prove that the performance of least-squares is no worse than 0.997 and 0.98 times the optimal one. Numerical simulations corroborate our theoretical findings and suggest they are accurate even for relatively small problem dimensions.

We consider a decentralized stochastic learning problem where data points are distributed among computing nodes communicating over a directed graph. As the model size gets large, decentralized learning faces a major bottleneck that is the heavy communication load due to each node transmitting large messages (model updates) to its neighbors. To tackle this bottleneck, we propose the quantized decentralized stochastic learning algorithm over directed graphs that is based on the push-sum algorithm in decentralized consensus optimization. We prove that our algorithm achieves the same convergence rates of the decentralized stochastic learning algorithm with exact-communication for both convex and non-convex losses. Numerical evaluations corroborate our main theoretical results and illustrate significant speed-up compared to the exact-communication methods.

We consider the problem of learning the coefficients of a linear classifier through Empirical Risk Minimization with a convex loss function in the high-dimensional setting. In particular, we introduce an approach to characterize the best achievable classification risk among convex losses, when data points follow a standard Gaussian-mixture model. Importantly, we prove that the square loss function achieves the minimum classification risk for this data model. Our numerical illustrations verify the theoretical results and show that they are accurate even for relatively small problem dimensions.

We consider a decentralized learning problem, where a set of computing nodes aim at solving a non-convex optimization problem collaboratively. It is well-known that decentralized optimization schemes face two major system bottlenecks: stragglers' delay and communication overhead. In this paper, we tackle these bottlenecks by proposing a novel decentralized and gradient-based optimization algorithm named as QuanTimed-DSGD. Our algorithm stands on two main ideas: (i) we impose a deadline on the local gradient computations of each node at each iteration of the algorithm, and (ii) the nodes exchange quantized versions of their local models. The first idea robustifies to straggling nodes and the second alleviates communication efficiency. The key technical contribution of our work is to prove that with non-vanishing noises for quantization and stochastic gradients, the proposed method exactly converges to the global optimal for convex loss functions, and finds a first-order stationary point in non-convex scenarios. Our numerical evaluations of the QuanTimed-DSGD on training benchmark datasets, MNIST and CIFAR-10, demonstrate speedups of up to 3x in run-time, compared to state-of-the-art decentralized optimization methods.

Language-Centered Intelligence (LCI) is an approach to artificial intelligence currently under development at the University of Idaho. In this approach, the existing intervehicle communication language and associated logics are harnessed for parallel use to enable more advanced cognitive abilities, such as anticipation and hypothetical reasoning, and expand the behavioral repertoire of unmanned underwater vehicles (UUVs). We begin with a brief background of relevant UUV research and continue by defining LCI. Next, we propose an architecture for LCI and describe a few applications of LCI to collaborative UUV behaviors. Here, we detail the three primary sub-modules of LCI, namely, the Look-Ahead Inspections Module, the Imagination Replacement Approach, and the newly-proposed Message Anticipation Module (MAM). We conclude that the LCI approach represents a sophisticated extension of cognition in artificial agents and systems that use language to communicate.