In this paper, we revisit and improve the convergence of policy gradient (PG), natural PG (NPG) methods, and their variance-reduced variants, under general smooth policy parametrizations. More specifically, with the Fisher information matrix of the policy being positive definite: i) we show that a state-of-the-art variance-reduced PG method, which has only been shown to converge to stationary points, converges to the globally optimal value up to some inherent function approximation error due to policy parametrization; ii) we show that NPG enjoys a lower sample complexity; iii) we propose SRVR-NPG, which incorporates variance-reduction into the NPG update. Our improvements follow from an observation that the convergence of (variance-reduced) PG and NPG methods can improve each other: the stationary convergence analysis of PG can be applied to NPG as well, and the global convergence analysis of NPG can help to establish the global convergence of (variance-reduced) PG methods. Our analysis carefully integrates the advantages of these two lines of works. Thanks to this improvement, we have also made variance-reduction for NPG possible, with both global convergence and an efficient finite-sample complexity.

Symmetric positive definite matrices (symmetric matrices with positive eigenvalues) are ubiquitous, not least because they arise in the solution of many minimization problems. However, a matrix that is supposed to be positive definite may fail to be so for a variety of reasons. Missing or inconsistent data in forming a covariance matrix or a correlation matrix can cause a loss of definiteness, and rounding errors can cause a tiny positive eigenvalue to go negative. The best way to check definiteness is to compute a Cholesky factorization, which is often needed anyway. The MATLAB function chol returns an error message if the factorization fails, and a second output argument can be requested, which is set to the number of the stage on which the factorization failed, or to zero if the factorization succeeded.

Subgraph isomorphism or subgraph matching is generally considered as an NP-complete problem, made more complex in practical applications where the edge weights take real values and are subject to measurement noise and possible anomalies. To the best of our knowledge, almost all subgraph matching methods utilize node labels to perform node-node matching. In the absence of such labels (in applications such as image matching and map matching among others), these subgraph matching methods do not work. We propose a method for identifying the node correspondence between a subgraph and a full graph in the inexact case without node labels in two steps - (a) extract the minimal unique topology preserving subset from the subgraph and find its feasible matching in the full graph, and (b) implement a consensus-based algorithm to expand the matched node set by pairing unique paths based on boundary commutativity. Going beyond the existing subgraph matching approaches, the proposed method is shown to have realistically sub-linear computational efficiency, robustness to random measurement noise, and good statistical properties. Our method is also readily applicable to the exact matching case without loss of generality. To demonstrate the effectiveness of the proposed method, a simulation and a case study is performed on the Erdos-Renyi random graphs and the image-based affine covariant features dataset respectively.

Given a Lipschitz or smooth convex function $\, f:K \to \mathbb{R}$ for a bounded polytope $K \subseteq \mathbb{R}^d$ defined by $m$ inequalities, we consider the problem of sampling from the log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to $K$. Interest in this problem derives from its applications to Bayesian inference and differentially private learning. Our main result is a generalization of the Dikin walk Markov chain to this setting that requires at most $O((md + d L^2 R^2) \times md^{\omega-1}) \log(\frac{w}{\delta}))$ arithmetic operations to sample from $\pi$ within error $\delta>0$ in the total variation distance from a $w$-warm start. Here $L$ is the Lipschitz-constant of $f$, $K$ is contained in a ball of radius $R$ and contains a ball of smaller radius $r$, and $\omega$ is the matrix-multiplication constant. Our algorithm improves on the running time of prior works for a range of parameter settings important for the aforementioned learning applications. Technically, we depart from previous Dikin walks by adding a "soft-threshold" regularizer derived from the Lipschitz or smoothness properties of $f$ to the log-barrier function for $K$ that allows our version of the Dikin walk to propose updates that have a high Metropolis acceptance ratio for $f$, while at the same time remaining inside the polytope $K$.

Forecasting and decision-making are generally modeled as two sequential steps with no feedback, following an open-loop approach. In this paper, we present application-driven learning, a new closed-loop framework in which the processes of forecasting and decision-making are merged and co-optimized through a bilevel optimization problem. We present our methodology in a general format and prove that the solution converges to the best estimator in terms of the expected cost of the selected application. Then, we propose two solution methods: an exact method based on the KKT conditions of the second-level problem and a scalable heuristic approach suitable for decomposition methods. The proposed methodology is applied to the relevant problem of defining dynamic reserve requirements and conditional load forecasts, offering an alternative approach to current \emph{ad hoc} procedures implemented in industry practices. We benchmark our methodology with the standard sequential least-squares forecast and dispatch planning process. We apply the proposed methodology to an illustrative system and to a wide range of instances, from dozens of buses to large-scale realistic systems with thousands of buses. Our results show that the proposed methodology is scalable and yields consistently better performance than the standard open-loop approach.

Transformers can learn to perform numerical computations from examples only. I study nine problems of linear algebra, from basic matrix operations to eigenvalue decomposition and inversion, and introduce and discuss four encoding schemes to represent real numbers. On all problems, transformers trained on sets of random matrices achieve high accuracies (over 90%). The models are robust to noise, and can generalize out of their training distribution. In particular, models trained to predict Laplace-distributed eigenvalues generalize to different classes of matrices: Wigner matrices or matrices with positive eigenvalues. The reverse is not true.

Minimalist Data Wrangling with Python is envisaged as a student's first introduction to data science, providing a high-level overview as well as discussing key concepts in detail. We explore methods for cleaning data gathered from different sources, transforming, selecting, and extracting features, performing exploratory data analysis and dimensionality reduction, identifying naturally occurring data clusters, modelling patterns in data, comparing data between groups, and reporting the results. This textbook is a non-profit project. Its online and PDF versions are freely available at https://datawranglingpy.gagolewski.com/.

Stochastic first-order methods such as Stochastic Extragradient (SEG) or Stochastic Gradient Descent-Ascent (SGDA) for solving smooth minimax problems and, more generally, variational inequality problems (VIP) have been gaining a lot of attention in recent years due to the growing popularity of adversarial formulations in machine learning. However, while high-probability convergence bounds are known to reflect the actual behavior of stochastic methods more accurately, most convergence results are provided in expectation. Moreover, the only known highprobability complexity results have been derived under restrictive sub-Gaussian (light-tailed) noise and bounded domain assumption [Juditsky et al., 2011a]. In this work, we prove the first high-probability complexity results with logarithmic dependence on the confidence level for stochastic methods for solving monotone and structured non-monotone VIPs with non-sub-Gaussian (heavy-tailed) noise and unbounded domains. In the monotone case, our results match the best-known ones in the light-tails case [Juditsky et al., 2011a], and are novel for structured non-monotone problems such as negative comonotone, quasi-strongly monotone, and/or star-cocoercive ones. We achieve these results by studying SEG and SGDA with clipping. In addition, we numerically validate that the gradient noise of many practical GAN formulations is heavy-tailed and show that clipping improves the performance of SEG/SGDA.

This paper studies stochastic control problems with the action space taken to be the space of measures, regularized by the relative entropy. We identify suitable metric space on which we construct a gradient flow for the measure-valued control process along which the cost functional is guaranteed to decrease. It is shown that any invariant measure of this gradient flow satisfies the Pontryagin optimality principle. If the problem we work with is sufficiently convex, the gradient flow converges exponentially fast. Furthermore, the optimal measure-valued control admits Bayesian interpretation which means that one can incorporate prior knowledge when solving stochastic control problem. This work is motivated by a desire to extend the theoretical underpinning for the convergence of stochastic gradient type algorithms widely used in the reinforcement learning community to solve control problems.

Abstract: The brain is a highly complex system. Most of such complexity stems from the intermingled connections between its parts, which give rise to rich dynamics and to the emergence of high-level cognitive functions. Disentangling the underlying network structure is crucial to understand the brain functioning under both healthy and pathological conditions. Yet, analyzing brain networks is challenging, in part because their structure represents only one possible realization of a generative stochastic process which is in general unknown. Having a formal way to cope with such intrinsic variability is therefore central for the characterization of brain network properties.