We show that a function is strongly convex with respect to some norm if and only if its conjugate function is strongly smooth with respect to the dual norm. This result has already been found to be a key component in deriving and analyzing several learning algorithms. Utilizing this duality, we isolate a single inequality which seamlessly implies both generalization bounds and online regret bou...

Linear convergence rates of descent methods for unconstrained minimization are usually proven under the assumption that the objective function is strongly convex. Recently it was shown that the weaker assumption of restricted strong convexity suffices for linear convergence of the ordinary gradient descent method. A decisive difference to strong convexity is that the set of minimizers of a rest...

We introduce the notion of strongly h-convex functions (defined on a normed space) and present some properties and representations of such functions. We obtain a characterization of inner product spaces involving the notion of strongly h-convex functions. Finally, a Hermite–Hadamard–type inequality for strongly h-convex functions is given.

The solution to a Nash or a nonsymmetric bargaining game is obtained by maximizing a concave function over a convex set, i.e., it is the solution to a convex program. We show that each 2-player game whose convex program has linear constraints, admits a rational solution and such a solution can be found in polynomial time using only an LP solver. If in addition, the game is succinct, i.e., the c...

Abstract We consider alternating minimization procedures for convex and non-convex optimization problems with the vector of variables divided into several blocks, each block being amenable respect to its while maintaining other blocks constant. In case two we prove a linear convergence rate an procedure under Polyak–Łojasiewicz (PL) condition, which can be seen as relaxation strong convexity as...

The iterative shrinkage/thresholding algorithm (ISTA) and its faster version FISTA have been widely used in the literature. In this paper, we consider general versions of the ISTA and FISTA in the more general “strongly + semi” convex setting, i.e., minimizing the sum of a strongly convex function and a semiconvex function; and conduct convergence analysis for them. The consideration of a semic...

The solution to a Nash or a nonsymmetric bargaining game is obtained by maximizing a concave function over a convex set, i.e., it is the solution to a convex program. We show that each 2-player game whose convex program has linear constraints, admits a rational solution and such a solution can be found in polynomial time using only an LP solver. If in addition, the game is succinct, i.e., the c...

We show that the exact worst-case performance of fixed-step first-order methods for unconstrained optimization of smooth (possibly strongly) convex functions can be obtained by solving convex programs. Finding the worst-case performance of a black-box first-order method is formulated as an optimization problem over a set of smooth (strongly) convex functions and initial conditions. We develop c...