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We prove risk bounds for binary classification in high-dimensional settingswhen the sample size is allowed to be smaller than the dimensionality of thetraining set observations. In particular, we prove upper bounds for both'compressive learning' by empirical risk minimization (ERM) (that is when theERM classifier is learned from data that have been projected fromhigh-dimensions onto a randomly selected low-dimensional subspace) as well asuniform upper bounds in the full high-dimensional space. A novel tool we employin both settings is the 'flipping probability' of Durrant and Kaban (ICML 2013)which we use to capture benign geometric structures that make a classificationproblem 'easy' in the sense of demanding a relatively low sample size forguarantees of good generalization. Furthermore our bounds also enable us toexplain or draw connections between several existing successful classificationalgorithms. Finally we show empirically that our bounds are informative enoughin practice to serve as the objective function for learning a classifier (byusing them to do so).

Structure-aware error bounds for linear classification with the zero-one loss