# Maximum Likelihood Computation using Homomorphic Computation

#### Balasubramanian Narasimhan

#### 2025-04-08

## Introduction

We present a toy example of a homomorphic computation involving maximum likelihood estimation.

Consider the following data motivated by an example from the `mle` function in the `stats4` R package: we wish to estimate \(\lambda\), the Poisson parameter \(\lambda\) for randomly generated count data `y` below:

    library(stats4)
    set.seed(17822)
    y <- rpois(n = 40, lambda=10)
    # Easy one-dimensional MLE:
    nLL <- function(lambda) -sum(stats::dpois(y, lambda, log = TRUE))
    fit0 <- mle(nLL, start = list(lambda = 5), nobs = NROW(y))

The function `nLL` is the negative log-likelihood of the data and the function `mle` computes the maximum likelihood estimate that can be printed out.

    summary(fit0)

    ## Maximum likelihood estimation
    ##
    ## Call:
    ## mle(minuslogl = nLL, start = list(lambda = 5), nobs = NROW(y))
    ##
    ## Coefficients:
    ##        Estimate Std. Error
    ## lambda    9.175  0.4789311
    ##
    ## -2 log L: 199.5328

    logLik(fit0)

    ## 'log Lik.' -99.76641 (df=1)

## Distributed Computation

Assume now that the data `y` is distributed between three sites, none of whom want to share actual data among each other or even with a master computation process. They wish to keep their data secret but are willing, together, to provide the sum of their local negative log-likelihoods. They need to do this in a way so that the master process will not be able to associate the contribution to the likelihood from each site. To simulate this, let’s partition the data `y` as follows.

    y1 <- y[1:20]
    y2 <- y[21:27]
    y3 <- y[28:40]

The overall likelihood function \(l(\lambda)\) for the entire data is therefore the sum of the likelihoods at each site: \(l(\lambda) = l_1(\lambda)+l_2(\lambda)+l_3(\lambda).\) How can this likelihood be computed while maintaining privacy?

Assuming that every site including the master has access to a homomorphic computation library such as `homomorpheR`, the likelihood can be computed in a privacy-preserving manner using the following scheme. We use \(E(x)\) and \(D(x)\) to denote the encrypted and decrypted values of \(x\) respectively.

0. Master generates a public/private key pair. Master distributes the public key to all sites. (The private key is not distributed and kept only by the master.)
1. Master generates a random offset \(r\) to obfuscate the intial likelihood.
2. Master sends \(E(r)\) and a guess \(\lambda_0\) to site 1. Note that \(\lambda\) is not encrypted.
3. Site 1 computes \(l_1 = l(\lambda_0, y_1)\), the local likelihood for local data \(y_1\) using parameter \(\lambda_0\). It then sends on \(\lambda_0\) and \(E(r) + E(l_1)\) to site 2.
4. Site 2 computes \(l_2 = l(\lambda_0, y_2)\), the local likelihood for local data \(y_2\) using parameter \(\lambda_0\). It then sends on \(\lambda_0\) and \(E(r) + E(l_1) + E(l_2)\) to site 3.
5. Site 3 computes \(l_3 = l(\lambda_0, y_3)\), the local likelihood for local data \(y_3\) using parameter \(\lambda_0\). It then sends on \(E(r) + E(l_1) + E(l_2) + E(l_3)\) back to master.
6. Master retrieves \(E(r) + E(l_1) + E(l_2) + E(l_3)\) which, due to the homomorphism, is exactly \(E(r+l_1+l_2+l_3) = E(r+l).\) So the master computes \(D(E(r+l)) - r\) to obtain the value of the overall likelihood at \(\lambda_0\).
7. Master updates \(\lambda_0\) with a new guess \(\lambda_1\) and repeats steps 1-5. This process is iterated to convergence. For added security, even steps 0-5 can be repeated, at additional computational cost.

This is pictorially shown below.

## Implementation

The above implementation assumes that the encryption and decryption can happen with real numbers which is not the actual situation. Instead, we use rational approximations using a large denominator, \(2^{256}\), say. In the future, of course, we need to build an actual library is built with rigorous algorithms guaranteeing precision and overflow/undeflow detection. For now, this is just an ad hoc implementation.

Also, since we are only using homomorphic additive properties, a partial homomorphic scheme such as the Paillier Encryption system will be sufficient for our computations.

We define a class to encapsulate our sites that will compute the Poisson likelihood on site data given a parameter \(\lambda\). Note how the `addNLLAndForward` method takes care to split the result into an integer and fractional part while performing the arithmetic operations. (The latter is approximated by a rational number.)

    library(gmp)
    library(homomorpheR)
    Site <- R6::R6Class("Site",
                        private = list(
                            ## name of the site
                            name = NA,
                            ## only master has this, NA for workers
                            privkey = NA,
                            ## local data
                            data = NA,
                            ## The next site in the communication: NA for master
                            nextSite = NA,
                            ## is this the master site?
                            iAmMaster = FALSE,
                            ## intermediate result variable
                            intermediateResult = NA
                        ),
                        public = list(
                            ## Common denominator for approximate real arithmetic
                            den = NA,
                            ## The public key; everyone has this
                            pubkey = NA,
                            initialize = function(name, data, den) {
                                private$name <- name
                                private$data <- data
                                self$den <- den
                            },
                            setPublicKey = function(pubkey) {
                                self$pubkey <- pubkey
                            },
                            setPrivateKey = function(privkey) {
                                private$privkey <- privkey
                            },
                            ## Make me master
                            makeMeMaster = function() {
                                private$iAmMaster <- TRUE
                            },
                            ## add neg log lik and forward to next site
                            addNLLAndForward = function(lambda, enc.offset) {
                                if (private$iAmMaster) {
                                    ## We are master, so don't forward
                                    ## Just store intermediate result and return
                                    private$intermediateResult <- enc.offset
                                } else {
                                    ## We are workers, so add and forward
                                    ## add negative log likelihood and forward result to next site
                                    ## Note that offset is encrypted
                                    nllValue <- self$nLL(lambda)
                                    result.int <- floor(nllValue)
                                    result.frac <- nllValue - result.int
                                    result.fracnum <- as.bigq(numerator(as.bigq(result.frac) * self$den))
                                    pubkey <- self$pubkey
                                    enc.result.int <- pubkey$encrypt(result.int)
                                    enc.result.fracnum <- pubkey$encrypt(result.fracnum)
                                    result <- list(int = pubkey$add(enc.result.int, enc.offset$int),
                                                   frac = pubkey$add(enc.result.fracnum, enc.offset$frac))
                                    private$nextSite$addNLLAndForward(lambda, enc.offset = result)
                                }
                                ## Return a TRUE result for now.
                                TRUE
                            },
                            ## Set the next site in the communication graph
                            setNextSite = function(nextSite) {
                                private$nextSite <- nextSite
                            },
                            ## The negative log likelihood
                            nLL = function(lambda) {
                                if (private$iAmMaster) {
                                    ## We're master, so need to get result from sites
                                    ## 1. Generate a random offset and encrypt it
                                    pubkey <- self$pubkey
                                    offset <- list(int = random.bigz(nBits = 256),
                                                   frac = random.bigz(nBits = 256))
                                    enc.offset <- list(int = pubkey$encrypt(offset$int),
                                                       frac = pubkey$encrypt(offset$frac))
                                    ## 2. Send off to next site
                                    throwaway <- private$nextSite$addNLLAndForward(lambda, enc.offset)
                                    ## 3. When the call returns, the result will be in
                                    ##    the field intermediateResult, so decrypt that.
                                    sum <- private$intermediateResult
                                    privkey <- private$privkey
                                    intResult <- as.double(privkey$decrypt(sum$int) - offset$int)
                                    fracResult <- as.double(as.bigq(privkey$decrypt(sum$frac) - offset$frac) / den)
                                    intResult + fracResult
                                } else {
                                    ## We're worker, so compute local nLL
                                    -sum(stats::dpois(private$data, lambda, log = TRUE))
                                }
                            })
                        )

We are now ready to use our sites in the computation.

### 1. Generate public and private key pair

We also choose a denominator for all our rational approximations.

    keys <- PaillierKeyPair$new(1024) ## Generate new public and private key.
    den <- gmp::as.bigq(2)^256  #Our denominator for rational approximations

### 2. Create sites

    site1 <- Site$new(name = "Site 1", data = y1, den = den)
    site2 <- Site$new(name = "Site 2", data = y2, den = den)
    site3 <- Site$new(name = "Site 3", data = y3, den = den)

The master process is also a site but has no data. So has to be thus designated.

    ## Master has no data!
    master <- Site$new(name = "Master", data = c(), den = den)
    master$makeMeMaster()

### 2. Distribute public key to sites

    site1$setPublicKey(keys$pubkey)
    site2$setPublicKey(keys$pubkey)
    site3$setPublicKey(keys$pubkey)
    master$setPublicKey(keys$pubkey)

Only master has private key for decryption.

    master$setPrivateKey(keys$getPrivateKey())

### 3. Define the communication graph

Master will always send to the first site, and then the others have to forward results in turn with the last site returning to the master.

    master$setNextSite(site1)
    site1$setNextSite(site2)
    site2$setNextSite(site3)
    site3$setNextSite(master)

### 4. Perform the likelihood estimation

    fit1 <- mle(master$nLL, start = list(lambda = 5))

Print the summary.

    summary(fit1)

    ## Maximum likelihood estimation
    ##
    ## Call:
    ## mle(minuslogl = master$nLL, start = list(lambda = 5))
    ##
    ## Coefficients:
    ##        Estimate Std. Error
    ## lambda    9.175  0.4789311
    ##
    ## -2 log L: 199.5328

    logLik(fit1)

    ## 'log Lik.' -99.76641 (df=1)

The results should be the same as above.

## Applications via `distcomp` and `opencpu`

One can imagine these sort of computations being constructed within the framework described the R package distcomp where the sites are opencpu servers and there is a master process executing the computation. Much work remains to be done to make this work in a seamless manner; however, as this proof-of-concept example shows, the technical hurdles are quite surmountable.

## CAVEAT

This is an initial proof-of-concept implementation that has to substantially improved for real-world use. You’ve been warned.