Recent studies have provided both empirical and theoretical evidence illustrating that heavy tails can emerge in stochastic gradient descent (SGD) in various scenarios. Such heavy tails potentially result in iterates with diverging variance, which hinders the use of conventional convergence analysis techniques that rely on the existence of the second-order moments. In this paper, we provide convergence guarantees for SGD under a state-dependent and heavy-tailed noise with a potentially infinite variance, for a class of strongly convex objectives. In the case where the p -th moment of the noise exists for some p\in [1,2), we first identify a condition on the Hessian, coined p -positive (semi-)definiteness', that leads to an interesting interpolation between the positive semi-definite cone ( p 2) and the cone of diagonally dominant matrices with non-negative diagonal entries ( p 1). Under this condition, we provide a convergence rate for the distance to the global optimum in L p .

Neural Architecture Search (NAS) has fostered the automatic discovery of state-of-the-art neural architectures. Despite the progress achieved with NAS, so far there is little attention to theoretical guarantees on NAS. In this work, we study the generalization properties of NAS under a unifying framework enabling (deep) layer skip connection search and activation function search. To this end, we derive the lower (and upper) bounds of the minimum eigenvalue of the Neural Tangent Kernel (NTK) under the (in)finite-width regime using a certain search space including mixed activation functions, fully connected, and residual neural networks. We use the minimum eigenvalue to establish generalization error bounds of NAS in the stochastic gradient descent training.

We consider linear prediction with a convex Lipschitz loss, or more generally, stochastic convex optimization problems of generalized linear form, i.e. We show that in this setting, early stopped Gradient Descent (GD), without any explicit regularization or projection, ensures excess error at most \varepsilon (compared to the best possible with unit Euclidean norm) with an optimal, up to logarithmic factors, sample complexity of \tilde{O}(1/\varepsilon 2) and only \tilde{O}(1/\varepsilon 2) iterations. This contrasts with general stochastic convex optimization, where \Omega(1/\varepsilon 4) iterations are needed Amir et al. 2021. The lower iteration complexity is ensured by leveraging uniform convergence rather than stability. But instead of uniform convergence in a norm ball, which we show can guarantee suboptimal learning using \Theta(1/\varepsilon 4) samples, we rely on uniform convergence in a distribution-dependent ball.

Despite the non-convex optimization landscape, over-parametrized shallow networks are able to achieve global convergence under gradient descent. The picture can be radically different for narrow networks, which tend to get stuck in badly-generalizing local minima. Here we investigate the cross-over between these two regimes in the high-dimensional setting, and in particular investigate the connection between the so-called mean-field/hydrodynamic regime and the seminal approach of Saad \& Solla. Focusing on the case of Gaussian data, we study the interplay between the learning rate, the time scale, and the number of hidden units in the high-dimensional dynamics of stochastic gradient descent (SGD). Our work builds on a deterministic description of SGD in high-dimensions from statistical physics, which we extend and for which we provide rigorous convergence rates.

Communication cost is one major bottleneck for the scalability for distributed learning. One approach to reduce the communication cost is to compress the gradient during communication. However, directly compressing the gradient decelerates the convergence speed, and the resulting algorithm may diverge for biased compression. Recent work addressed this problem for stochastic gradient descent by adding back the compression error from the previous step. This idea was further extended to one class of variance reduced algorithms, where the variance of the stochastic gradient is reduced by taking a moving average over all history gradients.

Sampling from an unnormalized target distribution is an essential problem with many applications in probabilistic inference. Stein Variational Gradient Descent (SVGD) has been shown to be a powerful method that iteratively updates a set of particles to approximate the distribution of interest. Furthermore, when analysing its asymptotic properties, SVGD reduces exactly to a single-objective optimization problem and can be viewed as a probabilistic version of this single-objective optimization problem. A natural question then arises: Can we derive a probabilistic version of the multi-objective optimization?''. To answer this question, we propose Stochastic Multiple Target Sampling Gradient Descent (MT-SGD), enabling us to sample from multiple unnormalized target distributions.

The loss terrains of over-parameterized neural networks have multiple global minima. However, it is well known that stochastic gradient descent (SGD) can stably converge only to minima that are sufficiently flat w.r.t. In this paper we study the effect that this mechanism has on the function implemented by the trained model. First, we extend the existing knowledge on minima stability to non-differentiable minima, which are common in ReLU nets. We then use our stability results to study a single hidden layer univariate ReLU network.

Complex-valued neural networks potentially possess better representations and performance than real-valued counterparts when dealing with some complicated tasks such as acoustic analysis, radar image classification, etc. Despite empirical successes, it remains unknown theoretically when and to what extent complex-valued neural networks outperform real-valued ones. We take one step in this direction by comparing the learnability of real-valued neurons and complex-valued neurons via gradient descent. We show that a complex-valued neuron can efficiently learn functions expressed by any one real-valued neuron and any one complex-valued neuron with convergence rate O(t {-3}) and O(t {-1}) where t is the iteration index of gradient descent, respectively, whereas a two-layer real-valued neural network with finite width cannot learn a single non-degenerate complex-valued neuron. We prove that a complex-valued neuron learns a real-valued neuron with rate \Omega (t {-3}), exponentially slower than the O(\mathrm{e} {- c t}) rate of learning one real-valued neuron using a real-valued neuron with a constant c .

Nesterov's accelerated gradient method (NAG) marks a pivotal advancement in gradient-based optimization, achieving faster convergence compared to the vanilla gradient descent method for convex functions. However, its algorithmic complexity when applied to strongly convex functions remains unknown, as noted in the comprehensive review by Chambolle and Pock [2016]. This issue, aside from the critical step size, was addressed by Li et al. [2024b], with the monotonic case further explored by Fu and Shi [2024]. In this paper, we introduce a family of controllable momentum coefficients for forward-backward accelerated methods, focusing on the critical step size $s=1/L$. Unlike traditional linear forms, the proposed momentum coefficients follow an $\alpha$-th power structure, where the parameter $r$ is adaptively tuned to $\alpha$. Using a Lyapunov function specifically designed for $\alpha$, we establish a controllable $O\left(1/k^{2\alpha} \right)$ convergence rate for the NAG-$\alpha$ method, provided that $r > 2\alpha$. At the critical step size, NAG-$\alpha$ achieves an inverse polynomial convergence rate of arbitrary degree by adjusting $r$ according to $\alpha > 0$. We further simplify the Lyapunov function by expressing it in terms of the iterative sequences $x_k$ and $y_k$, eliminating the need for phase-space representations. This simplification enables us to extend the controllable $O \left(1/k^{2\alpha} \right)$ rate to the monotonic variant, M-NAG-$\alpha$, thereby enhancing optimization efficiency. Finally, by leveraging the fundamental inequality for composite functions, we extended the controllable $O\left(1/k^{2\alpha} \right)$ rate to proximal algorithms, including the fast iterative shrinkage-thresholding algorithm (FISTA-$\alpha$) and its monotonic counterpart (M-FISTA-$\alpha$).

Stochastic Gradient Descent (SGD) is arguably the most important single algorithm in modern machine learning. Although SGD with unbiased gradient estimators has been studied extensively over at least half a century, SGD variants relying on biased estimators are rare. Nevertheless, there has been an increased interest in this topic in recent years. However, existing literature on SGD with biased estimators lacks coherence since each new paper relies on a different set of assumptions, without any clear understanding of how they are connected, which may lead to confusion. We address this gap by establishing connections among the existing assumptions, and presenting a comprehensive map of the underlying relationships. Additionally, we introduce a new set of assumptions that is provably weaker than all previous assumptions, and use it to present a thorough analysis of BiasedSGD in both convex and non-convex settings, offering advantages over previous results.