ConcordantBarriers

In this section we provide the necessary background on self-concordant barriers. Let fi be self-concordant ondomfi, for all i [[k]]. In the sequel we will tacitly assume thatf isnondegenerate, in the sense that the Hessian 2f(x)is positive definite, for anyx domf. Tolighten our notation, we will oftentimes simply write u x when the underlying self-concordant function is clearfromthecontext. As usual, for the purpose of our analysis we consider the auxiliarybe the leader (BTL) sequence, definedfort N {0}asfollows. Similarly, we have that x(t) g(t 1) g(t 1) = g(t 1) argmin( Φ(t)) g(t 1),bydefinitionof (OFTRL).