We consider applying stochastic approximation (SA) methods to solve nonsmooth variational inclusion problems. Existing studies have shown that the averaged iterates of SA methods exhibit asymptotic normality, with an optimal limiting covariance matrix in the local minimax sense of H\'ajek and Le Cam. However, no methods have been proposed to estimate this covariance matrix in a nonsmooth and potentially non-monotone (nonconvex) setting. In this paper, we study an online batch-means covariance matrix estimator introduced in Zhu et al.(2023). The estimator groups the SA iterates appropriately and computes the sample covariance among batches as an estimate of the limiting covariance. Its construction does not require prior knowledge of the total sample size, and updates can be performed recursively as new data arrives. We establish that, as long as the batch size sequence is properly specified (depending on the stepsize sequence), the estimator achieves a convergence rate of order $O(\sqrt{d}n^{-1/8+\varepsilon})$ for any $\varepsilon>0$, where $d$ and $n$ denote the problem dimensionality and the number of iterations (or samples) used. Although the problem is nonsmooth and potentially non-monotone (nonconvex), our convergence rate matches the best-known rate for covariance estimation methods using only first-order information in smooth and strongly-convex settings. The consistency of this covariance estimator enables asymptotically valid statistical inference, including constructing confidence intervals and performing hypothesis testing.

We develop and analyze algorithms for instrumental variable regression by viewing the problem as a conditional stochastic optimization problem. In the context of least-squares instrumental variable regression, our algorithms neither require matrix inversions nor mini-batches and provides a fully online approach for performing instrumental variable regression with streaming data. When the true model is linear, we derive rates of convergence in expectation, that are of order $\mathcal{O}(\log T/T)$ and $\mathcal{O}(1/T^{1-\iota})$ for any $\iota>0$, respectively under the availability of two-sample and one-sample oracles, respectively, where $T$ is the number of iterations. Importantly, under the availability of the two-sample oracle, our procedure avoids explicitly modeling and estimating the relationship between confounder and the instrumental variables, demonstrating the benefit of the proposed approach over recent works based on reformulating the problem as minimax optimization problems. Numerical experiments are provided to corroborate the theoretical results.

We investigate the online overlapping batch-means covariance estimator for Stochastic Gradient Descent (SGD) under Markovian sampling. Convergence rates of order $O\big(\sqrt{d}\,n^{-1/8}(\log n)^{1/4}\big)$ and $O\big(\sqrt{d}\,n^{-1/8}\big)$ are established under state-dependent and state-independent Markovian sampling, respectively, where $d$ is the dimensionality and $n$ denotes observations or SGD iterations. These rates match the best-known convergence rate for independent and identically distributed (i.i.d) data. Our analysis overcomes significant challenges that arise due to Markovian sampling, leading to the introduction of additional error terms and complex dependencies between the blocks of the batch-means covariance estimator. Moreover, we establish the convergence rate for the first four moments of the $\ell_2$ norm of the error of SGD dynamics under state-dependent Markovian data, which holds potential interest as an independent result. Numerical illustrations provide confidence intervals for SGD in linear and logistic regression models under Markovian sampling. Additionally, our method is applied to the strategic classification with logistic regression, where adversaries adaptively modify features during training to affect target class classification.

In multi-objective optimization, a single decision vector must balance the trade-offs between many objectives. Solutions achieving an optimal trade-off are said to be Pareto optimal: these are decision vectors for which improving any one objective must come at a cost to another. But as the set of Pareto optimal vectors can be very large, we further consider a more practically significant Pareto-constrained optimization problem, where the goal is to optimize a preference function constrained to the Pareto set. We investigate local methods for solving this constrained optimization problem, which poses significant challenges because the constraint set is (i) implicitly defined, and (ii) generally non-convex and non-smooth, even when the objectives are. We define notions of optimality and stationarity, and provide an algorithm with a last-iterate convergence rate of $O(K^{-1/2})$ to stationarity when the objectives are strongly convex and Lipschitz smooth.

Fairness-aware machine learning has garnered significant attention in recent years because of extensive use of machine learning in sensitive applications like judiciary systems. Various heuristics, and optimization frameworks have been proposed to enforce fairness in classification \cite{del2020review} where the later approaches either provides empirical results or provides fairness guarantee for the exact minimizer of the objective function \cite{celis2019classification}. In modern machine learning, Stochastic Gradient Descent (SGD) type algorithms are almost always used as training algorithms implying that the learned model, and consequently, its fairness properties are random. Hence, especially for crucial applications, it is imperative to construct Confidence Interval (CI) for the fairness of the learned model. In this work we provide CI for test unfairness when a group-fairness-aware, specifically, Disparate Impact (DI), and Disparate Mistreatment (DM) aware linear binary classifier is trained using online SGD-type algorithms. We show that asymptotically a Central Limit Theorem holds for the estimated model parameter of both DI and DM-aware models. We provide online multiplier bootstrap method to estimate the asymptotic covariance to construct online CI. To do so, we extend the known theoretical guarantees shown on the consistency of the online bootstrap method for unconstrained SGD to constrained optimization which could be of independent interest. We illustrate our results on synthetic and real datasets.

For a full-stack web or app development, it requires a software firm or more specifically a team of experienced developers to contribute a large portion of their time and resources to design the website and then convert it to code. As a result, the efficiency of the development team is significantly reduced when it comes to converting UI wireframes and database schemas into an actual working system. It would save valuable resources and fasten the overall workflow if the clients or developers can automate this process of converting the pre-made full-stack website design to get a partially working if not fully working code. In this paper, we present a novel approach of generating the skeleton code from sketched images using Deep Learning and Computer Vision approaches. The dataset for training are first-hand sketched images of low fidelity wireframes, database schemas and class diagrams. The approach consists of three parts. First, the front-end or UI elements detection and extraction from custom-made UI wireframes. Second, individual database table creation from schema designs and lastly, creating a class file from class diagrams.

We study stochastic optimization algorithms for constrained nonconvex stochastic optimization problems with Markovian data. In particular, we focus on the case when the transition kernel of the Markov chain is state-dependent. Such stochastic optimization problems arise in various machine learning problems including strategic classification and reinforcement learning. For this problem, we study both projection-based and projection-free algorithms. In both cases, we establish that the number of calls to the stochastic first-order oracle to obtain an appropriately defined $\epsilon$-stationary point is of the order $\mathcal{O}(1/\epsilon^{2.5})$. In the projection-free setting we additionally establish that the number of calls to the linear minimization oracle is of order $\mathcal{O}(1/\epsilon^{5.5})$. We also empirically demonstrate the performance of our algorithm on the problem of strategic classification with neural networks.

Easy-to-use programming interfaces paired with cloud-scale processing engines have enabled big data system users to author arbitrarily complex analytical jobs over massive volumes of data. However, as the complexity and scale of analytical jobs increase, they encounter a number of unforeseen problems, hotspots with large intermediate data on temporary storage, longer job recovery time after failures, and worse query optimizer estimates being examples of issues that we are facing at Microsoft. To address these issues, we propose Phoebe, an efficient learning-based checkpoint optimizer. Given a set of constraints and an objective function at compile-time, Phoebe is able to determine the decomposition of job plans, and the optimal set of checkpoints to preserve their outputs to durable global storage. Phoebe consists of three machine learning predictors and one optimization module. For each stage of a job, Phoebe makes accurate predictions for: (1) the execution time, (2) the output size, and (3) the start/end time taking into account the inter-stage dependencies. Using these predictions, we formulate checkpoint optimization as an integer programming problem and propose a scalable heuristic algorithm that meets the latency requirement of the production environment. We demonstrate the effectiveness of Phoebe in production workloads, and show that we can free the temporary storage on hotspots by more than 70% and restart failed jobs 68% faster on average with minimum performance impact. Phoebe also illustrates that adding multiple sets of checkpoints is not cost-efficient, which dramatically reduces the complexity of the optimization.

In this work, we establish risk bounds for the Empirical Risk Minimization (ERM) with both dependent and heavy-tailed data-generating processes. We do so by extending the seminal works of Mendelson [Men15, Men18] on the analysis of ERM with heavy-tailed but independent and identically distributed observations, to the strictly stationary exponentially $\beta$-mixing case. Our analysis is based on explicitly controlling the multiplier process arising from the interaction between the noise and the function evaluations on inputs. It allows for the interaction to be even polynomially heavy-tailed, which covers a significantly large class of heavy-tailed models beyond what is analyzed in the learning theory literature. We illustrate our results by deriving rates of convergence for the high-dimensional linear regression problem with dependent and heavy-tailed data.

In this paper, we show that under over-parametrization several standard stochastic optimization algorithms escape saddle-points and converge to local-minimizers much faster. One of the fundamental aspects of over-parametrized models is that they are capable of interpolating the training data. We show that, under interpolation-like assumptions satisfied by the stochastic gradients in an over-parametrization setting, the first-order oracle complexity of Perturbed Stochastic Gradient Descent (PSGD) algorithm to reach an $\epsilon$-local-minimizer, matches the corresponding deterministic rate of $\tilde{\mathcal{O}}(1/\epsilon^{2})$. We next analyze Stochastic Cubic-Regularized Newton (SCRN) algorithm under interpolation-like conditions, and show that the oracle complexity to reach an $\epsilon$-local-minimizer under interpolation-like conditions, is $\tilde{\mathcal{O}}(1/\epsilon^{2.5})$. While this obtained complexity is better than the corresponding complexity of either PSGD, or SCRN without interpolation-like assumptions, it does not match the rate of $\tilde{\mathcal{O}}(1/\epsilon^{1.5})$ corresponding to deterministic Cubic-Regularized Newton method. It seems further Hessian-based interpolation-like assumptions are necessary to bridge this gap. We also discuss the corresponding improved complexities in the zeroth-order settings.