Title: Only tails matter: Average-Case Universality and Robustness in the Convex Regime. Abstract: Recent works have studied the average convergence properties of first-order optimization methods on distributions of quadratic problems. The average-case framework allows a more fine-grained and representative analysis of convergence than usual worst-case results, in exchange for a more precise hypothesis over the data generating process, namely assuming knowledge of the expected spectral distribution (e.s.d) of the random matrix associated with the problem. In this work, we show that a problem's asymptotic average complexity is determined by the concentration of eigenvalues near the edges of the e.s.d. We argue that having à priori information on this concentration is a more grounded assumption than complete knowledge of the e.s.d.,  and that basing our analysis on the approximate concentration is effectively a middle ground between the coarseness of the worst-case convergence and this more unrealistic hypothesis. We introduce the Generalized Chebyshev method, asymptotically optimal under a hypothesis on this concentration, and globally optimal when the e.s.d. follows  a Beta distribution. We compare its performance to classical optimization algorithms, such as Gradient Descent or Nesterov's scheme, and we show that, asymptotically, Nesterov's method is universally nearly-optimal in the average-case.