arXiv:1403.4696 [math.OC]
Design and Analysis of Distributed Averaging with Quantized Communication
Mahmoud El Chamie, Ji Liu, Tamer Başar

The goal of this paper is to analyze the “performance of a subclass of deterministic distributed averaging algorithms where the information exchange between neighboring nodes (agents) is subject to uniform quantization.” I was interested in the connections to Lavaei and Murray’s TAC paper. Here though, they consider a standard consensus setup with a doubly stochastic weight matrix $W$ and deterministic, rather than randomized, quantization. They consider two types — rounding and truncation (essentially the floor operation). The update rule is $x_i(t+1) = x_i(t) + \sum_{j} W_{ij} (Q(x_j(t)) - Q(x_i(t)))$ where $Q$ is the quantization operation. They shoe that in finite time the agents either reach a consensus on the floor of the average of their initial values, or that the cycle indefinitely in a neighborhood around the average. They they show how to control the size of the neighborhood in a decentralized way. There are a lot of works on quantized consensus that have appeared in the last 5 years, and to be honest I haven’t really kept up on the recent literature, so I’m not sure how to compare this to the other works that have appeared, but perhaps some of the readers of the blog have…

arXiv:1403.4699 [math.OC]
A Proximal Stochastic Gradient Method with Progressive Variance Reduction
Lin Xiao, Tong Zhang

This paper looks at convex optimization problems of the form

$\min_{x \in \mathbb{R}^d} F(x) + R(x)$

where the overall objective $P(x) + R(x)$ is strongly convex, the regularizer $R(x)$ is lower semicontinuous and convex, and the term $F(x) = \frac{1}{n} \sum_{i=1}^{n} f_i(x)$ separates into a sum of function $f_i(x)$ which are Lipschitz continuous. The proximal gradient method is an iterative procedure for solving this program that does the following:

$x_{t} = \mathrm{argmin}_{x \in \mathbb{R}^d} \left\{ \nabla F(x_{t})^{\top} x + \frac{1}{2 \eta_t} \| x - x_{t-1} \|^2 + R(x) \right\}$

If we define the $\mathrm{prox}_{R}$ function as

$\mathrm{prox}_R(y) = \mathrm{argmin}_{x \in \mathbb{R}^d} \left\{ \frac{1}{2} \|x - y\|^2 + R(x) \right\}$

then the step looks like:

$x_t = \mathrm{prox}_{\eta_t R}(x_{t-1} - \eta_{t} \nabla F(x_{t-1}))$

A stochastic gradient (SG) version of this is

$x_t = \mathrm{prox}_{\eta_t R}(x_{t-1} - \eta_{t} \nabla f_{i_t}(x_{t-1}))$

where $i_t$ is sampled uniformly from $\{1,2,\ldots, n\}$ at each time. The advantage of the SG variant is that it takes less time to do one iteration, but each iteration is much noisier. The goal of this paper is to adapt a previous method/approach to variance reduction to improve the performance of the Prox-SG algorithm. The approach is one or resampling points according to the Lipschitz constants. This sort of “sampling based adaptivity” was also used by my ex-colleague Samory Kpotufe and collaborators in their NIPS paper from 2012 (a longer version is under review). At least I think they’re related.

arXiv:1403.5341v1 [cs.LG]
An Information-Theoretic Analysis of Thompson Sampling
Daniel Russo, Benjamin Van Roy

In a multi-armed bandit problem we have a set of actions (arms) $A$ and at each time the learner picks an action $a$ and observes an outcome $Y_t(a) \in \mathcal{Y}$ which is assigned a reward by a function $R: \mathcal{Y} \to \mathbb{R}$. The rewards are assumed to be i.i.d. across time for each action with distributions $p(y | a)$ that are unknown to the learner. The goal is to maximize the reward, which is the same as finding the arm with the largest expected reward. This leads to a classical explore/exploit tradeoff where the learner has to decide whether to explore new arms which may have higher expected reward, or continue exploiting the reward offered by the current arm. Thompson sampling is a Bayesian approach where the learner starts with a prior on the best action and then samples actions at each time according to its posterior belief on the best arm. The authors here analyze the regret of such a policy in terms of what they call the information gain of the system. This gain depends on the ratio between two quantities that are functions of the outcome distributions $p(y | a)$. One is what they call the “divergence in mean,” namely the difference in expected reward between arms, and the other is the KL divergence.

One of the things I’m always asked when giving a talk on differential privacy is “how should we interpret $\epsilon$?” There a lot of ways of answering this but one way that seems to make more sense to people who actually think about risk, hypothesis testing, and prediction error is through the “area under the curve” metric, or AUC. This post came out of a discussion from a talk I gave recently at Boston University, and I’d like to thank Clem Karl for the more detailed questioning.

One thing that strikes me about US graduate programs in electrical engineering is that the student population is overwhelmingly international. For most of these students, English is a second or third language, and so we need to adopt more “ESL”-friendly pedagogical approaches to teaching writing. I came across a blog post from ATTW by Meg Morgan from UNC Charlotte that raises a number of interesting issues. For one, the term “ESL” is perhaps problematic. The linguistic and social differences in pedagogy between other countries and the US mean that we need to use different methods for engaging the students.

In terms of teaching technical writing at the graduate level, the issues may be similar but the students are generally older — they may have even had some writing experience from undergraduate or masters-level research. How should the “ESL” issue affect how we teach technical writing?

I saw a paper on ArXiV yesterday called Kalman meets Shannon, which got me thinking: in how many papers has someone met Shannon, anyway? Krish blogged about this a few years ago, but since then Shannon has managed to meet some more people. I plugged “meets Shannon” into Google Scholar, and out popped:

Sometimes people are meeting Shannon, and sometimes he is meeting them, but each meeting produces at least one paper.

A bit of the new, a bit of the old, for this Maundy Thursday.

1. You Can Never Hold Back Spring (Tom Waits)
2. Le Gars qui vont à la fête (Stutzmann/Södergren, by Poulenc)
3. Judas mercator pessimus (King’s Singers, by Gesualdo)
4. Calling (Snorri Helgason)
7. A Little Lost (Nat Baldwin)
8. Gun Has No Trigger (Dirty Projectors)
9. Stranger to My Happiness (Sharon Jones & The Dap-Kings)
10. Dama Dam Mast Qalandar (Red Baraat)
11. Libra Stripes (Polyrhythmics)
12. Jaan Pehechan Ho (The Bombay Royale)
13. Jolie Coquine (Caravan Palace)
14. The Natural World (CYMBALS)
15. Je Ne Vois Que Vous (Benjamin Schoos feat. Laetitia Sadier)
16. Romance (Wild Flag)

I think it would be great to have a more formal way of teaching technical writing for graduate students in engineering. It’s certainly not being taught at (most) undergraduate institutions, and the mistakes are so common across the examples that I’ve seen that there must be a way to formalize the process for students. Since we tend to publish smaller things a lot earlier in our graduate career, having a “checklist” approach to writing/editing could be very helpful to first-time authors. There are several coupled problems here:

• students often don’t have a clear line of thought before they write,
• they don’t think of who their audience is,
• they don’t know how to rewrite, or indeed how important it is.

Adding to all of this is that they don’t know how to read a paper. In particular, they don’t know what to be reading for in terms of content or form. This makes the experience of reading “related work” sections incredibly frustrating.

What I was thinking was a class where students learn to write a literature review (a small one) on a topic of their choosing. The first part will be how to read papers and make connections between them. What is the point of a literature review, anyway? The first objective is to develop a more systematic way of reading and processing papers. I think everyone I know professionally, myself included, learned how to do this in an ad-hoc way. I believe that developing a formula would help improve my own literature surveying. The second part of the course would be teaching about rewriting (rather than writing). That is, instead of providing rules like “don’t use the passive voice so much” we could focus on “how to revise your sentences to be more active.” I would also benefit from a systematic approach to this for my own writing.

I was thinking of a kind of once-a-week writing seminar style class. Has anyone seen a class like this in engineering programs? Are there tips/tricks from other fields/departments which do have such classes that could be useful in such a class? Even though it is “for social scientists”, Harold Becker’s book is a really great resource.

I always end up bookmarking a bunch of papers from ArXiV and then looking at them a bit later than I want. So here are a few notes on some papers from the last month. I have a backlog of reading to catch up on, so I’ll probably split this into a couple of posts.

arXiv:1403.3465v1 [cs.LG]: Analysis Techniques for Adaptive Online Learning
H. Brendan McMahan
This is a nice survey on online learning/optimization algorithms that adapt to the data. These are all variants of the Follow-The-Regularized-Leader algorithms. The goal is to provide a more unified analysis of online algorithms where the regularization is data dependent. The intuition (as I see it) is that you’re doing a kind of online covariance estimation and then regularizing with respect to the distribution as you are learning it. Examples include the McMahan and Streeter (2010) paper and the Duchi et al. (2011) paper. Such adaptive regularizers also appear in dual averaging methods, where they are called “prox-functions.” This is a useful survey, especially if, like me, you’ve kind of checked in and out with the online learning literature and so may be missing the forest for the trees. Or is that the FoReL for the trees?

arXiv:1403.4011 [cs.IT]: Whose Opinion to follow in Multihypothesis Social Learning? A Large Deviation Perspective
Wee Peng Tay
This is a sort of learning from expert advice problem, though not in the setting that machine learners would consider it. The more control-oriented folks would recognize it as a multiple-hypothesis test. The model is that there is a single agent (agent $0$) and $K$ experts (agents $1, 2, \ldots, K$). The agent is trying to do an $M$-ary hypothesis test. The experts (and the agent) have access to local (private) observations $Y_k[1], Y_k[2], \ldots, Y_k[n_k]$ for $k \in \{0,1,2,\ldots,K\}$. The observations come from a family of distributions determined by the true hypothesis $m$. The agent $0$ needs to pick one of the $K$ experts to hire — the analogy is that you are an investor picking an analyst to hire. Each expert has its own local loss function $C_k$ which is a function of the amount of data it has as well as the true hypothesis and the decision it makes. This is supposed to model a “bias” for the expert — for example, they may not care to distinguish between two hypotheses. The rest of the paper looks at finding policies/decision rules for the agents that optimize the exponents with respect to their local loss functions, and then looking at how agent $0$ should act to incorporate that advice. This paper is a little out of my wheelhouse, but it seemed interesting enough to take a look at. In particular, it might be interesting to some readers out there.

arXiv:1403.3862 [math.OC] Asynchronous Stochastic Coordinate Descent: Parallelism and Convergence Properties
Ji Liu, Stephen J. Wright
This is another paper on lock-free optimization (c.f. HOGWILD!). The key difference, as stated in the introduction, is that they “do not assume that the evaluation vector $\hat{x}$ is a version of $x$ that actually existed in the shared memory at some point in time.” What does this mean? It means that a local processor, when it reads the current state of the iterate, may be performing an update with respect to a point not on the sample path of the algorithm. They do assume that the delay between reading and updating the common state is bounded. To analyze this method they need to use a different analysis technique. The analysis is a bit involved and I’ll have to take a deeper look to understand it better, but from a birds-eye view this would make sense as long as the step size is chosen properly and the “hybrid” updates can be shown to be not too far from the original sample path. That’s the stochastic approximator in me talking though.