Federated learning aims at conducting inference when data are decentralised and locally stored on several clients, under two main constraints: data ownership and communication overhead. In this paper, we address these issues under the Bayesian paradigm. To this end, we propose a novel Markov chain Monte Carlo algorithm coined \texttt{QLSD} built upon quantised versions of stochastic gradient Langevin dynamics. To improve performance in a big data regime, we introduce variance-reduced alternatives of our methodology referred to as \texttt{QLSD}$^\star$ and \texttt{QLSD}$^{++}$. We provide both non-asymptotic and asymptotic convergence guarantees for the proposed algorithms and illustrate their benefits on several federated learning benchmarks.

Accelerated gradient-based methods are being extensively used for solving non-convex machine learning problems, especially when the data points are abundant or the available data is distributed across several agents. Two of the prominent accelerated gradient algorithms are AdaGrad and Adam. AdaGrad is the simplest accelerated gradient method, which is particularly effective for sparse data. Adam has been shown to perform favorably in deep learning problems compared to other methods. In this paper, we propose a new fast optimizer, Generalized AdaGrad (G-AdaGrad), for accelerating the solution of potentially non-convex machine learning problems. Specifically, we adopt a state-space perspective for analyzing the convergence of gradient acceleration algorithms, namely G-AdaGrad and Adam, in machine learning. Our proposed state-space models are governed by ordinary differential equations. We present simple convergence proofs of these two algorithms in the deterministic settings with minimal assumptions. Our analysis also provides intuition behind improving upon AdaGrad's convergence rate. We provide empirical results on MNIST dataset to reinforce our claims on the convergence and performance of G-AdaGrad and Adam.

In this paper, we propose an energy-efficient federated meta-learning framework. The objective is to enable learning a meta-model that can be fine-tuned to a new task with a few number of samples in a distributed setting and at low computation and communication energy consumption. We assume that each task is owned by a separate agent, so a limited number of tasks is used to train a meta-model. Assuming each task was trained offline on the agent's local data, we propose a lightweight algorithm that starts from the local models of all agents, and in a backward manner using projected stochastic gradient ascent (P-SGA) finds a meta-model. The proposed method avoids complex computations such as computing hessian, double looping, and matrix inversion, while achieving high performance at significantly less energy consumption compared to the state-of-the-art methods such as MAML and iMAML on conducted experiments for sinusoid regression and image classification tasks.

A Spiking Neural Network (SNN) (Maass, 1997; Gerstner and Kistler, 2002b; Pfeiffer and Pfeil, 2018) is a neuro-inspired machine learning (ML) model that mimics the spike-based operation of the human brain (Bi and Poo, 1998). The Spike Time Dependent Plasticity (STDP) is a policy for unsupervised learning in rate-encoded SNNs (Bell et al., 1997; Magee and Johnston, 1997; Gerstner and Kistler, 2002a). The STDP relates the expected change in synaptic weights to the timing difference between postsynaptic spikes and presynaptic spikes (Feldman, 2012). Recent works using STDP trained SNNs have demonstrated promising results as an unsupervised learning paradigm for various tasks such as object classification and recognition (She et al., 2021; Diehl and Cook, 2015; Kheradpisheh et al., 2018). The generalizability is a measure of how well an ML model performs on test data that lies outside of the distribution of the training samples (Kawaguchi et al., 2017; Neyshabur et al., 2017). The generalization properties of Stochastic Gradient Descent (SGD) based training for deep neural network (DNN) have received significant attention in recent years (Poggio et al., 2019; Allen-Zhu et al., 2018; Allen-Zhu and Li, 2019). The dynamics of SGD have been studied via models of stochastic gradient Langevin dynamics with an assumption that gradient noise is Gaussian (Simsekli et al., 2020b). Here SGD is considered to be driven by a Brownian motion. Chen et al. showed that SGD dynamics commonly exhibit highly anisotropic and dynamic-changing properties (Chen et al., 2020), suggesting the presence of

We posit a new mechanism for cooperation in multi-agent reinforcement learning (MARL) based upon any nonlinear function of the team's long-term state-action occupancy measure, i.e., a \emph{general utility}. This subsumes the cumulative return but also allows one to incorporate risk-sensitivity, exploration, and priors. % We derive the {\bf D}ecentralized {\bf S}hadow Reward {\bf A}ctor-{\bf C}ritic (DSAC) in which agents alternate between policy evaluation (critic), weighted averaging with neighbors (information mixing), and local gradient updates for their policy parameters (actor). DSAC augments the classic critic step by requiring agents to (i) estimate their local occupancy measure in order to (ii) estimate the derivative of the local utility with respect to their occupancy measure, i.e., the "shadow reward". DSAC converges to $\epsilon$-stationarity in $\mathcal{O}(1/\epsilon^{2.5})$ (Theorem \ref{theorem:final}) or faster $\mathcal{O}(1/\epsilon^{2})$ (Corollary \ref{corollary:communication}) steps with high probability, depending on the amount of communications. We further establish the non-existence of spurious stationary points for this problem, that is, DSAC finds the globally optimal policy (Corollary \ref{corollary:global}). Experiments demonstrate the merits of goals beyond the cumulative return in cooperative MARL.

Finding parameters in a deep neural network (NN) that fit training data is a nonconvex optimization problem, but a basic first-order optimization method (gradient descent) finds a global optimizer with perfect fit (zero-loss) in many practical situations. We examine this phenomenon for the case of Residual Neural Networks (ResNet) with smooth activation functions in a limiting regime in which both the number of layers (depth) and the number of neurons in each layer (width) go to infinity. First, we use a mean-field-limit argument to prove that the gradient descent for parameter training becomes a gradient flow for a probability distribution that is characterized by a partial differential equation (PDE) in the large-NN limit. Next, we show that the solution to the PDE converges in the training time to a zero-loss solution. Together, these results imply that the training of the ResNet gives a near-zero loss if the ResNet is large enough. We give estimates of the depth and width needed to reduce the loss below a given threshold, with high probability.

The reinforcement learning problem of finding a control policy that minimizes the minimum time objective for the Mountain Car environment is considered. Particularly, a class of parameterized nonlinear feedback policies is optimized over to reach the top of the highest mountain peak in minimum time. The optimization is carried out using quasi-Stochastic Gradient Descent (qSGD) methods. In attempting to find the optimal minimum time policy, a new parameterized policy approach is considered that seeks to learn an optimal policy parameter for different regions of the state space, rather than rely on a single macroscopic policy parameter for the entire state space. This partitioned parameterized policy approach is shown to outperform the uniform parameterized policy approach and lead to greater generalization than prior methods, where the Mountain Car became trapped in circular trajectories in the state space.

Gradient-based methods for two-player games produce rich dynamics that can solve challenging problems, yet can be difficult to stabilize and understand. Part of this complexity originates from the discrete update steps given by simultaneous or alternating gradient descent, which causes each player to drift away from the continuous gradient flow -- a phenomenon we call discretization drift. Using backward error analysis, we derive modified continuous dynamical systems that closely follow the discrete dynamics. These modified dynamics provide an insight into the notorious challenges associated with zero-sum games, including Generative Adversarial Networks. In particular, we identify distinct components of the discretization drift that can alter performance and in some cases destabilize the game. Finally, quantifying discretization drift allows us to identify regularizers that explicitly cancel harmful forms of drift or strengthen beneficial forms of drift, and thus improve performance of GAN training.

We study discrete-time mirror descent applied to the unregularized empirical risk in matrix sensing. In both the general case of rectangular matrices and the particular case of positive semidefinite matrices, a simple potential-based analysis in terms of the Bregman divergence allows us to establish convergence of mirror descent -- with different choices of the mirror maps -- to a matrix that, among all global minimizers of the empirical risk, minimizes a quantity explicitly related to the nuclear norm, the Frobenius norm, and the von Neumann entropy. In both cases, this characterization implies that mirror descent, a first-order algorithm minimizing the unregularized empirical risk, recovers low-rank matrices under the same set of assumptions that are sufficient to guarantee recovery for nuclear-norm minimization. When the sensing matrices are symmetric and commute, we show that gradient descent with full-rank factorized parametrization is a first-order approximation to mirror descent, in which case we obtain an explicit characterization of the implicit bias of gradient flow as a by-product.

Most MCMC algorithms contain user-controlled hyperparameters which need to be carefully selected to ensure that the MCMC algorithm explores the posterior distribution efficiently. Optimal tuning rates for many popular MCMC algorithms such the random-walk (Gelman et al., 1997) or Metropolis-adjusted Langevin algorithms (Roberts and Rosenthal, 1998) rely on setting the tuning parameters according to the Metropolis-Hastings acceptance rate. Using metrics such as the acceptance rate, hyperparameters can be optimized on-the-fly within the MCMC algorithm using adaptive MCMC (Andrieu and Thoms, 2008; Vihola, 2012). However, in the context of stochastic gradient MCMC (SGMCMC), there is no acceptance rate to tune against and the trade-off between bias and variance for a fixed computational budget means that tuning approaches designed for target invariant MCMC algorithms are not applicable. Related work Previous adaptive SGMCMC algorithms have focused on embedding ideas from the optimization literature within the SGMCMC framework, e.g.