We consider online learning of ensembles of portfolio selection algorithms and aim to regularize risk by encouraging diversification with r espect to a predefined risk-driven grouping of stocks. Our procedure uses online convex optimization to control capital allocation to underlying investment alg orithms while encouraging non-sparsity over the given grouping. We prove a logarithmic regret for this procedure with respect to the best-in-hindsight ensemble. We applied the procedure with known mean-reversion portfolio selection algorithms using the standard GICS industry sector grouping. Empirical Experimental results showed an impressive percentage increase of risk-adjusted return (Sharpe r atio).
Although our procedure relies on the same @math group norm to encourage diversification, our algorithm differs from that of Johnson and Banerjee in two ways: first, rather than generating portfolios on the stock themselves, we generate a weighted ensemble over investment algorithms. Second, our learning algorithm exploits the exp-concavity of our loss function, allowing the use of online Newton steps as in @cite_16 to guarantee @math regret w.r.t. the best fixed ensemble in hindsight. A direct application of our procedure over the stocks themselves yields exponential improvement in regret relative to the result of JohnsonB2015 . Also worth mentioning is that our implementation utilizes a fixed grouping of the stocks given by the standard GICS industry taxonomy, whereas JohnsonB2015 employ a correlation-based heuristic to group the stocks on the fly. Our numerical examples in Section 3 include a direct comparison with the method of JohnsonB2015 , showing an overwhelming advantage to our method (see, e.g., Figure ).
@cite_16
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We present an algorithm for the statistical learning setting with a bounded exp-concave loss in @math dimensions that obtains excess risk @math with probability at least @math . The core technique is to boost the confidence of recent in-expectation @math excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability @math the standard ERM method obtains excess risk @math . We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate. We study resource-limited online learning, motivated by the problem of conditional-branch outcome prediction in computer architecture. In particular, we consider (parallel) time and space-efficient ensemble learners for online settings, empirically demonstrating benefits similar to those shown previously for offline ensembles. Our learning algorithms are inspired by the previously published “boosting by filtering” framework as well as the offline Arc-x4 boosting-style algorithm. We train ensembles of online decision trees using a novel variant of the ID4 online decision-tree algorithm as the base learner, and show empirical results for both boosting and bagging-style online ensemble methods. Our results evaluate these methods on both our branch prediction domain and online variants of three familiar machine-learning benchmarks. Our data justifies three key claims. First, we show empirically that our extensions to ID4 significantly improve performance for single trees and additionally are critical to achieving performance gains in tree ensembles. Second, our results indicate significant improvements in predictive accuracy with ensemble size for the boosting-style algorithm. The bagging algorithms we tried showed poor performance relative to the boosting-style algorithm (but still improve upon individual base learners). Third, we show that ensembles of small trees are often able to outperform large single trees with the same number of nodes (and similarly outperform smaller ensembles of larger trees that use the same total number of nodes). This makes online boosting particularly useful in domains such as branch prediction with tight space restrictions (i.e., the available real-estate on a microprocessor chip). The probability that a user will click a search result depends both on its relevance and its position on the results page. The position based model explains this behavior by ascribing to every item an attraction probability, and to every position an examination probability. To be clicked, a result must be both attractive and examined. The probabilities of an item-position pair being clicked thus form the entries of a rank- @math matrix. We propose the learning problem of a Bernoulli rank- @math bandit where at each step, the learning agent chooses a pair of row and column arms, and receives the product of their Bernoulli-distributed values as a reward. This is a special case of the stochastic rank- @math bandit problem considered in recent work that proposed an elimination based algorithm Rank1Elim, and showed that Rank1Elim's regret scales linearly with the number of rows and columns on "benign" instances. These are the instances where the minimum of the average row and column rewards @math is bounded away from zero. The issue with Rank1Elim is that it fails to be competitive with straightforward bandit strategies as @math . In this paper we propose Rank1ElimKL which simply replaces the (crude) confidence intervals of Rank1Elim with confidence intervals based on Kullback-Leibler (KL) divergences, and with the help of a novel result concerning the scaling of KL divergences we prove that with this change, our algorithm will be competitive no matter the value of @math . Experiments with synthetic data confirm that on benign instances the performance of Rank1ElimKL is significantly better than that of even Rank1Elim, while experiments with models derived from real data confirm that the improvements are significant across the board, regardless of whether the data is benign or not. We consider the task of learning a classifier for semantic segmentation using weak supervision in the form of image labels specifying objects present in the image. Our method uses deep convolutional neural networks (cnns) and adopts an Expectation-Maximization (EM) based approach. We focus on the following three aspects of EM: (i) initialization; (ii) latent posterior estimation (E-step) and (iii) the parameter update (M-step). We show that saliency and attention maps, bottom-up and top-down cues respectively, of images with single objects (simple images) provide highly reliable cues to learn an initialization for the EM. Intuitively, given weak supervisions, we first learn to segment simple images and then move towards the complex ones. Next, for updating the parameters (M step), we propose to minimize the combination of the standard softmax loss and the KL divergence between the latent posterior distribution (obtained using the E-step) and the likelihood given by the cnn. This combination is more robust to wrong predictions made by the E step of the EM algorithm. Extensive experiments and discussions show that our method is very simple and intuitive, and outperforms the state-of-the-art method with a very high margin of 3.7 and 3.9 on the PASCAL VOC12 train and test sets respectively, thus setting new state-of-the-art results.