The convolutional neural networks (CNNs) are generally trained using stochastic gradient descent (SGD) based optimization techniques. The existing SGD optimizers generally suffer with the overshooting of the minimum and oscillation near minimum. In this paper, we propose a new approach, hereafter referred as AdaInject, for the gradient descent optimizers by injecting the second order moment into the first order moment. Specifically, the short-term change in parameter is used as a weight to inject the second order moment in the update rule. The AdaInject optimizer controls the parameter update, avoids the overshooting of the minimum and reduces the oscillation near minimum. The proposed approach is generic in nature and can be integrated with any existing SGD optimizer. The effectiveness of the AdaInject optimizer is explained intuitively as well as through some toy examples. We also show the convergence property of the proposed injection based optimizer. Further, we depict the efficacy of the AdaInject approach through extensive experiments in conjunction with the state-of-the-art optimizers, namely AdamInject, diffGradInject, RadamInject, and AdaBeliefInject on four benchmark datasets. Different CNN models are used in the experiments. A highest improvement in the top-1 classification error rate of $16.54\%$ is observed using diffGradInject optimizer with ResNeXt29 model over the CIFAR10 dataset. Overall, we observe very promising performance improvement of existing optimizers with the proposed AdaInject approach. The code is available at: \url{https://github.com/shivram1987/AdaInject}.

Averaging neural network weights sampled by a backbone stochastic gradient descent (SGD) is a simple yet effective approach to assist the backbone SGD in finding better optima, in terms of generalization. From a statistical perspective, weight averaging (WA) contributes to variance reduction. Recently, a well-established stochastic weight averaging (SWA) method is proposed, which is featured by the application of a cyclical or high constant (CHC) learning rate schedule (LRS) in generating weight samples for WA. Then a new insight on WA appears, which states that WA helps to discover wider optima and then leads to better generalization. We conduct extensive experimental studies for SWA, involving a dozen modern DNN model structures and a dozen benchmark open-source image, graph, and text datasets. We disentangle contributions of the WA operation and the CHC LRS for SWA, showing that the WA operation in SWA still contributes to variance reduction but does not always lead to wide optima. The experimental results indicate that there are global scale geometric structures in the DNN loss landscape. We then present an algorithm termed periodic SWA (PSWA) which makes use of a series of WA operations to discover the global geometric structures. PSWA outperforms its backbone SGD remarkably, providing experimental evidences for the existence of global geometric structures. Codes for reproducing the experimental results are available at https://github.com/ZJLAB-AMMI/PSWA.

Two aspects of neural networks that have been extensively studied in the recent literature are their function approximation properties and their training by gradient descent methods. The approximation problem seeks accurate approximations with a minimal number of weights. In most of the current literature these weights are fully or partially hand-crafted, showing the capabilities of neural networks but not necessarily their practical performance. In contrast, optimization theory for neural networks heavily relies on an abundance of weights in over-parametrized regimes. This paper balances these two demands and provides an approximation result for shallow networks in $1d$ with non-convex weight optimization by gradient descent. We consider finite width networks and infinite sample limits, which is the typical setup in approximation theory. Technically, this problem is not over-parametrized, however, some form of redundancy reappears as a loss in approximation rate compared to best possible rates.

Training deep neural networks often forces users to work in a distributed or outsourced setting, accompanied with privacy concerns. Split learning aims to address this concern by distributing the model among a client and a server. The scheme supposedly provides privacy, since the server cannot see the clients' models and inputs. We show that this is not true via two novel attacks. (1) We show that an honest-but-curious split learning server, equipped only with the knowledge of the client neural network architecture, can recover the input samples and obtain a functionally similar model to the client model, without being detected. (2) We show that if the client keeps hidden only the output layer of the model to "protect" the private labels, the honest-but-curious server can infer the labels with perfect accuracy. We test our attacks using various benchmark datasets and against proposed privacy-enhancing extensions to split learning. Our results show that plaintext split learning can pose serious risks, ranging from data (input) privacy to intellectual property (model parameters), and provide no more than a false sense of security.

Learning pair interactions from experimental or simulation data is of great interest for molecular simulations. We propose a general stochastic method for learning pair interactions from data using differentiable simulations (DiffSim). DiffSim defines a loss function based on structural observables, such as the radial distribution function, through molecular dynamics (MD) simulations. The interaction potentials are then learned directly by stochastic gradient descent, using backpropagation to calculate the gradient of the structural loss metric with respect to the interaction potential through the MD simulation. This gradient-based method is flexible and can be configured to simulate and optimize multiple systems simultaneously. For example, it is possible to simultaneously learn potentials for different temperatures or for different compositions. We demonstrate the approach by recovering simple pair potentials, such as Lennard-Jones systems, from radial distribution functions. We find that DiffSim can be used to probe a wider functional space of pair potentials compared to traditional methods like Iterative Boltzmann Inversion. We show that our methods can be used to simultaneously fit potentials for simulations at different compositions and temperatures to improve the transferability of the learned potentials.

Identification of linear time-invariant (LTI) systems plays an important role in control and reinforcement learning. Both asymptotic and finite-time offline system identification are well-studied in the literature. For online system identification, the idea of stochastic-gradient descent with reverse experience replay (SGD-RER) was recently proposed, where the data sequence is stored in several buffers and the stochastic-gradient descent (SGD) update performs backward in each buffer to break the time dependency between data points. Inspired by this work, we study distributed online system identification of LTI systems over a multi-agent network. We consider agents as identical LTI systems, and the network goal is to jointly estimate the system parameters by leveraging the communication between agents. We propose DSGD-RER, a distributed variant of the SGD-RER algorithm, and theoretically characterize the improvement of the estimation error with respect to the network size. Our numerical experiments certify the reduction of estimation error as the network size grows.

Multiple-objective optimization (MOO) aims to simultaneously optimize multiple conflicting objectives and has found important applications in machine learning, such as minimizing classification loss and discrepancy in treating different populations for fairness. At optimality, further optimizing one objective will necessarily harm at least another objective, and decision-makers need to comprehensively explore multiple optima (called Pareto front) to pinpoint one final solution. We address the efficiency of finding the Pareto front. First, finding the front from scratch using stochastic multi-gradient descent (SMGD) is expensive with large neural networks and datasets. We propose to explore the Pareto front as a manifold from a few initial optima, based on a predictor-corrector method. Second, for each exploration step, the predictor solves a large-scale linear system that scales quadratically in the number of model parameters and requires one backpropagation to evaluate a second-order Hessian-vector product per iteration of the solver. We propose a Gauss-Newton approximation that only scales linearly, and that requires only first-order inner-product per iteration. This also allows for a choice between the MINRES and conjugate gradient methods when approximately solving the linear system. The innovations make predictor-corrector possible for large networks. Experiments on multi-objective (fairness and accuracy) misinformation detection tasks show that 1) the predictor-corrector method can find Pareto fronts better than or similar to SMGD with less time; and 2) the proposed first-order method does not harm the quality of the Pareto front identified by the second-order method, while further reduce running time.

We consider the stochastic gradient descent (SGD) algorithm driven by a general stochastic sequence, including i.i.d noise and random walk on an arbitrary graph, among others; and analyze it in the asymptotic sense. Specifically, we employ the notion of `efficiency ordering', a well-analyzed tool for comparing the performance of Markov Chain Monte Carlo (MCMC) samplers, for SGD algorithms in the form of Loewner ordering of covariance matrices associated with the scaled iterate errors in the long term. Using this ordering, we show that input sequences that are more efficient for MCMC sampling also lead to smaller covariance of the errors for SGD algorithms in the limit. This also suggests that an arbitrarily weighted MSE of SGD iterates in the limit becomes smaller when driven by more efficient chains. Our finding is of particular interest in applications such as decentralized optimization and swarm learning, where SGD is implemented in a random walk fashion on the underlying communication graph for cost issues and/or data privacy. We demonstrate how certain non-Markovian processes, for which typical mixing-time based non-asymptotic bounds are intractable, can outperform their Markovian counterparts in the sense of efficiency ordering for SGD. We show the utility of our method by applying it to gradient descent with shuffling and mini-batch gradient descent, reaffirming key results from existing literature under a unified framework. Empirically, we also observe efficiency ordering for variants of SGD such as accelerated SGD and Adam, open up the possibility of extending our notion of efficiency ordering to a broader family of stochastic optimization algorithms.

Standard federated optimization methods successfully apply to stochastic problems with single-level structure. However, many contemporary ML problems -- including adversarial robustness, hyperparameter tuning, and actor-critic -- fall under nested bilevel programming that subsumes minimax and compositional optimization. In this work, we propose \fedblo: A federated alternating stochastic gradient method to address general nested problems. We establish provable convergence rates for \fedblo in the presence of heterogeneous data and introduce variations for bilevel, minimax, and compositional optimization. \fedblo introduces multiple innovations including federated hypergradient computation and variance reduction to address inner-level heterogeneity. We complement our theory with experiments on hyperparameter \& hyper-representation learning and minimax optimization that demonstrate the benefits of our method in practice. Code is available at https://github.com/ucr-optml/FedNest.

In this paper, we propose to solve a regularized distributionally robust learning problem in the decentralized setting, taking into account the data distribution shift. By adding a Kullback-Liebler regularization function to the robust min-max optimization problem, the learning problem can be reduced to a modified robust minimization problem and solved efficiently. Leveraging the newly formulated optimization problem, we propose a robust version of Decentralized Stochastic Gradient Descent (DSGD), coined Distributionally Robust Decentralized Stochastic Gradient Descent (DR-DSGD). Under some mild assumptions and provided that the regularization parameter is larger than one, we theoretically prove that DR-DSGD achieves a convergence rate of $\mathcal{O}\left(1/\sqrt{KT} + K/T\right)$, where $K$ is the number of devices and $T$ is the number of iterations. Simulation results show that our proposed algorithm can improve the worst distribution test accuracy by up to $10\%$. Moreover, DR-DSGD is more communication-efficient than DSGD since it requires fewer communication rounds (up to $20$ times less) to achieve the same worst distribution test accuracy target. Furthermore, the conducted experiments reveal that DR-DSGD results in a fairer performance across devices in terms of test accuracy.