Many real-world problems not only have complicated nonconvex functional constraints but also use a large number of data points. This motivates the design of efficient stochastic methods on finite-sum or expectation constrained problems. In this paper, we design and analyze stochastic inexact augmented Lagrangian methods (Stoc-iALM) to solve problems involving a nonconvex composite (i.e. smooth+nonsmooth) objective and nonconvex smooth functional constraints. We adopt the standard iALM framework and design a subroutine by using the momentum-based variance-reduced proximal stochastic gradient method (PStorm) and a postprocessing step. Under certain regularity conditions (assumed also in existing works), to reach an $\varepsilon$-KKT point in expectation, we establish an oracle complexity result of $O(\varepsilon^{-5})$, which is better than the best-known $O(\varepsilon^{-6})$ result. Numerical experiments on the fairness constrained problem and the Neyman-Pearson classification problem with real data demonstrate that our proposed method outperforms an existing method with the previously best-known complexity result.

A recent line of work has shown remarkable behaviors of the generalization error curves in simple learning models. Even the least-squares regression has shown atypical features such as the model-wise double descent, and further works have observed triple or multiple descents. Another important characteristic are the epoch-wise descent structures which emerge during training. The observations of model-wise and epoch-wise descents have been analytically derived in limited theoretical settings (such as the random feature model) and are otherwise experimental. In this work, we provide a full and unified analysis of the whole time-evolution of the generalization curve, in the asymptotic large-dimensional regime and under gradient-flow, within a wider theoretical setting stemming from a gaussian covariate model. In particular, we cover most cases already disparately observed in the literature, and also provide examples of the existence of multiple descent structures as a function of a model parameter or time. Furthermore, we show that our theoretical predictions adequately match the learning curves obtained by gradient descent over realistic datasets. Technically we compute averages of rational expressions involving random matrices using recent developments in random matrix theory based on "linear pencils". Another contribution, which is also of independent interest in random matrix theory, is a new derivation of related fixed point equations (and an extension there-off) using Dyson brownian motions.

In this paper, we carry out numerical analysis to prove convergence of a novel sample-wise back-propagation method for training a class of stochastic neural networks (SNNs). The structure of the SNN is formulated as discretization of a stochastic differential equation (SDE). A stochastic optimal control framework is introduced to model the training procedure, and a sample-wise approximation scheme for the adjoint backward SDE is applied to improve the efficiency of the stochastic optimal control solver, which is equivalent to the back-propagation for training the SNN. The convergence analysis is derived with and without convexity assumption for optimization of the SNN parameters. Especially, our analysis indicates that the number of SNN training steps should be proportional to the square of the number of layers in the convex optimization case. Numerical experiments are carried out to validate the analysis results, and the performance of the sample-wise back-propagation method for training SNNs is examined by benchmark machine learning examples.

The main difference with prior work is that the width of the network can be a fixed number instead of growing as some multiple or power of the number of data points. The convergence properties of gradient descent are well-understood when the objective function f is convex [14, 43], and it is known that finding local minima of nonconvex functions by gradient descent is an NPcomplete problem [42]. In spite of this, gradient descent is widely used in practice to find local and global minima in highly nonconvex problems, especially in high dimensions. For example, it has been observed that gradient descent can often find global minima of training loss in deep learning [27, 50], which is one of the reasons behind great success of the'deep learning revolution' [12, 36]. This article presents a novel criterion for convergence of gradient descent to a global minimum.

Transformers have become the state-of-the-art neural network architecture across numerous domains of machine learning. This is partly due to their celebrated ability to transfer and to learn in-context based on few examples. Nevertheless, the mechanisms by which Transformers become in-context learners are not well understood and remain mostly an intuition. Here, we argue that training Transformers on auto-regressive tasks can be closely related to well-known gradient-based meta-learning formulations. We start by providing a simple weight construction that shows the equivalence of data transformations induced by 1) a single linear self-attention layer and by 2) gradient-descent (GD) on a regression loss. Motivated by that construction, we show empirically that when training self-attention-only Transformers on simple regression tasks either the models learned by GD and Transformers show great similarity or, remarkably, the weights found by optimization match the construction. Thus we show how trained Transformers implement gradient descent in their forward pass. This allows us, at least in the domain of regression problems, to mechanistically understand the inner workings of optimized Transformers that learn in-context. Furthermore, we identify how Transformers surpass plain gradient descent by an iterative curvature correction and learn linear models on deep data representations to solve non-linear regression tasks. Finally, we discuss intriguing parallels to a mechanism identified to be crucial for in-context learning termed induction-head (Olsson et al., 2022) and show how it could be understood as a specific case of in-context learning by gradient descent learning within Transformers.

Effective decision making involves flexibly relating past experiences and relevant contextual information to a novel situation. In deep reinforcement learning (RL), the dominant paradigm is for an agent to amortise information that helps decision making into its network weights via gradient descent on training losses. Here, we pursue an alternative approach in which agents can utilise large-scale context sensitive database lookups to support their parametric computations. This allows agents to directly learn in an end-to-end manner to utilise relevant information to inform their outputs. In addition, new information can be attended to by the agent, without retraining, by simply augmenting the retrieval dataset. We study this approach for offline RL in 9x9 Go, a challenging game for which the vast combinatorial state space privileges generalisation over direct matching to past experiences. We leverage fast, approximate nearest neighbor techniques in order to retrieve relevant data from a set of tens of millions of expert demonstration states. Attending to this information provides a significant boost to prediction accuracy and game-play performance over simply using these demonstrations as training trajectories, providing a compelling demonstration of the value of large-scale retrieval in offline RL agents.

As organizations increasingly use machine learning in real world systems to provide insights on the data generated by real users (Team, 2017), issues of user data privacy have risen to the forefront of existing problems in machine learning. Differential privacy (DP) (Dwork et al., 2006) is the de facto standard for privacy preserving statistics. Common algorithms for privately training machine learning models are differentially private stochastic gradient descent (DP-SGD) (Song et al., 2013; Abadi et al., 2016) and differentially private empirical risk minimization (DP-ERM) (Chaudhuri et al., 2011). While advancements in deep learning can be partially attributed to scaling up the number of model parameters (Kaplan et al., 2020; Brown et al., 2020), as shown by (Kurakin et al., 2022; Tramèr and Boneh, 2020; Yu et al., 2021b; Shen et al., 2021) increasing the number of model parameters in DP-SGD often has an adverse impact on the privacy-utility tradeoff due to the curse of dimensionality present in DP-SGD. Briefly, the "curse" is that the magnitude of the noise added scales with d the square root of the number of parameters, and because the signal does not scale with the number of parameters, the signal to noise ratio (SNR) suffers at scale.

We introduce and analyze Structured Stochastic Zeroth order Descent (S-SZD), a finite difference approach which approximates a stochastic gradient on a set of $l\leq d$ orthogonal directions, where $d$ is the dimension of the ambient space. These directions are randomly chosen, and may change at each step. For smooth convex functions we prove almost sure convergence of the iterates and a convergence rate on the function values of the form $O(d/l k^{-c})$ for every $c<1/2$, which is arbitrarily close to the one of Stochastic Gradient Descent (SGD) in terms of number of iterations. Our bound also shows the benefits of using $l$ multiple directions instead of one. For non-convex functions satisfying the Polyak-{\L}ojasiewicz condition, we establish the first convergence rates for stochastic zeroth order algorithms under such an assumption. We corroborate our theoretical findings in numerical simulations where assumptions are satisfied and on the real-world problem of hyper-parameter optimization, observing that S-SZD has very good practical performances.

The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a quantitative central limit theorem for such SPDEs is derived, again, with optimal rate of convergence. The results apply, in particular, to the convergence in the mean-field scaling of stochastic gradient descent dynamics in overparametrized, shallow neural networks to solutions of SPDEs. It is shown that the inclusion of fluctuations in the limiting SPDE improves the rate of convergence, and retains information about the fluctuations of stochastic gradient descent in the continuum limit.

Differential privacy (DP) provides a formal privacy guarantee that prevents adversaries with access to machine learning models from extracting information about individual training points. Differentially private stochastic gradient descent (DPSGD) is the most popular training method with differential privacy in image recognition. However, existing DPSGD schemes lead to significant performance degradation, which prevents the application of differential privacy. In this paper, we propose a simulated annealing-based differentially private stochastic gradient descent scheme (SA-DPSGD) which accepts a candidate update with a probability that depends both on the update quality and on the number of iterations. Through this random update screening, we make the differentially private gradient descent proceed in the right direction in each iteration, and result in a more accurate model finally. In our experiments, under the same hyperparameters, our scheme achieves test accuracies 98.35%, 87.41% and 60.92% on datasets MNIST, FashionMNIST and CIFAR10, respectively, compared to the state-of-the-art result of 98.12%, 86.33% and 59.34%. Under the freely adjusted hyperparameters, our scheme achieves even higher accuracies, 98.89%, 88.50% and 64.17%. We believe that our method has a great contribution for closing the accuracy gap between private and non-private image classification.