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We present an adaptive stochastic variance reduced method with an implicit approach for adaptivity. As a variant of SARAH, our method employs the stochastic recursive gradient yet adjusts step-size based on local geometry. We provide convergence guarantees for finite-sum minimization problems and show a faster convergence than SARAH can be achieved if local geometry permits. Furthermore, we propose a practical, fully adaptive variant, which does not require any knowledge of local geometry and any effort of tuning the hyper-parameters. This algorithm implicitly computes step-size and efficiently estimates local Lipschitz smoothness of stochastic functions. The numerical experiments demonstrate the algorithm's strong performance compared to its classical counterparts and other state-of-the-art first-order methods.

This paper considers the problem of supervised learning with linear methods when both features and labels can be corrupted, either in the form of heavy tailed data and/or corrupted rows. We introduce a combination of coordinate gradient descent as a learning algorithm together with robust estimators of the partial derivatives. This leads to robust statistical learning methods that have a numerical complexity nearly identical to non-robust ones based on empirical risk minimization. The main idea is simple: while robust learning with gradient descent requires the computational cost of robustly estimating the whole gradient to update all parameters, a parameter can be updated immediately using a robust estimator of a single partial derivative in coordinate gradient descent. We prove upper bounds on the generalization error of the algorithms derived from this idea, that control both the optimization and statistical errors with and without a strong convexity assumption of the risk. Finally, we propose an efficient implementation of this approach in a new python library called linlearn, and demonstrate through extensive numerical experiments that our approach introduces a new interesting compromise between robustness, statistical performance and numerical efficiency for this problem.

Classical interior penalty discontinuous Galerkin (IPDG) methods for diffusion problems require a number of assumptions on the local variation of mesh-size, polynomial degree, and of the diffusion coefficient to determine the values of the, so-called, discontinuity-penalization parameter and/or to perform error analysis. Variants of IPDG methods involving weighted averages of the gradient of the approximate solution have been proposed in the context of high-contrast diffusion coefficients to mitigate the dependence of the contrast in the stability and in the error analysis. Here, we present a new IPDG method, involving carefully constructed weighted averages of the gradient of the approximate solution, which is shown to be robust even for the most extreme simultaneous local mesh, polynomial degree and diffusion coefficient variation scenarios, without resulting in unreasonably large penalization. The new method, henceforth termed as \emph{robust IPDG} (RIPDG), offers typically significantly better conditioning than the standard IPDG method when applied to scenarios with strong mesh/polynomial degree/diffusion local variation. On the other hand, when using uniform meshes, constant polynomial degree and for problems with constant diffusion coefficients, the RIPDG method is identical to the classical IPDG. Numerical experiments indicate the favourable performance of the new RIPDG method over the classical version in terms of conditioning and error.

We propose a novel method for training deep neural networks that are capable of interpolation, that is, driving the empirical loss to zero. At each iteration, our method constructs a stochastic approximation of the learning objective. The approximation, known as a bundle, is a pointwise maximum of linear functions. Our bundle contains a constant function that lower bounds the empirical loss. This enables us to compute an automatic adaptive learning rate, thereby providing an accurate solution. In addition, our bundle includes linear approximations computed at the current iterate and other linear estimates of the DNN parameters. The use of these additional approximations makes our method significantly more robust to its hyperparameters. Based on its desirable empirical properties, we term our method Bundle Optimisation for Robust and Accurate Training (BORAT). In order to operationalise BORAT, we design a novel algorithm for optimising the bundle approximation efficiently at each iteration. We establish the theoretical convergence of BORAT in both convex and non-convex settings. Using standard publicly available data sets, we provide a thorough comparison of BORAT to other single hyperparameter optimisation algorithms. Our experiments demonstrate BORAT matches the state-of-the-art generalisation performance for these methods and is the most robust.

In this paper, we study the finite-sum convex optimization problem focusing on the general convex case. Recently, the study of variance reduced (VR) methods and their accelerated variants has made exciting progress. However, the step size used in the existing VR algorithms typically depends on the smoothness parameter, which is often unknown and requires tuning in practice. To address this problem, we propose two novel adaptive VR algorithms: Adaptive Variance Reduced Accelerated Extra-Gradient (AdaVRAE) and Adaptive Variance Reduced Accelerated Gradient (AdaVRAG). Our algorithms do not require knowledge of the smoothness parameter. AdaVRAE uses $\mathcal{O}\left(n\log\log n+\sqrt{\frac{n\beta}{\epsilon}}\right)$ gradient evaluations and AdaVRAG uses $\mathcal{O}\left(n\log\log n+\sqrt{\frac{n\beta\log\beta}{\epsilon}}\right)$ gradient evaluations to attain an $\mathcal{O}(\epsilon)$-suboptimal solution, where $n$ is the number of functions in the finite sum and $\beta$ is the smoothness parameter. This result matches the best-known convergence rate of non-adaptive VR methods and it improves upon the convergence of the state of the art adaptive VR method, AdaSVRG. We demonstrate the superior performance of our algorithms compared with previous methods in experiments on real-world datasets.

The Boolean Hidden Matching (BHM) problem, introduced in a seminal paper of Gavinsky et. al. [STOC'07], has played an important role in the streaming lower bounds for graph problems such as triangle and subgraph counting, maximum matching, MAX-CUT, Schatten $p$-norm approximation, maximum acyclic subgraph, testing bipartiteness, $k$-connectivity, and cycle-freeness. The one-way communication complexity of the Boolean Hidden Matching problem on a universe of size $n$ is $\Theta(\sqrt{n})$, resulting in $\Omega(\sqrt{n})$ lower bounds for constant factor approximations to several of the aforementioned graph problems. The related (and, in fact, more general) Boolean Hidden Hypermatching (BHH) problem introduced by Verbin and Yu [SODA'11] provides an approach to proving higher lower bounds of $\Omega(n^{1-1/t})$ for integer $t\geq 2$. Reductions based on Boolean Hidden Hypermatching generate distributions on graphs with connected components of diameter about $t$, and basically show that long range exploration is hard in the streaming model of computation with adversarial arrivals. In this paper we introduce a natural variant of the BHM problem, called noisy BHM (and its natural noisy BHH variant), that we use to obtain higher than $\Omega(\sqrt{n})$ lower bounds for approximating several of the aforementioned problems in graph streams when the input graphs consist only of components of diameter bounded by a fixed constant. We also use the noisy BHM problem to show that the problem of classifying whether an underlying graph is isomorphic to a complete binary tree in insertion-only streams requires $\Omega(n)$ space, which seems challenging to show using BHM or BHH alone.

We study the numerical approximation by space-time finite element methods of a multi-physics system coupling hyperbolic elastodynamics with parabolic transport and modelling poro- and thermoelasticity. The equations are rewritten as a first-order system in time. Discretizations by continuous Galerkin methods in space and time with inf-sup stable pairs of finite elements for the spatial approximation of the unknowns are investigated. Optimal order error estimates of energy-type are proven. Superconvergence at the time nodes is addressed briefly. The error analysis can be extended to discontinuous and enriched Galerkin space discretizations. The error estimates are confirmed by numerical experiments.

Functional constrained optimization is becoming more and more important in machine learning and operations research. Such problems have potential applications in risk-averse machine learning, semisupervised learning, and robust optimization among others. In this paper, we first present a novel Constraint Extrapolation (ConEx) method for solving convex functional constrained problems, which utilizes linear approximations of the constraint functions to define the extrapolation (or acceleration) step. We show that this method is a unified algorithm that achieves the best-known rate of convergence for solving different functional constrained convex composite problems, including convex or strongly convex, and smooth or nonsmooth problems with a stochastic objective and/or stochastic constraints. Many of these rates of convergence were in fact obtained for the first time in the literature. In addition, ConEx is a single-loop algorithm that does not involve any penalty subproblems. Contrary to existing primal-dual methods, it does not require the projection of Lagrangian multipliers into a (possibly unknown) bounded set. Second, for nonconvex functional constrained problems, we introduce a new proximal point method that transforms the initial nonconvex problem into a sequence of convex problems by adding quadratic terms to both the objective and constraints. Under a certain MFCQ-type assumption, we establish the convergence and rate of convergence of this method to KKT points when the convex subproblems are solved exactly or inexactly. For large-scale and stochastic problems, we present a more practical proximal point method in which the approximate solutions of the subproblems are computed by the aforementioned ConEx method. To the best of our knowledge, most of these convergence and complexity results of the proximal point method for nonconvex problems also seem to be new in the literature.

This PhD thesis contains several contributions to the field of statistical causal modeling. Statistical causal models are statistical models embedded with causal assumptions that allow for the inference and reasoning about the behavior of stochastic systems affected by external manipulation (interventions). This thesis contributes to the research areas concerning the estimation of causal effects, causal structure learning, and distributionally robust (out-of-distribution generalizing) prediction methods. We present novel and consistent linear and non-linear causal effects estimators in instrumental variable settings that employ data-dependent mean squared prediction error regularization. Our proposed estimators show, in certain settings, mean squared error improvements compared to both canonical and state-of-the-art estimators. We show that recent research on distributionally robust prediction methods has connections to well-studied estimators from econometrics. This connection leads us to prove that general K-class estimators possess distributional robustness properties. We, furthermore, propose a general framework for distributional robustness with respect to intervention-induced distributions. In this framework, we derive sufficient conditions for the identifiability of distributionally robust prediction methods and present impossibility results that show the necessity of several of these conditions. We present a new structure learning method applicable in additive noise models with directed trees as causal graphs. We prove consistency in a vanishing identifiability setup and provide a method for testing substructure hypotheses with asymptotic family-wise error control that remains valid post-selection. Finally, we present heuristic ideas for learning summary graphs of nonlinear time-series models.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.