# Insensitive Intersections of Halfspaces – STOC 2014 Recaps (Part 11)

In the eleventh and final installment of our STOC 2014 recaps, Jerry Li tells us about a spectacularly elegant result by Daniel Kane. It’s an example of what I like to call a “one-page wonder” — this a bit of a misnomer, since Kane’s paper is slightly more than five pages long, but the term refers to any beautiful paper which is (relatively) short and readable.

We hope you’ve enjoyed our coverage of this year’s STOC. We were able to write about a few of our favorite results, but there’s a wealth of other interesting papers that deserve your attention. I encourage you to peruse the proceedings and discover some favorites of your own.

The Monday afternoon sessions kicked off with Daniel Kane presenting his work on the average sensitivity of an intersection of halfspaces. Usually, FOCS/STOC talks can’t even begin to fit all the technical details from the paper, but unusually, Daniel’s talk included a complete proof of his result, without omitting any details. Amazingly, his result is very deep and general, so something incredible is clearly happening somewhere.

At the highest level, Daniel deals with the study of a certain class of Boolean functions. When we classically think about Boolean functions, we think of things such as CNFs, DNFs, decision trees, etc., which map into things like $\mathbb{F}_2, \{0, 1\},$ or $\{\mbox{True}, \mbox{False}\}$, but today, and often in the study of Boolean analysis, we will think of functions as mapping $\{-1, 1\}^n$ to $\{-1, 1\}$, which is roughly equivalent for many purposes (O’Donnell has a nice rule of thumb as when to use one convention or the other here). Given a function $f:\{-1, 1\}^n \to \{-1, 1\}$, we can define two important measures of sensitivity. The first is the average sensitivity (or, for those of you like me who grew up with O’Donnell’s book, the total influence) of the function, namely,

$\mathbb{AS}(f) = \mathbb{E}_{x \sim \{-1, 1\}^n} [| \{i: f(x^{i \to 1}) \neq f(x^{i \to -1}) \} |]$

where $x^{i \to a}$ is simply $x$ with its $i$th coordinate set to $a$. The second is the noise sensitivity of the function, which is defined similarly: for a parameter $\varepsilon \in (0, 1)$, it is the probability that if we sample $x$ uniformly at random from $\{-1, 1\}^n$, then independently flip each of its bits with probability $\varepsilon$, the value of $f$ at these two inputs is different. We denote this quantity $\mathbb{NS}_\varepsilon (f)$. When we generate a string $y$ from a string $x$ in this way we say they are $1 - 2\varepsilon$-correlated. The weird function of $\varepsilon$ in that expression is because often we equivalently think of $y$ being generated from $x$ by independently keeping each coordinate of $x$ fixed with probability $1 - 2\varepsilon$, and uniformly rerandomizing that bit with probability $2 \varepsilon$.

Why are these two measures important? If we have a concept class of functions $\mathcal{F}$, then it turns out that bounds on these two quantities can often be translated directly into learning algorithms for these classes. By Fourier analytic arguments, good bounds on the noise sensitivity of a function immediately imply that the function has good Fourier concentration on low degree terms, which in turn imply that the so-called “low-degree algorithm” can efficiently learn the class of functions in the PAC model, with random access. Unfortunately, I can’t really give any more detail here without a lot more technical detail, see [2] for a good introduction to the topic.

Now why is the average sensitivity of a Boolean function important? First of all, trust me when I say that it’s a fundamental measure of the robustness of the function. If we identify $f$ with $f^{-1} (1)$, then the average sensitivity is how many edges cross from one subset into another (over $2^n$), so it is fundamentally related to the surface area of subsets of the hypercube, which comes up all over the place in Boolean analysis. Secondly, in some cases, we can translate between one measure and the other by considering restrictions of functions. To the best of my knowledge, this appears to be a technique first introduced by Peres in 1999, though his paper is from 2004 [3]. Let $f: \{-1, +1\}^n \to \mathbb{R}$. We wish to bound the noise sensitivity of $f$, so we need to see how it behaves when we generate $x$ uniformly at random, then $y$ as $1 - 2 \varepsilon$-correlated to $x$. Suppose $\varepsilon = 1/m$ for some integer $m$ (if not, just round it). Fix a partition of the $n$ coordinates into $m$ bins $B_1, \ldots, B_m$, and a $z \in \{-1, 1\}^n$. Then, for any string $s \in \{-1, 1\}^m$, we associate it with the string $x \in \{-1, 1\}^n$ whose $i$th coordinate is the $i$th coordinate of $z$ times the $j$th coordinate of $s$, if $i \in B_j$. Why are we doing this? Well, after some thought, it’s not too hard to convince yourself that if we choose the $m$ bins and the strings $z, s$ uniformly at random, then we get a uniformly random string $x$. Moreover, to generate a string $y$ which is $1 - 2 \varepsilon$-correlated with $x$, it suffices to, after having already randomly chosen the bins, $z$, and $s$, to randomly pick a coordinate of $s$ and flip its sign to produce a new string $s'$, and produce a new string $y$ with these choices of the bins, $z,$ and $s'$. Thus, importantly, we can reduce the process of producing $1 - 2 \varepsilon$-correlated strings to the process of randomly flipping one bit of some weird new function–but this is the process we consider when we consider the average sensitivity! Thus noise sensitivity of $f$ is exactly equal to the expected (over the random choice of the bins and $z$) average sensitivity of this weird restricted thing. Why this is useful will (hopefully) become clear later.

Since the title of the paper includes the phrase “intersection of halfspaces,” at some point I should probably define what an intersection of halfspaces is. First of all, a halfspace (or linear threshold function) is a Boolean function of the form $f(x) = \text{sgn}\left(w \cdot x - \theta\right)$ where $w \in \mathbb{R}^n, \theta \in \mathbb{R}$ and for concreteness let’s say $\text{sgn}\left(0\right) = 1$ (however, it’s not too hard to see that any halfspace has a representation so that the linear function inside the sign is never zero on the hypercube). Intuitively, take the hyperplane in $\mathbb{R}^n$ with normal vector $w$, then assign to all points which are in the same side as $w$ of the hyper plane the value $+1$, and the rest $-1$. Halfspaces are an incredibly rich family of Boolean functions which include arguably some of the important objects in Boolean analysis, such as the dictator functions, the majority function, etc. There is basically a mountain of work on halfspaces, due to their importance in learning theory, and as elementary objects which capture a surprising amount of structure.

Secondly, the intersection of $k$ functions $f_1, \ldots, f_k$ is the function which is $1$ at $x$ if and only $f_i (x) = 1$ for all $i$, and $-1$ otherwise. If we think of each $f_i$ as a $\{ \mbox{True}, \mbox{False}\}$ predicate on the boolean cube, then their intersection is simply their AND (or NOR, depending on your map from $\{-1, 1\}$ to $\{ \mbox{True}, \mbox{False}\}$).

Putting these together gives us the family of functions that Daniel’s work concerns. I don’t know what else to say other than they are a natural algebraic generalization of halfspaces. Hopefully you think these functions are interesting, but even if you don’t, it’s (kinda) okay, because, amazingly, it turns out Kane’s main result barely uses any properties of halfspaces! In fact, it only uses the fact that halfspaces are unate, that is, they are either monotone increasing or decreasing in each coordinate. In fact, he proves the following, incredibly general, theorem:

Theorem. [Kane14]
Let $f:\{-1, 1\}^n \to \{-1, 1\}$ be an intersection of $k$ unate functions. Then

$\mathbb{AS}(f) = O(\sqrt{n \log k})$.

I’m not going to go into too much detail about the proof; unfortunately, despite my best efforts there’s not much intuition I can compress out of it (in his talk, Daniel himself admitted that there was a lemma which was mysterious even to him). Plus it’s only roughly two pages of elementary (but extremely deep) calculations, just read it yourself! At a very, very high level, the idea is that intersecting a intersection of $k - 1$ halfspaces with one more can only increase the average sensitivity by a small factor.

The really attentive reader might have also figured out why I gave that strange reduction between noise sensitivity and average sensitivity. This is because, importantly, when we apply this weird process of randomly choosing bins to an intersection of halfspaces, the resulting function is still an intersection of halfspaces, just over fewer coordinates (besides their unate-ness, this is the only property of intersections of halfspaces that Daniel uses). Thus, since we now know how to bound the average sensitivity of halfspaces, we also know tight bounds for the noise sensitivities of intersection of halfspaces, namely, the following:

Theorem. [Kane14]
Let $f:\{-1, 1\}^n \to \{-1, 1\}$ be an intersection of $k$ halfspaces. Then

$\mathbb{NS}_\varepsilon (f) = O(\sqrt{\varepsilon \log k})$.

Finally, this gives us good learning algorithms for intersections of halfspaces.

The paper is remarkable; there had been previous work by Nazarov (see [4]) proving optimal bounds for sensitivities of intersections of halfspaces in the Gaussian setting, which is a more restricted setting than the Boolean setting (intuitively because we can simulate Gaussians by sums of independent Boolean variables), and there were some results in the Boolean setting, but they were fairly suboptimal [5]. Furthermore, all these proofs were scary: they were incredibly involved, used powerful machinery from real analysis, drew heavily on the properties of halfspaces, etc. On the other hand, Daniel’s proof of his main result (which I would say builds on past work in the area, except it doesn’t use anything besides elementary facts), well, I think Erdos would say this proof is from “the Book”.

[1] The Average Sensitivity of an Intersection of Half Spaces, Kane, 2014.
[2] Analysis of Boolean Functions, O’Donnell, 2014.
[3] Noise Stability of Weighted Majority, Peres, 2004.
[4] On the maximal perimeter of a convex set in $\mathbb{R}^n$ with respect to a Gaussian measure, Nazarov, 2003.
[5] An Invariance Principle for Polytopes, Harsha, Klivans, Meka, 2010.

# An Encore: More Learning and Testing – STOC 2014 Recaps (Part 8)

Thought you were rid of us? Not quite: in a last hurrah, Clément and I come back with a final pair of distribution estimation recaps — this time on results from the actual conference!

Density estimation is the question on everyone’s mind. It’s as simple as it gets – we receive samples from a distribution and want to figure out what the distribution looks like. The problem rears its head in almost every setting you can imagine — fields as diverse as medicine, advertising, and compiler design, to name a few. Given its ubiquity, it’s embarrassing to admit that we didn’t have a provably good algorithm for this problem until just now.

Let’s get more precise. We’ll deal with the total variation distance metric (AKA statistical distance). Given distributions with PDFs $f$ and $g$, their total variation distance is $d_{\mathrm{TV}}(f,g) = \frac12\|f - g\|_1$. Less formally but more intuitively, it upper bounds the difference in probabilities for any given event. With this metric in place, we can define what it means to learn a distribution: given sample access to a distribution $X$, we would like to output a distribution $\hat X$ such that $d_{\mathrm{TV}}(f_X,f_{\hat X}) \leq \varepsilon$.

This paper presents an algorithm for learning $t$-piecewise degree-$d$ polynomials. Wow, that’s a mouthful — what does it mean? A $t$-piecewise degree-$d$ polynomial is a function where the domain can be partitioned into $t$ intervals, such that the function is a degree-$d$ polynomial on each of these intervals. The main result says that a distribution with a PDF described by a $t$-piecewise degree-$d$ polynomial can be learned to accuracy $\varepsilon$ using $O((d+1)kt/\varepsilon^2)$ samples and polynomial time. Moreover, the sample complexity is optimal up to logarithmic factors.

A 4-piecewise degree-3 polynomial.
Lifted from Ilias’ slides.

Now this is great and all, but what good are piecewise polynomials? How many realistic distributions are described by something like “$-x^2 + 1$ for $x \in [0,1)$ but $\frac12x - 1$ for $x \in [1,2)$ and $2x^4 - x^2 +$…”? The answer turns out to be a ton of distributions — as long as you squint at them hard enough.

The wonderful thing about this result is that it’s semi-agnostic. Many algorithms in the literature are God-fearing subroutines, and will sacrifice their first-born child to make sure they receive samples from the class of distributions they’re promised — otherwise, you can’t make any guarantees about the quality of their output. But our friend here is a bit more skeptical. He deals with a funny class of distributions, and knows true piecewise polynomial distributions are few and far between — if you get one on the streets, who knows if it’s pure? Our friend is resourceful: no matter the quality, he makes it work.

Let’s elaborate, in slightly less blasphemous terms. Suppose you’re given sample access to a distribution $\mathcal{D}$ which is at total variation distance $\leq \tau$ from some $t$-piecewise degree-$d$ polynomial (you don’t need to know which one). Then the algorithm will output a $(2t-1)$-piecewise degree-$d$ polynomial which is at distance $\leq 4\tau + O(\varepsilon)$ from $\mathcal{D}$. In English: even if the algorithm isn’t given a piecewise polynomial, it’ll still produce something that’s (almost) as good as you could hope for.

With this insight under our cap, let’s ask again — where do we see piecewise polynomials? They’re everywhere: this algorithm can handle distributions which are log-concave, bounded monotone, Gaussian, $t$-modal, monotone hazard rate, and Poisson Binomial. And the kicker is that it can handle mixtures of these distributions too. Usually, algorithms fail catastrophically when considering mixtures, but this algorithm keeps chugging and handles them all — and near optimally, most of the time.

The analysis is tricky, but I’ll try to give a taste of some of the techniques. One of the key tools is the Vapnik-Chervonenkis (VC) inequality. Without getting into the details, the punchline is that if we output a piecewise polynomial which is “close” to the empirical distribution (under a weaker metric than total variation distance), it’ll give us our desired learning result. In this setting, “close” means (roughly) that the CDFs don’t stray too far from each (though in a sense that is stronger than the Kolmogorov distance metric).

Let’s start with an easy case – what if the distribution is a $1$-piecewise polynomial? By the VC inequality, we just have to match the empirical CDF. We can do this by setting up a linear program which outputs a linear combination of the Chebyshev polynomials, constrained to resemble the empirical distribution.

It turns out that this subroutine is the hardest part of the algorithm. In order to deal with multiple pieces, we first discretize the support into small intervals which are roughly equal in probability mass. Next, in order to discover a good partition of these intervals, we run a dynamic program. This program uses the subroutine from the previous paragraph to compute the best polynomial approximation over each contiguous set of the intervals. Then, it stitches the solutions together in the minimum cost way, with the constraint that it uses fewer than $2t - 1$ pieces.

In short, this result essentially closes the problem of density estimation for an enormous class of distributions — they turn existential approximations (by piecewise polynomials) into approximation algorithms. But there’s still a lot more work to do — while this result gives us improper learning, we yearn for proper learning algorithms. For example, this algorithm lets us approximate a mixture of Gaussians using a piecewise polynomial, but can we output a mixture of Gaussians as our hypothesis instead? Looking at the sample complexity, the answer is yes, but we don’t know of any computationally efficient way to solve this problem yet. Regardless, there’s many exciting directions to go — I’m looking forward to where the authors will take this line of work!

-G

Clément Canonne on $L_p$-Testing, by Piotr Berman, Sofya Raskhodnikova, and Grigory Yaroslavtsev [1])

Almost every — if not all — work in property testing of functions are concerned with the Hamming distance between functions, that is the fraction of inputs on which they disagree. Very natural when we deal for instance with Boolean functions $f\colon\{0,1\}^d\to\{0,1\}$, this distance becomes highly arguable when the codomain is, say, the real line: sure, $f\colon x\mapsto x^2-\frac{\sin^2 x}{2^{32}}$ and $g\colon x\mapsto x^2$ technically disagree on almost every single input, but should they be considered two completely different functions?

This question, Grigory answered by the negative; and presented (joint work with Piotr Berman and Sofya Raskhodnikova [2]) a new framework for testing real-valued functions $f\colon X^d\to[0,1]$, less sensitive to this sort of annoying “technicalities” (i.e., noise). Instead of the usual Hamming/$L_0$ distance between function, they suggest the more robust $L_p$ ($p\geq 1$) distance

$\mathrm{d}_p(f,g) = \frac{ \left(\int_{X^d} \lvert f(x)-g(x) \rvert^p dx\right)^{\frac{1}{p}} }{ \left(\int_{X^d} \mathbf{1} dx \right)^{\frac{1}{p}} } = \mathbb{E}_{x\sim\mathcal{U}(X^d)}\left[\lvert f(x)-g(x) \rvert^p\right]^{\frac{1}{p}}\in [0,1]$

(think of $X^d$ as being the hypercube $\{0,1\}^d$ or the hypergrid $[n]^d$, and $p$ being 1 or 2. In this case, the denominator is just a normalizing factor $\lvert X\rvert$ or $\sqrt{\lvert X\rvert}$)

Now, erm… why?

• because it is much more robust to noise in the data;
• because it is much more robust to outliers;
• because it plays well (as a preprocessing step for model selection) with existing variants of PAC-learning under $L_p$ norms;
• because $L_1$ and $L_2$ are pervasive in (machine) learning;
• because they can.

Their results and methods turn out to be very elegant: to outline only a few, they

• give the first example of testing monotonicity testing (de facto, for the $L_1$ distance) when adaptivity provably helps; that is, a testing algorithm that selects its future queries as a function of the answers it previously got can outperform any tester that commits in advance to all its queries. This settles a longstanding question for testing monotonicity with respect to Hamming distance;
• improve several general results for property testing, also applicable to Hamming testing (e.g. Levin’s investment strategy [3]);
• provide general relations between sample complexity of testing (and tolerant testing) for various norms ($L_0, L_1,L_2,L_p$);
• have quite nice and beautiful algorithms (e.g., testing via partial learning) for testing monotonicity and Lipschitz property;
• give close-to-tight bounds for the problems they consider;
• have slides in which the phrase “Big Data” and a mention to stock markets appear (!);
• have an incredibly neat reduction between $L_1$ and Hamming testing of monotonicity.

I will hereafter only focus on the last of these bullets, one which really tickled my fancy (gosh, my fancy is so ticklish) — for the other ones, I urge you to read the paper. It is a cool paper. Really.

Here is the last bullet, in a slightly more formal fashion — recall that a function $f$ defined on a partially ordered set is monotone if for all comparable inputs $x,y$ such that $x\preceq y$, one has $f(x)\leq f(y)$; and that a one-sided tester is an algorithm which will never reject a “good” function: it can only err on “bad” functions (that is, it may sometimes accept, with small probability, a function far from monotone, but will never reject a monotone function).

Theorem.
Suppose one has a one-sided, non-adaptive tester $T$ for monotonicity of Boolean functions $f\colon X\to \{0,1\}$ with respect to Hamming distance, with query complexity $q$. Then the very same $T$ is also a tester for monotonicity of real-valued functions $f\colon X\to [0,1]$ with respect to $L_1$ distance.

Almost too good to be true: we can recycle testers! How? The idea is to express our real-valued $f$ as some “mixture” of Boolean functions, and use $T$ as if we were accessing these. More precisely, let $f\colon [n]^d \to [0,1]$ be a function which one intends to test for monotonicity. For all thresholds $t\in[0,1]$, the authors define the Boolean function $f_t$ by

$f_t(x) = \begin{cases} 1 & \text{ if } f(x) \geq t\\ 0 & \text{ if } f(x) < t \end{cases}$

All these $f_t$ are Boolean; and one can verify that for all $x$, $f(x)=\int_0^1 f_t(x) dt$. Here comes the twist: one can also show that the distance of $f$ to monotone satisfies

$\mathrm{d}_1(f,\mathcal{M}) = \int_0^1 \mathrm{d}_0(f_t,\mathcal{M}) dt$

i.e. the $L_1$ distance of $f$ to monotone is the integral of the Hamming distances of the $f_t$‘s to monotone. And by a very simple averaging argument, if $f$ is far from monotone, then at least one of the $f_t$‘s must be…
How does that help? Well, take your favorite Boolean, Hamming one-sided (non-adaptive) tester for monotonicity, $T$: being one-sided, it can only reject a function if it has some “evidence” it is not monotone — indeed, if it sees some violation: i.e., a pair $x,y$ with $x\prec y$ but $f(x) > f(y)$.

Feed this tester, instead of the Boolean function it expected, our real-valued $f$; as one of the $f_{t^\ast}$‘s is far from monotone, our tester would reject $f_{t^\ast}$; so it would find a violation of monotonicity by $f_{t^\ast}$ if it were given access to $f_{t^\ast}$. But being non-adaptive, the tester does exactly the same queries on $f$ as it would have done on this $f_{t^\ast}$! And it is not difficult to see that a violation for $f_{t^\ast}$ is still a violation for $f$: so the tester finds a proof that $f$ is not monotone, and rejects. $\square$

Wow.

— Clément.

Final, small remark: one may notice a similarity between $L_1$ testing of functions $f\colon[n]\to[0,1]$ and the “usual” testing (with relation to total variation distance, $\propto L_1$) of distributions $D\colon [n]\to[0,1]$. There is actually a quite important difference, as in the latter the distance is not normalized by $n$ (because distributions have to sum to $1$ anyway). In this sense, there is no direct relation between the two, and the work presented here is indeed novel in every respect.

Edit: thanks to Sofya Raskhodnikova for spotting an imprecision in the original review.

[1] Slides available here: http://grigory.github.io/files/talks/BRY-STOC14.pptm
[2] http://dl.acm.org/citation.cfm?id=2591887
[3] See e.g. Appendix A.2 in “On Multiple Input Problems in Property Testing”, Oded Goldreich. 2013. http://eccc-preview.hpi-web.de/report/2013/067/

# Efficient Distribution Estimation 4: Greek Afternoon – STOC 2014 Recaps (Part 7)

This post is the finale on our series on the Efficient Distribution Estimation workshop. See also, part 1, part 2, and part 3.

After one last caffeination, we gathered once more for the exciting finale to the day — we were in for a very Greek afternoon, with back-to-back tutorials by Ilias Diakonikolas and Costis Daskalakis. As a wise man once said a few blog posts ago, “if you can’t beat them… change what ‘them’ means”. That was the running theme of this session – they showed us how to exploit the structure of a distribution to design more efficient algorithms.

Gautam Kamath on Beyond Histograms: Exploiting Structure in Distribution Estimation

Ilias focused on the latter of these – in particular, $k$-flat distributions. A $k$-flat distribution can be described by $k$ intervals, over which the probability mass function is constant – it literally looks like $k$ flat pieces. Warm up: what if we knew where each piece started and ended? Then the learning problem would be easy: by grouping samples that fall into each interval together, we reduce the support size from $n$ to $k$. Now we can use the same algorithm as for the general case – only this time, it takes $O(k/\varepsilon^2)$ samples.

But we’re rarely lucky enough to know the correct intervals. What should we do? Guess the intervals? Too expensive. Guess the intervals approximately? A bit better, but still pricey. Make the problem a bit easier and allow ourselves to use $k/\varepsilon$ intervals, instead of only $k$? This actually works — while lesser mortals would be satisfied with this solution and call it a day, Ilias refused to leave us with anything less than the grand prize. Can we efficiently output a $k$-flat distribution using only $O(k/\varepsilon^2)$ samples?

Yes, we can, using some tools from Vapnik-Chervonenkis (VC) theory. The VC inequality is a useful tool which allows us to relate the empirical distribution and the true distribution, albeit under a weaker metric than the total variation distance. Skipping some technical details, the key idea is that we want to output a $k$-flat distribution that is close to the empirical distribution under this weaker metric. Using the triangle inequality, specific properties of this metric, and the fact that we’re comparing two $k$-flat distributions, this will give us a $k$-flat distribution which is close to the target in total variation distance, as desired. Computing a $k$-flat distribution that’s close to the empirical one isn’t actually too tough – a careful dynamic program works.

We can learn $k$-flat distributions – so what? This class might strike you as rather narrow, but this result leads to algorithms for a variety of classes of distributions, including monotone, $t$-modal, monotone hazard rate, and log-concave distributions. These classes are all close to $k$-flat, and this algorithm is fine with that. In this sense, this tool captures all these classes at the same time — One algorithm to rule them all, so to speak. This algorithm even directly generalizes to mixtures of these distributions, which is huge — studying mixtures usually makes the problem much more difficult.

Alright, but what’s the catch? Not all distributions are that close to $k$-flat. For example, this algorithm requires $\tilde O(1/\varepsilon^3)$ samples to learn a log-concave distribution, even though the optimal sample-complexity is $O(1/\varepsilon^{5/2})$. It turns out that log-concave distributions are close to $k$-linear, rather than $k$-flat, and we must use a $k$-linear approximation if we want a near-optimal sample complexity.

“Please sir, I want some more.”

You can have it, Oliver — it’s actually possible to learn distributions which are close to mixtures of piecewise polynomials! But this topic is juicy enough that it deserves its own blog post.

Open problems

• The perennial question – what can we do in high dimensions?
• Most of these results are fundamentally improper – they approximate a distribution with a distribution which may be from a different class. Can these techniques lead to computationally efficient proper learning algorithms, where we output a hypothesis from the same class? We already know sample-efficient algorithms, the trouble is the running time.

Gautam Kamath on Beyond Berry Esseen: Structure and Learning of Sums of Random Variables

Finally, we had Costis, who talked about sums of random variables. We would like to understand a class of distributions $\mathcal{D}$ in three (highly related) ways:

• Structure: What “simpler” class of distributions approximates $\mathcal{D}$?
• Covering: Can we generate a small set of distributions $S$, such that for any distribution $D \in \mathcal{D}$, we have a distribution $D^* \in S$ such that $D^*$ is $\varepsilon$-close to $D$?
• Learning: Given sample access to a distribution $D \in \mathcal{D}$, can we output a distribution $D^*$ such that $D^*$ is $\varepsilon$-close to $D$?

Understanding the structure often implies covering results, since we can usually enumerate the simpler class of distributions. And covering implies learning, since generic results allow us to find a “good” distribution from $S$ at the cost of $O(\log |S|)$ samples. It’s not that easy though, since either the details are not obvious, or we’d like to go beyond these generic results.

Consider Poisson Binomial Distributions*, or PBDs for short. A PBD is the sum $X = \sum_i X_i$ of $n$ independent Bernoulli random variables, where $X_i$ is $\mathrm{Bernoulli}(p_i)$. This is like a Binomial distribution on steroids: the Binomial distribution is the special case when all $p_i$ are equal.

PBDs pop up everywhere in math and computer science, so there’s a plethora of classical structural results – with a few catches. For example, check out a result by Le Cam, which approximates a PBD by a Poisson distribution with the same mean:

$d_{TV}\left(\sum_i X_i, \mathrm{Poisson}\left(\sum_i p_i\right)\right) \leq \sum_i p_i^2$

Catch one: this structural result describes only a small fraction of PBDs – it gives a good approximation when the $p_i$ values are really small. Catch two: we’re approximating a PBD with a Poisson, a distribution from a different family – we’d ideally like to have a proper approximation, that is, approximate the PBD with another PBD.

Recent results from our community show that you can get around both of these issues at once (woo, chalk one up for the computer scientists!). Structurally, we know the following: any PBD is $\varepsilon$-close to either a Binomial distribution or a shifted PBD with the parameter $n$ replaced by $O(1/\varepsilon^3)$ — a much smaller number of “coin flips”. While the original distribution had $n$ different $p_i$ parameters, the approximating distribution has only $O(1/\varepsilon^3)$ parameters. Since $n$ is usually way bigger than $1/\varepsilon$, this is a huge reduction in complexity. By carefully enumerating the distributions in this structural result, we can get a small cover. Specifically, the size of the cover is polynomial in $n$ and quasi-polynomial in $1/\varepsilon$, while the naive cover is exponential in $n$. Now, using the generic reduction mentioned before, we can learn the distribution with $\mathrm{polylog}(n)$ samples. Again though, let’s pretend that $n$ is so big that it makes André the Giant feel insecure – can we take it out of the picture? If you’ve been paying any attention to my rhetorical style, you might predict the answer is yes (and you’d be right): $O(1/\varepsilon^2)$ samples suffice for learning.

Is that the end of the story? No, we need to go deeper — let’s generalize PBDs once more: Sums of Independent Integer Random Variables (SIIRVs)**. A $k$-SIIRV is the sum $X = \sum_i X_i$ of $n$ independent random variables ($k$-IRVs), where each $X_i \in \{1, \dots, k\}$. Note that when $k = 2$, this is essentially equivalent to a PBD. PBDs are actually quite tame, and have many nice properties – for example, they’re unimodal and log-concave. However, even a $3$-SIIRV is far from being $k$-modal or log-concave. After some careful analysis and modular arithmetic, it turns out that a $k$-SIIRV with sufficiently large variance is approximated by $cZ + Y$, where $c$ is some integer $\in \{1, \dots, k-1\}$, $Z$ is a discretized Gaussian, $Y$ is a $c$-IRV, and $Y$ and $Z$ are independent. On the other hand, if the variance is not large, the distribution has a sparse support – it can be approximated by a $(k/\varepsilon)$-IRV. This structural observation leads to a learning result with sample complexity which is polynomial in $k$ and $1/\varepsilon$, but once again independent of $n$.

A particularly sinister 3-SIIRV, far from being 3-modal or log-concave. Image taken from Costis’ slides.

Open problems

• Current results for PBDs are quasi-polynomial in $1/\varepsilon$, for both the cover size and the running time for the proper learning result. Can we make this polynomial?
• For $k$-SIIRVs, our current understanding is fundamentally improper. Can we obtain proper covers or learning algorithms for this class?
• What about higher dimensions? We have some preliminary results on the multidimensional generalization of PBDs (joint work with Costis and Christos Tzamos), but we’re totally in the dark when it comes to generalizing $k$-SIIRVs.

-G

*For maximum confusion, Poisson Binomial Distributions have nothing to do with Poisson Distributions, except that they were both named after our good friend Siméon.
**For another perspective on the SIIRV result, see this post by my partner in crime, Clément.

[1] Learning k-modal distributions via testing, Daskalakis, Diakonikolas, Servedio (2012).
[2] Approximating and Testing k-Histogram Distributions in Sub-Linear time, Indyk, Levi, Rubinfeld (2012).
[3] Learning Poisson Binomial Distributions, Daskalakis, Diakonikolas, Servedio (2012).
[4] Learning Mixtures of Structured Distributions over Discrete Domains, Chan, Diakonikolas, Servedio, Sun (2013).
[5] Testing k-modal Distributions: Optimal Algorithms via Reductions, Daskalakis, Diakonikolas, Servedio, Valiant, Valiant (2013).
[6] Learning Sums of Independent Integer Random Variables, Daskalakis, Diakonikolas, Servedio (2013).
[7] Efficient Density Estimation via Piecewise Polynomial Approximation, Chan, Diakonikolas, Servedio, Sun (2013).
[9] Faster and Sample Near-Optimal Algorithms for Proper Learning Mixtures of Gaussians, Daskalakis, Kamath (2014).
[10] Near-optimal-sample estimators for spherical Gaussian mixtures, Acharya, Jafarpour, Orlitsky, Suresh (2014).

# Efficient Distribution Estimation 3: Rocco’s Modern Life – STOC 2014 Recaps (Part 5)

The road twists, as Clément takes us down a slightly different path in the penultimate post of our Efficient Distribution Estimation workshop summary series. See also: Part 1 and Part 2.

Clément Canonne on A complexity theoretic perspective on unsupervised learning

And now, for something completely different… Rocco Servedio presented a (joint with Ilias Diakonikolas and Anindya De [1]) work, not dealing with distribution testing. Showing there is a world outside distribution testing, and that this world is filled with wonder, probabilities and Boolean functions. A world of inverse problems.

I will only try to give the main idea, maybe the main ideas, of the problem they consider, why it is hard, and how they overcame the difficulties. To start, fix an unknown Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ in some class $\mathcal{C}$ — think of the class of linear threshold functions, for instance (LTFs, a.k.a. weighted majorities, or halfspaces). $f$ is unknown, but it brings with it the set of its satisfying inputs, $I\stackrel{\rm{}def}{=} f^{-1}(\{1\})$; and you can get uniform samples from this $I$. You press a button, you get $x\in I$.

Can you do this a few time, and then output a new device with its own button, that will generate on demand i.i.d. samples from some distribution $D$ close (in total variation distance, as usual) to the uniform distribution on $I$? In other terms, can you approximate the uniform distribution on $I$, by observing i.i.d. samples from $I$ (but without knowing $f$)?

Of course, one way to do this would be to use a learning algorithm for $\mathcal{C}$ (but using only what we get: only positive examples): learn an approximate version $\hat{f}$ of $f$, and then study it “offline” to find out exactly what $\hat{f}^{-1}(\{1\})$ is, before spitting uniform samples from it. Clearly, if $\hat{f}$ is close enough to ${f}$, this will work!

But does it always? Recall that here, we want to be close to uniform on $I$, that is we only consider the error with relation to the positive examples of the function; while the general results for (PAC-)learning of functions under the uniform distribution only care about the overall error. This means $\hat{f}$ may need to be very close to $f$ for this approach to succeed… As an example, take $f$ to be $\mathrm{AND}_n$, i.e. 0 everywhere but on the single point $x=1^n$. The function $\hat{f}= \lnot\mathrm{OR}_n$ is a very good approximation: it has error only $1/2^{n-1}$; and yet, this would lead to a distribution supported on the single point $0^n$, which is thus 1-far from the actual distribution (supported on $1^n$)!

Intuitively, the problem here is that PAC-learning gives a sort of additive guarantee:

$\mathbb{P}\{\hat{f}(x) \neq f(x)\} \leq \varepsilon$

while we need some some of multiplicative one:

$\mathbb{P}\{\hat{f}(x) \neq f(x)\} \leq \varepsilon\cdot \mathbb{P}\{f(x) = 1\}$

which is much harder to obtain when $f$ has very few satisfying assignments.

This being said, the discussion above shows that when we are in the dense case, things look simpler: i.e., when $f$ has “many” (an inverse polynomial fraction of the hypercube) satisfying inputs. In this case, the idea is to combine two steps — and show they can be performed:
(1) SQ Learning: get a hypothesis $\hat{f}$. Upshot: one has to show that it’s possible to use a SQ learner, even from positive examples only.
(2) Hypothesis testing: test if $\hat{f}$ is good enough (if not, try again)
before then using the now explicitly known “good” $\hat{f}$ to approximate the set of satisfying inputs of $f$.

But then, once this “simple” (erm) case is dealt with, how to handle the general case, where $f$ could have an exponentially small fraction of satisfying inputs? Rocco explained one of their main ingredient — what they called a “densifier”, i.e. an algorithm that will start from positive examples from $f$ and output a “densified function” $g$ which agrees with $f$ on most of $f$‘s satisfying inputs, and does not have many more (so that $f^{-1}(\{1\})$ is “dense” in $g^{-1}(\{1\})$). Provided such a densifier exists (and can be efficiently simulated) for a class $\mathcal{C}$, they show how to combine it with (1) and (2) above to iteratively “densify” the sparse set $f^{-1}(\{1\})$ and reduce to the “easy”, dense case.

Now that all the bricks are in place, the next step in order to play Lego is, of course, to show that such densifiers do exist… for interesting classes, and in particular classes who also have SQ learning algorithms (otherwise, brick (1) is still missing!). In the paper, the authors describe how to build such a densifier for (for instance) the class of LTFs. Combining this with the ingredients sketched above, this gives them a polynomial time algorithm for learning the uniform distribution over satisfying assignments to an unknown LTF. As another instantiation of their approach, the authors also give a densifier for DNF formulas and a resulting quasi-polynomial time distribution learning algorithm.

Two remarks here:

• maybe surprisingly, the hardness of this problem stems from the Boolean setting — indeed, the same question, dealing with functions $f\colon\mathbb{R}^n\to\{0,1\}$ on the Gaussian space, becomes way easier;
• the related “direct” problem (on input the full specification of $f$, generate close-to-uniform samples from $f^{-1}(\{1\})$), is neither harder nor simpler: the authors provide examples where (under some suitable well-believed assumption) the former is easy and the latter hard, and vice-versa. In particular, if $f^{-1}(\{1\})$ is a finite cyclic group, getting even one positive example allows the algorithm to get easily the whole $f^{-1}(\{1\})$ “for free”.

[1] Inverse Problems in Approximate Uniform Generation, De, Diakonikolas, Servedio, 2012.

# Efficient Distribution Estimation 2: The Valiant Sequel – STOC 2014 Recaps (Part 4)

This is a continuation of Efficient Distribution Estimation STOC workshop summary series by G and Clément (see the Part 1 here).

Gautam Kamath on Distributional Property Testing and Estimation: Past, Present, and Future

After lunch, we were lucky enough to witness back-to-back talks by the Valiant brothers, Greg and Paul. If you aren’t familiar with their work, first of all, how have you been dealing with all of your everyday property testing needs? But second of all, over the past few years, they’ve touched a number of classic statistical problems and left most of them with matching optimal upper and lower bounds.

Greg kicked off the show and focused on property estimation. Given a property of interest and access to independent draws from a fixed distribution $\mathcal{D}$ over $\{1, \dots, n\}$, how many draws are necessary to estimate the property accurately?

Of course, this is an incredibly broad problem. For example, properties of a distribution which could be of interest include the mean, the $L^{3}$ norm, or the probability that the samples will spell out tomorrow’s winning lottery numbers. In order to tame this mighty set of problems, we focus on a class of properties which are slightly easier to understand — symmetric properties. These are properties which are invariant to permuting the labels on the support — for instance, if we decided to say 3 was now called 12 and vice-versa, the property wouldn’t change. Notable examples of symmetric properties include the support size*, entropy, and distance between two distributions. Still a broad class of properties, but surprisingly, we can make statements about all of them at once.

What’s the secret? For symmetric properties, it turns out that the fingerprint of a set of samples contains all the relevant information. The fingerprint of a set of samples $X$ is a vector $\mathcal{F}_X$ whose $i$th component is the number of elements in the domain which occur exactly $i$ times in $X$. The CliffNotes version of the rest of this post is that using linear programming on the fingerprint gives us an unreasonable amount of power**.

First off, Greg talked about a surprising dichotomy which demonstrates the power of linear estimators. A linear estimator of a property is an estimator which outputs a linear combination of the fingerprint. Given a property $\pi$, a number of samples $k$, and a distance $\varepsilon$, exactly one of the following holds:

• There exist distributions $y_1$ and $y_2$ such that $|\pi(y_1) - \pi(y_2)| > \varepsilon$ but no estimator (linear or not) that uses only $k$ samples can distinguish the two with probability $\geq 51\%$.
• There exists a linear estimator which requires $k(1 + o(1))$ samples and estimates the property to within $\varepsilon(1 + o(1))$ with probability $\geq 99\%$.

In other words, if there exists an estimator for a property, there is a linear estimator which is almost as good. Even better, their result is constructive, for both cases! A linear program is used to construct the estimator, where the variables are the estimator coefficients. Roughly, this linear program attempts to minimize the bias of the estimator while keeping the variance relatively small. On the other hand, when we take the dual of this linear program, we construct two distributions which have similar fingerprint expectations, but differ in their value for the property — we construct an explicit counter-example to any estimator which claims to use $k$ samples and estimate the property to accuracy $\varepsilon$.

But here’s the \$64 question – do these estimators work in practice? Sadly, the answer is no, which actually isn’t that surprising here. The estimator is defined in terms of the worst-case instance for each property — in other words, it’s oblivious to the particular distribution it receives samples from, which can be wasteful in many cases.

Let’s approach the problem again. What if we took the fingerprint of our samples and computed the property for the empirical distribution? This actually works great – the empirical distribution optimally estimates the portion of the distribution which we have seen. The downside is that we have to see the entire distribution, which takes $\Omega(n)$ samples. If we want to break past this linear barrier, we must estimate the unseen portion. It turns out that, once again, linear programming saves the day. We solve a program which describes all distributions whose expected fingerprint is close to the observed fingerprint. It can be shown that the diameter of this set is small, meaning that any distribution in this space will approximate the target distribution for all symmetric properties at the same time! Furthermore, this estimator takes only $O(n/\log n)$ samples, which turns out to be optimal.

Again, we ask the same question – do these estimators work in practice? This time, the answer is yes! While performance plots are often conspicuously missing from theory papers, Greg was happy to compare their results to estimators used in practice — their approach outperformed the benchmarks in almost all instances and regimes.

Finally, Greg mentioned a very recent result on instance-by-instance optimal identity testing. Recall that in identity testing, given a known distribution $P$ and an unknown distribution $Q$, we wish to test if they are equal or $\varepsilon$-far (in total variation distance). As mentioned before, when $P$ is uniform, $\Theta(\sqrt{n}/\varepsilon^2)$ are necessary and sufficient for this problem. But what about when $P$ is $\mathrm{Bin}(n,\frac12)$, or when $P$ assigns a probability to $i$ which is proportional to the $i$th digit of $\pi$ — do we have to design an algorithm by scratch for every $P$?

Thankfully, the answer is no. The Valiants provide an algorithm which is optimal up to constant factors for all but the most pathological instances of $P$. Somewhat unexpectedly, the optimal number of samples turns out to be $\Theta(\|P\|_{2/3}/\varepsilon^2)$ – aside from the fact it matches the uniform case, it’s not obvious why the $2/3$-norm is the “correct” norm here. Their estimator is reasonably simple – it involves a small tweak to Pearson’s classical $\chi$-squared test.

-G

Clément Canonne on Lower Bound Techniques for Statistical Estimation Tasks

At this point, Greg tagged out, allowing Paul to give us a primer in “Lower Bound Techniques for Statistical Estimation Tasks”: roughly speaking, the other side of Greg’s coin. Indeed, all the techniques used to derive upper bounds (general, systematic testers for symmetric properties) can also been turned into their evil counterparts: how to show no algorithm supposed to test a given property can be “too efficient”.

The dish is very different, yet the ingredients similar: without going into too much details, let’s sketch the recipe. In the following, keep in mind that we will mostly cook against symmetric properties, that is properties $\mathcal{P}$ invariant by relabeling of the domain elements: if $D\in\mathcal{P}$, then for any permutation $\pi$ of $[n]$ one has $D\circ\pi\in\mathcal{P}$.

• elements do not matter: all information any tester can use can be found in the fingerprint of the samples it obtained. In other terms, we can completely forget the actual samples for the sake of our lower bound, and only prove things about their fingerprint, fingerprint of fingerprint, and so on.
• Poissonization is our friend: if we take $m$ samples from a distribution, then the fingerprint, a tuple of $m$ integer random variables $X_1,\dots X_m$, lacks a very nice feature: the $X_i$‘s are not independent (they do kind of have to sum to $m$, for instance). But if instead of this, we were to take $m^\prime\sim\mathrm{Poisson}(m)$ samples, then the individual components of the fingerprint will be independent! Even better, many nice properties of the Poisson distribution will apply: the random variable $m^\prime$ will be tightly concentrated around its mean $m$, for instance. (this applies whether the property is symmetric or not. When in doubt, Poissonize. Independence!)
• Central Limit Theorems: the title ain’t witty here, but the technique is. As a very high-level, imagine we get to random variables $T, S$ corresponding to the “transcript” of what a tester/algorithm sees from taking $m$ samples from two possible distributions it should distinguish. We would like to argue that the distributions of $T$ and $S$ are close, so close (in a well-defined sense) that it is information-theoretically impossible to do so. This can be very difficult; a general idea would be to prove that there exists another distribution $G$, depending on very few parameters — think of a Gaussian: only the mean and (co)variance (matrix) — such that both $S$ and $T$ are close to $G$. Then, by the triangle inequality, they must be close, and we are done.Sweeping a lot of details under the rug, what Paul presented is this very elegant idea of proving new Central Limit Theorems that exactly give that: theorems that state that asymptotically (with quantitative bounds on the convergence rate), some class of distributions of interest will “behave like” a multivariate Gaussian or Multinomial with the “right parameters” — where “behave like” refers to a convient metric on distributions (e.g. Earthmover or Total Variation), and “right parameters” means “as few as possible”, in order to keep as many degrees of freedom when designing the distributions of $S$ and $T$).Proving such CLT’s is no easy task, but — on the bright side — they have many applications, even outside the field of distribution testing.

There would be much, much more to cover here, but I’m past 600 words already, and was only allowed $\mathrm{Poisson}(500)$.

-Clément

*In order to avoid pathological cases for estimating support size, we assume none of the probability are in the range $(0,\frac{1}{n})$.

**At least, when channeled by sufficiently talented theoreticians.

[1] Estimating the unseen: A sublinear-sample canonical estimator of distributions, Valiant, Valiant (2010).
[2] A CLT and tight lower bounds for estimating entropy, Valiant, Valiant (2010).
[3] Estimating the Unseen: An n/log(n)-sample Estimator for Entropy and Support Size, Shown Optimal via New CLTs, Valiant, Valiant (2011).
[4] Estimating the Unseen: Improved Estimators for Entropy and other Properties, Valiant, Valiant (2013).
[5] An Automatic Inequality Prover and Instance Optimal Identity Testing, Valiant, Valiant (2014).
[6] Testing Symmetric Properties of Distributions, Valiant (2011).

# Efficient Distribution Estimation – STOC 2014 Recaps (Part 2)

The STOC 2014 saga continues (check out Part 1 here), with Clément Canonne and Gautam “G” Kamath covering the day-long workshop on Efficient Distribution Estimation. There’s a wealth of interesting stuff from the workshop, so we’ve broken it up into episodes. Enjoy this one, and stay tuned for the next installment!

Clément Canonne on Taming distributions over big domains

Talk slides here

Before STOC officially began, before the talks and the awe and the New Yorker Hotel and all the neat and cool results the reviews on this blog will try to give an idea and a sense of, before even this very long and cumbersome and almost impossible — almost, only? to understand sentence, before all this came Saturday.
Workshops.

Three simultaneous sessions, all three very tempting: one covering “Recent Advances on the Lovasz Local Lemma”, one giving tools and insight on how to “[overcome] the intractability bottleneck in unsupervised machine learning”, and finally one — yay! — on “Efficient Distribution Estimation“. I will hereby focus on the third one, organized by Ilias Diakonikolas and Gregory Valiant, as it is the one I attended, and in particular 3 of the 6 talks of the workshop; G (Gautam) will review the other ones, I bet with shorter sentences and more relevance.

It all started with Ronitt Rubinfeld, from MIT and Tel Aviv University, describing the flavour of problems this line of work was dealing with, what exactly were the goals, and why should one care about this. Imagine we have samples from a space of data — a Big space of Data; whether it be records from car sales, or the past ten years of results for the New Jersey lottery, or even logs of network traffic; and we want to know something about the underlying distribution: has the trend in cars changed recently? Is the lottery fixed? Is someone doing unusual requests to our server?

How can we tell?

Let us rephrase slightly the problem: there exists a probability distribution $D$ on a set of $n$ elements $[n]=\{1,\dots,n\}$; given i.i.d. (independent, identically distributed) samples from $D$, can we decide with as few of these samples as possible if, for instance,

• $D$ is uniform, or “far”* from it?
• $D$ is equal to a known, fully specified distribution (say, the distribution of car sales from 10 years ago), or far from it?
• has $D$ big entropy, or not?

Of course, there exist many tools in statistics for this type of tasks: the Chi-Square test, for instance. Yet, they usually have a fatal flaw: namely, to give good guarantees they require a number of samples at least linear in $n$. And $n$ is big. Like, huge. (and anyway, one could always actually learn $D$ to high accuracy with $O(n)$ samples) No, the question is really can one do anything interesting with $o(n)$ samples?

As Ronitt explained: yes, one can. Testing uniformity can be done with (only!) $O(\sqrt{n})$ samples**; one can test identity to a known distribution with $O(\sqrt{n})$ as well; and testing whether two unknown distributions are equal or far from each other only requires $O(n^{2/3})$ samples from each. All these results (see e.g. [1] for a survey of this huge body of research) are quite recent; and come from works in TCS spanning the last 15 years or so.

“And they saw they could test distributions in $o(n)$ samples; and they saw that this was good, and they also saw that most of these results were tight”. But as always, the same question arose: can we do better? This may sound strange: if the sample complexities are tight, how could we even hope to do better? If there is a lower bound, how can we even ask to beat it?

This was the second part of Ronitt’s talk: bypassing impossibility results. She presented two different approaches, focusing on the second:

• “if you can’t beat them… change what ‘them’ means”. Indeed, getting better algorithms for general, arbitrary distributions may be impossible; but in practice, things are rarely arbitrary. What if we knew something more about this unknown $D$ — for instance, it is a mixture of simple distributions? Or it has a particular “shape” (e.g., its probability mass function is monotone)? Or it is obtained by summing many “nice” and simple random variables? Under these assumptions, can the structure of $D$ leveraged to get better, more sample efficients algorithms? (in short: yes. See for instance [2], where the sample complexity, from $n^c$ in general, becomes $\mathrm{poly}(\log n)$)
• “if playing by the rules is too hard, change the rules”. All the results and lower bound above suppose the algorithms have (only) access to i.i.d. samples from the distribution $D$. But in many cases, they could get more: for instance, there are scenarii where a tester can focus on some subset of the domain, and only get samples from it (think about “stratified sampling”, or just the situation where a pollster targets in particular voters between age 25 and 46). Given this new type of access to the “conditional distributions” induced by $D$, can algorithms be more efficient? And if the data comes from a known, preprocessed (even sorted!) dataset, as it is the case for example with Google and the $n$-grams (frequencies and statistics of words in a vast body of English literature), the testing algorithm can actually ask the exact frequency (equivalently, the probability value) of any given element of its choosing, in addition to just sampling a random word! This type of access, with additionial or modified types of query to $D$, has been studied for instance in [3,4,5,6]; and, in a nutshell, the answer is positive.

Yes, Virginia, there is a Santa Claus. And he has constant sample complexity!

Open problems

• Study more types of structures or assumptions on the distribution: namely, what type of natural assumptions on $D$ can we get, for which the testing algorithms get a significant speedup? (in terms of sample complexity)
• What is the relation between the new type of access? Are they related? Can one simulate the other? What other natural sorts of access can one consider?
• What about learning with these new types of access? (so far, we were interested in testing whether $D$ had some property of interest; but — maybe — learning the distribution also became easier?)

Such excitement — and this was only the beginning!

Clément

* Far in total variation (or statistical) distance: this is essentially (up to a factor 2) the $\ell_1$ distance between two probability distributions.
** and this is tight.

(this bibliography is far from comprehensive; if any reader wants additional pointers, I’ll be happy to provide more!)

[1] Taming big probability distributions, R. Rubinfeld (2012).
[2] Testing k-Modal Distributions: Optimal Algorithms via Reductions, Daskalakis, Diakonikolas, Servedio, Valiant, Valiant (2011).
[3] On the Power of Conditional Samples in Distribution Testing, Chakraborty, Fischer, Goldhirsh, Matsliah (2012).
[4] Testing probability distributions using conditional samples, Canonne, Ron, Servedio (2012).
[5] Streaming and Sublinear Approximation of Entropy and Information Distances, Guha, McGregor, Venkatasubramanian (2005).
[6] Testing probability distributions underlying aggregated data, Canonne, Rubinfeld (2014).