Many high-dimensional statistical inference problems are believed to possess inherent computational hardness. Various frameworks have been proposed to give rigorous evidence for such hardness, including lower bounds against restricted models of computation (such as low-degree functions), as well as methods rooted in statistical physics that are based on free energy landscapes. This paper aims to make a rigorousconnectionbetween the seeminglydifferent low-degreeand free-energybased approaches. We define a free-energybasedcriterionfor hardnessand formallyconnectit to the well-establishednotionof low-degree hardness for a broad class of statistical problems, namely all Gaussian additive models and certain models with a sparse planted signal.

Wefocusonsixmethods:(i)discriminative K-means (DisKmeans) in Ye et al. (2008); (ii) a discriminative clustering formulation described inBach andHarchaoui (2008); Flammarion etal.(2017); We compare two classesF of feature mappings: linear functions and fully-connected neural networks with one hidden layer that has 100 nodes. An epoch refers ton/B = 12 consecutive iterations. The learning curves in Figure 1 shows the advantage of neural network and demonstrates the flexibility of CURE with nonlinear function classes. One of the main obstacles is the complicated piecewise definition off, which prevent us from obtaining closed form formulae.

To model the demand as a function of price and context, the existing literature either adopts a parametric model or a non-parametric model.

Such a question is very relevant: boostinghas quickly evolvedasatechnique requiring first-order information about the loss optimized [6, Section 10.3], [41, Section 7.2.2]

Gaussian entries, thegoal isto recover thek-sparse unit vectorx Rn. The structure we enforce on x is sparsity: |supp(x)| k.

Gaussian entries, thegoal isto recover thek-sparse unit vectorx Rn. The structure we enforce on x is sparsity: |supp(x)| k.

These algorithms inject (carefully tuned) random noise to value function to encourage exploration. UCB-type algorithms enjoy well-established theoretical guarantees but suffer from difficult implementation since an upper confidence bound isusually infeasible for manypractical models like neural networks. Instead, practitioners prefer randomized exploration such as noisy networks in [19], and algorithms with randomized exploration have been widely used in practice [37,13,11,35].

These algorithms inject (carefully tuned) random noise to value function to encourage exploration. UCB-type algorithms enjoy well-established theoretical guarantees but suffer from difficult implementation since an upper confidence bound isusually infeasible for manypractical models like neural networks. Instead, practitioners prefer randomized exploration such as noisy networks in [19], and algorithms with randomized exploration have been widely used in practice [37,13,11,35].

Forexample,30) and 32) do Therefore, attention 8].

Let and bethepushforwardofPunderX andY respectively, i.e.,X andY .