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 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. This is much better than the previously anticipated linear dependency in $g$ and is much better than the previously expected square root dependency on the total number of rounds within $E$ epochs which is generally much more than $\sqrt{E}$.
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Federated learning is an emerging machine learning paradigm that enables devices to train collaboratively without exchanging their local data. The clients participating in the training process are a random subset selected from the pool of clients. The above procedure is called client selection which is an important area in federated learning as it highly impacts the convergence rate, learning efficiency, and generalization. In this work, we introduce client filtering in federated learning (FilFL), a new approach to optimize client selection and training. FilFL first filters the active clients by choosing a subset of them that maximizes a specific objective function; then, a client selection method is applied to that subset. We provide a thorough analysis of its convergence in a heterogeneous setting. Empirical results demonstrate several benefits to our approach, including improved learning efficiency, accelerated convergence, $2$-$3\times$ faster, and higher test accuracy, around $2$-$10$ percentage points higher.
The $K$-armed dueling bandits problem, where the feedback is in the form of noisy pairwise preferences, has been widely studied due its applications in information retrieval, recommendation systems, etc. Motivated by concerns that user preferences/tastes can evolve over time, we consider the problem of dueling bandits with distribution shifts. Specifically, we study the recent notion of significant shifts (Suk and Kpotufe, 2022), and ask whether one can design an adaptive algorithm for the dueling problem with $O(\sqrt{K\tilde{L}T})$ dynamic regret, where $\tilde{L}$ is the (unknown) number of significant shifts in preferences. We show that the answer to this question depends on the properties of underlying preference distributions. Firstly, we give an impossibility result that rules out any algorithm with $O(\sqrt{K\tilde{L}T})$ dynamic regret under the well-studied Condorcet and SST classes of preference distributions. Secondly, we show that $\text{SST} \cap \text{STI}$ is the largest amongst popular classes of preference distributions where it is possible to design such an algorithm. Overall, our results provides an almost complete resolution of the above question for the hierarchy of distribution classes.
We investigate the contraction coefficients derived from strong data processing inequalities for the $E_\gamma$-divergence. By generalizing the celebrated Dobrushin's coefficient from total variation distance to $E_\gamma$-divergence, we derive a closed-form expression for the contraction of $E_\gamma$-divergence. This result has fundamental consequences in two privacy settings. First, it implies that local differential privacy can be equivalently expressed in terms of the contraction of $E_\gamma$-divergence. This equivalent formula can be used to precisely quantify the impact of local privacy in (Bayesian and minimax) estimation and hypothesis testing problems in terms of the reduction of effective sample size. Second, it leads to a new information-theoretic technique for analyzing privacy guarantees of online algorithms. In this technique, we view such algorithms as a composition of amplitude-constrained Gaussian channels and then relate their contraction coefficients under $E_\gamma$-divergence to the overall differential privacy guarantees. As an example, we apply our technique to derive the differential privacy parameters of gradient descent. Moreover, we also show that this framework can be tailored to batch learning algorithms that can be implemented with one pass over the training dataset.
Mobile parcel lockers have been recently proposed by logistics operators as a technology that could help reduce traffic congestion and operational costs in urban freight distribution. Given their ability to relocate throughout their area of deployment, they hold the potential to improve customer accessibility and convenience. In this study, we formulate the Mobile Parcel Locker Problem (MPLP) , a special case of the Location-Routing Problem (LRP) which determines the optimal stopover location for MPLs throughout the day and plans corresponding delivery routes. A Hybrid Q Learning Network based Method (HQM) is developed to resolve the computational complexity of the resulting large problem instances while escaping local optima. In addition, the HQM is integrated with global and local search mechanisms to resolve the dilemma of exploration and exploitation faced by classic reinforcement learning methods. We examine the performance of HQM under different problem sizes (up to 200 nodes) and benchmarked it against the exact approach and Genetic Algorithm (GA). Our results indicate that HQM achieves better optimisation performance with shorter computation time than the exact approach solved by the Gurobi solver in large problem instances. Additionally, the average reward obtained by HQM is 1.96 times greater than GA, which demonstrates that HQM has a better optimisation ability. Further, we identify critical factors that contribute to fleet size requirements, travel distances, and service delays. Our findings outline that the efficiency of MPLs is mainly contingent on the length of time windows and the deployment of MPL stopovers. Finally, we highlight managerial implications based on parametric analysis to provide guidance for logistics operators in the context of efficient last-mile distribution operations.
$L^1$ based optimization is widely used in image denoising, machine learning and related applications. One of the main features of such approach is that it naturally provide a sparse structure in the numerical solutions. In this paper, we study an $L^1$ based mixed DG method for second-order elliptic equations in the non-divergence form. The elliptic PDE in nondivergence form arises in the linearization of fully nonlinear PDEs. Due to the nature of the equations, classical finite element methods based on variational forms can not be employed directly. In this work, we propose a new optimization scheme coupling the classical DG framework with recently developed $L^1$ optimization technique. Convergence analysis in both energy norm and $L^{\infty}$ norm are obtained under weak regularity assumption. Such $L^1$ models are nondifferentiable and therefore invalidate traditional gradient methods. Therefore all existing gradient based solvers are no longer feasible under this setting. To overcome this difficulty, we characterize solutions of $L^1$ optimization as fixed-points of proximity equations and utilize matrix splitting technique to obtain a class of fixed-point proximity algorithms with convergence analysis. Various numerical examples are displayed to illustrate the numerical solution has sparse structure with careful choice of the bases of the finite dimensional spaces. Numerical examples in both smooth and nonsmooth settings are provided to validate the theoretical results.
We provide sharp path-dependent generalization and excess risk guarantees for the full-batch Gradient Descent (GD) algorithm on smooth losses (possibly non-Lipschitz, possibly nonconvex). At the heart of our analysis is an upper bound on the generalization error, which implies that average output stability and a bounded expected optimization error at termination lead to generalization. This result shows that a small generalization error occurs along the optimization path, and allows us to bypass Lipschitz or sub-Gaussian assumptions on the loss prevalent in previous works. For nonconvex, convex, and strongly convex losses, we show the explicit dependence of the generalization error in terms of the accumulated path-dependent optimization error, terminal optimization error, number of samples, and number of iterations. For nonconvex smooth losses, we prove that full-batch GD efficiently generalizes close to any stationary point at termination, and recovers the generalization error guarantees of stochastic algorithms with fewer assumptions. For smooth convex losses, we show that the generalization error is tighter than existing bounds for SGD (up to one order of error magnitude). Consequently the excess risk matches that of SGD for quadratically less iterations. Lastly, for strongly convex smooth losses, we show that full-batch GD achieves essentially the same excess risk rate as compared with the state of the art on SGD, but with an exponentially smaller number of iterations (logarithmic in the dataset size).
Federated Learning (FL) has emerged as a new paradigm for training machine learning models distributively without sacrificing data security and privacy. Learning models on edge devices such as mobile phones is one of the most common use cases for FL. However, Non-identical independent distributed~(non-IID) data in edge devices easily leads to training failures. Especially, over-parameterized machine learning models can easily be over-fitted on such data, hence, resulting in inefficient federated learning and poor model performance. To overcome the over-fitting issue, we proposed an adaptive dynamic pruning approach for FL, which can dynamically slim the model by dropping out unimportant parameters, hence, preventing over-fittings. Since the machine learning model's parameters react differently for different training samples, adaptive dynamic pruning will evaluate the salience of the model's parameter according to the input training sample, and only retain the salient parameter's gradients when doing back-propagation. We performed comprehensive experiments to evaluate our approach. The results show that our approach by removing the redundant parameters in neural networks can significantly reduce the over-fitting issue and greatly improves the training efficiency. In particular, when training the ResNet-32 on CIFAR-10, our approach reduces the communication cost by 57\%. We further demonstrate the inference acceleration capability of the proposed algorithm. Our approach reduces up to 50\% FLOPs inference of DNNs on edge devices while maintaining the model's quality.
We study a variant of online convex optimization where the player is permitted to switch decisions at most $S$ times in expectation throughout $T$ rounds. Similar problems have been addressed in prior work for the discrete decision set setting, and more recently in the continuous setting but only with an adaptive adversary. In this work, we aim to fill the gap and present computationally efficient algorithms in the more prevalent oblivious setting, establishing a regret bound of $O(T/S)$ for general convex losses and $\widetilde O(T/S^2)$ for strongly convex losses. In addition, for stochastic i.i.d.~losses, we present a simple algorithm that performs $\log T$ switches with only a multiplicative $\log T$ factor overhead in its regret in both the general and strongly convex settings. Finally, we complement our algorithms with lower bounds that match our upper bounds in some of the cases we consider.
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.
A core capability of intelligent systems is the ability to quickly learn new tasks by drawing on prior experience. Gradient (or optimization) based meta-learning has recently emerged as an effective approach for few-shot learning. In this formulation, meta-parameters are learned in the outer loop, while task-specific models are learned in the inner-loop, by using only a small amount of data from the current task. A key challenge in scaling these approaches is the need to differentiate through the inner loop learning process, which can impose considerable computational and memory burdens. By drawing upon implicit differentiation, we develop the implicit MAML algorithm, which depends only on the solution to the inner level optimization and not the path taken by the inner loop optimizer. This effectively decouples the meta-gradient computation from the choice of inner loop optimizer. As a result, our approach is agnostic to the choice of inner loop optimizer and can gracefully handle many gradient steps without vanishing gradients or memory constraints. Theoretically, we prove that implicit MAML can compute accurate meta-gradients with a memory footprint that is, up to small constant factors, no more than that which is required to compute a single inner loop gradient and at no overall increase in the total computational cost. Experimentally, we show that these benefits of implicit MAML translate into empirical gains on few-shot image recognition benchmarks.
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