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Small ReLU networks are powerful memorizers: a tight analysis of memorization capacity
Chulhee Yun · Suvrit Sra · Ali Jadbabaie

Wed Dec 11 04:20 PM -- 04:25 PM (PST) @ West Exhibition Hall C + B3
in Track 1 Session 4 »
We study finite sample expressivity, i.e., memorization power of ReLU networks. Recent results require $N$ hidden nodes to memorize/interpolate arbitrary $N$ data points. In contrast, by exploiting depth, we show that 3-layer ReLU networks with $\Omega(\sqrt{N})$ hidden nodes can perfectly memorize most datasets with $N$ points. We also prove that width $\Theta(\sqrt{N})$ is necessary and sufficient for memorizing $N$ data points, proving tight bounds on memorization capacity. The sufficiency result can be extended to deeper networks; we show that an $L$-layer network with $W$ parameters in the hidden layers can memorize $N$ data points if $W = \Omega(N)$. Combined with a recent upper bound $O(WL\log W)$ on VC dimension, our construction is nearly tight for any fixed $L$. Subsequently, we analyze memorization capacity of residual networks under a general position assumption; we prove results that substantially reduce the known requirement of $N$ hidden nodes. Finally, we study the dynamics of stochastic gradient descent (SGD), and show that when initialized near a memorizing global minimum of the empirical risk, SGD quickly finds a nearby point with much smaller empirical risk.
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Author Information

Chulhee Yun (MIT)
Suvrit Sra (MIT)

Suvrit Sra is a faculty member within the EECS department at MIT, where he is also a core faculty member of IDSS, LIDS, MIT-ML Group, as well as the statistics and data science center. His research spans topics in optimization, matrix theory, differential geometry, and probability theory, which he connects with machine learning --- a key focus of his research is on the theme "Optimization for Machine Learning” (http://opt-ml.org)

Ali Jadbabaie (MIT)

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