Affine policies (or control) are widely used as a solution approach in dynamic optimization where computing an optimal adjustable solution is usually intractable. While the worst case performance of affine policies can be significantly bad, the empirical performance is observed to be near-optimal for a large class of problem instances. For instance, in the two-stage dynamic robust optimization problem with linear covering constraints and uncertain right hand side, the worst-case approximation bound for affine policies is $O(\sqrt m)$ that is also tight (see Bertsimas and Goyal (2012)), whereas observed empirical performance is near-optimal. In this paper, we aim to address this stark-contrast between the worst-case and the empirical performance of affine policies. In particular, we show that affine policies give a good approximation for the two-stage adjustable robust optimization problem with high probability on random instances where the constraint coefficients are generated i.i.d.

Hiererachical clustering, that is computing a recursive partitioning of a dataset to obtain clusters at increasingly finer granularity is a fundamental problem in data analysis. Although hierarchical clustering has mostly been studied through procedures such as linkage algorithms, or top-down heuristics, rather than as optimization problems, recently Dasgupta [1] proposed an objective function for hierarchical clustering and initiated a line of work developing algorithms that explicitly optimize an objective (see also [2, 3, 4]). In this paper, we consider a fairly general random graph model for hierarchical clustering, called the hierarchical stochastic blockmodel (HSBM), and show that in certain regimes the SVD approach of McSherry [5] combined with specific linkage methods results in a clustering that give an O(1)-approximation to Dasgupta's cost function. We also show that an approach based on SDP relaxations for balanced cuts based on the work of Makarychev et al. [6], combined with the recursive sparsest cut algorithm of Dasgupta, yields an O(1) approximation in slightly larger regimes and also in the semi-random setting, where an adversary may remove edges from the random graph generated according to an HSBM. Finally, we report empirical evaluation on synthetic and real-world data showing that our proposed SVD-based method does indeed achieve a better cost than other widely-used heurstics and also results in a better classification accuracy when the underlying problem was that of multi-class classification.

Generative models at times produce "invalid" outputs, such as images with generation artifacts and unnatural sounds. Validity-constrained distribution learning attempts to address this problem by requiring that the learned distribution have a provably small fraction of its mass in invalid parts of space -- something which standard loss minimization does not always ensure. To this end, a learner in this model can guide the learning via "validity queries", which allow it to ascertain the validity of individual examples. Prior work on this problem takes a worst-case stance, showing that proper learning requires an exponential number of validity queries, and demonstrating an improper algorithm which -- while generating guarantees in a wide-range of settings -- makes a relatively large polynomial number of validity queries. In this work, we take a first step towards characterizing regimes where guaranteeing validity is easier than in the worst-case.

This paper studies the problem of hierarchical clustering in a beyond-the-worst-case setting, where the data is a random graph generated by a Hierarchical Stochastic Block Model (HSBM). It proposes an SVD linkage algorithm and proves that in certain regimes it returns a solution that is a constant approximation to the cost function of hierarchical clustering proposed by Dasgupta. It also considers an algorithm that combines the SDP relaxation approach and recursive sparsest cut approach in previous works to get a constant approximation in larger regimes. Roughly speaking, the HSBM model considered assumes some underlying tree T* over k leaves (each leaf corresponding to a bottom cluster containing a certain number of vertices). Each node in the tree has some weight in (0,1) and a node's weight should not be larger than its descendant's.

We study worst-case bounds on the quality of any fixed point assignment of the max-product algorithm for Markov Random Fields (MRF). We start proving a bound independent of the MRF structure and parameters. Afterwards, we show how this bound can be improved for MRFs with particular structures such as bipartite graphs or grids. Our results provide interesting insight into the behavior of max-product. For example, we prove that max-product provides very good results (at least 90% of the optimal) on MRFs with large variable-disjoint cycles (MRFs in which all cycles are variable-disjoint, namely that they do not share any edge and in which each cycle contains at least 20 variables).

We study worst-case bounds on the quality of any fixed point assignment of the max-product algorithm for Markov Random Fields (MRF). We start proving a bound independent of the MRF structure and parameters. Afterwards, we show how this bound can be improved for MRFs with particular structures such as bipartite graphs or grids. Our results provide interesting insight into the behavior of max-product. For example, we prove that max-product provides very good results (at least 90% of the optimal) on MRFs with large variable-disjoint cycles (MRFs in which all cycles are variable-disjoint, namely that they do not share any edge and in which each cycle contains at least 20 variables).

Hiererachical clustering, that is computing a recursive partitioning of a dataset to obtain clusters at increasingly finer granularity is a fundamental problem in data analysis. Although hierarchical clustering has mostly been studied through procedures such as linkage algorithms, or top-down heuristics, rather than as optimization problems, recently Dasgupta [1] proposed an objective function for hierarchical clustering and initiated a line of work developing algorithms that explicitly optimize an objective (see also [2, 3, 4]). In this paper, we consider a fairly general random graph model for hierarchical clustering, called the hierarchical stochastic blockmodel (HSBM), and show that in certain regimes the SVD approach of McSherry [5] combined with specific linkage methods results in a clustering that give an O(1)-approximation to Dasgupta's cost function. We also show that an approach based on SDP relaxations for balanced cuts based on the work of Makarychev et al. [6], combined with the recursive sparsest cut algorithm of Dasgupta, yields an O(1) approximation in slightly larger regimes and also in the semi-random setting, where an adversary may remove edges from the random graph generated according to an HSBM. Finally, we report empirical evaluation on synthetic and real-world data showing that our proposed SVD-based method does indeed achieve a better cost than other widely-used heurstics and also results in a better classification accuracy when the underlying problem was that of multi-class classification.

Affine policies (or control) are widely used as a solution approach in dynamic optimization where computing an optimal adjustable solution is usually intractable. While the worst case performance of affine policies can be significantly bad, the empirical performance is observed to be near-optimal for a large class of problem instances. For instance, in the two-stage dynamic robust optimization problem with linear covering constraints and uncertain right hand side, the worst-case approximation bound for affine policies is $O(\sqrt m)$ that is also tight (see Bertsimas and Goyal (2012)), whereas observed empirical performance is near-optimal. In this paper, we aim to address this stark-contrast between the worst-case and the empirical performance of affine policies. In particular, we show that affine policies give a good approximation for the two-stage adjustable robust optimization problem with high probability on random instances where the constraint coefficients are generated i.i.d.