Andrei Patrascu

Andrei Patrascu, Ciprian Paduraru, Paul Irofti

Stochastic optimization lies at the core of most statistical learning models. The recent great development of stochastic algorithmic tools focused significantly onto proximal gradient iterations, in order to find an efficient approach for nonsmooth (composite) population risk functions. The complexity of finding optimal predictors by minimizing regularized risk is largely understood for simple regularizations such as $\ell_1/\ell_2$ norms. However, more complex properties desired for the predictor necessitates highly difficult regularizers as used in grouped lasso or graph trend filtering. In this chapter we develop and analyze minibatch variants of stochastic proximal gradient algorithm for general composite objective functions with stochastic nonsmooth components. We provide iteration complexity for constant and variable stepsize policies obtaining that, for minibatch size $N$, after $\mathcal{O}(\frac{1}{Nε})$ iterations $ε-$suboptimality is attained in expected quadratic distance to optimal solution. The numerical tests on $\ell_2-$regularized SVMs and parametric sparse representation problems confirm the theoretical behaviour and surpasses minibatch SGD performance.

Andrei Patrascu, Paul Irofti

Supported by the recent contributions in multiple branches, the first-order splitting algorithms became central for structured nonsmooth optimization. In the large-scale or noisy contexts, when only stochastic information on the smooth part of the objective function is available, the extension of proximal gradient schemes to stochastic oracles is based on proximal tractability of the nonsmooth component and it has been deeply analyzed in the literature. However, there remained gaps illustrated by composite models where the nonsmooth term is not proximally tractable anymore. In this note we tackle composite optimization problems, where the access only to stochastic information on both smooth and nonsmooth components is assumed, using a stochastic proximal first-order scheme with stochastic proximal updates. We provide $\mathcal{O}\left( \frac{1}{k} \right)$ the iteration complexity (in expectation of squared distance to the optimal set) under the strong convexity assumption on the objective function. Empirical behavior is illustrated by numerical tests on parametric sparse representation models.

Andra Baltoiu, Andrei Patrascu, Paul Irofti

Anomaly detection in networks often boils down to identifying an underlying graph structure on which the abnormal occurrence rests on. Financial fraud schemes are one such example, where more or less intricate schemes are employed in order to elude transaction security protocols. We investigate the problem of learning graph structure representations using adaptations of dictionary learning aimed at encoding connectivity patterns. In particular, we adapt dictionary learning strategies to the specificity of network topologies and propose new methods that impose Laplacian structure on the dictionaries themselves. In one adaption we focus on classifying topologies by working directly on the graph Laplacian and cast the learning problem to accommodate its 2D structure. We tackle the same problem by learning dictionaries which consist of vectorized atomic Laplacians, and provide a block coordinate descent scheme to solve the new dictionary learning formulation. Imposing Laplacian structure on the dictionaries is also proposed in an adaptation of the Single Block Orthogonal learning method. Results on synthetic graph datasets comprising different graph topologies confirm the potential of dictionaries to directly represent graph structure information.

Paul Irofti, Andrei Patrascu, Andra Baltoiu

In general, anomaly detection is the problem of distinguishing between normal data samples with well defined patterns or signatures and those that do not conform to the expected profiles. Financial transactions, customer reviews, social media posts are all characterized by relational information. In these networks, fraudulent behaviour may appear as a distinctive graph edge, such as spam message, a node or a larger subgraph structure, such as when a group of clients engage in money laundering schemes. Most commonly, these networks are represented as attributed graphs, with numerical features complementing relational information. We present a survey on anomaly detection techniques used for fraud detection that exploit both the graph structure underlying the data and the contextual information contained in the attributes.

Andrei Patrascu

Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is an iterative scheme born from the adaptation of proximal point algorithm to noisy stochastic optimization, with a resulting iteration related to stochastic alternating projections. Inspired by the scalability of alternating projection methods, we start from the (linear) regularity assumption, typically used in convex feasiblity problems to guarantee the linear convergence of stochastic alternating projection methods, and analyze a general weak linear regularity condition which facilitates convergence rate boosts in stochastic proximal point schemes. Our applications include many non-strongly convex functions classes often used in machine learning and statistics. Moreover, under weak linear regularity assumption we guarantee $\mathcal{O}\left(\frac{1}{k}\right)$ convergence rate for SPP, in terms of the distance to the optimal set, using only projections onto a simple component set. Linear convergence is obtained for interpolation setting, when the optimal set of the expected cost is included into the optimal sets of each functional component.