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Gradient Descent (GD) is a powerful workhorse of modern machine learning thanks to its scalability and efficiency in high-dimensional spaces. Its ability to find local minimisers is only guaranteed for losses with Lipschitz gradients, where it can be seen as a `bona-fide' discretisation of an underlying gradient flow. Yet, many ML setups involving overparametrised models do not fall into this problem class, which has motivated research beyond the so-called ``Edge of Stability'' (EoS), where the step-size crosses the admissibility threshold inversely proportional to the Lipschitz constant above. Perhaps surprisingly, GD has been empirically observed to still converge regardless of local instability and oscillatory behavior. The incipient theoretical analysis of this phenomena has mainly focused in the overparametrised regime, where the effect of choosing a large learning rate may be associated to a `Sharpness-Minimisation' implicit regularisation within the manifold of minimisers, under appropriate asymptotic limits. In contrast, in this work we directly examine the conditions for such unstable convergence, focusing on simple, yet representative, learning problems, via analysis of two-step gradient updates. Specifically, we characterize a local condition involving third-order derivatives that guarantees existence and convergence to fixed points of the two-step updates, and leverage such property in a teacher-student setting, under population loss. Finally, starting from Matrix Factorization, we provide observations of period-2 orbit of GD in high-dimensional settings with intuition of its dynamics, along with exploration into more general settings.

Stochastic gradient descent (SGD) is the simplest deep learning optimizer with which to train deep neural networks. While SGD can use various learning rates, such as constant or diminishing rates, the previous numerical results showed that SGD performs better than other deep learning optimizers using when it uses learning rates given by line search methods. In this paper, we perform a convergence analysis on SGD with a learning rate given by an Armijo line search for nonconvex optimization. The analysis indicates that the upper bound of the expectation of the squared norm of the full gradient becomes small when the number of steps and the batch size are large. Next, we show that, for SGD with the Armijo-line-search learning rate, the number of steps needed for nonconvex optimization is a monotone decreasing convex function of the batch size; that is, the number of steps needed for nonconvex optimization decreases as the batch size increases. Furthermore, we show that the stochastic first-order oracle (SFO) complexity, which is the stochastic gradient computation cost, is a convex function of the batch size; that is, there exists a critical batch size that minimizes the SFO complexity. Finally, we provide numerical results that support our theoretical results. The numerical results indicate that the number of steps needed for training deep neural networks decreases as the batch size increases and that there exist the critical batch sizes that can be estimated from the theoretical results.

Classical differential private DP-SGD implements individual clipping with random subsampling, which forces a mini-batch SGD approach. We provide a general differential private algorithmic framework that goes beyond DP-SGD and allows any possible first order optimizers (e.g., classical SGD and momentum based SGD approaches) in combination with batch clipping, which clips an aggregate of computed gradients rather than summing clipped gradients (as is done in individual clipping). The framework also admits sampling techniques beyond random subsampling such as shuffling. Our DP analysis follows the $f$-DP approach and introduces a new proof technique which allows us to derive simple closed form expressions and to also analyse group privacy. In particular, for $E$ epochs work and groups of size $g$, we show a $\sqrt{g E}$ DP dependency for batch clipping with shuffling.

Let $\hat\Sigma=\frac{1}{n}\sum_{i=1}^n X_i\otimes X_i$ denote the sample covariance operator of centered i.i.d. observations $X_1,\dots,X_n$ in a real separable Hilbert space, and let $\Sigma=\mathbf{E}(X_1\otimes X_1)$. The focus of this paper is to understand how well the bootstrap can approximate the distribution of the operator norm error $\sqrt n\|\hat\Sigma-\Sigma\|_{\text{op}}$, in settings where the eigenvalues of $\Sigma$ decay as $\lambda_j(\Sigma)\asymp j^{-2\beta}$ for some fixed parameter $\beta>1/2$. Our main result shows that the bootstrap can approximate the distribution of $\sqrt n\|\hat\Sigma-\Sigma\|_{\text{op}}$ at a rate of order $n^{-\frac{\beta-1/2}{2\beta+4+\epsilon}}$ with respect to the Kolmogorov metric, for any fixed $\epsilon>0$. In particular, this shows that the bootstrap can achieve near $n^{-1/2}$ rates in the regime of large $\beta$--which substantially improves on previous near $n^{-1/6}$ rates in the same regime. In addition to obtaining faster rates, our analysis leverages a fundamentally different perspective based on coordinate-free techniques. Moreover, our result holds in greater generality, and we propose a new model that is compatible with both elliptical and Mar\v{c}enko-Pastur models in high-dimensional Euclidean spaces, which may be of independent interest.

We study distributed estimation and learning problems in a networked environment in which agents exchange information to estimate unknown statistical properties of random variables from their privately observed samples. By exchanging information about their private observations, the agents can collectively estimate the unknown quantities, but they also face privacy risks. The goal of our aggregation schemes is to combine the observed data efficiently over time and across the network, while accommodating the privacy needs of the agents and without any coordination beyond their local neighborhoods. Our algorithms enable the participating agents to estimate a complete sufficient statistic from private signals that are acquired offline or online over time, and to preserve the privacy of their signals and network neighborhoods. This is achieved through linear aggregation schemes with adjusted randomization schemes that add noise to the exchanged estimates subject to differential privacy (DP) constraints. In every case, we demonstrate the efficiency of our algorithms by proving convergence to the estimators of a hypothetical, omniscient observer that has central access to all of the signals. We also provide convergence rate analysis and finite-time performance guarantees and show that the noise that minimizes the convergence time to the best estimates is the Laplace noise, with parameters corresponding to each agent's sensitivity to their signal and network characteristics. Finally, to supplement and validate our theoretical results, we run experiments on real-world data from the US Power Grid Network and electric consumption data from German Households to estimate the average power consumption of power stations and households under all privacy regimes.

Distributed collaborative machine learning (DCML) is a promising method in the Internet of Things (IoT) domain for training deep learning models, as data is distributed across multiple devices. A key advantage of this approach is that it improves data privacy by removing the necessity for the centralized aggregation of raw data but also empowers IoT devices with low computational power. Among various techniques in a DCML framework, federated split learning, known as splitfed learning (SFL), is the most suitable for efficient training and testing when devices have limited computational capabilities. Nevertheless, when resource-constrained IoT devices have only positive labeled data, multiclass classification deep learning models in SFL fail to converge or provide suboptimal results. To overcome these challenges, we propose splitfed learning with positive labels (SFPL). SFPL applies a random shuffling function to the smashed data received from clients before supplying it to the server for model training. Additionally, SFPL incorporates the local batch normalization for the client-side model portion during the inference phase. Our results demonstrate that SFPL outperforms SFL: (i) by factors of 51.54 and 32.57 for ResNet-56 and ResNet-32, respectively, with the CIFAR-100 dataset, and (ii) by factors of 9.23 and 8.52 for ResNet-32 and ResNet-8, respectively, with CIFAR-10 dataset. Overall, this investigation underscores the efficacy of the proposed SFPL framework in DCML.

It is commonplace to use data containing personal information to build predictive models in the framework of empirical risk minimization (ERM). While these models can be highly accurate in prediction, results obtained from these models with the use of sensitive data may be susceptible to privacy attacks. Differential privacy (DP) is an appealing framework for addressing such data privacy issues by providing mathematically provable bounds on the privacy loss incurred when releasing information from sensitive data. Previous work has primarily concentrated on applying DP to unweighted ERM. We consider an important generalization to weighted ERM (wERM). In wERM, each individual's contribution to the objective function can be assigned varying weights. In this context, we propose the first differentially private wERM algorithm, backed by a rigorous theoretical proof of its DP guarantees under mild regularity conditions. Extending the existing DP-ERM procedures to wERM paves a path to deriving privacy-preserving learning methods for individualized treatment rules, including the popular outcome weighted learning (OWL). We evaluate the performance of the DP-wERM application to OWL in a simulation study and in a real clinical trial of melatonin for sleep health. All empirical results demonstrate the viability of training OWL models via wERM with DP guarantees while maintaining sufficiently useful model performance. Therefore, we recommend practitioners consider implementing the proposed privacy-preserving OWL procedure in real-world scenarios involving sensitive data.

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.

Promoting behavioural diversity is critical for solving games with non-transitive dynamics where strategic cycles exist, and there is no consistent winner (e.g., Rock-Paper-Scissors). Yet, there is a lack of rigorous treatment for defining diversity and constructing diversity-aware learning dynamics. In this work, we offer a geometric interpretation of behavioural diversity in games and introduce a novel diversity metric based on \emph{determinantal point processes} (DPP). By incorporating the diversity metric into best-response dynamics, we develop \emph{diverse fictitious play} and \emph{diverse policy-space response oracle} for solving normal-form games and open-ended games. We prove the uniqueness of the diverse best response and the convergence of our algorithms on two-player games. Importantly, we show that maximising the DPP-based diversity metric guarantees to enlarge the \emph{gamescape} -- convex polytopes spanned by agents' mixtures of strategies. To validate our diversity-aware solvers, we test on tens of games that show strong non-transitivity. Results suggest that our methods achieve much lower exploitability than state-of-the-art solvers by finding effective and diverse strategies.

Federated learning is a new distributed machine learning framework, where a bunch of heterogeneous clients collaboratively train a model without sharing training data. In this work, we consider a practical and ubiquitous issue in federated learning: intermittent client availability, where the set of eligible clients may change during the training process. Such an intermittent client availability model would significantly deteriorate the performance of the classical Federated Averaging algorithm (FedAvg for short). We propose a simple distributed non-convex optimization algorithm, called Federated Latest Averaging (FedLaAvg for short), which leverages the latest gradients of all clients, even when the clients are not available, to jointly update the global model in each iteration. Our theoretical analysis shows that FedLaAvg attains the convergence rate of $O(1/(N^{1/4} T^{1/2}))$, achieving a sublinear speedup with respect to the total number of clients. We implement and evaluate FedLaAvg with the CIFAR-10 dataset. The evaluation results demonstrate that FedLaAvg indeed reaches a sublinear speedup and achieves 4.23% higher test accuracy than FedAvg.