This paper studies the generalization performance of iterates obtained by Gradient Descent (GD), Stochastic Gradient Descent (SGD) and their proximal variants in high-dimensional robust regression problems. The number of features is comparable to the sample size and errors may be heavy-tailed. We introduce estimators that precisely track the generalization error of the iterates along the trajectory of the iterative algorithm. These estimators are provably consistent under suitable conditions. The results are illustrated through several examples, including Huber regression, pseudo-Huber regression, and their penalized variants with non-smooth regularizer.

Stochastic gradient descent with backpropagation is the workhorse of artificial neural networks. It has long been recognized that backpropagation fails to be a biologically plausible algorithm. Fundamentally, it is a non-local procedure---updating one neuron's synaptic weights requires knowledge of synaptic weights or receptive fields of downstream neurons. This limits the use of artificial neural networks as a tool for understanding the biological principles of information processing in the brain. Lillicrap et al. (2016) propose a more biologically plausible "feedback alignment" algorithm that uses random and fixed backpropagation weights, and show promising simulations.

In multi-output regression, we identify a previously neglected challenge that arises from the inability of training distribution to cover all combinations of input features, leading to combinatorial distribution shift (CDS). To the best of our knowledge, this is the first work to formally define and address this problem. We tackle it through a novel tensor decomposition perspective, proposing the Functional t-Singular Value Decomposition (Ft-SVD) theorem which extends the classical tensor SVD to infinite and continuous feature domains, providing a natural tool for representing and analyzing multi-output functions. Within the Ft-SVD framework, we formulate the multi-output regression problem under CDS as a low-rank tensor estimation problem under the missing not at random (MNAR) setting, and introduce a series of assumptions about the true functions, training and testing distributions, and spectral properties of the ground-truth embeddings, making the problem more tractable.To address the challenges posed by CDS in multi-output regression, we develop a tailored Double-Stage Empirical Risk Minimization (ERM-DS) algorithm that leverages the spectral properties of the embeddings and uses specific hypothesis classes in each frequency component to better capture the varying spectral decay patterns. We provide rigorous theoretical analyses that establish performance guarantees for the ERM-DS algorithm.

Annealed importance sampling (AIS) and related algorithms are highly effective tools for marginal likelihood estimation, but are not fully differentiable due to the use of Metropolis-Hastings correction steps. Differentiability is a desirable property as it would admit the possibility of optimizing marginal likelihood as an objective using gradient-based methods. To this end, we propose Differentiable AIS (DAIS), a variant of AIS which ensures differentiability by abandoning the Metropolis-Hastings corrections. As a further advantage, DAIS allows for mini-batch gradients. We provide a detailed convergence analysis for Bayesian linear regression which goes beyond previous analyses by explicitly accounting for the sampler not having reached equilibrium.

The ability of learning useful features is one of the major advantages of neural networks. Although recent works show that neural network can operate in a neural tangent kernel (NTK) regime that does not allow feature learning, many works also demonstrate the potential for neural networks to go beyond NTK regime and perform feature learning. Recently, a line of work highlighted the feature learning capabilities of the early stages of gradient-based training. In this paper we consider another mechanism for feature learning via gradient descent through a local convergence analysis. We show that once the loss is below a certain threshold, gradient descent with a carefully regularized objective will capture ground-truth directions.

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 .

Understanding generalization in deep learning has been one of the major challenges in statistical learning theory over the last decade. While recent work has illustrated that the dataset and the training algorithm must be taken into account in order to obtain meaningful generalization bounds, it is still theoretically not clear which properties of the data and the algorithm determine the generalization performance. In this study, we approach this problem from a dynamical systems theory perspective and represent stochastic optimization algorithms as \emph{random iterated function systems} (IFS). Well studied in the dynamical systems literature, under mild assumptions, such IFSs can be shown to be ergodic with an invariant measure that is often supported on sets with a \emph{fractal structure}. As our main contribution, we prove that the generalization error of a stochastic optimization algorithm can be bounded based on the complexity' of the fractal structure that underlies its invariant measure.

Gradient descent and its variants are de facto standard algorithms for training machine learning models. As gradient descent is sensitive to its hyperparameters, we need to tune the hyperparameters carefully using a grid search. However, the method is time-consuming, particularly when multiple hyperparameters exist. Therefore, recent studies have analyzed parameter-free methods that adjust the hyperparameters on the fly. However, the existing work is limited to investigations of parameter-free methods for the stepsize, and parameter-free methods for other hyperparameters have not been explored. For instance, although the gradient clipping threshold is a crucial hyperparameter in addition to the stepsize for preventing gradient explosion issues, none of the existing studies have investigated parameter-free methods for clipped gradient descent.

Generalized eigenvalue problems (GEPs) find applications in various fields of science and engineering. For example, principal component analysis, Fisher's discriminant analysis, and canonical correlation analysis are specific instances of GEPs and are widely used in statistical data processing. In this work, we study GEPs under generative priors, assuming that the underlying leading generalized eigenvector lies within the range of a Lipschitz continuous generative model. Under appropriate conditions, we show that any optimal solution to the corresponding optimization problems attains the optimal statistical rate. Moreover, from a computational perspective, we propose an iterative algorithm called the Projected Rayleigh Flow Method (PRFM) to approximate the optimal solution.

In this work, we study the mean-field flow for learning subspace-sparse polynomials using stochastic gradient descent and two-layer neural networks, where the input distribution is standard Gaussian and the output only depends on the projection of the input onto a low-dimensional subspace. We establish a necessary condition for SGD-learnability, involving both the characteristics of the target function and the expressiveness of the activation function. In addition, we prove that the condition is almost sufficient, in the sense that a condition slightly stronger than the necessary condition can guarantee the exponential decay of the loss functional to zero.