Conditional stochastic optimization covers a variety of applications ranging from invariant learning and causal inference to meta-learning. However, constructing unbiased gradient estimators for such problems is challenging due to the composition structure. As an alternative, we propose a biased stochastic gradient descent (BSGD) algorithm and study the bias-variance tradeoff under different structural assumptions. We establish the sample complexities of BSGD for strongly convex, convex, and weakly convex objectives under smooth and non-smooth conditions. Our lower bound analysis shows that the sample complexities of BSGD cannot be improved for general convex objectives and nonconvex objectives except for smooth nonconvex objectives with Lipschitz continuous gradient estimator.

We study online change point detection problems under the constraint of local differential privacy (LDP) where, in particular, the statistician does not have access to the raw data. As a concrete problem, we study a multivariate nonparametric regression problem. At each time point t, the raw data are assumed to be of the form (X_t, Y_t), where X_t is a d -dimensional feature vector and Y_t is a response variable. Our primary aim is to detect changes in the regression function m_t(x) \mathbb{E}(Y_t X_t x) as soon as the change occurs. We provide algorithms which respect the LDP constraint, which control the false alarm probability, and which detect changes with a minimal (minimax rate-optimal) delay.

Stochastic gradient descent (SGD) and its variants have established themselves as the go-to algorithms for large-scale machine learning problems with independent samples due to their generalization performance and intrinsic computational advantage. However, the fact that the stochastic gradient is a biased estimator of the full gradient with correlated samples has led to the lack of theoretical understanding of how SGD behaves under correlated settings and hindered its use in such cases. In this paper, we focus on the Gaussian process (GP) and take a step forward towards breaking the barrier by proving minibatch SGD converges to a critical point of the full loss function, and recovers model hyperparameters with rate O(\frac{1}{K}) up to a statistical error term depending on the minibatch size. Numerical studies on both simulated and real datasets demonstrate that minibatch SGD has better generalization over state-of-the-art GP methods while reducing the computational burden and opening a new, previously unexplored, data size regime for GPs.

Dimensionality reduction is a classical technique widely used for data analysis. One foundational instantiation is Principal Component Analysis (PCA), which minimizes the average reconstruction error. In this paper, we introduce the multi-criteria dimensionality reduction problem where we are given multiple objectives that need to be optimized simultaneously. As an application, our model captures several fairness criteria for dimensionality reduction such as the Fair-PCA problem introduced by Samadi et al. [NeurIPS18] and the Nash Social Welfare (NSW) problem. In the Fair-PCA problem, the input data is divided into k groups, and the goal is to find a single d-dimensional representation for all groups for which the maximum reconstruction error of any one group is minimized.

Stochastic compositional optimization arises in many important machine learning tasks such as reinforcement learning and portfolio management. The objective function is the composition of two expectations of stochastic functions, and is more challenging to optimize than vanilla stochastic optimization problems. In this paper, we investigate the stochastic compositional optimization in the general smooth non-convex setting. We employ a recently developed idea of \textit{Stochastic Recursive Gradient Descent} to design a novel algorithm named SARAH-Compositional, and prove a sharp Incremental First-order Oracle (IFO) complexity upper bound for stochastic compositional optimization: \mathcal{O}((n m) {1/2} \varepsilon {-2}) in the finite-sum case and \mathcal{O}(\varepsilon {-3}) in the online case. Such a complexity is known to be the best one among IFO complexity results for non-convex stochastic compositional optimization.

This article suggests that deterministic Gradient Descent, which does not use any stochastic gradient approximation, can still exhibit stochastic behaviors. In particular, it shows that if the objective function exhibit multiscale behaviors, then in a large learning rate regime which only resolves the macroscopic but not the microscopic details of the objective, the deterministic GD dynamics can become chaotic and convergent not to a local minimizer but to a statistical distribution. In this sense, deterministic GD resembles stochastic GD even though no stochasticity is injected. A sufficient condition is also established for approximating this long-time statistical limit by a rescaled Gibbs distribution, which for example allows escapes from local minima to be quantified. Both theoretical and numerical demonstrations are provided, and the theoretical part relies on the construction of a stochastic map that uses bounded noise (as opposed to Gaussian noise).

We argue that regularizing terms in standard regression methods not only help against overfitting finite data, but sometimes also help in getting better causal models. We first consider a multi-dimensional variable linearly influencing a target variable with some multi-dimensional unobserved common cause, where the confounding effect can be decreased by keeping the penalizing term in Ridge and Lasso regression even in the population limit. The reason is a close analogy between overfitting and confounding observed for our toy model. In the case of overfitting, we can choose regularization constants via cross validation, but here we choose the regularization constant by first estimating the strength of confounding, which yielded reasonable results for simulated and real data. Further, we show a'causal generalization bound' which states (subject to our particular model of confounding) that the error made by interpreting any non-linear regression as causal model can be bounded from above whenever functions are taken from a not too rich class.

In the context of statistical supervised learning, the noiseless linear model assumes that there exists a deterministic linear relation Y \langle \theta_*, \Phi(U) \rangle between the random output Y and the random feature vector \Phi(U), a potentially non-linear transformation of the inputs U . We analyze the convergence of single-pass, fixed step-size stochastic gradient descent on the least-square risk under this model. The convergence of the iterates to the optimum \theta_* and the decay of the generalization error follow polynomial convergence rates with exponents that both depend on the regularities of the optimum \theta_* and of the feature vectors \Phi(U) . We interpret our result in the reproducing kernel Hilbert space framework. As a special case, we analyze an online algorithm for estimating a real function on the unit hypercube from the noiseless observation of its value at randomly sampled points; the convergence depends on the Sobolev smoothness of the function and of a chosen kernel.

We consider the robust linear regression problem in the online setting where we have access to the data in a streaming manner, one data point after the other. More specifically, for a true parameter \theta *, we consider the corrupted Gaussian linear model y \varepsilon b where the adversarial noise b can take any value with probability \eta and equals zero otherwise. We consider this adversary to be oblivious (i.e., b independent of the data) since this is the only contamination model under which consistency is possible. Current algorithms rely on having the whole data at hand in order to identify and remove the outliers. In contrast, we show in this work that stochastic gradient descent on the l1 loss converges to the true parameter vector at a \tilde{O}( 1 / (1 - \eta) 2 n) rate which is independent of the values of the contaminated measurements.

We study the problem of list-decodable linear regression, where an adversary can corrupt a majority of the examples. Specifically, we are given a set T of labeled examples (x, y) \in \mathbb{R} d \times \mathbb{R} and a parameter 0 \alpha 1/2 such that an \alpha -fraction of the points in T are i.i.d. The goal is to output a small list of hypothesis vectors such that at least one of them is close to the target regression vector. Our main result is a Statistical Query (SQ) lower bound of d {\mathrm{poly}(1/\alpha)} for this problem. Our SQ lower bound qualitatively matches the performance of previously developed algorithms, providing evidence that current upper bounds for this task are nearly best possible.