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Register a new userExponential Weights on the Hypercube in Polynomial Time
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We study a general online linear optimization problem(OLO). At each round, a subset of objects from a fixed universe of $n$ objects is chosen, and a linear cost associated with the chosen subset is incurred. To measure the performance of our algorithms, we use the notion of regret which is the difference between the total cost incurred over all iterations and the cost of the best fixed subset in hindsight. We consider Full Information and Bandit feedback for this problem. This problem is equivalent to OLO on the ${0,1}^n$ hypercube. The Exp2 algorithm and its bandit variant are commonly used strategies for this problem. It was previously unknown if it is possible to run Exp2 on the hypercube in polynomial time. In this paper, we present a polynomial time algorithm called PolyExp for OLO on the hypercube. We show that our algorithm is equivalent Exp2 on ${0,1}^n$, Online Mirror Descent(OMD), Follow The Regularized Leader(FTRL) and Follow The Perturbed Leader(FTPL) algorithms. We show PolyExp achieves expected regret bound that is a factor of $sqrt{n}$ better than Exp2 in the full information setting under $L_infty$ adversarial losses. Because of the equivalence of these algorithms, this implies an improvement on Exp2s regret bound in full information. We also show matching regret lower bounds. Finally, we show how to use PolyExp on the ${-1,+1}^n$ hypercube, solving an open problem in Bubeck et al (COLT 2012).
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A standard introduction to online learning might place Online Gradient Descent at its center and then proceed to develop generalizations and extensions like Online Mirror Descent and second-order methods. Here we explore the alternative approach of putting Exponential Weights (EW) first. We show that many standard methods and their regret bounds then follow as a special case by plugging in suitable surrogate losses and playing the EW posterior mean. For instance, we easily recover Online Gradient Descent by using EW with a Gaussian prior on linearized losses, and, more generally, all instances of Online Mirror Descent based on regular Bregman divergences also correspond to EW with a prior that depends on the mirror map. Furthermore, appropriate quadratic surrogate losses naturally give rise to Online Gradient Descent for strongly convex losses and to Online Newton Step. We further interpret several recent adaptive methods (iProd, Squint, and a variation of Coin Betting for experts) as a series of closely related reductions to exp-concave surrogate losses that are then handled by Exponential Weights. Finally, a benefit of our EW interpretation is that it opens up the possibility of sampling from the EW posterior distribution instead of playing the mean. As already observed by Bubeck and Eldan, this recovers the best-known rate in Online Bandit Linear Optimization.
We study random walks on the dilute hypercube using an exact enumeration Master equation technique, which is much more efficient than Monte Carlo methods for this problem. For each dilution $p$ the form of the relaxation of the memory function $q(t)$ can be accurately parametrized by a stretched exponential $q(t)=exp(-(t/tau)^beta)$ over several orders of magnitude in $q(t)$. As the critical dilution for percolation $p_c$ is approached, the time constant $tau(p)$ tends to diverge and the stretching exponent $beta(p)$ drops towards 1/3. As the same pattern of relaxation is observed in wide class of glass formers, the fractal like morphology of the giant cluster in the dilute hypercube is a good representation of the coarse grained phase space in these systems. For these glass formers the glass transition can be pictured as a percolation transition in phase space.
Modeling interactions between features improves the performance of machine learning solutions in many domains (e.g. recommender systems or sentiment analysis). In this paper, we introduce Exponential Machines (ExM), a predictor that models all interactions of every order. The key idea is to represent an exponentially large tensor of parameters in a factorized format called Tensor Train (TT). The Tensor Train format regularizes the model and lets you control the number of underlying parameters. To train the model, we develop a stochastic Riemannian optimization procedure, which allows us to fit tensors with 2^160 entries. We show that the model achieves state-of-the-art performance on synthetic data with high-order interactions and that it works on par with high-order factorization machines on a recommender system dataset MovieLens 100K.
Optimizing the reliability and the robustness of a design is important but often unaffordable due to high sample requirements. Surrogate models based on statistical and machine learning methods are used to increase the sample efficiency. However, for higher dimensional or multi-modal systems, surrogate models may also require a large amount of samples to achieve good results. We propose a sequential sampling strategy for the surrogate based solution of multi-objective reliability based robust design optimization problems. Proposed local Latin hypercube refinement (LoLHR) strategy is model-agnostic and can be combined with any surrogate model because there is no free lunch but possibly a budget one. The proposed method is compared to stationary sampling as well as other proposed strategies from the literature. Gaussian process and support vector regression are both used as surrogate models. Empirical evidence is presented, showing that LoLHR achieves on average better results compared to other surrogate based strategies on the tested examples.
We show experimentally that the accuracy of a trained neural network can be predicted surprisingly well by looking only at its weights, without evaluating it on input data. We motivate this task and introduce a formal setting for it. Even when using simple statistics of the weights, the predictors are able to rank neural networks by their performance with very high accuracy (R2 score more than 0.98). Furthermore, the predictors are able to rank networks trained on different, unobserved datasets and with different architectures. We release a collection of 120k convolutional neural networks trained on four different datasets to encourage further research in this area, with the goal of understanding network training and performance better.
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