Explanations as Messages over a Communication Channel

I have made a cartoon illustration in Figure 1, which shows how we can see the task of explainability as being analogous to setting up a communication channel in information theory: the object to communicate is (some aspect of) a complicated function \(f\) that is learned by a neural network or other machine learning model; then explanations encode this object, and they are transmitted to a user. But users can only absorb a limited amount of information, so there is a bandwidth limit on how much information explanations can maximally contain. We may perhaps stretch this limit slightly by designing good user interfaces, but the central challenge is to restrict what we want to communicate, so that explanations can be sufficiently concise. As a consequence, we will never be able to communicate millions of internal parameters from e.g. a neural network in their raw form.

Limiting the Amount of Information

The conclusion from the communication channel perspective is that we must limit the amount of information that we want to communicate. So what are possible ways do this?

Option 1: Restrict Attention to Simple Models

The most attractive solution, whenever possible, is to start out with a model \(f\) that is sufficiently simple. For instance, if \(f\) can be represented by a small decision tree, then we can describe the structure of the tree in a limited number of bits, so we can communicate \(f\) in full.

This is the approach advocated by Rudin and collaborators, who have put much effort into identifying high-stakes machine learning applications where interpretable models like small decision trees can provide high accuracy (missing reference). Even though such models are very restricted (by definition), there still turn out to be surprisingly many such applications [1]. Relatedly, Bressan et al. [2] have recently developed a mathematically sophisticated characterization that captures how well models \(f\) can be approximated by decision trees.

Option 2: Select Information that is Relevant to a Specific Task

Can we still do something when \(f\) is inherently complicated? Yes, but we will have to choose which aspects of \(f\) we are going to communicate. How do we select the right aspects? This may be possible if we know that the user is trying to accomplish a specific task, because then we can communicate only the specific aspects of \(f\) that are relevant to that particular task.

For instance, maybe the user only cares about verifying that people similar to them get approximately the same predictions. In this case we may communicate the range of values taken by \(f\) in a neighborhood of the user’s features, instead of communicating all of \(f\). Another example comes from algorithmic recourse [3]: if a user’s bank loan is denied by a credit scoring algorithm \(f\), they may want to know how to improve their loan application to get the loan when they reapply. In this case, it may be sufficient to communicate a so-called counterfactual option to the user that tells them the easiest changes they can make to their application to get a loan.¹

Option 3: Exploit Common Background Knowledge

Finally, a third option to consider is that the user may have domain knowledge that is relevant to understanding \(f\). In this case it is possible that \(f\) is very complicated by itself, but, conditional on the user’s background knowledge, \(f\) can be described with a limited number of bits. This fits, for instance, with concept bottleneck models [6], which train \(f\) to be a simple function \(g\) of a given set of concepts that the user already understands.² In this case, we do not need to communicate the concepts, because the user already knows them, and it is sufficient to communicate \(g\), which is feasible.

Conclusion

In this post, I have described a general perspective on explainability that gives a way to think about the fundamental limitations of what can be explained. An important issue I haven’t touched upon, is that it is not necessarily sufficient to transmit the desired information over a communication channel, because the receiving party (the user) also needs to be able to decode the message. For explainability, this means that certain users will have limitations on the types of messages they can understand, which may force the explanations to use a less efficient coding scheme than would be information-theoretically possible.

References

1. L. Semenova, C. Rudin, and R. Parr, “On the Existence of Simpler Machine Learning Models,” in Proceedings of the 2022 ACM Conference on Fairness, Accountability, and Transparency, New York, NY, USA, 2022, pp. 1827–1858, doi: 10.1145/3531146.3533232.
2. M. Bressan, N. Cesa-Bianchi, E. Esposito, Y. Mansour, S. Moran, and M. Thiessen, “A Theory of Interpretable Approximations,” in Proceedings of Thirty Seventh Conference on Learning Theory (COLT), 2024, vol. 247, pp. 648–668, [Online]. Available at: https://proceedings.mlr.press/v247/bressan24a.html.
3. A.-H. Karimi, G. Barthe, B. Schölkopf, and I. Valera, “A survey of algorithmic recourse: definitions, formulations, solutions, and prospects,” arXiv preprint arXiv:2010.04050, 2021.
4. H. Fokkema, D. Garreau, and T. van Erven, “The Risks of Recourse in Binary Classification,” in Proceedings of the 27th International Conference on Artificial Intelligence and Statistics (AISTATS), 2024, vol. 238, pp. 550–558, [Online]. Available at: https://proceedings.mlr.press/v238/fokkema24a.html.
5. S. Milli, J. Miller, A. D. Dragan, and M. Hardt, “The Social Cost of Strategic Classification,” 2019.
6. P. W. Koh et al., “Concept Bottleneck Models,” in Proceedings of the 37th International Conference on Machine Learning, 2020, vol. 119, pp. 5338–5348, [Online]. Available at: https://proceedings.mlr.press/v119/koh20a.html.
7. Y. Goyal, A. Feder, U. Shalit, and B. Kim, “Explaining classifiers with causal concept effect (CaCE),” arXiv preprint arXiv:1907.07165, 2019.
8. A. Margeloiu, M. Ashman, U. Bhatt, Y. Chen, M. Jamnik, and A. Weller, “Do Concept Bottleneck Models Learn as Intended?,” arXiv preprint arXiv:2106.13314, 2021.
9. A. Mahinpei, J. Clark, I. Lage, F. Doshi-Velez, and W. Pan, “Promises and Pitfalls of Black-Box Concept Learning Models,” arXiv preprint arXiv:2106.13314, 2021.
10. M. E. Zarlenga et al., “Towards robust metrics for concept representation evaluation,” 2023.
11. H. Fokkema, T. van Erven, and S. Magliacane, “Sample-efficient Learning of Concepts with Theoretical Guarantees: from Data to Concepts without Interventions,” in Advances in Neural Information Processing Systems (NeurIPS), 2025, vol. 38, pp. 111783–111843, [Online]. Available at: https://proceedings.neurips.cc/paper_files/paper/2025/file/a23fa41edb52c314c058fd5ce97217d5-Paper-Conference.pdf.

Setting: Online Convex Optimization

We consider the following online learning setting, which is called \(\DeclareMathOperator{\E}{\mathbb{E}} \DeclareMathOperator{\argmin}{arg\,min} \DeclareMathOperator{\diameter}{diameter} \DeclareMathOperator{\proj}{Proj} \newcommand{\u}{\boldsymbol u} \newcommand{\v}{\boldsymbol v} \newcommand{\w}{\boldsymbol w} \newcommand{\grad}{\boldsymbol g} \newcommand{\clipgrad}{\bar \grad} \newcommand{\tf}{\tilde f} \newcommand{\tR}{\tilde R} \newcommand{\domain}{\mathcal{W}} \newcommand{\reals}{\mathbb{R}} \newcommand{\Dmax}{D_\text{max}} \newcommand{\Gmax}{G_\text{max}} \newcommand{\Done}{D_1}\) Online Convex Optimization (OCO): Each round \(t=1,\ldots,T\), a learner predicts a point \(\w_t\) from a convex domain \(\domain \subset \reals^d\) of diameter at most \(\diameter(\domain) \leq D\). The environment then selects a convex loss function \(f_t : \domain \to \reals\) and the learner incurs loss \(f_t(\w_t)\). Most online learning algorithms update their predictions efficiently, using only the gradient \(\grad_t = \nabla f_t(\w_t)\), and the objective is for the learner to control the regret

for all \(\u \in \domain\) simultaneously.

Two Flavors of Online Gradient Descent

For simplicity, I will focus on online gradient descent (OGD), but the same discussion equally applies to other instances of OMD and FTRL. I am also considering here the version of FTRL with linearized losses, which makes it line up well with OMD. Given a starting point \(\w_1 = 0\) and a sequence of stepsizes \(\eta_1,\eta_2,\ldots\), the two versions of OGD are defined as follows:

OMD Version: \(\hspace{3em} \begin{align*} \w_{t+1} &= \argmin_{w \in \domain} \w^\top \grad_t + \frac{1}{2\eta_t} \|\w - \w_t\|^2\\ &= \proj(\w_t - \eta_t \grad_t) \end{align*}\)

FTRL Version: \(\hspace{5em} \begin{align*} \w_{t+1} &= \argmin_{w \in \domain} \sum_{s=1}^t \w^\top \grad_s + \frac{1}{2\eta_t} \|\w - \w_1\|^2\\ &= \proj(\w_1 - \eta_t \sum_{s=1}^t \grad_s), \end{align*}\)

where \(\proj(\tilde \w) = \argmin_{\w \in \domain} \|\w - \tilde \w\|\) is the Euclidean projection of \(\tilde \w\) onto the allowed domain. (All norms in this post are standard Euclidean norms.)

The difference between the two versions is that OMD updates from the previous iterate \(\w_t\) using only the latest gradient \(\grad_t\), whereas FTRL updates from the starting point \(\w_1\) using all the gradients. Computationally, both versions are equally efficient, because you end up storing a sum of gradients in both cases.

In fact, if the step size \(\eta_t = \eta\) is constant and there are no projections, then the two methods even coincide! This is highly suboptimal in practice, however, where we always want to use decreasing step sizes. To illustrate this, suppose we would set \(\eta \leq \text{constant}/\sqrt{T}\), then the algorithm would essentially be stuck at its starting point \(\w_1\) for roughly the first \(\sqrt{T}\) rounds. But if we set \(\eta \gg 1/\sqrt{T}\), then the iterates can become too unstable for large \(t\), and suffer large regret as a result.

Regret Bounds

On to regret bounds! The regret of OGD has been extensively analyzed in the literature. For instance, from the proof of Theorem 1 in Zinkevich’s well-known paper it can be seen that the OMD version satisfies the following guarantee:

Theorem 1 (OMD Version): For any non-increasing step sizes \(\eta_1\geq \eta_2 \geq \ldots\), the OMD version of OGD guarantees regret at most

Consequently, if we set \(\eta_t = \frac{\Dmax}{\sqrt{\sum_{s=1}^t \|\grad_s\|^2}}\) for \(\Dmax \geq \max_{t \in \{1,\ldots,T\}} \|\w_t - \u\|\), then

where \(\Gmax = \max_{t \in \{1,\ldots,T\}} \|\grad_t\|\).

A good reference to look up the standard analysis of FTRL is Corollary 7.9 in Orabona’s very nice introduction to online learning, which specializes as follows to OGD:

Theorem 2 (FTRL Version): For any non-increasing step sizes \(\eta_1\geq \eta_2 \geq \ldots\), the FTRL version of OGD guarantees regret at most

Consequently, if we set \(\eta_t = \frac{\Done}{\sqrt{\Gmax^2 + \sum_{s=1}^t \|\grad_s\|^2}}\) for \(\Done \geq \|\w_1 - \u\|\) with \(\Gmax\) as in Theorem 1, then

Two Big Differences

Comparing \(\eqref{eqn:mdbound}\) and \(\eqref{eqn:ftrlbound}\), we see that the bounds coincide when \(\eta_t = \eta\) is a constant, but they start to differ significantly in the common case that the step sizes are decreasing. The two main differences between Theorems 1 and 2 are as follows:

1. Advantage of FTRL over OMD: \(\Done \ll \Dmax\)

Comparing the two theorems, we see that the guarantee for OMD depends on \(\Dmax\) whereas the guarantee for FTRL only depends on \(\Done \leq \Dmax\). This matters, because, after tuning the step sizes, \(\Done\) and \(\Dmax\) end up as multiplicative factors in front of the regret bounds, so they have a big impact on the regret. To see that the difference can be significant, note that, if the domain \(\domain\) is large compared to \(\Done\) (or even unbounded) and the data are non-stationary, then the intermediate iterates can move all over the place and consequently we will have that \(\Dmax \gg \Done\), implying that OMD is much worse than FTRL. Moreover, we can directly influence \(\Done\) by changing the starting position \(\w_1\) of the algorithm, but \(\Dmax\) is not easy to control, because it depends on all the iterates of the algorithm, which are heavily dependent on the data. Now we might still hope that the difference between \(\Done\) and \(\Dmax\) is just an artifact of the bounds and that things won’t be as bad in practice, but this hope turns out to be vain, because Orabona and Pál (Theorem 3) construct a counterexample where the difference is real.

2. Advantage of OMD over FTRL: Lipschitz-adaptivity

There is another difference between Theorems 1 and 2, which is so subtle that it is easily overlooked when reading the literature: compared to \(\eqref{eqn:mdbound}\) the step sizes in the sum \(\sum_{t=1}^T \eta_{t-1} \|\grad_t\|^2\) in \(\eqref{eqn:ftrlbound}\) are one time step behind.¹ At first sight, this may seem like a minor difference, but it is in fact crucial if we do not know the sizes of our gradients in advance (and in practice we never do!). To see this, note that for OMD the sum can be controlled by setting

But getting the same control in FTRL would require

which is impossible because \(\|\grad_{t+1}\|\) is unknown at the time of choosing \(\eta_t\). The solution shown in Theorem 2 is to fall back on

which replaces \(\|\grad_{t+1}\|\) by the upper bound \(\Gmax\). But this is not really a solution at all, because it leaves us stuck with \(\Gmax\) as a hyper-parameter that we need to set. Now a seasoned machine learning practitioner may not easily be intimidated by having to tune yet another hyper-parameter, but \(\Gmax\) is particularly nasty. Indeed, \(\Gmax\) is a moving target, because changing its estimate changes the iterates \(\w_t\) of the algorithm, which in turn changes the gradients \(\grad_t\) that it encounters, which changes \(\Gmax\)!

Solutions

Are we then to live with the choice between a nasty dependence on \(\Dmax\) (for OMD) or an awful hyper-parameter \(\Gmax\) (for FTRL)? Luckily not, and there are even two possible ways out! In line with the title of this post, both of these build on FTRL, by resolving the hyper-parameter tuning issue.

Solution 1: Improve the Analysis of FTRL

The first solution, which is perhaps the most comforting, is due to Orabona and Pál. It turns out that there is really nothing wrong with FTRL per se, but the problem arises only in the analysis. As shown in the proof of Orabona and Pál’s Theorem 1, it is possible to refine \(\eqref{eqn:ftrlbound}\) as follows:

where \(\Dmax^* = \max_{\v,\w \in \domain} \|\v - \w\|\). The result of the new minimum in the bound is that the effect of not including \(\|\grad_{t+1}\|\) in \(\eta_t\) is mitigated, and that we can get away with step sizes

Plugging this into \(\eqref{eqn:opbound}\), a standard summation analogous to the one used in Theorem 1 leads to

To handle the remaining sum, Orabona and Pál provide the following useful inequality (their Lemma 3):

Lemma 3: Let \(C, a_1, a_2, \ldots, a_T \geq 0\). Then

Applying this with \(a_t = \|\grad_t\|\) and \(C = 2\Dmax^*/\Done\), we find that

This is great, because we tuned the step sizes without prior knowledge of the gradient lengths, so we got Lipschitz-adaptivity, and the leading part of the bound also scales with \(\Done\) as desired! The fact that \(\Dmax\) does appear in the constant term at the end is also fine as long as our domain is not exceedingly large.² The only downside is that our leading constant has increased by slightly more than a factor of \(2\) from \(3/2\) to \(13/4\). Are we being too greedy if we complain about such a minor remaining imperfection? Certainly not, because even this increase can be avoided using the second solution presented below!

Solution 2: Cutkosky Clipping

The second solution does not require any improvement to the analysis of Theorem 2. Instead, we will combine \(\eqref{eqn:ftrlbound}\) with a technique to clip the observed gradients, which we call Cutkosky clipping. I first learned about it from Ashok Cutkosky on the bus at COLT 2018, and he later published it in a paper about combining Lipschitz adaptivity with unbounded domains. The technique is both amazingly simple and super useful to have in your toolbox at the same time, and I remember being blown away that such a simple idea had not been discovered before.

To explain the approach, let \(G_t = \max_{s \leq t} \|\grad_s\|\) with boundary case \(G_0 = 0\), and define the clipped gradients

These are just the original gradients, but with their lengths truncated to the previous largest observed length \(G_{t-1}\). The result is that, for \(\clipgrad_t\), we know one time-step in advance what its maximum length is going to be.

How does this help us? Well, many online learning algorithms, including the FTRL version of OGD, are based on linear approximations \(\tf_t(\w) = \w^\top \grad_t\) of the losses, which enter in the analysis as

Summing over \(t\), it follows that

and all the bounds from Theorems 1 and 2 are also upper bounds on \(\tR_T(\u)\). Moreover, these bounds do not make any assumptions about where the vectors \(\grad_t\) are coming from, as long as the same vectors are used consistently in the algorithms and their analysis. But this means that we can also apply the algorithms and bounds to the clipped gradients \(\clipgrad_t\). Keeping track of the approximation error when replacing \(\grad_t\) by \(\clipgrad_t\), inequality \(\eqref{eqn:ftrlbound}\) for the FTRL version of OGD then gives us

Since the lengths of the clipped gradients satisfy \(\|\clipgrad_t\| \leq G_{t-1}\), we can tune the step sizes as

to get

It remains to control the overhead introduced by clipping, which can be done as follows:

Putting everything together, we find that:

Theorem 4: Running the FTRL version of OGD on the clipped gradients \(\clipgrad_t\) with step sizes as in \(\eqref{eqn:clipstep}\) guarantees regret at most

This is the same as \(\eqref{eqn:refinedbound}\), but now even the constant factors turn out as nice as we could hope for!

Conclusion

We have seen that the FTRL version of OGD is better than the OMD version, because its regret depends only on the distance \(\Done\) of the first iterate from the optimal parameters, as opposed to the maximum distance over all the iterates \(\Dmax\) for OMD. Comparing the standard regret bounds for FTRL and OMD suggests that OMD might still have the advantage in adapting to the sizes of the gradients, but it turns out that this can be remedied for FTRL, either by improving the standard analysis (Solution 1) or by combining the standard analysis with Cutkosky clipping of the gradients (Solution 2).

Skeletons in the Closet

There is one topic that I have completely glossed over so far, which is how to adapt to the parameter \(\Done\) that still appears in the tuning of the step sizes. This is actually not a straightforward issue, but through a long series of works it is by now quite well understood. The upshot and a selection of key references are as follows:

• As discussed in this post, you can have Lipschitz-adaptivity without adaptivity to \(\Done\).
• You can also have adaptivity to \(\Done\) if you give up on Lipschitz-adaptivity [Cutkosky and Orabona, 2018].
• Having both Lipschitz-adaptivity and adaptivity to \(\Done\) at the same time is not possible [Cutkosky and Boahen, 2017]…
• … except if you pay an additional cubic additive constant of order \(O(\Done^3)\) in the rates… [Cutkosky, 2019], [Mhammedi and Koolen, 2020]
• … or if you make extra structural assumptions about your losses \(f_t\) [Kempka, Kotłowski and Warmuth, 2019].

That’s it. Thanks for reading!

But if you do not enforce a bound on the gradient lengths, then you make yourself vulnerable to outliers: a single extreme data point (e.g. with a large gradient) can completely break the online learner. How do we defend against such outliers? In this post, I describe a simple (but optimal) filtering strategy that can be applied to any existing online learning algorithm. This is based on our recent COLT paper [4] with Sarah Sachs, Wouter Koolen and Wojciech Kotłowski. There are also the COLT recorded videos if you prefer.

Motivation: Reasons for Outliers

Before getting into the technical side of things, it is helpful to consider several reasons why outliers might occur:

1. Naturally Heavy-tailed Data: The distribution of your data might be heavy-tailed, meaning that extreme data points occur naturally with small probability.
2. Malicious Users: If each data-point corresponds to a different user, then a small number of users may be trying to poison the data stream by deliberately generating adversarial inputs. Think, e.g., of spammers generating malformatted e-mails.
3. Sensor Glitches: With the increasing prominence of cheap sensors, measurement errors can occasionally generate extreme inputs.

To deal with all such cases simultaneously, we want to avoid making any assumptions about the nature of the outliers. This motivates the game-theoretic approach described below. \(\DeclareMathOperator{\E}{\mathbb{E}} \DeclareMathOperator{\argmin}{arg\,min} \DeclareMathOperator{\diameter}{diameter} \DeclareMathOperator{\risk}{Risk} \newcommand{\u}{\boldsymbol u} \newcommand{\w}{\boldsymbol w} \newcommand{\grad}{\boldsymbol g} \newcommand{\domain}{\mathcal{W}} \newcommand{\reals}{\mathbb{R}} \newcommand{\cS}{\mathcal{S}} \newcommand{\gradlist}{\mathcal{L}}\)

The Robust Regret Setting

Let’s start by fixing the setting, which will be a variant of the standard online convex optimization framework: Each round \(t=1,\ldots,T\), the learner predicts a point \(\w_t\) from a convex domain \(\domain \subset \reals^d\) of diameter at most \(\diameter(\domain) \leq D\). The environment then selects a convex loss function \(f_t : \domain \to \reals\) and the learner incurs loss \(f_t(\w_t)\). Most online learning algorithms update their predictions efficiently, using only the gradient \(\grad_t = \nabla f_t(\w_t)\), and the standard objective is for the learner to control the regret

for all \(\u \in \domain\) simultaneously.

Suppose now that only a subset of rounds \(\cS \subseteq [T] := \{1,\ldots,T\}\) consist of reliable inliers and the other rounds \([T] \setminus \cS\) are all outliers that our learning algorithm should ignore. Then the objective will be to control the robust regret:

Note the difference, which is that we are measuring performance only on inlier rounds in \(\cS\).

Main Challenges

The robust regret setup creates the following challenges:

1. \(\cS\) is unknown, and may in fact be chosen adversarially to maximally confuse the learner.
2. We will assume that the maximum length \(G(\cS) = \max_{t \in \cS} \|\grad_t\|\) of the inlier gradients is reasonable, but we make no assumptions whatsoever about the (lengths of) outlier gradients \(\grad_t\) for \(t \not \in \cS\).

Since we do not assume that outliers behave reasonably in any way, it follows that bounds on the robust regret can only depend on the scale \(G(\cS)\) of the inliers, but not on the lengths of the outliers!

Approach: Top-k Filtering

So what can we do? We will take a very general approach, in which we modify any existing online learner ALG by filtering out some rounds with extreme gradients. By filtering a round, I mean that we pretend to ALG that the round did not happen, so we do not feed it any gradient and ALG’s prediction remains unchanged.

We will assume that there are at most \(k\) outliers, so

Then the filtering strategy, which we call top-k filter, works as follows:

• Maintain a list \(\gradlist_t\) of the \(k+1\) largest gradient lengths seen during rounds \(1,\ldots,t\).
• Filter out round \(t\) if \(\|\grad_t\| \geq 2 \min \gradlist_t\); otherwise pass the round on to ALG.

Note the factor 2 in front of \(\min \gradlist_t\) in the filtering threshold!

Analysis: Are We Happy?

One thing to like about top-k filter is that it is a nice and simple approach, but of course true happiness can only be found if its performance is any good. In the paper, we first show the following general guarantee for top-k filtering when the losses are linear, i.e. \(f_t(\w) = \w^\top \grad_t\).

Theorem 1 (Linear Losses). Suppose losses are linear, and ALG achieves regret bound \(B_T(G)\) when running on at most \(T\) rounds with gradients of length at most \(G\). Then ALG + top-k filter guarantees that the robust regret does not exceed

See below for the main ideas behind the proof.

The first term, \(B_T(2 G(\cS))\), comes from feeding ALG gradients of length at most \(2 G(\cS)\), which it turns out is guaranteed by top-k filter. The second term of order \(O(G(\cS) k)\) is the price we pay for achieving robustness. Note that it scales with the size of the inlier gradients \(G(\cS)\) only, and does not depend on the outliers at all!

At first sight, it might appear restrictive that the bound only applies to linear losses, but we can easily get around this using a standard reduction from general convex losses to the linear case via the inequality \(f_t(\w_t) - f_t(\u) \leq (\w_t - \u)^\top \grad_t\). There exists a similar reduction for so-called strongly convex losses. By instantiating ALG to be the standard online gradient descent algorithm, this leads to the following guarantees on the robust regret:

These rates turn out to be optimal, because we prove matching lower bounds both for general convex losses and for strongly convex losses. In fact, for general convex losses the lower bound construction uses independent, identically distributed (i.i.d.) losses, so even if we assume that the losses are coming from a nice fixed distribution, then our upper bound is still optimal. We see that the optimal price of robustness is an additive \(O(k)\) in quite some generality, and top-k filter achieves it.

So are we happy? Well, the pursuit of happiness is a fool’s errant. We should rather aim to live a fulfilling life, but certainly there is nothing more fulfilling than optimality!

Proof Ideas Behind Theorem 1

The main ideas behind top-k filtering and Theorem 1 are as follows:

1. Top-k filter ensures that we never pass ALG any gradients whose length exceeds \(2 G(\cS)\).

To see this, note that:

• \(\gradlist_t\) must contain at least one inlier, because it has length \(k+1\) and there are at most \(k\) outliers.
• Hence \(\min \gradlist_t \leq G(\cS)\), and our filtering threshold is \(2\min \gradlist_t\).

2. The overhead for filtering is \(O(k)\).

We can account for incorrectly filtered rounds \(t \in \cS\) as follows:

• Every filtered round must have a large gradient, whose length is added to \(\gradlist_t\).
• By the factor 2 in the filtering threshold, \(\min \gradlist_t\) therefore (at least) doubles every \(k+1\) filtered rounds.
• Because of this exponential growth of \(\min \gradlist_t\), the last \(k+1\) incorrectly filtered rounds dominate in the bound.

Robustified Online-to-Batch Conversion for Huber ε-Contamination

There is one more result from the paper that I would like to highlight, which is that the robust regret can be used for robust learning in the Huber ε-contamination setting via a robustified variant of the usual online-to-batch conversion trick.

In the Huber ε-contamination model, our losses are of the form \(f_t(\w) = f(\w,\xi_t)\), where the underlying data \(\xi_1,\ldots,\xi_T\) are sampled i.i.d. from a mixture distribution \(P_\epsilon\):

The interpretation is that observations from the distribution of interest \(P\) are corrupted by observations from some unknown outlier distribution \(Q\) with probability \(\epsilon\). This is a batch-learning setting, in which the goal is to output a single set of parameters \(\bar \w_T\) that achieve small risk under \(P\):

When \(\epsilon = 0\) (i.e., there are no outliers), the standard online-to-batch conversion approach [5] allows us to average the predictions \(\w_1,\ldots,\w_T\) of any online learning algorithm according to

with the guarantee that

where \(\u_P = \argmin_{\u \in \domain} \risk_P(\u)\).

It turns out that this can be generalized to the case \(\epsilon > 0\), in which there are outliers, if we replace the standard regret by the robust regret:

where the inliers \(\cS^*\) are those rounds in which we receive a sample from the distribution of interest \(P\). Notice that now the expectations are under the mixture distribution \(P_\epsilon\), but the risk is measured with respect to \(P\)! If the losses or gradients of the inliers are bounded \(P\)-almost surely, then an analogous result also holds with high probability.

For instance, we show the following corollary, which achieves rates for Huber ε-contamination that are optimal under its assumptions:

Corollary. Suppose \(\|\nabla f(\w,\xi)\| \leq G\) when \(\xi \sim P\) is an inlier. Then online gradient descent with top-k filtering achieves

with \(P_\epsilon\)-probability at least \(1-\delta\), for \(k\) tuned suitably depending on \(\epsilon,T\) and \(\delta \in (0,1]\).

Conclusion

In this post, I have described a filtering approach that provides a way to robustify any online learning algorithm ALG by filtering out large gradients using top-k filter. This approach turns out to be optimal in the new robust regret setting, which further allows a type of robustified online-to-batch conversion for the Huber ε-contamination model.

References

1. T. van Erven, W. M. Koolen, and D. van der Hoeven, “MetaGrad: Adaptation using Multiple Learning Rates in Online Learning,” Journal of Machine Learning Research, vol. 22, no. 161, pp. 1–61, 2021, [Online]. Available at: http://jmlr.org/papers/v22/20-1444.html.
2. J. Duchi, E. Hazan, and Y. Singer, “Adaptive Subgradient Methods for Online Learning and Stochastic Optimization,” Journal of Machine Learning Research, vol. 12, pp. 2121–2159, 2011.
3. S. de Rooij, T. van Erven, P. D. Grünwald, and W. M. Koolen, “Follow the Leader If You Can, Hedge If You Must,” Journal of Machine Learning Research, vol. 15, pp. 1281–1316, 2014.
4. T. van Erven, S. Sachs, W. M. Koolen, and W. Kotłowski, “Robust Online Convex Optimization in the Presence of Outliers,” in Proceedings of the 34th Conference on Learning Theory (COLT), 2021, vol. 134, pp. 4174–4194, [Online]. Available at: https://proceedings.mlr.press/v134/vanerven21a.html.
5. N. Cesa-Bianchi, A. Conconi, and C. Gentile, “On the generalization ability of on-line learning algorithms,” IEEE Transactions on Information Theory, vol. 50, no. 9, pp. 2050–2057, 2004.

NB On the website I can only provide a high-level sneak preview of the tutorial for brevity. For the full version, which is still quite short, but includes the proofs and fills in the missing details, see PAC-BayesMiniTutorial.pdf. \(\DeclareMathOperator{\E}{\mathbb{E}} \DeclareMathOperator*{\argmin}{arg\,min}\)

The Cramér-Chernoff Method

I will start by outlining the Cramér-Chernoff method, from which Hoeffding’s and Bernstein’s inequalities and many others follow. This method is incredibly well explained in Appendix A of the textbook by Cesa-Bianchi and Lugosi [1], but I will have to change the presentation a little to easily connect with the PAC-Bayesian bounds later on.

Let \(D =((X_1,Y_1),\ldots,(X_n,Y_n))\) be independent, identically distributed (i.i.d.) examples, and let \(h\) be a hypothesis from a set of hypotheses \(\mathcal{H}\), which gets loss \(\ell(X_i,Y_i,h)\) on the \(i\)-th example. For example, we might think of the squared loss \(\ell(X_i,Y_i,h) = (Y_i - h(X_i))^2\). We also define the empirical error ¹ of \(h\)

and our goal is to prove that the empirical error is close to the expected error

with high probability. To do this, we define the function

which will act as a surrogate for \(R(h)\). Now the Cramér-Chernoff method tells us that:

Lemma 1. For any \(\eta > 0\), \(\delta \in (0,1]\), \begin{equation}\label{eqn:chernoff} M_\eta(h) \leq R_n(h,D) + \frac{1}{\eta n}\ln \frac{1}{\delta} \end{equation}with probability at least \(1-\delta\).

Many standard concentration inequalities can then be derived by first relating \(M_\eta(h)\) to \(R(h)\), and then optimizing \(\eta\). This includes, for example, Hoeffding’s inequality and inequalities involving the variance like Bernstein’s inequality. See the full version of this post for details.

The Union Bound

Now suppose we use an estimator \(\hat{h} \equiv \hat{h}(D) \in \mathcal{H}\) to pick a hypothesis based on the data, for example using empirical risk minimization: \(\hat{h} = \argmin_{h \in \mathcal{H}} R_n(D,h)\). To get a bound for \(\hat{h}\) instead of a fixed \(h\), we want \eqref{eqn:chernoff} to hold for all \(h \in \mathcal{H}\) simultaneously. If \(\mathcal{H}\) is countable, this can be done using the union bound, which leads to:

Lemma 4. Suppose \(\mathcal{H}\) is countable. For \(h \in \mathcal{H}\), let \(\pi(h)\) be any numbers such that \(\pi(h) \geq 0\) and \(\sum_h \pi(h)= 1\). Then, for any \(\eta > 0\), \(\delta \in (0,1]\), \begin{equation} M_\eta(\hat{h}) \leq R_n(D,\hat{h}) + \frac{1}{\eta n}\ln \frac{1}{\pi(\hat{h})\delta} \end{equation} with probability at least \(1-\delta\).

In this context, the function \(\pi\) is often referred to as a prior distribution, even though it need not have anything to do with prior beliefs.

Just like for Lemma 1, we can then again relate \(M_\eta(h)\) to \(R(h)\) to obtain a bound on the expected error. This shows, in a nutshell, how one can combine the Cramér-Chernoff method with the union bound to obtain concentration inequalities for estimators \(\hat{h}\). The use of the union bound, however, is quite crude when there are multiple hypotheses in \(\mathcal{H}\) with very similar losses, and the current proof breaks down completely if we want to extend it to continuous classes \(\mathcal{H}\). This is where PAC-Bayesian bounds come to the rescue: in the next section I will explain the PAC-Bayesian generalisation of Lemma 4 to continuous hypothesis classes \(\mathcal{H}\), which will require replacing \(\hat{h}\) by a randomized estimator.

PAC-Bayesian Concentration

Let \(\hat{\pi} \equiv \hat{\pi}(D)\) be a distribution on \(\mathcal{H}\) that depends on the data \(D\), which we will interpret as a randomized estimator: instead of choosing \(\hat{h}\) deterministically, we will sample \(h \sim \hat{\pi}\) randomly. The distribution \(\hat{\pi}\) is often called the PAC-Bayesian posterior distribution. Now the result that the PAC-Bayesians have, may be expressed as follows:

Lemma 6. Let \(\pi\) be a (prior) distribution on \(\mathcal{H}\) that does not depend on \(D\), and let \(\hat{\pi}\) be a randomized estimator that is allowed to depend on \(D\). Then, for any \(\eta > 0\), \(\delta \in (0,1]\), \begin{equation}\label{eqn:pacbayes} \E_{h \sim \hat{\pi}}[M_\eta(h)] \leq \E_{h \sim \hat{\pi}}[R_n(D,h)] + \frac{1}{\eta n}\Big(D(\hat{\pi}|\pi) + \ln \frac{1}{\delta}\Big) \end{equation} with probability at least \(1-\delta\). Moreover, \begin{equation}\label{eqn:pacbayesexp} \E_D \E_{h \sim \hat{\pi}}[M_\eta(h)] \leq \E_D\Big[ \E_{h \sim \hat{\pi}}[R_n(D,h)] + \frac{1}{\eta n}D(\hat{\pi}|\pi)\Big]. \end{equation}

Here \(D(\hat{\pi}\|\pi) = \int \hat{\pi}(h) \ln \frac{\hat{\pi}(h)}{\pi(h)} \mathrm{d} h\) denotes the Kullback-Leibler divergence of \(\hat{\pi}\) from \(\pi\).

To see that Lemma 6 generalises Lemma 4, suppose that \(\hat{\pi}\) is a point-mass on \(\hat{h}\). Then \(D(\hat{\pi}\|\pi) = \ln (1/\pi(\hat{h}))\), and we recover Lemma 4 as a special case of \eqref{eqn:pacbayes}. An important difference with Lemma 4, however, is that Lemma 6 does not require \(\mathcal{H}\) to be countable, and in fact in many PAC-Bayesian applications it is not.

Keep Reading…

We have seen how PAC-Bayesian inequalities naturally extend standard concentration inequalities based on the Cramér-Chernoff method by generalising the union bound to a continuous version. I’m afraid that’s all that will reasonably fit into a single blog post on my website. If you want more, simply continue with the full version, which covers several more issues, including:

• How to relate \(M_\eta(h)\) to \(R(h)\) to obtain, for example, Hoeffding’s inequality.
• How to optimize \(\eta\), which comes “for free” in Lemma 4, but requires an additional union bound in the general PAC-Bayesian case of Lemma 6.
• How the PAC-Bayesians choose their prior \(\pi\) and posterior \(\hat{\pi}\). (There’s an unexpected trick called localisation.)
• Proofs
• Literature references

References

1. N. Cesa-Bianchi and G. Lugosi, Prediction, learning, and games. Cambridge University Press, 2006.

In the 1990’s, people familiar with the work of Vovk called these special loss functions mixable losses, but nowadays the notion of mixability appears to be mostly forgotten, and the geometric concept of exp-concavity has taken its place. This raises the question of how the two are related, which strangely does not appear to be answered in very much detail in the literature. As I have been studying mixability quite a bit in my recent work, I was wondering about this, so here are some thoughts. Update: In particular, I will construct a parameterization of the squared loss in which it is \(1/2\)-exp-concave instead of only \(1/8\)-exp-concave like in its usual parameterization.

This post is also available as FromExpConcavityToMixability.pdf.

Mixability and Exp-concavity

Suppose we predict an outcome \(y \in \mathcal{Y}\) by specifying a prediction \(a \in \mathcal{A}\). The better our prediction, the smaller our loss \(\ell(y,a)\). (I will assume \(\ell(y,a)\) is nonnegative, but that does not really matter.) For example, if \(y\) and \(a\) both take values in \(\{0,1\}\), then the \(0/1\)-loss \(\ell(y,a) = |y-a|\) is \(0\) if we predict correctly and \(1\) otherwise. Alternatively, if \(y\) and \(a\) are both real-valued, then the squared loss is \(\ell(y,a) = (y-a)^2\). And finally, if \(a\) specifies a probability density \(f_a\) on \(\mathcal{Y}\), then our loss may be the log loss \(\ell(y,a) = -\ln f_a(y)\).

Mixability

For \(\eta > 0\), a loss function is called \(\eta\)-mixable [1] if for any probability distribution \(\pi\) on \(\mathcal{A}\) there exists a prediction \(a_\pi \in \mathcal{A}\) such that

The constant in the \(O(1)\) overhead compared to the best expert is proportional to \(1/\eta\), so the bigger \(\eta\) the better.

Exp-concavity

For \(\eta > 0\), a loss function is called \(\eta\)-exp-concave if for any distribution \(\pi\) on \(\mathcal{A}\) the prediction \(a_\pi = \int a \,\pi(\mathrm{d} a)\) satisfies \eqref{eqn:mixable}. So exp-concavity is just mixability with \(a_\pi\) fixed to be the mean.

This choice is appropriate in the case of log loss. In this case, for \(\eta = 1\), the numbers \(e^{-\eta \ell(y,a)} = f_a(y)\) just equal probability densities and \eqref{eqn:mixable} holds with equality. For squared loss, however, the appropriate choice for \(a_\pi\) is not the mean. Suppose that \(y\) and \(a\) both take values in \([-1,+1]\). Then, while the squared loss is \(1/2\)-mixable for \(a_\pi = \frac{h_{1/2}(-1) - h_{1/2}(1)}{4}\) with \(h_\eta(y) = \frac{-1}{\eta} \ln \int e^{-\eta (y-a)^2} \pi(\mathrm{d} a)\), it is only \(1/8\)-exp-concave when parameterized by \(a\) [2], [3]. This does not rule out, however, that the squared loss might be \(1/2\)-exp-concave in a different parameterization. As we shall see, such a parameterization indeed exists if we restrict \(y\) to take only two values \(\{-1,+1\}\), but I have not been able to find a suitable reparameterization in general.

Relations

Clearly, exp-concavity implies mixability: it just makes the choice for \(a_\pi\) explicit. What is not so obvious, is when the implication also goes the other way. It turns out that in some cases it actually does if we reparameterize our predictions in a clever (one might also say: complicated) way by the elements of a certain set \(\mathcal{B}_\eta\).

Theorem Suppose a loss \(\ell \colon \mathcal{Y} \times \mathcal{A} \to [0,\infty]\) satisfies Conditions 1 and 2 below for some \(\eta > 0\). Then \(\ell\) is \(\eta\)-mixable if and only if it can be parameterized in such a way that it is \(\eta\)-exp-concave.

The technical conditions I need are the following:

1. All predictions in \(\mathcal{A}\) should be admissible.
2. For any element \(g\) on the north-east boundary of the set \(\mathcal{B}_\eta\), there should exist a prediction \(a \in \mathcal{A}\) such that \(g(y) = e^{-\eta \ell(y,a)}\) for all \(y\).

It remains to explain what these conditions mean, and discuss their severity. I will argue that Condition 1 is very mild. Condition 2 also appears to be generally satisfied if the dimensionality of the set of predictions equals the number of possible predictions minus one, i.e. \(\dim(\mathcal{A}) = |\mathcal{Y}|-1\), but not in general. For example, for the squared loss we predict by a single number \(a\), so \(\dim(\mathcal{A}) = 1\) and hence we have \(\dim(\mathcal{A}) = |\mathcal{Y}|-1\) if \(y\) only takes two different values, but not if \(\mathcal{Y}\) is the whole range \([-1,+1]\). Update: We can work around this, though. See below.

The Technical Conditions

Admissibility

Condition 1 is the easiest of the two. I will call a prediction \(a \in \mathcal{A}\) admissible if there exists no other prediction \(b \in \mathcal{A}\) that is always at least as good in the sense that \(\ell(y,b) \leq \ell(y,a)\) for all \(y \in \mathcal{Y}\). If \(a\) is inadmissible, then we could just remove it from the set of available predictions \(\mathcal{A}\), because predicting \(b\) is always at least as good anyway. So admissibility seems more of an administrative requirement (get rid of all predictions that make no sense) than a real restriction.

Condition 2

To explain the second condition, we define the new parameterization \(\mathcal{B}_\eta\) as the set of functions

Note that the set \(\mathcal{B}_\eta\) is convex by construction.

Let \(\mathbb{1}(y) = 1\) be the constant function that is \(1\) on all \(y \in \mathcal{Y}\), and for any \(g \in \mathcal{B}_\eta\) let \(c(g) = \sup \{c \geq 0 \mid (g + c \cdot \mathbb{1}) \in \mathcal{B}_\eta\}\). By the north-east boundary of \(\mathcal{B}_\eta\), I mean the set of points \(\{g + c(g) \mid g \in \mathcal{B}_\eta \}\). That is, if we move `south-east' from any point in this set (in the direction of \(-\mathbb{1}\)), we are inside \(\mathcal{B}_\eta\), but if we move further `north-east' (in the direction of \(\mathbb{1}\)) we are outside.

Condition 2 implies that the north-east boundary of \(\mathcal{B}_\eta\) should be equal to the set \(\{e^{-\eta \ell(\cdot,a)} \mid a \in \mathcal{A}\}\), which appears to be quite typical if \(\dim(A) = |\mathcal{Y}| - 1\), but not in general.

Construction of the Parameterization and Proof

As we have already seen that \(\eta\)-exp-concavity trivially implies \(\eta\)-mixability, it remains to construct the parameterization in which \(\ell\) is \(\eta\)-exp-concave given that it is \(\eta\)-mixable.

The parameterization we choose is indexed by the elements of \(\mathcal{B}_\eta\), which we map onto \(\mathcal{A}\), with multiple elements in \(\mathcal{B}_\eta\) mapping to the same element of \(\mathcal{A}\). So let \(g\) be an arbitrary element of \(\mathcal{B}_\eta\). How do we map it to a prediction \(a \in \mathcal{A}\)? We do this by choosing the prediction \(a\) such that \(g(y) + c(g) = e^{-\eta \ell(y,a)}\) for all \(y\). As \(g + c(g)\cdot \mathbb{1}\) lies on the north-east boundary of \(\mathcal{B}_\eta\), such a prediction exists by Condition 2.

Our construction ensures there exists a \(g \in \mathcal{B}_\eta\) that maps to \(a\) for any \(a \in \mathcal{A}\). To see this, suppose there was an \(a\) for which this was not the case, and let \(g_a = e^{-\eta \ell(\cdot,a)}\). Then we must have \(c(g_a) > 0\), because otherwise we would have \(c(g) = 0\) and \(g_a\) would map to \(a\). But then the prediction \(b \in \mathcal{A}\) such that \(e^{-\eta \ell(\cdot,b)} = g + c(g) \cdot \mathbb{1}\) would satisfy \(e^{-\eta \ell(y,b)} > e^{-\eta \ell(y,a)}\) for all \(y\), and hence \(\ell(y,b) < \ell(y,a)\) for all \(y\), so that \(a\) would be inadmissible, which we have ruled out by assumption.

We are now ready to prove that the loss is \(\eta\)-exp-concave in our parameterization. To show this, let \(\pi\) be an arbitrary probability distribution on \(\mathcal{B}_\eta\). Then we need to show that

where \(g_\pi = \int g\ \pi(\mathrm{d} g)\). To this end, observe that

where \(c_\pi = \int c(g)\ \pi(\mathrm{d} g)\). Now convexity of \(\mathcal{B}_\eta\) ensures that \(\int e^{-\eta \ell(\cdot,g)} \pi(\mathrm{d} g) \in \mathcal{B}_\eta\), so that we must have \(c_\pi \leq c(g_\pi)\). But then

for all \(y\), which was to be shown.

Squared Loss

So how do things play out for the squared loss? We know that it is \(1/2\)-mixable, so we would like to find a parameterization in which it is also \(1/2\)-exp-concave. Suppose first that \(a\) takes values in \([-1,+1]\) and \(y\) takes only two values \(\{-1,+1\}\). Then Condition 1 is clearly satisfied. The set \(\mathcal{B}_{1/2}\) consists of all the functions \(g \colon \{-1,+1\} \to [e^{-2},1]\) such that

for some distribution \(\pi\) on \(\mathcal{A}\). So to verify Condition 2, we need to check that for any \(g \in \mathcal{B}_{1/2}\) there exists a prediction \(a_g \in \mathcal{A}\) that satisfies

Solving this we find that \(a_g\) indeed exists and equals

where \(f^{-1}\) is the inverse of \(f(\alpha) = e^{-\frac{1}{2}(1-\alpha)^2} - e^{-\frac{1}{2}(\alpha+1)^2}\) (see the figure below). The existence of \(a_g\) for all \(g\) implies that Condition 2 is satisfied, and by Theorem 1 we have found a parameterization in which the squared loss is \(1/2\)-exp-concave, provided that \(y\) only takes the values \(\{-1,+1\}\).

So what happens if we allow \(y\) to vary over the whole range \([-1,+1]\)? In this case I believe that no choice of \(a_g\) will satisfy \eqref{eqn:squaredlosstosolve} for all \(y\), and consequently Condition 2 does not hold. Update: However, it turns out that any parametrization that is \(\eta\)-exp-concave for \(y \in \{-1,+1\}\) is also \(\eta\)-exp-concave for the whole range \(y \in [-1,+1]\). This is a special property, proved by Haussler, Kivinen and Warmuth [2, Lemma 4.1], [3, Lemma 3], that only holds for certain loss functions, including the squared loss. Thus we have found a parameterization of the squared loss with \(y \in [-1,+1]\) in which it is \(1/2\)-exp-concave (instead of only \(1/8\)-exp-concave like in the standard parameterization): parameterize by the functions \(g\) defined in \eqref{eqn:newparam}, and map them to original parameters via the mapping \(a_g\) defined in \eqref{eqn:backtoorig}.

Discussion

We have seen that exp-concavity trivially implies mixability. Conversely, mixability also implies exp-concavity roughly when the dimensionality of the set of predictions \(\dim(\mathcal{A})\) equals the number of outcomes \(|\mathcal{Y}|\) minus one. In general, however, it remains unknown whether any \(\eta\)-mixable loss can be reparameterized to make it \(\eta\)-exp-concave with the same \(\eta\).

As exp-concavity is a stronger requirement than mixability and introduces these complicated reparameterization problems, one might ask: why bother with it at all? One answer to this is that taking \(a_\pi\) to be the mean reduces the requirement \eqref{eqn:mixable} to ordinary concavity, which has a nice geometrical interpretation. Nevertheless, the extra flexibility offered by mixability can make it easier to satisfy (for example, for the squared loss), so in general mixability would appear to be the most convenient of the two properties.

Afterthought

It seems the proof of Theorem 1 would still work if we replaced \(\mathbb{1}\) by any other positive function. I wonder whether this extra flexibility might make Condition 2 easier to satisfy.

Acknowledgements

Update: I would like to thank Sébastien Gerchinovitz for pointing me to Lemma 4.1 of Haussler, Kivinen and Warmuth [2].

References

1. N. Cesa-Bianchi and G. Lugosi, Prediction, learning, and games. Cambridge University Press, 2006.
2. D. Haussler, J. Kivinen, and M. K. Warmuth, “Sequential prediction of individual sequences under general loss functions,” IEEE Transactions on Information Theory, vol. 44, no. 5, pp. 1906–1925, 1998, doi: 10.1109/18.705569.
3. V. G. Vovk, “Competitive On-line Statistics,” International Statistical Review, vol. 69, no. 2, pp. 213–248, 2001.

It turns out that the two are more or less equivalent, but while Sanov’s theorem has the nicest information theoretic interpretation, the Cramér-Chernoff theorem seems to introduce the fewest measure-theoretic complications. Let me explain…

The Cramér-Chernoff Theorem

Let \(X, X_1, ..., X_n\) be independent, identically distributed random variables, and \(\DeclareMathOperator{\E}{\mathbb{E}} \DeclareMathOperator*{\argmin}{arg\,min}\) let \(\mu(\lambda) = \log \E[e^{\lambda X}]\) be their cumulant generating function. Then many standard concentration inequalities (Hoeffding, Bernstein, Bennett) can be derived [1, Appendix A.1] from a single basic result:

This result can easily be proved using Markov’s inequality and is non-trivial as long as \(a > \E[X]\). Its ubiquity is explained by the Cramér-Chernoff theorem [2], which states that it asymptotically achieves the best possible exponent if we optimize our choice of \(\lambda\):

Sanov’s Theorem

The empirical distribution \(P_n\) of \(X_1, \ldots, X_n\) gives probability \(P_n(A) = \frac{1}{n}\sum_{i=1}^n {\bf 1}_{\{X_i \in A\}}\) to any event \(A\), which is the fraction of variables taking their value in \(A\). If the distribution of the variables is \(P^*\), then asymptotically we would expect \(P_n\) to be close to \(P^*\). So then if \(\mathcal{P}\) is a set of distributions that is far away from \(P^*\) in some sense, the probability that \(P_n \in \mathcal{P}\) should be small. This intuition is quantified by Sanov’s theorem [3], [4].

To avoid measure-theoretic complications, let us assume that the variables \(X_1, \ldots, X_n\) are discrete, with a finite number of possible values. Suppose \(\mathcal{P}\) is a convex set of distributions and assume that the information projection

of \(P^*\) on \(\mathcal{P}\) exists, where \(D(P\|P^*) = \sum_x P(x) \log \frac{P(x)}{P^*(x)}\) is the Kullback-Leibler divergence. Then \(D(Q^*\|P^*)\) provides an information-theoretic measure of the “distance” of \(\mathcal{P}\) from \(P^*\), and indeed [3]

Moreover, Sanov’s theorem tells us that this bound is asymptotically tight in the exponent:

Csiszár [3, p. 790] has an elegant information-theoretic proof of the upper bound. He also works out sufficient measure-theoretic conditions for the theorem to hold in continuous spaces, which are quite clean, but still require considerable care to verify.

One may extend Sanov’s theorem to non-convex sets \(\mathcal{P}\) by a union bound argument. For example, Cover and Thomas [4] take a union bound over all possible values for \(P_n\), which they call types. By discretization arguments one may further extend the theorem to infinite spaces [5], but then things get a bit too asymptotic for my taste.

From Sanov to Cramér-Chernoff

It turns out that the Cramér-Chernoff theorem can be obtained from Sanov’s theorem. (This is called contraction.) The trick is to observe that

so if we define the convex set \(\mathcal{P} = \{P \mid \E_{X \sim P}[X] \geq a\}\), then

It remains to evaluate \(D(Q^* \| P^*)\) in this case, which can be done as follows:

1. Introduce a Lagrange multiplier to obtain:

   \[\min_{P \in \mathcal{P}} D(P\|P^*) = \min_{\{P \mid \E_{X \sim P}[X] \geq a\}} D(P\|P^*) = \min_P \sup_{\lambda > 0} \Big\{D(P\|P^*) - \lambda (\E_P[X] - a)\Big\}.\]

2. Use Sion’s minimax theorem to swap the order of the min and the sup:

   \[\min_P \sup_{\lambda > 0} \Big\{D(P\|P^*) - \lambda (\E_P[X] - a)\Big\} = \sup_{\lambda > 0}\inf_P \Big\{D(P\|P^*) - \lambda (\E_P[X] - a)\Big\}.\]

3. Recognize the following convex duality for Kullback-Leibler divergence:

   \[\sup_P\ \big(\E_{X \sim P}[\lambda X] - D(P||P^*\big) = \mu(\lambda),\]

where \(\mu(\lambda) = \log \E_{P^*}[e^{\lambda X}]\) is the cumulant generating function defined above. We get:

Chaining everything together we exactly recover the Cramér-Chernoff theorem, and we see that the upper bounds have exactly the same constants.

From Cramér-Chernoff to Sanov

One may also view things the other way around. The Cramér-Chernoff theorem bounds the probability that the value of the empirical mean \(\frac{1}{n} \sum_{i=1}^n X_i\) lies in the set \(A = [a,\infty)\). As discussed by [2, pp. 208-210], both the notion of empirical mean and the set \(A\) can be generalized. In particular, one may regard the empirical distribution \(P_n\) as the mean of \(n\) point-masses (i.e. Dirac measures) on the points \(X_1, \ldots, X_n\). Van der Vaart then presents Sanov’s theorem as just one instance of such generalized Cramér-Chernoff theorems.

Conclusion

We have seen the close similarities between the Cramér-Chernoff theorem and Sanov’s theorem. For me Sanov’s theorem seems easier to interpret, but in continuous spaces one has to deal with the more complicated measure-theoretic conditions of Csiszár [3]. For technical reasons it may therefore be preferable to use the Cramér-Chernoff result.

Last Words

It turns out that the upper bound in the Cramér-Chernoff theorem does leave some slack in the order of \(1/\sqrt{n}\), which is negligible compared to the term in the exponent.

References

1. N. Cesa-Bianchi and G. Lugosi, Prediction, learning, and games. Cambridge University Press, 2006.
2. A. W. van der Vaart, Asymptotic Statistics. Cambridge University Press, 1998.
3. I. Csiszár, “Sanov Property, Generalized I-Projection and a Conditional Limit Theorem,” The Annals of Probability, vol. 12, no. 3, pp. 768–793, 1984.
4. T. M. Cover and J. A. Thomas, Elements of Information Theory, Second Edition. John Wiley & Sons, 2006.
5. F. den Hollander, “Large deviations,” in Large Deviations, vol. 14, American Mathematical Society, 2000.