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1. Stability and optimization error of stochastic gradient descent for pairwise learning

   https://doi.org/10.1142/S0219530519400062  

   Shen, Wei ; Yang, Zhenhuan ; Ying, Yiming ; Yuan, Xiaoming ( July 2019 , Analysis and Applications)

   In this paper, we study the stability and its trade-off with optimization error for stochastic gradient descent (SGD) algorithms in the pairwise learning setting. Pairwise learning refers to a learning task which involves a loss function depending on pairs of instances among which notable examples are bipartite ranking, metric learning, area under ROC curve (AUC) maximization and minimum error entropy (MEE) principle. Our contribution is twofolded. Firstly, we establish the stability results for SGD for pairwise learning in the convex, strongly convex and non-convex settings, from which generalization errors can be naturally derived. Secondly, we establish the trade-off between stability and optimization error of SGD algorithms for pairwise learning. This is achieved by lower-bounding the sum of stability and optimization error by the minimax statistical error over a prescribed class of pairwise loss functions. From this fundamental trade-off, we obtain lower bounds for the optimization error of SGD algorithms and the excess expected risk over a class of pairwise losses. In addition, we illustrate our stability results by giving some specific examples of AUC maximization, metric learning and MEE. 

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2. Uniform-in-Time Wasserstein Stability Bounds for (Noisy) Stochastic Gradient Descent

   Zhu, Lingjiong ; Gurbuzbalaban, Mert ; Raj, Anant ; Simsekli, Umut ( December 2023 , Advances in neural information processing systems)

   Algorithmic stability is an important notion that has proven powerful for deriving generalization bounds for practical algorithms. The last decade has witnessed an increasing number of stability bounds for different algorithms applied on different classes of loss functions. While these bounds have illuminated various properties of optimization algorithms, the analysis of each case typically required a different proof technique with significantly different mathematical tools. In this study, we make a novel connection between learning theory and applied probability and introduce a unified guideline for proving Wasserstein stability bounds for stochastic optimization algorithms. We illustrate our approach on stochastic gradient descent (SGD) and we obtain time-uniform stability bounds (i.e., the bound does not increase with the number of iterations) for strongly convex losses and non-convex losses with additive noise, where we recover similar results to the prior art or extend them to more general cases by using a single proof technique. Our approach is flexible and can be generalizable to other popular optimizers, as it mainly requires developing Lyapunov functions, which are often readily available in the literature. It also illustrates that ergodicity is an important component for obtaining time uniform bounds – which might not be achieved for convex or non-convex losses unless additional noise is injected to the iterates. Finally, we slightly stretch our analysis technique and prove time-uniform bounds for SGD under convex and non-convex losses (without additional additive noise), which, to our knowledge, is novel. 

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3. The geometry of adversarial training in binary classification

   https://doi.org/10.1093/imaiai/iaac029  

   Bungert, Leon ; García Trillos, Nicolás ; Murray, Ryan ( February 2023 , Information and Inference: A Journal of the IMA)

   We establish an equivalence between a family of adversarial training problems for non-parametric binary classification and a family of regularized risk minimization problems where the regularizer is a nonlocal perimeter functional. The resulting regularized risk minimization problems admit exact convex relaxations of the type $L^1+\text{(nonlocal)}\operatorname{TV}$, a form frequently studied in image analysis and graph-based learning. A rich geometric structure is revealed by this reformulation which in turn allows us to establish a series of properties of optimal solutions of the original problem, including the existence of minimal and maximal solutions (interpreted in a suitable sense) and the existence of regular solutions (also interpreted in a suitable sense). In addition, we highlight how the connection between adversarial training and perimeter minimization problems provides a novel, directly interpretable, statistical motivation for a family of regularized risk minimization problems involving perimeter/total variation. The majority of our theoretical results are independent of the distance used to define adversarial attacks.

    

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4. Renyi Differentially Private ERM for Smooth Objectives

   Chen, Chen ; Lee, Jaewoo ; Kifer, Daniel ( January 2019 , AISTATS)

   In this paper, we present a Renyi Differentially Private stochastic gradient descent (SGD) algorithm for convex empirical risk minimization. The algorithm uses output perturbation and leverages randomness inside SGD, which creates a “randomized sensitivity”, in order to reduce the amount of noise that is added. One of the benefits of output perturbation is that we can incorporate a periodic averaging step that serves to further reduce sensitivity while improving accuracy (reducing the well known oscillating behavior of SGD near the optimum). Renyi Differential Privacy can be used to provide (epsilon, delta)-differential privacy guarantees and hence provide a comparison with prior work. An empirical evaluation demonstrates that the proposed method outperforms prior methods on differentially private ERM. 

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5. Parameter-Free Locally Differentially Private Stochastic Subgradient Descent

   Jun, Kwang-Sung ; Orabona, Francesco ( January 2019 , Workshop on Privacy in Machine Learning at NeurIPS'19)

   null (Ed.)

   We consider the problem of minimizing a convex risk with stochastic subgradients guaranteeing $\epsilon$-locally differentially private ($\epsilon$-LDP). While it has been shown that stochastic optimization is possible with $\epsilon$-LDP via the standard SGD, its convergence rate largely depends on the learning rate, which must be tuned via repeated runs. Further, tuning is detrimental to privacy loss since it significantly increases the number of gradient requests. In this work, we propose BANCO (Betting Algorithm for Noisy COins), the first $\epsilon$-LDP SGD algorithm that essentially matches the convergence rate of the tuned SGD without any learning rate parameter, reducing privacy loss and saving privacy budget. 

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