We initiate the study of a new notion of adversarial loss which we call distributional adversarial loss. In this notion, we assume for each original example, the allowed adversarial perturbation set is a family of distributions, and the adversarial loss over each example is the maximum loss over all the associated distributions. The goal is to minimize the overall adversarial loss. We show sample complexity bounds in the PAC-learning setting for our notion of adversarial loss. Our notion of adversarial loss contrasts the prior work on robust learning that considers a set of points, not distributions, as the perturbation set of each clean example. As an application of our approach, we show how to unify the two lines of work on randomized smoothing and robust learning in the PAC-learning setting and derive sample complexity bounds for randomized smoothing methods. Furthermore, we investigate the role of randomness in achieving robustness against adversarial attacks. We show a general derandomization technique that preserves the extent of a randomized classifier’s robustness against adversarial attacks and show its effectiveness empirically.

A wide range of machine learning applications such as privacy-preserving learning, algorithmic fairness, and domain adaptation/generalization among others, involve learning invariant representations of the data that aim to achieve two competing goals: (a) maximize information or accuracy with respect to a target response, and (b) maximize invariance or independence with respect to a set of protected features (e.g.\ for fairness, privacy, etc). Despite their wide applicability, theoretical understanding of the optimal tradeoffs --- with respect to accuracy, and invariance --- achievable by invariant representations is still severely lacking. In this paper, we provide an information theoretic analysis of such tradeoffs under both classification and regression settings. More precisely, we provide a geometric characterization of the accuracy and invariance achievable by any representation of the data; we term this feasible region the information plane. We provide an inner bound for this feasible region for the classification case, and an exact characterization for the regression case, which allows us to either bound or exactly characterize the Pareto optimal frontier between accuracy and invariance. Although our contributions are mainly theoretical, a key practical application of our results is in certifying the potential sub-optimality of any given representation learning algorithm for either classification or regression tasks. Our results shed new light on the fundamental interplay between accuracy and invariance, and may be useful in guiding the design of future representation learning algorithms.

Simulated annealing is an effective and general means of optimization. It is in fact inspired by metallurgy, where the temperature of a material determines its behavior in thermodynamics. Likewise, in simulated annealing, the actions that the algorithm takes depend entirely on the value of a variable which captures the notion of temperature. Typically, simulated annealing starts with a high temperature, which makes the algorithm pretty unpredictable, and gradually cools the temperature down to become more stable. A key component that plays a crucial role in the performance of simulated annealing is the criteria under which the temperature changes namely, the cooling schedule. Motivated by this, we study the following question in this work: "Given enough samples to the instances of a specific class of optimization problems, can we design optimal (or approximately optimal) cooling schedules that minimize the runtime or maximize the success rate of the algorithm on average when the underlying problem is drawn uniformly at random from the same class?" We provide positive results both in terms of sample complexity and simulation complexity. For sample complexity, we show that O (m^1/2) samples suffice to find an approximately optimal cooling schedule of length m. We complement this result by giving a lower bound of Ω (m^1/3) on the sample complexity of any learning algorithm that provides an almost optimal cooling schedule. These results are general and rely on no assumption. For simulation complexity, however, we make additional assumptions to measure the success rate of an algorithm. To this end, we introduce the monotone stationary graph that models the performance of simulated annealing. Based on this model, we present polynomial time algorithms with provable guarantees for the learning problem.

Many machine learning tasks involve subpopulation shift where the testing data distribution is a subpopulation of the training distribution. For such settings, a line of recent work has proposed the use of a variant of empirical risk minimization(ERM) known as distributionally robust optimization (DRO). In this work, we apply DRO to real, large-scale tasks with subpopulation shift, and observe that DRO performs relatively poorly, and moreover has severe instability. We identify one direct cause of this phenomenon: sensitivity of DRO to outliers in the datasets. To resolve this issue, we propose the framework of DORO, for Distributional and Outlier Robust Optimization. At the core of this approach is a refined risk function which prevents DRO from overfitting to potential outliers. We instantiate DORO for the Cressie-Read family of Rényi divergence, and delve into two specific instances of this family: CVaR and $\chi^2$-DRO. We theoretically prove the effectiveness of the proposed method, and empirically show that DORO improves the performance and stability of DRO with experiments on large modern datasets, thereby positively addressing the open question raised by Hashimoto et al., 2018. Codes are available at https://github.com/RuntianZ/doro.

Many modern machine learning tasks require models with high tail performance, i.e. high performance over the worst-off samples in the dataset. This problem has been widely studied in fields such as algorithmic fairness, class imbalance, and risk-sensitive decision making. A popular approach to maximize the model's tail performance is to minimize the CVaR (Conditional Value at Risk) loss, which computes the average risk over the tails of the loss. However, for classification tasks where models are evaluated by the zero-one loss, we show that if the classifiers are deterministic, then the minimizer of the average zero-one loss also minimizes the CVaR zero-one loss, suggesting that CVaR loss minimization is not helpful without additional assumptions. We circumvent this negative result by minimizing the CVaR loss over randomized classifiers, for which the minimizers of the average zero-one loss and the CVaR zero-one loss are no longer the same, so minimizing the latter can lead to better tail performance. To learn such randomized classifiers, we propose the Boosted CVaR Classification framework which is motivated by a direct relationship between CVaR and a classical boosting algorithm called LPBoost. Based on this framework, we design an algorithm called $\alpha$-AdaLPBoost. We empirically evaluate our proposed algorithm on four benchmark datasets and show that it achieves higher tail performance than deterministic model training methods.

We develop a framework for learning sparse nonparametric directed acyclic graphs (DAGs) from data. Our approach is based on a recent algebraic characterization of DAGs that led to the first fully continuous optimization for score-based learning of DAG models parametrized by a linear structural equation model (SEM). We extend this algebraic characterization to nonparametric SEM by leveraging nonparametric sparsity based on partial derivatives, resulting in a continuous optimization problem that can be applied to a variety of nonparametric and semiparametric models including GLMs, additive noise models, and index models as special cases. Unlike existing approaches that require specific modeling choices, loss functions, or algorithms, we present a completely general framework that can be applied to general nonlinear models (e.g. without additive noise), general differentiable loss functions, and generic black-box optimization routines.

Adversarial robustness has become a fundamental requirement in modern machine learning applications. Yet, there has been surprisingly little statistical understanding so far. In this paper, we provide the first result of the \emph{optimal} minimax guarantees for the excess risk for adversarially robust classification, under Gaussian mixture model proposed by \cite{schmidt2018adversarially}. The results are stated in terms of the \emph{Adversarial Signal-to-Noise Ratio (AdvSNR)}, which generalizes a similar notion for standard linear classification to the adversarial setting. For the Gaussian mixtures with AdvSNR value of $r$, we prove an excess risk lower bound of order $\Theta(e^{-(\frac{1}{2}+o(1)) r^2} \frac{d}{n})$ and design a computationally efficient estimator that achieves this optimal rate. Our results built upon minimal assumptions while cover a wide spectrum of adversarial perturbations including $\ell_p$ balls for any $p \ge 1$.

We consider imbalanced classification, the problem in which a label may have low marginal probability relative to other labels, by weighting losses according to the correct class. First, we examine the convergence rates of the expected excess weighted risk of plug-in classifiers where the weighting for the plug-in classifier and the risk may be different. This leads to irreducible errors that do not converge to the weighted Bayes risk, which motivates our consideration of robust risks. We define a robust risk that minimizes risk over a set of weightings, show excess risk bounds for this problem, and demonstrate that particular choices of the weighting set leads to a special instance of conditional value at risk (CVaR) from stochastic programming, which we call label conditional value at risk (LCVaR). Additionally, we generalize this weighting to derive a new robust risk problem that we call label heterogeneous conditional value at risk (LHCVaR). Finally, we empirically demonstrate the efficacy of LCVaR and LHCVaR on improving class conditional risks.

We show that for the problem of $\ell_p$ rank-$k$ approximation of any given matrix over $R^{n\times m}$ and $C^{n\times m}$, the algorithm of column subset selection enjoys approximation ratio $(k+1)^{1/p}$ for $1\le p\le 2$ and $(k+1)^{1-1/p}$ for $p\ge 2$. This improves upon the previous $O(k+1)$ bound (Chierichetti et al.,2017) for $p\ge 1$. We complement our analysis with lower bounds; these bounds match our upper bounds up to constant 1 when $p\geq 2$. At the core of our techniques is an application of Riesz-Thorin interpolation theorem from harmonic analysis, which might be of independent interest to other algorithmic designs and analysis more broadly. Our analysis results in improvements on approximation guarantees of several other algorithms with various time complexity. For example, to make the algorithm of column subset selection computationally efficient, we analyze a polynomial time bi-criteria algorithm which selects $O(k\log m)$ number of columns. We show that this algorithm has an approximation ratio of $O((k+1)^{1/p})$ for $1\le p\le 2$ and $O((k+1)^{1-1/p})$ for $p\ge 2$. This improves over the bound in (Chierichetti et al.,2017) with an $O(k+1)$ approximation ratio. Our bi-criteria algorithm also implies an exact-rank method in polynomial time with a slightly larger approximation ratio.

Low rank matrix approximation is an important tool in machine learning. Given a data matrix, low rank approximation helps to find factors, patterns and provides concise representations for the data. Research on low rank approximation usually focus on real matrices. However, in many applications data are binary (categorical) rather than continuous. This leads to the problem of low rank approximation of binary matrix. Here we are given a $d \times n$ binary matrix $A$ and a small integer $k$. The goal is to find two binary matrices $U$ and $V$ of sizes $d \times k$ and $k \times n$ respectively, so that the Frobenius norm of $A - U V$ is minimized. There are two models of this problem, depending on the definition of the dot product of binary vectors: The $\mathrm{GF}(2)$ model and the Boolean semiring model. Unlike low rank approximation of real matrix which can be efficiently solved by Singular Value Decomposition, approximation of binary matrix is $NP$-hard even for $k=1$. In this paper, we consider the problem of Column Subset Selection (CSS), in which one low rank matrix must be formed by $k$ columns of the data matrix. We characterize the approximation ratio of CSS for binary matrices. For $GF(2)$ model, we show the approximation ratio of CSS is bounded by $\frac{k}{2}+1+\frac{k}{2(2^k-1)}$ and this bound is asymptotically tight. For Boolean model, it turns out that CSS is no longer sufficient to obtain a bound. We then develop a Generalized CSS (GCSS) procedure in which the columns of one low rank matrix are generated from Boolean formulas operating bitwise on columns of the data matrix. We show the approximation ratio of GCSS is bounded by $2^{k-1}+1$, and the exponential dependency on $k$ is inherent.

We study the sample complexity of semi-supervised learning (SSL) and introduce new assumptions based on the mismatch between a mixture model learned from unlabeled data and the true mixture model induced by the (unknown) class conditional distributions. Under these assumptions, we establish an $\Omega(K\log K)$ labeled sample complexity bound without imposing parametric assumptions, where $K$ is the number of classes. Our results suggest that even in nonparametric settings it is possible to learn a near-optimal classifier using only a few labeled samples. Unlike previous theoretical work which focuses on binary classification, we consider general multiclass classification ($K>2$), which requires solving a difficult permutation learning problem. This permutation defines a classifier whose classification error is controlled by the Wasserstein distance between mixing measures, and we provide finite-sample results characterizing the behaviour of the excess risk of this classifier. Finally, we describe three algorithms for computing these estimators based on a connection to bipartite graph matching, and perform experiments to illustrate the superiority of the MLE over the majority vote estimator.