We exemplify our result in two tasks of interest in statistical learning: a) classification for a mixture with sparse means, wherewestudytheefficiencyof `1penaltywithrespectto `2;b)max-marginmulticlass classification, where we characterise the phase transition on the existence ofthemulti-class logistic maximum likelihood estimator forK >2.

However, existing quantization methods usually have inferior performancewhenusedonLSTMs.

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Let k be the collection of distributions over the alphabet[k]:= {1,2,...,k}, and [k] be the set of finite sequences over[k].

Rd d is coefficient;s t: Rd!

See Theorem 2 in Section 3.1 formoredetail.

Specifically, we extend the Catalyst approach originally designed fordeterministic objectives tothe stochastic setting.

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See Appendix Aforanoteontrain/testsplitfor Task 3. loss Testloss Energy Baseline HNNBaseline HNNBaseline HNN mass-spring170 20.38 .1 pendulum 42 10 25 5 pendulum 390 7 14 5 (6.3e4 3e4 39 5 pendulum.3

Deep Compression: Compressing Deep Neural Networkswith Trained Quantizationand Huffman Coding.