We consider the problem of PAC-learning decision trees, i.e., learning a decision tree over the n-dimensional hypercube from independent random labeled examples. Despite significant effort, no polynomial-time algorithm is known for learning polynomial-sized decision trees (even trees of any super-constant size), even when examples are assumed to be drawn from the uniform distribution on 0,1 ^n. We give an algorithm that learns arbitrary polynomial-sized decision trees for most product distributions . In particular, consider a random product distribution where the bias of each bit is chosen independently and uniformly from, say, [.49,.51]. Then with high probability over the parameters of the product distribution and the random examples drawn from it, the algorithm will learn any tree. More generally, in the spirit of smoothed analysis, we consider an arbitrary product distribution whose parameters are specified only up to a [-c,c] accuracy (perturbation), for an arbitrarily small positive constant c.
We consider a smoothed learning'' model inspired by Smoothed Analysis, which Spielman and Teng introduced to explain why the simplex method for linear programming (LP) usually runs in polynomial time @cite_9 . Roughly speaking, they show that if each parameter of an LP is perturbed by a small amount, then the simplex method will run in polynomial time with high probability (in fact, the expected run-time will be polynomial). For LP's arising from nature or business (as opposed to reduction from another computational problem), the parameters are measurements or estimates that have some inherent inaccuracy or uncertainty. Hence, the model is reasonable for a large class of interesting LP's.
@cite_9
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We describe an end-to-end framework for learning parameters of min-cost flow multitarget tracking problem with quadratic trajectory interactions including suppression of overlapping tracks and contextual cues about co-occurrence of different objects. Our approach utilizes structured prediction with a tracking-specific loss function to learn the complete set of model parameters. Under our learning framework, we evaluate two different approaches to finding an optimal set of tracks under quadratic model objective based on an LP relaxation and a novel greedy extension to dynamic programming that handles pairwise interactions. We find the greedy algorithm achieves almost equivalent accuracy to the LP relaxation while being 2-7x faster than a commercial solver. We evaluate trained models on the challenging MOT and KITTI benchmarks. Surprisingly, we find that with proper parameter learning, our simple data-association model without explicit appearance motion reasoning is able to outperform many state-of-the-art methods that use far more complex motion features and affinity metric learning.