On the Random

Subset Sum Problem

and Neural Networks

Emanuele Natale jointly with A. Da Cunha & L. Viennot

23 March 2023

Deep Learning on the Edge

Turing test (1950)


1st AI winter (1974–1980)

2nd AI winter (1974–1980)

"A hand lifts a cup"

Use of GPUs in AI (2011)

Today, most AI heavy lifting is done in the cloud due to the concentration of large data sets and dedicated compute, especially when it comes to the training of machine learning (ML) models. But when it comes to the application of those models in real-world inferencing near the point where a decision is needed, a cloud-centric AI model struggles. [...] When time is of the essence, it makes sense to distribute the intelligence from the cloud to the edge.


1. ANN pruning

2. The Strong Lottery Ticket Hypothesis

3. SLTH for CNNs

4. Neuromorphic hardware

Dense ANNs

Feed-forward homogeneous dense ANN \(f\):

Application of \(\ell\) layers \(N_i\) with

\(N_i(x) = \sigma(W_i x)\),

ReLu activation \(\sigma(x)=\max(0,x)\)

and weight matrices \(W_i\), so that


Compressing ANN

Quantization techniques

Pruning techniques

Neural Network Pruning

Blalock et al. (2020): iterative magnitude pruning still SOTA pruning technique.






The Lottery Ticket Hypothesis

Frankle & Carbin (ICLR 2019):

Large random networks contains sub-networks that reach comparable accuracy when trained


sparse random network


bad network

..., train&prune

train&prune, ...,

large random network

sparse good network


sparse "ticket" network


good network



1. ANN pruning

2. The Strong Lottery Ticket Hypothesis

3. SLTH for CNNs

4. Neuromorphic hardware

The Strong LTH

Ramanujan et al. (CVPR 2020) find a good subnetwork without changing weights (train by pruning!)

A network with random weights contains sub-networks that can approximate any given sufficiently-smaller neural network (without training)

Formalizing the SLTH

Random network \(R_0\) with \(h\cdot d\) parameters

lottery ticket

\(N_{L}\subseteq N_0\)

Target network that solves task \(N_T\)

with \(d\) parameters


Proving the SLTH

Malach et al. (ICML 2020)


Find random weight

close to \(w\)

Idea: Find patterns in the random networks which are equivalent to sampling a weight until you are lucky.

Q: How many uniform\((-1,1)\) sample to approximate \(z\) up to \(\epsilon\)?

Malach et al.'s Idea

Suppose \(x\) and all \(w'_i\)s are positive, then

\[y=\sum_i w_i\sigma(w'_i x) = \sum_i w_i w'_i x \]

For general \(x\), use the ReLu trick \(x=\sigma(x)-\sigma(-x)\):

\[y= \sum_{i:w'_i\geq 0} w_i\sigma(w'_i x)+\sum_{i:w'_i<0} w_i\sigma(w'_i x) \]

\[= \sum_{i:w'_i\geq 0} w_i w'_i x {\mathbb 1}_{x\geq 0}+\sum_{i:w'_i<0} w_i w'_i x {\mathbb 1}_{x< 0}\]


Better Bound for SLTH

(assume \(x\) and \(w'_i\)s are positive)

\(y=  \sum_{i} w_i w'_i x \)

Pensia et al. (NeurIPS 2020)


Find combination of random weights close to \(w\)

alternative in Orseau et al. (Neurips 2020)


RSSP. Given \(X_1,...,X_n\) i.i.d. random variables, with prob. \(1-\epsilon\) for each \(z\in [-1,1]\) find a subset \(S\subseteq\{1,...,n\}\) such that \(|z-\sum_{i\in S} X_i |\leq \epsilon.\)

Lueker '98. Solution exists with prob. \(1-\epsilon\) if \(n=O(\log \frac 1{\epsilon})\).

RSS - Proof Idea 1/2

If \(n=O(\log \frac 1{\epsilon})\), given \(X_1,...,X_n\) i.i.d. random variables, with prob. \(1-\epsilon\) for each \(z\in [-\frac 12, \frac 12]\) there is \(S\subseteq\{1,...,n\}\) such that \(|z-\sum_{i\in S} X_i |\leq \epsilon.\)

Let \(f_t(z)=\mathbf 1(z\in (-\frac 12, \frac 12),\exists S\subseteq\{1,...,t\}: |z-\sum_{i\in S} X_i |\leq \epsilon)\)

then \(f_t(z)=f_{t-1}(z)+(1-f_{t-1}(z))f_{t-1}(z-X_t)\).

Observation: If we can approximate any \(z\in (a,b)\) and we add \(X'\) to the sample, then we can approximate any

\(z\in (a,b) \cup (a+X',b+X')\).

RSS - Proof Idea 2/2

\(z\in(-\frac 12, \frac 12), f_t(z)=f_{t-1}(z)+(1-f_{t-1}(z))f_{t-1}(z-X_t)\).

\(\int_{-\frac 12}^{\frac 12}f_{t-1}(z)dz+\mathbb E[\int_{-\frac 12}^{\frac 12}(1-f_{t-1}(z))f_{t-1}(z-X_t)dz|\,X_{t-1},...,X_1]\)

\(=v_{t-1}+\frac 12 (1-v_{t-1})v_{t-1}.\)

\(=v_{t-1}+\frac 12 \int_{-1}^{1}[\int_{-\frac 12}^{\frac 12}(1-f_{t-1}(z))f_{t-1}(z-x)dz]dx\)

\(=v_{t-1}+\frac 12 \int_{-\frac 12}^{\frac 12}(1-f_{t-1}(z))[\int_{-1}^{1}f_{t-1}(z-x)dx]dz\)

\(=v_{t-1}+\frac 12 \int_{-\frac 12}^{\frac 12}(1-f_{t-1}(z))[\int_{z-1}^{z+1}f_{t-1}(s)ds]dz\)

\(=v_{t-1}+\frac 12 \int_{-\frac 12}^{\frac 12}(1-f_{t-1}(z))[\int_{-\frac 12}^{\frac 12}f_{t-1}(s)ds]dz\)

\(\mathbb E[v_t\,|\,X_{t-1},...,X_1]=\)

Let \(v_t=\int_{-\frac 12}^{\frac 12}f_t(z)dz\), then 

"Revisiting the Random Subset Sum problem" https://hal.science/hal-03654720/

s = z-x


1. ANN pruning

2. The Strong Lottery Ticket Hypothesis

3. SLTH for CNNs

4. Neuromorphic hardware

Convolutional Neural Network

The convolution of \(K\in\reals^{d\times d\times c}\) and \(X\in\reals^{D\times D\times c}\) is \( \left(K * X\right)_{i,j\in\left[D\right]}=\)\[\sum_{i',j'\in\left[d\right],k\in\left[c\right]}K_{i',j',k}\cdot X_{i-i'+1,j-j'+1,k}, \]

where \(X\) is zero-padded.

A simple CNN \(N:\left[0,1\right]^{D\times D\times c_{0}}\rightarrow\mathbb{R}^{D\times D\times c_{\ell}}\) is defined as

\[ N\left(X\right)= \sigma\left( K^{(\ell)}*\sigma\left(K^{(\ell-1)}*\sigma\left(\cdots * \sigma\left(K^{(1)} * X\right)\right)\right)\right)\]

where \(K^{(i)} \in\mathbb R^{d_{i}\times d_{i}\times c_{i-1}\times c_{i}}\).

2D Discrete Convolution

If \(K\in\reals^{d\times d\times c_{0}\times c_{1}}\) and \(X\in\reals^{D\times D\times c_{0}}\)

\[ \left(K * X\right)_{i,j\in\left[D\right],\ell\in\left[c_{1}\right]}=\sum_{i',j'\in\left[d\right],k\in\left[c_{0}\right]}K_{i',j',k,\ell}\cdot X_{i-i'+1,j-j'+1,k}.\]

SLTH for Convolutional Neural Networks

Theorem (da Cunha et al., ICLR 2022).

Given \(\epsilon,\delta>0\), any CNN with \(k\) parameters and \(\ell\) layers, and kernels with \(\ell_1\) norm at most 1, can be approximated within error \(\epsilon\) by pruning a random CNN with \(O\bigl(k\log \frac{k\ell}{\min\{\epsilon,\delta\}}\bigr)\) parameters and \(2\ell\) layers with probability at least \(1-\delta\).

Proof Idea 1/2

For any \(K\in [-1,1]^{d\times d\times c\times1}\) with \(\|K\|_{1}\leq1\) and \(X\in [0,1]^{D\times D\times c}\) we want to approximate \(K*X\) with \(V*\sigma(U*X)\) where \(U\) and \(V\) are tensors with i.i.d. \(\text{Uniform}(-1,1)\) entries.

Let \(U\) be \(d\times d \times c\times n\) and \(V\) be \(1\times 1 \times n \times 1\).

Proof Idea 2/2

\left(V*\left(U* X\right)\right)_{r,s,1} =\sum_{t=1}^{n}V_{1,1,t,1}\cdot\left(U* X\right)_{r,s,t}\\ =\sum_{t=1}^{n}V_{1,1,t,1}\cdot\left(\sum_{i,j\in\left[d\right],k\in\left[c\right]}U_{i,j,k,t}\cdot X_{r-i+1,s-j+1,k}\right)_{r,s,t}\\ =\sum_{t=1}^{n}\sum_{i,j\in\left[d\right],k\in\left[c\right]}\left(V_{1,1,t,1}\cdot U_{i,j,k,t}\right)\cdot X_{r-i+1,s-j+1,k}\\ =\sum_{i,j\in\left[d\right],k\in\left[c\right]}\left(\sum_{t=1}^{n}V_{1,1,t,1}\cdot U_{i,j,k,t}\right)\cdot X_{r-i+1,s-j+1,k}\\ =\sum_{i,j\in\left[d\right],k\in\left[c\right]}L_{i,j,k,1}\cdot X_{r-i+1,s-j+1,k}

where \(L_{i,j,k,1}=\sum_{t=1}^{n}V_{1,1,t,1}\cdot U_{i,j,k,t}\)

Prune negative entries of \(U\) so that \(\sigma(U*X)=U*X\).


1. ANN pruning

2. The Strong Lottery Ticket Hypothesis

3. SLTH for CNNs

4. Neuromorphic hardware

Reducing Energy: the Hardware Way

Letting Physics Do The Math

Ohm's law


To multiply \(w\) and \(x\), set \(V_{in}=x\) and \(R=\frac 1w\), then \(I_{out}=wx\).

Resistive Crossbar Device

Analog MVM via crossbars of programmable resistances

Problem: Making precise programmable resistances is hard

Cfr. ~10k flops for digital 100x100 MVM

"Résistance équivalente modulable

à partir de résistances imprécises"

INRIA Patent deposit FR2210217

Leverage noise itself

to increase precision



effective resistance

RSS in Practice

bits of precision  for any target value are linear w.r.t. number of resistances

Worst case among

2.5k instances


RSS theory provides a fundamental tool for understanding the relation between noise and sparsification.

Open problem. SLTH for neural pruning, i.e. removing entire neurons in dense ANNs.

Candidate tool: Multidimensional RSS

See e.g. hal-03738204v2

(WIP for channel pruning in CNNs)

Thank you

GSSI 2023

By Emanuele Natale