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The minimax strategy for gaussian density estimation
 In COLT
, 2000
"... We consider online density estimation with a Gaussian of unit variance. In each trial t the learner predicts a mean θt. Then it receives an instance xt chosen by the adversary and incurs loss 1 2 (θt − xt) 2. The performance of the learner is measured by the regret de£ned as the total loss of the l ..."
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Cited by 10 (2 self)
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is an upper bound of the 2norm of instances. We also consider the standard algorithm that predicts with θt = ∑ t−1 q=1 xq/(t − 1 + a) for a £xed a. We show that the regret of this algorithm is 1 2 X2 (ln T − O(1)) regardless of the choice of a. This work was done while Eiji Takimoto was on a sabbatical
Random Projection and Its Application to Learning
"... Random projection is a technique of mapping a number of points in a highdimensional space into a low dimensional space with the property that the Euclidean distance of any two points is approximately preserved through the projection. A random projection from Rn to Rk is typically defined as a rand ..."
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Random projection is a technique of mapping a number of points in a highdimensional space into a low dimensional space with the property that the Euclidean distance of any two points is approximately preserved through the projection. A random projection from Rn to Rk is typically defined as a random n×k matrix R = (rij) with each entry chosen randomly and independently according to some probability distribution: an ndimensional vector u ∈ Rn is projected to u ′ = 1√ k
Online Rank Aggregation
"... We consider an online learning framework where the task is to predict a permutation which represents a ranking of n fixed objects. At each trial, the learner incurs a loss defined as Kendall tau distance between the predicted permutation and the true permutation given by the adversary. This setting ..."
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Cited by 2 (0 self)
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We consider an online learning framework where the task is to predict a permutation which represents a ranking of n fixed objects. At each trial, the learner incurs a loss defined as Kendall tau distance between the predicted permutation and the true permutation given by the adversary. This setting is quite natural in many situations such as information retrieval and recommendation tasks. We prove a lower bound of the cumulative loss and hardness results. Then, we propose an algorithm for this problem and prove its relative loss bound which shows our algorithm is close to optimal.
Smooth Boosting for MarginBased Ranking
"... Abstract. We propose a new boosting algorithm for bipartite ranking problems. Our boosting algorithm, called SoftRankBoost, is a modification of RankBoost which maintains only smooth distributions over data. SoftRankBoost provably achieves approximately the maximum soft margin over all pairs of po ..."
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Abstract. We propose a new boosting algorithm for bipartite ranking problems. Our boosting algorithm, called SoftRankBoost, is a modification of RankBoost which maintains only smooth distributions over data. SoftRankBoost provably achieves approximately the maximum soft margin over all pairs of positive and negative examples, which implies high AUC score for future data. 1
Improved Bounds for Online Learning Over the Permutahedron and Other Ranking Polytopes
"... Consider the following game: There is a fixed set V of n items. At each step an adversary chooses a score function st: V 7 → [0, 1], a learner outputs a ranking of V, and then st is revealed. The learner’s loss is the sum over v ∈ V, of st(v) times v’s position (0th, 1st, 2nd,...) in the ranking. ..."
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Cited by 2 (1 self)
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Consider the following game: There is a fixed set V of n items. At each step an adversary chooses a score function st: V 7 → [0, 1], a learner outputs a ranking of V, and then st is revealed. The learner’s loss is the sum over v ∈ V, of st(v) times v’s position (0th, 1st, 2nd,...) in the ranking. This problem captures, for example, online systems that iteratively present ranked lists of items to users, who then respond by choosing one (or more) sought items. The loss measures the users ’ burden, which increases the further the sought items are from the top. It also captures a version of online rank aggregation. We present an algorithm of expected regret O(n OPT + n2), where OPT is the loss of the best (single) ranking in hindsight. This improves the previously best known algorithm of Suehiro et. al (2012) by saving a factor of Ω( log n). We also reduce the perstep running time fromO(n2) toO(n log n). We provide matching lower bounds. 1
LETTER Communicated by Pekka Orponen On theComputational Power ofThresholdCircuitswithSparse Activity
"... Circuits composed of threshold gates (McCullochPitts neurons, or perceptrons) are simplifiedmodels of neural circuits with the advantage that they are theoretically more tractable than their biological counterparts. However, when such threshold circuits are designed to perform a specific computati ..."
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Circuits composed of threshold gates (McCullochPitts neurons, or perceptrons) are simplifiedmodels of neural circuits with the advantage that they are theoretically more tractable than their biological counterparts. However, when such threshold circuits are designed to perform a specific computational task, they usually differ in one important respect from computations in the brain: they require very high activity. On average every second threshold gate fires (sets a 1 as output) during a computation. By contrast, the activity of neurons in the brain is much sparser, with only about 1 % of neurons firing. This mismatch between threshold and neuronal circuits is due to the particular complexitymeasures (circuit size and circuit depth) that have beenminimized in previous threshold circuit constructions. In this letter, we investigate a new complexity measure for threshold circuits, energy complexity,whoseminimization yields computations with sparse activity. We prove that all computations by threshold circuits of polynomial size with entropy O(log n) can be restructured so
Results 1  10
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