This paper presents two models of neural-networks and their training applicable to neural networks of arbitrary width, depth and topology, assuming only finite-energy neural activations; and a novel representor theory for neural networks in terms of a matrix-valued kernel. The first model is exact (un-approximated) and global, casting the neural network as an elements in a reproducing kernel Banach space (RKBS); we use this model to provide tight bounds on Rademacher complexity. The second model is exact and local, casting the change in neural network function resulting from a bounded change in weights and biases (ie. a training step) in reproducing kernel Hilbert space (RKHS) in terms of a local-intrinsic neural kernel (LiNK). This local model provides insight into model adaptation through tight bounds on Rademacher complexity of network adaptation. We also prove that the neural tangent kernel (NTK) is a first-order approximation of the LiNK kernel. Finally, and noting that the LiNK does not provide a representor theory for technical reasons, we present an exact novel representor theory for layer-wise neural network training with unregularized gradient descent in terms of a local-extrinsic neural kernel (LeNK). This representor theory gives insight into the role of higher-order statistics in neural network training and the effect of kernel evolution in neural-network kernel models. Throughout the paper (a) feedforward ReLU networks and (b) residual networks (ResNet) are used as illustrative examples.

State-of-the-art neural network training methods depend on the gradient of the network function. Therefore, they cannot be applied to networks whose activation functions do not have useful derivatives, such as binary and discrete-time spiking neural networks. To overcome this problem, the activation function's derivative is commonly substituted with a surrogate derivative, giving rise to surrogate gradient learning (SGL). This method works well in practice but lacks theoretical foundation. The neural tangent kernel (NTK) has proven successful in the analysis of gradient descent. Here, we provide a generalization of the NTK, which we call the surrogate gradient NTK, that enables the analysis of SGL. First, we study a naive extension of the NTK to activation functions with jumps, demonstrating that gradient descent for such activation functions is also ill-posed in the infinite-width limit. To address this problem, we generalize the NTK to gradient descent with surrogate derivatives, i.e., SGL. We carefully define this generalization and expand the existing key theorems on the NTK with mathematical rigor. Further, we illustrate our findings with numerical experiments. Finally, we numerically compare SGL in networks with sign activation function and finite width to kernel regression with the surrogate gradient NTK; the results confirm that the surrogate gradient NTK provides a good characterization of SGL.

Optimal Transport (OT) based distances are powerful tools for machine learning to compare probability measures and manipulate them using OT maps. In this field, a setting of interest is semi-discrete OT, where the source measure $\mu$ is continuous, while the target $\nu$ is discrete. Recent works have shown that the minimax rate for the OT map is $\mathcal{O}(t^{-1/2})$ when using $t$ i.i.d. subsamples from each measure (two-sample setting). An open question is whether a better convergence rate can be achieved when the full information of the discrete measure $\nu$ is known (one-sample setting). In this work, we answer positively to this question by (i) proving an $\mathcal{O}(t^{-1})$ lower bound rate for the OT map, using the similarity between Laguerre cells estimation and density support estimation, and (ii) proposing a Stochastic Gradient Descent (SGD) algorithm with adaptive entropic regularization and averaging acceleration. To nearly achieve the desired fast rate, characteristic of non-regular parametric problems, we design an entropic regularization scheme decreasing with the number of samples. Another key step in our algorithm consists of using a projection step that permits to leverage the local strong convexity of the regularized OT problem. Our convergence analysis integrates online convex optimization and stochastic gradient techniques, complemented by the specificities of the OT semi-dual. Moreover, while being as computationally and memory efficient as vanilla SGD, our algorithm achieves the unusual fast rates of our theory in numerical experiments.

Neural networks can identify low-dimensional relevant structures within high-dimensional noisy data, yet our mathematical understanding of how they do so remains scarce. Here, we investigate the training dynamics of two-layer shallow neural networks trained with gradient-based algorithms, and discuss how they learn pertinent features in multi-index models, that is target functions with low-dimensional relevant directions. In the high-dimensional regime, where the input dimension $d$ diverges, we show that a simple modification of the idealized single-pass gradient descent training scenario, where data can now be repeated or iterated upon twice, drastically improves its computational efficiency. In particular, it surpasses the limitations previously believed to be dictated by the Information and Leap exponents associated with the target function to be learned. Our results highlight the ability of networks to learn relevant structures from data alone without any pre-processing. More precisely, we show that (almost) all directions are learned with at most $O(d \log d)$ steps. Among the exceptions is a set of hard functions that includes sparse parities. In the presence of coupling between directions, however, these can be learned sequentially through a hierarchical mechanism that generalizes the notion of staircase functions. Our results are proven by a rigorous study of the evolution of the relevant statistics for high-dimensional dynamics.

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, but it is time-consuming, especially when multiple hyperparameters exist. Recently, parameter-free methods that adjust the hyperparameters on the fly have been studied. However, the existing work only studied parameter-free methods for the stepsize, and parameter-free methods for other hyperparameters have not been explored. For instance, the gradient clipping threshold is also a crucial hyperparameter in addition to the stepsize to prevent gradient explosion issues, but none of the existing studies investigated the parameter-free methods for clipped gradient descent. In this work, we study the parameter-free methods for clipped gradient descent. Specifically, we propose Inexact Polyak Stepsize, which converges to the optimal solution without any hyperparameters tuning, and its convergence rate is asymptotically independent of L under L-smooth and $(L_0, L_1)$-smooth assumptions of the loss function as that of clipped gradient descent with well-tuned hyperparameters. We numerically validated our convergence results using a synthetic function and demonstrated the effectiveness of our proposed methods using LSTM, Nano-GPT, and T5.

We study theoretical properties of a broad class of regularized algorithms with vector-valued output. These spectral algorithms include kernel ridge regression, kernel principal component regression, various implementations of gradient descent and many more. Our contributions are twofold. First, we rigorously confirm the so-called saturation effect for ridge regression with vector-valued output by deriving a novel lower bound on learning rates; this bound is shown to be suboptimal when the smoothness of the regression function exceeds a certain level. Second, we present the upper bound for the finite sample risk general vector-valued spectral algorithms, applicable to both well-specified and misspecified scenarios (where the true regression function lies outside of the hypothesis space) which is minimax optimal in various regimes. All of our results explicitly allow the case of infinite-dimensional output variables, proving consistency of recent practical applications.

We consider the three parameter solvable neural scaling model introduced by Maloney, Roberts, and Sully. The model has three parameters: data complexity, target complexity, and model-parameter-count. We use this neural scaling model to derive new predictions about the compute-limited, infinite-data scaling law regime. To train the neural scaling model, we run one-pass stochastic gradient descent on a mean-squared loss. We derive a representation of the loss curves which holds over all iteration counts and improves in accuracy as the model parameter count grows. We then analyze the compute-optimal model-parameter-count, and identify 4 phases (+3 subphases) in the data-complexity/target-complexity phase-plane. The phase boundaries are determined by the relative importance of model capacity, optimizer noise, and embedding of the features. We furthermore derive, with mathematical proof and extensive numerical evidence, the scaling-law exponents in all of these phases, in particular computing the optimal model-parameter-count as a function of floating point operation budget.

We develop new algorithms for Riemannian bilevel optimization. We focus in particular on batch and stochastic gradient-based methods, with the explicit goal of avoiding second-order information such as Riemannian hyper-gradients. We propose and analyze $\mathrm{RF^2SA}$, a method that leverages first-order gradient information to navigate the complex geometry of Riemannian manifolds efficiently. Notably, $\mathrm{RF^2SA}$ is a single-loop algorithm, and thus easier to implement and use. Under various setups, including stochastic optimization, we provide explicit convergence rates for reaching $\epsilon$-stationary points. We also address the challenge of optimizing over Riemannian manifolds with constraints by adjusting the multiplier in the Lagrangian, ensuring convergence to the desired solution without requiring access to second-order derivatives.

Score-based generative models have demonstrated significant practical success in data-generating tasks. The models establish a diffusion process that perturbs the ground truth data to Gaussian noise and then learn the reverse process to transform noise into data. However, existing denoising methods such as Langevin dynamic and numerical stochastic differential equation solvers enjoy randomness but generate data slowly with a large number of score function evaluations, and the ordinary differential equation solvers enjoy faster sampling speed but no randomness may influence the sample quality. To this end, motivated by the Stochastic Gradient Descent (SGD) optimization methods and the high connection between the model sampling process with the SGD, we propose adaptive momentum sampling to accelerate the transforming process without introducing additional hyperparameters. Theoretically, we proved our method promises convergence under given conditions. In addition, we empirically show that our sampler can produce more faithful images/graphs in small sampling steps with 2 to 5 times speed up and obtain competitive scores compared to the baselines on image and graph generation tasks.

Gradient based optimization algorithms deployed in Machine Learning (ML) applications are often analyzed and compared by their convergence rates or regret bounds. While these rates and bounds convey valuable information they don't always directly translate to stability guarantees. Stability and similar concepts, like robustness, will become ever more important as we move towards deploying models in real-time and safety critical systems. In this work we build upon the results in Gaudio et al. 2021 and Moreu and Annaswamy 2022 for second order gradient descent when applied to explicitly time varying cost functions and provide more general stability guarantees. These more general results can aid in the design and certification of these optimization schemes so as to help ensure safe and reliable deployment for real-time learning applications. We also hope that the techniques provided here will stimulate and cross-fertilize the analysis that occurs on the same algorithms from the online learning and stochastic optimization communities.