An efficient algorithm for learning with semi-bandit feedback

We consider the problem of online combinatorial optimization under semi-bandit feedback. The goal of the learner is to sequentially select its actions from a combinatorial decision set so as to minimize its cumulative loss. We propose a learning algorithm for this problem based on combining the Follow-the-Perturbed-Leader (FPL) prediction method with a novel loss estimation procedure called Geometric Resampling (GR). Contrary to previous solutions, the resulting algorithm can be efficiently implemented for any decision set where efficient offline combinatorial optimization is possible at all. Assuming that the elements of the decision set can be described with d-dimensional binary vectors with at most m non-zero entries, we show that the expected regret of our algorithm after T rounds is O(m sqrt(dT log d)). As a side result, we also improve the best known regret bounds for FPL in the full information setting to O(m^(3/2) sqrt(T log d)), gaining a factor of sqrt(d/m) over previous bounds for this algorithm.Comment: submitted to ALT 201

Dacron vs Polytetrafluoroethylene Grafts for Femoropopliteal Bypass:a Prospective Randomised Multicentre Trial

AbstractObjectives: to compare the patency of PTFE (Polytetrafluoroethylene) and unsealed knitted Dacron femoro-popliteal bypasses. Design: multi-centre prospective randomised trial. Materials and methods: of 203 patients randomised, 194 were included in the final analysis (103 Dacron grafts and 91 PTFE grafts). The median follow-up was 36 months (range: 6–72 months); the distal anastomosis was above-knee in 141 and below-knee in 53 cases. Univariate comparisons of patency were made by the Kaplan–Meier method, multivariate calculations on the effects of covariables by a Cox regression analysis. Results: there was no difference regarding primary and secondary patency or limb salvage between Dacron and PTFE. The primary 3-year patency for Dacron grafts was 64% (95% confidence interval [C.I.] 55–74%) and for PTFE grafts 61% (C.I. 49–72%). The corresponding 3-year secondary patency was 81% (C.I. 73–89%) and 75% (C.I. 65–86%) respectively, the limb salvage rate 90% (C.I. 84–96%) and 91% (C.I. 84–97%). Upon multivariate analysis below-knee anastomosis was the principal independent predictor of primary graft failure (risk ratio 1.7 [C.I. 1.05–2.8]), impaired secondary patency was associated with infragenicular bypass (risk ratio 3.3 [C.I. 1.8–6.3]) and distal gangrene (risk ratio [C.I. 1.01–3.8]p=0.048), major amputation was independently predicted by below-knee bypass, tissue necrosis, and poor run-off index. Conclusions: PTFE and Dacron are equally suitable for femoro-popliteal bypass