We study the problem of safe online convex optimization, where the action at each time step must satisfy a set of linear safety constraints. The goal is to select a sequence of actions to minimize the regret without violating the safety constraints at any time step (with high probability). The parameters that specify the linear safety constraints are unknown to the algorithm. The algorithm has access to only the noisy observations of constraints for the chosen actions. We propose an algorithm, called the Safe Online Projected Gradient Descent (SO-PGD) algorithm, to address this problem. We show that, under the assumption of availability of a safe baseline action, the SO-PGD algorithm achieves a regret O(T^{2/3}). While there are many algorithms for online convex optimization (OCO) problems with safety constraints available in the literature, they allow constraint violations during learning/optimization, and the focus has been on characterizing the cumulative constraint violations. To the best of our knowledge, ours is the first work that provides an algorithm with provable guarantees on the regret, without violating the linear safety constraints (with high probability) at any time step.

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Advancements in medicine have increased the longevity of humans, resulting in a higher incidence of chronic diseases. Due to the rise in the elderly population, age-dependent neurodegenerative disorders are becoming increasingly prevalent. The available treatment options only provide symptomatic relief and do not cure the underlying cause of the disease. Therefore, it has become imperative to discover new markers and therapies to modulate the course of disease progression and develop better treatment options for the affected individuals. Growing evidence indicates that neuroinflammation is a common factor and one of the main inducers of neuronal damage and degeneration. Galectins (Gals) are a class of β-galactoside-binding proteins (lectins) ubiquitously expressed in almost all vital organs. Gals modulate various cellular responses and regulate significant biological functions, including immune response, proliferation, differentiation, migration, and cell growth, through their interaction with glycoproteins and glycolipids. In recent years, extensive research has been conducted on the Gal superfamily, with Gal-1, Gal-3, and Gal-9 in prime focus. Their roles have been described in modulating neuroinflammation and neurodegenerative processes. In this review, we discuss the role of Gals in the causation and progression of neurodegenerative disorders. We describe the role of Gals in microglia and astrocyte modulation, along with their pro- and anti-inflammatory functions. In addition, we discuss the potential use of Gals as a novel therapeutic target for neuroinflammation and restoring tissue damage in neurodegenerative diseases.

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