The partial coloring method is one of the most powerful and widely used method in combinatorial discrepancy problems. However, in many cases it leads to sub-optimal bounds as the partial coloring step must be iterated a logarithmic number of times, and the errors can add up in an adversarial way.We give a new and general algorithmic framework that overcomes the limitations of the partial coloring method and can be applied in a black-box manner to various problems. Using this framework, we give new improved bounds and algorithms for several classic problems in discrepancy. In particular, for Tusnady's problem, we give an improved O(log 2 n) bound for discrepancy of axis-parallel rectangles and more generally an O d (log d n) bound for d-dimensional boxes in R d . Previously, even non-constructively, the best bounds were O(log 2.5 n) and O d (log d+0.5 n) respectively. Similarly, for the Steinitz problem we give the first algorithm that matches the best known non-constructive bounds due to Banaszczyk [Ban12] in the ℓ ∞ case, and improves the previous algorithmic bounds substantially in the ℓ 2 case. Our framework is based upon a substantial generalization of the techniques developed recently in the context of the Komlós discrepancy problem [BDG16]. project no. 022.005.025. 0 Let (V, S) be a finite set system, with V = {1, . . . , n} and S = {S 1 , . . . , S m } a collection of subsets of V . For a two-coloring χ : V → {−1, 1}, the discrepancy of χ for a set S is defined as χ(S) = | j∈S χ(j)| and measures the imbalance from an even-split for S. The discrepancy of the system (V, S) is defined as disc(S) = minThat is, it is the minimum imbalance for all sets in S, over all possible two-colorings χ. More generally for any matrix A, its discrepancy is defined as disc(A) = min x∈{−1,1} n Ax ∞ . Discrepancy is a widely studied topic and has applications to many areas in mathematics and computer science. In particular in computer science, it arises naturally in computational geometry, data structure lower bounds, rounding in approximation algorithms, combinatorial optimization, communication complexity and pseudorandomness. For much more on these connections we refer the reader to the books [Cha00, Mat09, CST + 14].Partial Coloring Method: One of the most important and widely used technique in discrepancy is the partial coloring method developed in the early 80's by Beck, and its refinement by Spencer to the entropy method [Bec81b, Spe85]. An essentially similar approach, but based on ideas from convex geometry was developed independently by Gluskin [Glu89]. Besides being powerful, an important reason for its success is that it can be applied easily to many problems in a black-box manner and for most problems in discrepancy the best known bounds are achieved using this method. While these original arguments were based on the pigeonhole principle and were nonalgorithmic, in recent years several new algorithmic versions of the partial coloring method have been developed [Ban10, LM12, Rot14a, HSS14, ES14]. In particular, all ...