# Fenchel duality, entropy, and the log partition function

[Update: As Max points out in the comments, this is really a specialized version of the Donsker-Varadhan formula, also mentioned by Mokshay in a comment here. I think the difficulty with concepts like these is that they are true for deeper reasons than the ones given when you learn them — this is a special case that requires undergraduate probability and calculus, basically.]

One of my collaborators said to me recently that it’s well known that the “negative entropy is the Fenchel dual of the log-partition function.” Now I know what these words meant, but it somehow was not a fact that I had learned elsewhere, and furthermore, a sentence like that is frustratingly terse. If you already know what it means, then it’s a nice shorthand, but for those trying to figure it out, it’s impenetrable jargon. I tried running it past a few people here who are generally knowledgeable but are not graphical model experts, and they too were unfamiliar with it. While this is just a simple thing about conjugate duality, I think it doesn’t really show up in information theory classes unless the instructor talks a bit more about exponential family distributions, maximum entropy distributions, and other related concepts. Bert Huang has a post on Jensen’s inequality as a justification.

We have a distribution in the exponential family:

$p(x; \theta) = \exp( \langle \theta, \phi(x) \rangle - A(\theta) )$

As a side note, I often find that the exponential family is not often covered in systems EE courses. Given how important it is in statistics, I think it should be a bit more of a central concept — I’m definitely going to try and work it in to the detection and estimation course.

For the purposes of this post I’m going to assume $x$ takes values in a discrete alphabet $\mathcal{X}$ (say, n-bit strings). The function $\phi(x)$ is a vector of statistics calculated from $x$, and $\theta$ is a vector of parameters. the function $A(\theta)$ is the log partition function:

$A(\theta) = \log\left( \sum_{x} \exp( \langle \theta, \phi(x) \rangle ) \right)$

Where the partition function is

$Z(\theta) = \sum_{x} \exp( \langle \theta, \phi(x) \rangle )$

The entropy of the distribution is easy to calculate:

$H(p) = \mathbb{E}[ - \log p(x; \theta) ] = A(\theta) - \langle \theta, \mathbb{E}[\phi(x)] \rangle$

The Fenchel dual of a function $f(\theta)$ is the function

$g(\psi) = \sup_{\theta} \left\{ \langle \psi, \theta \rangle - f(\theta) \right\}$.

So what’s the Fenchel dual of the log partition function? We have to take the gradient:

$\nabla_{\theta} \left( \langle \psi, \theta \rangle - A(\theta) \right) = \psi - \frac{1}{Z(\theta)} \sum_{x} \exp( \langle \theta, \phi(x) \rangle ) \phi(x) = \psi - \mathbb{E}[ \phi(x) ]$

So now setting this equal to zero, we see that at the optimum $\theta^*$:

$\langle \psi, \theta^* \rangle = \langle \mathbb{E}[ \phi(x) ], \theta^* \rangle$

And the dual function is:

$g(\psi) = \langle \mathbb{E}[ \phi(x) ], \theta^* \rangle - A(\theta^*) = - H(p)$

The standard approach seems to go the other direction by computing the dual of the negative entropy, but that seems more confusing to me (perhaps inspiring Bert’s post above). Since the log partition function and negative entropy are both convex, it seems easier to exploit the duality to prove it in one direction only.

# Allerton 2014: hypercontracting interference channels while noisily feeding back what you detected on the graph

Before the expiration window passes, here are few more short takes from Allerton… for some talks I couldn’t take notes because I didn’t get a seat or I missed half the talk shuttling between sessions.

The Gaussian Channel with Noisy Feedback: Near-Capacity Performance Via Simple Interaction
Assaf Ben-Yishai, Ofer Shayevitz
This was a really nice talk by Ofer on trying to get practical codes for AWGN channels with noisy feedback by using the intuition given by the Schalkwijk-Kailath scheme plus some tricks from using the mod operation. This is reminiscent of lattices (which may be an interesting future direction). The SK scheme has a problem with noise accumulation, which they deal with using these mode operations, and can get to errors around 10^(-6) with around 19 rounds, or blocklength 19 at reasonable SNRs. Blocklength is misleading here since there is feedback every symbol. The other catch is that the feedback link must have much higher SNR than the forward link, but this is true in applications such as sensing, where the receiver may be plugged into the wall, but the transmitter may be on a swallowable medical monitoring device.

Point-To-Point Codes for Interference Channels: A Journey Toward High Performance at Low Complexity
Young-Han Kim
Continuing with my UCSD bias, I also wanted to mention Young-Han’s talk, which was on using COTS (commercial, off-the-shelf) coding schemes on the interference channel (in particular, the 2 user IC). He talked about rate splitting approaches and block Markov schemes. Much of this work is with Lele Wang, who may be graduating soon…

Signal Detection on Graphs
Venkatesh Saligrama
This was a hypothesis testing problem where the observations come from nodes on graph. Under the null, they are Gaussian noise, and under the other hypothesis, there is a connected subgraph with an elevated mean. How should we do detection in this scenario? This is a compound hypothesis testing problem because there are (too) many possible connected subgraphs to consider. He gets around this by looking at a convex program parameterized by a measure of the size/shape of the connected component. This is where my notes get messy though, so you might want to look at the paper if it sounds interesting to you…

Hypercontractivity in Hamming Space
Yury Polyanskiy
I’ve hypercontractivity before, and Yury talked about his paper on ArXiV, which is about functions on the binary hypercube. This talk felt more like a tour of results on hypercontractivity and less like a “here is my new result” talk, which I actually appreciated because I felt it tied together ideas well and made me realize how strange the hypercontractivity parameter of an operator is.

# Allerton 2014: privately charging your information odometer while realizing your group has pretenders

Here are a few papers that I saw at Allerton — more to come later.

Group Testing with Unreliable Elements
Arya Mazumdar, Soheil Mohajer
This was a generalization of the group testing problem: items are either positive or null, and can be tested in groups such that if any element of the group is positive, the whole group will test positive. The item states can be thought of as a binary column vector $\mathbf{x}$ and each test is the row or a matrix $A$: the $(i,j)$-the entry of $A$ is 1 if the $i$-th item is part of the $j$-th group. The multiplication $A \mathbf{x}$ is taken using Boolean OR. The twist in this paper is that they consider a situation where $\tau$ elements can “pretend” to be positive in each test, with a possibly different group in each test. This is different than “noisy group testing” which was considered previously, which is more like $A \mathbf{x} + \mathrm{noise}$. They show achievable rates for detecting the positives using random coding methods (i.e. a random $A$ matrix). There was some stuff at the end about the non-i.i.d. case but my notes were sketchy at that point.

The Minimal Realization Problems for Hidden Markov Models
Qingqing Huang, Munther Dahleh, Rong Ge, Sham Kakade
The realization problem for an HMM is this: given the exact joint probability distribution on length $N$ strings $(Y_1, Y_2, \ldots, Y_N)$ from an HMM, can we create a multilinear system whose outputs have the same joint distribution? A multilinear system looks like this:
$w_{t+1} = A^{(\ell_t)} w_t$ for $w_t \in \mathbb{R}^k$ with $w_0 = v$
$z_{t} = u^t w_t$
We think of $A^{(\ell_t)} \in \mathbb{R}^{k \times k}$ as the transition for state $\ell_t$ (e.g. a stochastic matrix) and we want the output to be $z_t$. This is sort of at the nexus of control/Markov chains and HMMs and uses some of the tensor ideas that are getting hot in machine learning. As the abstract puts it, the results are that they can efficiently construct realizations if $N > max(4 log_d(k) + 1; 3)$, where $d$ is the size of the output alphabet size, and $k$ is the minimal order of the realization.”

Differentially Private Distributed Protocol for Electric Vehicle Charging
Shuo Han, Ufuk Topcu, George Pappas
This paper was about a central aggregator trying to get multiple electric vehicle users to report their charging needs. The central allocator has a utility maximization problem over the rates of charging:
$\min_{r_i} U(\sum r_i)$
$\mathrm{s.t.}\ \le r_i \le \bar{r}_i$
$\qquad \mathbf{1}^{\top} r_i = z_i$
They focus on the case where the charge demands $z_i$ are private but the charging rates $r_i$ can be public. This is in contrast to the mechanism design literature popularized by Aaron Roth and collaborators. They do an analysis of a projected gradient descent procedure with Laplace noise added for differential privacy. It’s a stochastic gradient method but seems to converge in just a few time steps — clearly the number of iterations may impact the privacy parameter, but for the examples they showed it was only a handful of time steps.

An Interactive Information Odometer and Applications
Mark Braverman and Omri Weinstein
This was a communication complexity talk about trying to maintain an estimate of the information complexity of a protocol $\Pi$ for computing a function $f(X,Y)$ interactively, where Alice has $X$, Bob has $Y$, and $(X,Y) \sim \mu$:
$\mathsf{IC}_{\mu}(\Pi) = I( \Pi ; X | Y ) + I( \Pi ; Y | X)$
Previous work has shown that the (normalized) communication complexity of computing $f$ on $n$-tuples of variables $\epsilon$-accurately approaches the minimum information complexity over all protocols for computing $f$ once $\epsilon$-accurately. At least I think this is what was shown previously — it’s not quite my area. The result in this paper is a strong converse for this — the goal is to maintain an online estimate of $\mathsf{IC}$ during the protocol. The mechanism seeems a bit like communication with feedback a la Horstein, but I got a bit confused as to what was going on — basically it seems that Alice and Bob need to be able to agree on the estimate during the protocol and use a few extra bits added to the communication to maintain this estimate. If I had an extra $n$ hours in the week I would read up more about this. Maybe on my next plane ride…

I’ve been a bit bogged down upon getting back from traveling, but here are a few interesting technical tidbits that came through.

Aaron Roth and Cynthia Dwork’s Foundation and Trends monograph on differential privacy is now available.

Speaking of differential privacy, Shiva Kasiviswanathan and Adam Smith have a paper in the Journal of Privacy and Confidentiality on Bayesian interpretations of differential privacy risk.

Deborah Mayo has a post up on whether p-values are error probabilities.

Raymond Yeung is offering a Coursera course on information theory (via the IT Society).

A CS Theory take on Fano’s inequality from Suresh over at the GeomBlog.

# SPCOM 2014: some more talks (and a plenary)

I did catch Greg Wornell’s plenary at SPCOM, which was called When Bits Absolutely, Positively, Have to be There as Soon as Possible, a riff on this FedEx commercial, which is older than I am. The talk was on link-aware PHY-layer design– basically looking at how ARQ enables incremental redundancy, and how to do a sort of layered superposition + incremental redundancy scheme in the sequential setting as well as a “multi-path” setting where blocks can arrive out of order. This was really digging into the signal issues in a way that a lot of non-communication engineering information theorists may get squeamish about. The nice thing is that I think the engineering problem is approachable without knowing a lot of heavy-duty math, but still requires some careful analysis.

Communication and Compression Via Sparse Linear Regression
Ramji Venkataramanan
This was on building codewords and codebooks out of a lower-complexity code dictionary $A \in \mathbb{R}^{n \times ML}$ where each codeword is a superposition of $L$ columns, one each from groups of size $M$. Thus encoding is $A \beta$ where $\beta$ is a sparse vector. I saw a talk by Barron and Joseph from a previous ISIT about this, but the framework extends to rate distortion (achieving the rate distortion function), and channel coding. The main point is to lower the complexity of the code at the expense of the gap to optimal rate — encoding and decoding are polynomial time but the rate gap for rate-distortion goes to zero as $1/\log n$. Ramji gave a really nice and clear talk on this — I hope he puts the slides up!

An Optimal Varentropy Bound for Log-Concave Distributions
Mokshay’s talk was also really clear and excellent. For a distribution $f(X)$ on $\mathbb{R}^n$, we can define $\tilde{h}(X) = - \log f(X)$. The entropy is the expectation of this random variable, and the varentropy is the variance. Their main result is a upper bound on the varentropu of log concave distributions $f(X)$. To wit, $\mathrm{Var}(\tilde{h}) \le n$. This bound doesn’t depend on the distribution and is sharp if $f$ is a product of exponentials. They then use this to prove a universal bound on the deviation of $\tilde{h}$ from its expectation, which gives a AEP that doesn’t really assume anything about the joint distribution of the variables except for log-concavity. There was more in the talk, but I eagerly await the paper.

Event-triggered Sampling and Reconstruction of Sparse Real-valued Trigonometric Polynomials
Neeraj Sharma; Thippur V. Sreenivas
This was on non-uniform sampling where the sampler tries to detect level crossings of the analog signal and samples at that point — the rate may not be uniform enough to use existing nonuniform sampling techniques. They come up with a method for reconstructing signals which are real-valued trigonometric polynomials with a few nonzero coefficients (e.g. sparse) and it seems to work pretty decently in experiments.

Removing Sampling Bias in Networked Stochastic Approximation
Vivek Borkar; Raaz Dwivedi
In networked stochastic approximation, the intermittent communication between nodes may mean that the system tracks a different ODE than the one we want. By modifying the method to account for “local clocks” on each edge, we can correct for this, but we end up with new conditions on the step size to make things work. I am pretty excited about this paper, but as usual, my notes were not quite up to getting the juicy bits. That’s what paper reading is for.

On Asymmetric Insertion and Deletion Errors
Ankur A. Kulkarni
The insertion/deletion channel model is notoriously hard. Ankur proposed a new model where $0$‘s are “indestructible” — they cannot be inserted or deleted. This asymmetric model leads to new asymptotic bounds on the capacity. I don’t really work on this channel model so I can’t get the finer points of the results, but once nice takeaway was that asymptotically, each indestructible $0$ in the codeword lets us correct around $1/2$ a deletion more.

# A teaser for ITAVision 2015

As part of ITAVision 2015 we are soliciting individuals and groups to submit videos documenting their love of information theory and/or its applications. During ISIT we put together a little example with our volunteers (it sounded better in rehearsal than at the banquet, alas). The song was Entropy is Awesome based on this, obviously. If you want to sing along, here is the Karaoke version:

The lyrics (so far) are:

Entropy is awesome!
Entropy is sum minus p log p
Entropy is awesome!
When you work on I.T.

Blockwise error vanishes as n gets bigger
Maximize I X Y
Polarize forever
Let’s party forever

I.I.D.
I get you, you get me
Communicating at capacity

Entropy is awesome…

This iteration of the lyrics is due to a number of contributors — truly a group effort. If you want to help flesh out the rest of the song, please feel free to email me and we’ll get a group effort going.

More details on the contest will be forthcoming!

# SPCOM 2014: some talks

Relevance Singular Vector Machine for Low-­rank Matrix Sensing
Martin Sundin; Saikat Chatterjee; Magnus Jansson; Cristian Rojas
This talk was on designing Bayesian priors for sparse-PCA problems — the key is to find a prior which induces a low-rank structure on the matrix. The model was something like $y = A \mathrm{vec}(X) + n$ where $X$ is a low-rank matrix and $n$ is noise. The previous state of the art is by Babacan et al., a paper which I obviously haven’t read, but the method they propose here (which involved some heavy algebra/matrix factorizations) appears to be competitive in several regimes. Probably more of interest to those working on Bayesian methods…

Non-Convex Sparse Estimation for Signal Processing
David Wipf
More Bayesian methods! Although David (who I met at ICML) was not trying to say that the priors are particularly “correct,” but rather that the penalty functions that they induce on the problems he is studying actually make sense. More of an algorithmist’s approach, you might say. He set up the problem a bit more generally, to minimize problems of the form
$\min_{X_i} \sum_{i} \alpha_i \mathrm{rank}[X_i] \ \ \ \ \ \ \ Y = \sum_{i} A_i(X_i)$
where $A_i$ are some operators. He made the case that convex relaxations of many of these problems, while analytically beautiful, have restrictions which are not satisfied in practice, and indeed they often have poor performance. His approach is via Empirical Bayes, but this leads to non-convex problems. What he can show is that the algorithm he proposes is competitive with any method that tries to separate the error from the “low-rank” constraint, and that the new optimization is “smoother.” I’m sure more details are in his various papers, for those who are interested.

PCA-HDR: A Robust PCA Based Solution to HDR Imaging
Vinod showed some information theoretic approaches to understanding how much communication is needed for secure computation protocols like remote oblivious transfer: Xavier has $\{X_0, X_1\}$, Yvonne has $Y \in \{0,1\}$ and Zelda wants $Z = X_Y$, but nobody should be able to infer each other’s values. Feige, Killian, and Naor have a protocol for this, which Vinod and Co. can show is communication-optimal. There were several ingredients here, including cut-set bounds, distribution switching, data processing inequalities, and special bounds for 3-party protocols. More details in his CRYPTO paper (and others).
In a MIMO wiretap setting, if the receiver has more antennas than the transmitter, then the transmitter can send noise in the nullspace of the channel matrix of the direct channel — as long as the eavesdropper has fewer antennas than the transmitter then secure transmission is possible. In this paper they show that positive secrecy capacity is possible even when the eavesdropper has more antennas, but as the number of eavesdropper antennas grows, the achievable rate goes to $0$. Perhaps a little bit of a surprise here!