Langevin algorithms are gradient descent methods augmented with additive noise, and are widely used in Markov Chain Monte Carlo (MCMC) sampling, optimization, and machine learning. In recent years, the non-asymptotic analysis of Langevin algorithms for non-convex learning has been extensively explored. For constrained problems with non-convex losses over a compact convex domain with IID data variables, the projected Langevin algorithm achieves a deviation of $O(T^{-1/4} (\log T)^{1/2})$ from its target distribution \cite{lamperski2021projected} in $1$-Wasserstein distance. In this paper, we obtain a deviation of $O(T^{-1/2} \log T)$ in $1$-Wasserstein distance for non-convex losses with $L$-mixing data variables and polyhedral constraints (which are not necessarily bounded). This improves on the previous bound for constrained problems and matches the best-known bound for unconstrained problems.

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.

This paper studies a risk minimization problem with decision dependent data distribution.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This paper provides some theoretical analysis on Positive and Unlabeled Data learning (PU), when only the positive instances and unlabeled instances are available. The authors show that learning from PU is equivalent to a cost-sensitive classification task if the label prior is known. The main contribution of the paper is that, the authors show that using any convex loss leads to inconsistent classifier for PU tasks. Instead, using a non-convex ramp loss gives a consistent estimator. This theoretical justification is supported by experiments, which demonstrate that adopting hinge loss of SVM may result in very bad classification error comparing to using the non-convex ramp loss.

Reviewer 2 raises some important points and I agree with some of the concerns about the clarity of the paper and the contrast to existing other work. The authors' reply contains some additional experiments, which is great. And I appreciate the clarification of my misunderstanding of the storage of parameters, not gradients. Data cleansing is an important step in machine learning systems and principled approaches for automating this step is a valuable contribution. The setting considered by the authors covers a substantial fraction of ML algorithms used today.

Langevin algorithms are gradient descent methods augmented with additive noise, and are widely used in Markov Chain Monte Carlo (MCMC) sampling, optimization, and machine learning. In recent years, the non-asymptotic analysis of Langevin algorithms for non-convex learning has been extensively explored. For constrained problems with non-convex losses over a compact convex domain with IID data variables, the projected Langevin algorithm achieves a deviation of O(T {-1/4} (\log T) {1/2}) from its target distribution \cite{lamperski2021projected} in 1 -Wasserstein distance. In this paper, we obtain a deviation of O(T {-1/2} \log T) in 1 -Wasserstein distance for non-convex losses with L -mixing data variables and polyhedral constraints (which are not necessarily bounded). This improves on the previous bound for constrained problems and matches the best-known bound for unconstrained problems.

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.

Regret has been widely adopted as the metric of choice for evaluating the performance of online optimization algorithms for distributed, multi-agent systems. However, data/model variations associated with agents can significantly impact decisions and requires consensus among agents. Moreover, most existing works have focused on developing approaches for (either strongly or non-strongly) convex losses, and very few results have been obtained regarding regret bounds in distributed online optimization for general non-convex losses. To address these two issues, we propose a novel composite regret with a new network regret-based metric to evaluate distributed online optimization algorithms. We concretely define static and dynamic forms of the composite regret. By leveraging the dynamic form of our composite regret, we develop a consensus-based online normalized gradient (CONGD) approach for pseudo-convex losses, and it provably shows a sublinear behavior relating to a regularity term for the path variation of the optimizer. For general non-convex losses, we first shed light on the regret for the setting of distributed online non-convex learning based on recent advances such that no deterministic algorithm can achieve the sublinear regret. We then develop the distributed online non-convex optimization with composite regret (DINOCO) without access to the gradients, depending on an offline optimization oracle. DINOCO is shown to achieve sublinear regret; to our knowledge, this is the first regret bound for general distributed online non-convex learning.

Over the past two decades, Support Vector Machine (SVM) has been a popular supervised machine learning model, and plenty of distinct algorithms are designed separately based on different KKT conditions of SVM model for classification/regression with the different losses, including the convex loss or non-convex loss. In this paper, we propose an algorithm that can train different SVM models in a \emph{unified} scheme. Firstly, we introduce a definition of the \emph{LS-DC} (least squares type of difference of convex) loss and show that the most commonly used losses in the SVM community are LS-DC loss or can be approximated by LS-DC loss. Then based on DCA (difference of convex algorithm), we propose a unified algorithm, called \emph{UniSVM} that can solve the SVM model with any convex or non-convex LS-DC loss, in which only a vector is computed especially by the specifically chosen loss. Particularly, for training robust SVM models with non-convex losses, UniSVM has a dominant advantage over all the existing algorithms, because it has a closed-form solution per iteration while the existing ones always need to solve an L1/L2-SVM per iteration. Furthermore, by the low-rank approximation of the kernel matrix, UniSVM can solve the large-scale nonlinear problems with efficiency. To verify the efficacy and feasibility of the proposed algorithm, experiments on large benchmark data sets with/without outliers for classification and regression are investigated. UniSVM can be easily grasped by users or researchers because its core code in Matlab is less than 10 lines.

We study the problem of online learning with non-convex losses, where the learner has access to an offline optimization oracle. We show that the classical Follow the Perturbed Leader (FTPL) algorithm achieves optimal regret rate of $O(T^{-1/2})$ in this setting. This improves upon the previous best-known regret rate of $O(T^{-1/3})$ for FTPL. We further show that an optimistic variant of FTPL achieves better regret bounds when the sequence of losses encountered by the learner is `predictable'.