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Regularized M-estimation—estimation of model parameters by minimizing a composite objective of loss plus regularizing penalty—-is a widely used framework for learning under high-dimensional or ill-posed conditions.  A prime example is the Lasso, which pairs a quadratic loss with an l1-norm penalty to learn high-dimensional linear models under sparsity assumptions.  While the Lasso is underpinned by elegant theory and its principles have been extended to other "sparse high-dimensional" settings, conceptual clarity in explaining mechanism often fades beyond the Lasso.  For example, existing theories for such extensions typically rely on ad hoc techniques—for instance, controlling "higher-order deviation terms" derived from Lasso proofs—leading to limited insight or suboptimal guarantees.  Moreover, to our knowledge, no overarching framework extends cleanly beyond a few specific settings.  Meanwhile, recent advances in machine learning have driven a shift in perspective: rather than positing a structure first and then crafting a tailored regularizer, practitioners increasingly choose a loss and regularizer upfront, then exploit the induced structure.  In this talk, we present preliminary results from our ongoing research (jointly with Venkat Chandrasekaran) that explores the geometry of regularized M-estimators beyond the Lasso paradigm, focusing on model-selection-consistency in a broad class of convex losses and norm regularizers.  We also propose a systematic approach for characterizing structures induced by norm-based regularizers.