SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The next dynamic focuses on obtaining a solution from a single timestep, rather than time-averaged solutions. To obtain this, we consider a dynamics in which the learner goes first and is no-regret, the adversary observes the learner's action and then best responds.
Lemma 3.4 (No-Regret vs. Best-Response (NRBR)). Consider a multi-objective learning problem  $(\mathcal{D},\mathcal{G},\mathcal{H})$ , where a learner chose  $p^{(1:T)}\in \Delta (\mathcal{H})$  and an adversary chose  $q^{(1:T)}\in \Delta (\mathcal{D}\times \mathcal{G})$ . If the learner is no-regret,  $\operatorname{Reg}_{weak}\left(p^{(1:T)},\{\mathcal{L}_{q^{(t)}}(\cdot)\}^{(1:T)}\right)\leq T\varepsilon$ , and the adversary  $\varepsilon$ -best-responded to the costs  $\{-\mathcal{L}_{(\cdot)}(p^{(t)})\}^{(1:T)}$  using  $q^{(1:T)}$ , then there is at  $t\in [T]$  where  $p^{(t)}$  is a  $2\varepsilon$ -optimal solution.
Once the existence of a single-round solution  $p^{(t)}$  is established by Lemma 3.4, it is easy to find which time step corresponds to this solution by using a few samples and testing all  $p^{(1:T)}$ , as follows.
Lemma 3.5. Suppose a set of hypotheses  $p^{(1:T)}$  contains an  $\varepsilon$ -optimal solution. We can find a  $5\varepsilon$ -optimal solution  $p^{(t)} \in p^{(1:T)}$  using  $O(\varepsilon^{-2}|\mathcal{D}|\ln(|\mathcal{D}| |\mathcal{G}| T / \delta))$  samples with probability  $1 - \delta$ .
The next dynamic considers difficult distribution-free problems, such as online multi-objective learning. To enable learning in these scenarios, we consider a no-regret adversary that chooses objectives  $q^{(1:T)} \in \Delta(\mathcal{G})$  and a learner who first observes  $q^{(t)}$  and plays a distribution-free best-response. The following lemma considers the consequences of these interactions.
Lemma 3.6 (Best-Response vs. No-Regret (BRNR)). Consider an online multi-objective learning problem  $(\mathcal{D},\mathcal{G},\mathcal{H})$ , where a learner chose  $p^{(1:T)} \in \Delta(\mathcal{H})$ , an adversary chose  $q^{(1:T)} \in \Delta(\mathcal{G})$ , and  $D^{(1:T)} \in \mathcal{D}$  is any sequence. Assume that the adversary is no-regret, i.e.,  $\operatorname{Reg}(q^{(1:T)},\{-\mathcal{L}_{D^{(t)},(\cdot)}(p^{(t)})\}^{(1:T)}) \leq T\varepsilon$ , and the learner's actions  $p^{(1:T)}$  are distribution-free  $\varepsilon$ -best-responses to the stochastic costs  $q^{(1:T)}$ , i.e.,  $\max_{x,y} \mathcal{L}_{(x,y),q^{(t)}}(p^{(t)}) \leq \max_{x,y} \min_{p^*} \mathcal{L}_{(x,y),q^{(t)}}(p^*) + \varepsilon$ . Then, the hypotheses  $p^{(1:T)}$  are  $2\varepsilon$ -optimal on  $D^{(1:T)}$ .
The question of which dynamic should be used and their implementation hinges on the type of solution desired and what online learning and best-response guarantees are possible for each player. NRNR dynamics often offer maximum sample efficiency, since calculating  $\varepsilon$ -best-responses may be more sample intensive, but produce a time-averaged solution. NRBR dynamics, though less sample-efficient due to the adversary's repeated best-response computation, provides a single timestep solution and is crucial for, e.g., deterministic multicalibration. BRNR dynamics, where learners follow (act after the adversary) and have greater ease being no-regret, are crucial for online settings.
3.1 No-regret and Best Response Computation in Multicalibration
In this section, we introduce algorithms for obtaining (weak) no-regret and best response guarantees in multicalibration, so that we can apply the previously discussed dynamics.
Since the adversary picks from the—usually, small—set of objectives  $\mathcal{G}_{\mathrm{mc}}$ , it can be no-regret using standard algorithms like Hedge. However, the learner picks from the—very large—space of all predictors  $\mathcal{P}$ ; if it used Hedge, its regret would grow linearly in domain size  $|\mathcal{X}|$ . An important aspect of multicalibration is that its complexity must be independent of the domain size  $\mathcal{X}$  (while it can depend on the complexity of the subgroups  $\mathcal{S}$ ). We leverage structural properties of multicalibration objectives (see Appendix C.2 for formal treatment) to obtain generic no-regret and best response algorithms that give the learner domain-independent guarantees. Our first theorem gives a no-(weak)-regret learning algorithm for the learner that provides three important properties simultaneously: 1) domain-independent regret-bound that is also logarithmic in  $k$ , 2) uses no samples (or knowledge of) the underlying distribution  $D$ , and 3) deterministically outputs a deterministic predictor per round. Properties 1-2 lead to fast convergence and low sample complexity in the aforementioned dynamics and property 3 is key for obtaining deterministic multicalibration guarantees (via NRBR).
Theorem 3.7. Consider  $\mathcal{P}$  the set of  $k$ -class predictors and any adversarial sequence of stochastic costs  $q^{(1:T)} \in \Delta(\mathcal{G}_{\mathrm{mc}})$ , where  $\mathcal{G}_{\mathrm{mc}}$  are the multicalibration objectives (3). There is a no-regret algorithm that outputs (deterministic) predictors  $h^{(1:T)} \in \mathcal{P}$  such that  $\operatorname{Reg}_{weak}(h^{(1:T)}, \{\mathcal{L}_{D,q^{(t)}}\}^{(1:T)}) \leq 2\sqrt{\ln(k)T}$  for every data distribution  $D$ . The algorithm does not need any samples from  $D$ .
Proof. Consider the following algorithm. At each feature  $x \in \mathcal{X}$ , initialize a Hedge algorithm that picks an action  $h^{(t)}(x) \in \Delta(\mathcal{Y})$  at each timestep  $t \in [T]$ . Aggregating each algorithm's action yields our learner's overall action  $h^{(t)} \in \mathcal{P}$ . For each  $x \in \mathcal{X}$ , let  $h^{(t+1)}(x)$  be the outcome of Hedge at step
$t + 1$  after observing linear loss functions  $f_{h^{(\tau),x}}^{(\tau)}:\mathbb{R}^k\to [0,1]$  for  $\tau \in [t]$ :
$$
f _ {h ^ {(\tau)}, x} ^ {(\tau)} (z) := 0. 5 + 0. 5 \sum_ {i \in \{\pm 1 \}, j \in [ k ], S \in \mathcal {S}, v \in V _ {\lambda} ^ {k}} z _ {j} \cdot q _ {i, j, S, v} ^ {(\tau)} \cdot i \cdot 1 [ h ^ {(\tau)} (x) \in v, x \in S ], \tag {4}
$$
where  $q_{i,j,S,v}^{(\tau)}$  is the probability  $q^{(\tau)}$  assigns to loss  $\ell_{i,j,S,v}$ . Hedge gives  $\sum_{t=1}^{T} f_{h^{(t)},x}^{(t)}(h^{(t)}(x)) - \min_{z^* \in \Delta(\mathcal{Y})} \sum_{t=1}^{T} f_{h^{(t)},x}^{(t)}(z^*) \leq 2\sqrt{\ln(k)T}$  (Lemma 3.1). Since this inequality holds for all  $x \in \mathcal{X}$ ,
$$
\begin{array}{l} 2 \sqrt {\ln (k) T} \geq \underset {(x, y) \sim D} {\mathbb {E}} \left[ \sum_ {t = 1} ^ {T} f _ {h (t), x} ^ {(t)} \left(h ^ {(t)} (x)\right) \right] - \underset {(x, y) \sim D} {\mathbb {E}} \left[ \min  _ {z ^ {*} \in \Delta (\mathcal {Y})} \sum_ {t = 1} ^ {T} f _ {h (t), x} ^ {(t)} \left(z ^ {*}\right) \right] \tag {5} \\ = \underset {(x, y) \sim D} {\mathbb {E}} \left[ \sum_ {t = 1} ^ {T} f _ {h ^ {(t)}, x} ^ {(t)} (h ^ {(t)} (x)) \right] - \min _ {h ^ {*} \in \mathcal {P}} \underset {(x, y) \sim D} {\mathbb {E}} \left[ \sum_ {t = 1} ^ {T} f _ {h ^ {(t)}, x} ^ {(t)} (h ^ {*} (x)) \right] \\ \end{array}
$$
by law of total expectation, where the last transition is by defining  $h^*$  such that  $h^*(x) = z^*$  for every  $x$ -dependent choice of  $z^*$  in (5). Add and subtract  $\sum_{t=1}^{T} \mathbb{E}_{(x,y) \sim D}\left[f_{h^{(t)},x}^{(t)}(\delta_y)\right]$  to obtain
$$
\sum_ {t = 1} ^ {T} \underset {(x, y) \sim D} {\mathbb {E}} \left[ f _ {h ^ {(t)}, x} ^ {(t)} \left(h ^ {(t)} (x) - \delta_ {y}\right) \right] \leq 2 \sqrt {\ln (k) T} + 0. 5 T, \tag {6}
$$
where the last transition is by the fact  $h^{*}(x) = \mathbb{E}[\delta_{y}|x]$ ,  $\sum_{t=1}^{T}\mathbb{E}\left[f_{h^{(t)},x}^{(t)}(h^{*}(x) - \delta_{y})\right] = 0.5T$ . Next, we show the LHS of (6) is equivalent to  $\mathrm{Reg}_{\mathrm{weak}}(h^{(1:T)},\{\mathcal{L}_{D,q^{(t)}}\}^{(1:T)})$ . First recall from multicalibration objectives that  $\ell_{i,j,S,v}(h,(x,y)) = 0.5 + 0.5\cdot i\cdot 1[h(x)\in v,x\in S]\cdot (h(x)_j - \delta_{y,j})$ . Since  $q_{i,j,S,v}^{(t)}$  is the probability  $q^{(t)}$  assigns to  $\ell_{i,j,S,v}(\cdot)$  in (4), we have that  $q^{(t)}(h^{(t)},(x,y)) = f_{h^{(t)},x}^{(t)}(h^{(t)}(x) - \delta_y)$ . Therefore, (6) implies that  $\sum_{t=1}^{T}\mathcal{L}_{D,q^{(t)}}(h^{(t)})\leq 0.5T + 2\sqrt{\ln(k)T}$ . It is left to establish that the min-max baseline of these losses is indeed at least 0.5. This is implied by Fact 2.5 and its proof is deferred to Appendix C.2. Thus  $\mathrm{Reg}_{\mathrm{weak}}(h^{(1:T)},\{\mathcal{L}_{D,q^{(t)}}\}^{(1:T)})\leq 2\sqrt{\ln(k)T}$ .
Our second theorem proves the existence of a distribution-free best-response algorithm for the learner, which is key for obtaining online multicalibration guarantees (via BRNR). Note that, as a distribution-free algorithm, it requires no samples to compute these best-responses.
Theorem 3.8. Consider the set of  $k$ -class predictors  $\mathcal{P}$ . Fix an  $\varepsilon > 0$  and let  $q \in \Delta(\mathcal{G}_{\mathrm{mc}})$  be a mixture of multicalibration objectives (3). There always exists a (non-deterministic) predictor  $p \in \Delta(\mathcal{P})$  that is a distribution-free  $\varepsilon$ -best-response (7) to the stochastic cost function  $q(\cdot, (x, y))$ .
Proof Sketch. At a high level, this statement and proof are similar to the min-max proof of calibration [10, 20], with additional details in Appendix C.2. Let  $\widetilde{\Delta}(\mathcal{Y})$  be a finite  $\varepsilon$ -covering of  $\Delta(\mathcal{Y})$  and  $\phi_x: \Delta(\widetilde{\Delta}(\mathcal{Y})) \times \Delta(\mathcal{Y}) \to [-1,1]$  the bilinear function  $\phi_x(a,b) = \mathbb{E}_{(i,j,S,v)\sim q,\widehat{a}\sim a}[i\cdot 1[\widehat{a}\in v,x\in s]\cdot (\widehat{a}_j - b_j)]$ . Consider  $p(x) = \operatorname*{argmin}_{a\in \Delta(\widetilde{\Delta}(\mathcal{Y}))} \max_{b\in \Delta(\mathcal{Y})} \phi_x(a,b)$ . By the minmax theorem,  $\max_{b\in \Delta(\mathcal{Y})} \phi_x(p(x),b) = \max_{b\in \Delta(\mathcal{Y})} \min_{a\in \widetilde{\Delta}(\mathcal{Y})} \phi_x(a,b) \leq \max_{b\in \Delta(\mathcal{Y})} \min_{a\in \Delta(\mathcal{Y})} \phi_x(a,b) + \varepsilon$ . Thus,  $\max_{y\in \mathcal{Y}} q(p,(x,y)) \leq 0.5 \cdot \varepsilon + \max_{y\in \mathcal{Y}} \min_{h^*\in \mathcal{P}} q(h^*,(x,y))$  for all  $x \in \mathcal{X}$ .
4 Multicalibration with Game Dynamics
We match and improve a broad set of previous results in multicalibration (See Table 1) that had received individualized and ad hoc treatments in the past. Our work establishes that not only is there a unified approach for obtaining these results but that it all comes back to game dynamics empowered by our no-regret and distribution-free-best response results for multicalibration—Theorems 3.7 and 3.8. Below we focus on three main results highlighting NRNR, NRBR, and BRNR dynamics. We defer the proofs, formal statement of algorithms, and additional results to Appendix D and E.
Multicalibration. Our first algorithm uses no-regret no-regret (NRNR) dynamics to find nondeterministic multicalibrated predictors. This algorithm matches the fastest known sample complexity rates for multicalibration [18, 35] of order  $O(\ln(|S| \lambda^k) / \varepsilon^2)$ . It also improves upon existing fast-rate algorithms of Gupta et al. [18], Noarov et al. [35] by producing predictors with a succinct support and small circuit size. These properties were previously only known to be attained by the less sample efficient multicalibration algorithms of [21]. In this way, Algorithm 1 simultaneously attains the best aspects of the algorithms of [18] and [21].
Theorem 4.1. Fix  $\varepsilon >0$ ,  $\lambda, k\in \mathbb{Z}_{+}$ , set of groups  $S\subseteq 2^{\mathcal{X}}$ , and data distribution  $D$ . The below algorithm, with probability  $1 - \delta$ , returns a non-deterministic  $k$ -class predictor that is  $(S,\varepsilon ,\lambda)$ -multicalibrated on  $D$  and takes no more than  $O\left(\varepsilon^{-2}(\ln (|S| / \delta) + k\ln (\lambda))\right)$  samples from  $D$ .
No-Regret vs No-Regret
Construct the problem  $(\{D\}, \mathcal{G}_{\mathrm{mc}}, \mathcal{P})$  from Fact 2.5 and let  $T = C\varepsilon^{-2}\ln(|S| \lambda^k / \delta)$  for some universal constant  $C$ . Over  $T$  rounds, have an adversary choose  $q^{(1:T)} \in \Delta(\mathcal{G}_{\mathrm{mc}})$  by applying Hedge to the costs  $\{1 - \ell_{(\cdot)}(h^{(t)}, (x^{(t)}, y^{(t)}))\}^{(1:T)}$  where  $(x^{(t)}, y^{(t)}) \stackrel{\mathrm{i.i.d.}}{\sim} D$ . In parallel, have a learner choose predictors  $h^{(1:T)} \in \mathcal{P}$  by applying the no-regret learning algorithm of Theorem 3.7 to the stochastic costs  $\ell^{(1:T)}$ , where  $\ell^{(t)} \stackrel{\mathrm{i.i.d.}}{\sim} q^{(t)}$ . Return the predictor  $p = \mathrm{Uniform}(h^{(1:T)})$ . This algorithm is written explicitly in Algorithm 1.
Theorem 4.1—along with all other results in this section—can be rewritten with  $\operatorname{VC}(S) \log(1 / \varepsilon)$  replacing  $\ln(|S|)$ ; this is done by taking a cover of  $S$ . We further note that Algorithm 1 can be instantiated with different choices of no-regret algorithms for the adversary and different versions of Theorem 3.7 for the learner. In Section 6, we empirically compare such variants of Algorithm 1.
Our second algorithm uses no-regret best-response (NRBR) dynamics to find deterministic multicalibrated predictors. This algorithm improves on Gopalan et al. [17]'s oracle complexity of  $O\left(k / \varepsilon^2\right)$  with  $O\left(\ln (k) / \varepsilon^2\right)$ , an exponential reduction of the dependence on the number of classes  $k$ .<sup>4</sup>
Theorem 4.2. Fix  $\varepsilon >0$ ,  $\lambda, k\in \mathbb{Z}_{+}$ , a set of groups  $S\subseteq 2^{\mathcal{X}}$ , and a data distribution  $D$ . The following algorithm returns a deterministic  $k$ -class predictor that is  $(S,\varepsilon ,\lambda)$ -multicalibrated on  $D$  and makes  $O(\ln (k) / \varepsilon^{2})$  calls to an agnostic learning oracle. Moreover, with probability  $1 - \delta$ , the oracle calls can be implemented with  $\widetilde{O}\left(\frac{1}{\varepsilon^3} (\sqrt{\ln(k)}\ln (k|S| / \delta) + k\ln (\lambda))\right)$  samples from  $D$ .
No-Regret vs Best-Response
Construct the problem  $(\{D\}, \mathcal{G}_{\mathrm{mc}}, \mathcal{P})$  from Fact 2.5 and let  $T = C\varepsilon^{-2}\ln(|S| \lambda^k \delta)$  for some universal constant  $C$ . Over  $T$  rounds, have a learner choose predictors  $h^{(1:T)} \in \mathcal{P}$  by applying the no-regret learning algorithm of Theorem 3.7 to the stochastic costs  $\ell^{(1:T)}$ . Have an adversary choose  $\ell^{(1:T)}$  by calling an agnostic learning oracle at each  $t \in [T]$ :  $\ell^{(t)} = \mathcal{A}_{\varepsilon/8}(1 - \mathcal{L}_{D,\langle \cdot \rangle}(h^{(t)}))$ . Using  $C\ln(T/\delta)/\varepsilon^2$  samples from  $D$ , return the predictor  $h^{(t^*)}$  with the lowest empirical multicalibration error. This algorithm is written explicitly in Algorithm 2.
Online multicalibration. Our next algorithm uses best-response no-regret (BRNR) dynamics for online multicalibration, matching the best known regret bounds for online multicalibration [18, 35]. As with Theorem 4.1, our analysis simplifies that of [18, 35] by avoiding exponential potential arguments in favor of no-regret dynamics. We state the following result for binary classification, where  $\mathcal{V} = \{0,1\}$  but note it trivially extends to multi-class settings. For convenience, we will say that predictors  $\mathcal{P}$  for binary classification output real-valued  $h(x)\in [0,1]$ , where  $h(x)$  is the predicted probability of class 1 and  $1 - h(x)$  is the predicted probability of class 0.
Theorem 4.3. Fix  $\varepsilon >0$ ,  $\lambda \in \mathbb{Z}_{+}$ , and a set of groups  $S\subseteq 2^{\mathcal{X}}$ . The following algorithm guarantees  $(S,\varepsilon ,\lambda)$ -online multicalibration with probability  $1 - \delta$ .
Best-Response vs No-Regret
Construct the online multi-objective learning problem  $(\mathcal{D},\mathcal{G}_{\mathrm{mc}},\mathcal{P})$  in Fact 2.6 and let  $T = C\varepsilon^{-2}\ln (|\mathcal{S}|\lambda \delta)$  for some universal constant  $C$ . Over  $T$  rounds, have an adversary choose  $q^{(1:T)}\in \Delta (\mathcal{G}_{\mathrm{mc}})$  by applying Hedge to the costs  $\{1 - \ell_{(\cdot)}(p^{(t)},(x^{(t)},y^{(t)}))\}^{(1:T)}$ , where
4In concurrent work, Dwork et al. [8] also showed that  $O\left(\ln (k) / \varepsilon^2\right)$  oracle calls are sufficient for multi-class multicalibration, with an algorithm similar to Algorithm 2.
$(x^{(t)},y^{(t)})\stackrel {\mathrm{i.i.d.}}{\sim}D^{(t)}$  . Have a learner best-respond to each stochastic cost  $q^{(t)}$  with the  $(\varepsilon /2)$  distribution-free best-response  $p^{(t)}\in \Delta (\mathcal{P})$  of Theorem 3.8. This algorithm is written explicitly in Algorithm 3.
The high-probability condition of Theorem 4.3 can be removed if we assume nature presents datapoints rather than data distributions, as is assumed in prior works. Interestingly, Algorithm 3's use of best-response no-regret dynamics exactly recovers the online multicalibration algorithm of [18, 35]. The analysis of Theorem 4.3 is, however, significantly simpler because we make explicit the role of the no-regret dynamics, whereas [18, 35] use potential arguments that ultimately prove no-regret dynamics and the multiplicative weights algorithm from scratch.
Additional results. We can also use game dynamics to improve guarantees for various other multicalibration settings and considerations. We can use Algorithms 1 and 2 to achieve domain-independent sample complexity rates for agnostic multicalibration (Theorem E.9), improving upon Shabat et al. [38]'s uniform convergence guarantees. Plugging in an online learning algorithm with second-order regret bounds allows us to achieve the first non-trivial sample complexity guarantees for conditional multicalibration in the batch setting (Theorem E.3) and a multicalibration algorithm that—with a minor less than cube-root increase in sample complexity—guarantees an error tolerance for each group that scales with the square-root of the group's probability mass (Theorem 4.4). We highlight the latter result in the following theorem.
Theorem 4.4. Fix  $\varepsilon >0$ ,  $\lambda ,k\in \mathbb{Z}_{+}$ , two sets of groups  $S\subseteq 2^{\mathcal{X}}$ , and a data distribution  $D$ . There is an algorithm that, with probability at least  $1 - \delta$ , takes  $O(\ln (k)\cdot (\ln (|S| / \varepsilon \delta) + k\ln (\lambda)) / \varepsilon^{4})$  samples from  $D$  and returns a deterministic  $k$ -class predictor  $h$  satisfying
$$
\left| \underset {(x, y) \sim D, h \sim p} {\mathbb {E}} \left[ (h (x) _ {j} - \delta_ {y, j}) \cdot 1 [ h (x) \in v, x \in S ] \right] \right| \leq \varepsilon \sqrt {\Pr (x \in S)},
$$
for all  $S\in \mathcal{S},v\in V_{\lambda}^{k},j\in [k]$
Plugging in an online learning algorithm with strongly adaptive regret bounds allows us to parallelize the moment multicalibration algorithm of Jung et al. [22] (Theorem E.13).
5 Other Fairness Notions
This general framework of approaching multi-objective learning with game dynamics can be extended beyond multicalibration to derive results on multi-group learning. Recall that, in agnostic multi-objective learning, the trade-off between objectives is arbitrated by the worst-off objective, which can be suboptimal when some objectives are inherently more difficult. An alternative is to, given a problem  $(\mathcal{D},\mathcal{G},\mathcal{H})$ , define a competitor class  $\mathcal{H}'$  and try to learn a hypothesis  $h$  so that there is no objective that a competitor performs significantly better on, i.e.  $\mathcal{L}_{\mathcal{H}'}^*(h) - \min_{h^* \in \mathcal{H}} \mathcal{L}_{\mathcal{H}'}^*(h^*) \leq \varepsilon$  where  $\mathcal{L}_{\mathcal{H}'}^*(h) := \max_{D \in \mathcal{D}} \max_{\ell \in \mathcal{G}} \mathcal{L}_{D,\ell}(h) - \min_{h^* \in \mathcal{H}'} \mathcal{L}_{D,\ell}(h^*)$ . We refer to such a solution  $h$  as being  $\varepsilon$ -competitive. A simple modification to multi-objective learning can find such a solution: replace the objectives  $\mathcal{G}$  with new objectives  $\mathcal{G}' := \left\{\frac{1}{2}(1 + \ell(\cdot) - \ell(h')) \mid \ell \in \mathcal{G}, h' \in \mathcal{H}'\right\}$  and solve as usual. Since the sample complexity of multi-objective learning is  $O(\varepsilon^{-2} (\ln(|\mathcal{H}|) + |\mathcal{D}| \ln(|\mathcal{D}| |\mathcal{G}| / \delta))$  [19], we have the below sample complexity bound on finding  $\varepsilon$ -competitive solutions.
Theorem 5.1. Consider a multi-objective learning problem  $(\mathcal{D},\mathcal{G},\mathcal{H})$  and competitor class  $\mathcal{H}'$ . There is an algorithm that takes  $O(\varepsilon^{-2}(\ln (|\mathcal{H}|) + |\mathcal{D}|\ln (|\mathcal{D}||\mathcal{H}'||\mathcal{G}| / \delta)))$  samples and with probability  $1 - \delta$  returns a solution  $p\in \Delta (\mathcal{H})$  that is  $\varepsilon$ -competitive against  $\mathcal{H}'$ .
Consider the multi-group learning problem, where we seek to simultaneously minimize a general loss function on different subsets of the domain [37].
Definition 5.2. Fix  $\varepsilon >0$ , a set of groups  $S\subseteq 2^{\mathcal{X}}$ , a hypothesis class  $\mathcal{H}$  and a loss  $\ell : \mathcal{H}\times (\mathcal{X}\times \mathcal{Y})\to [0,1]$ . An  $\varepsilon$ -optimal solution to the multi-group learning problem  $(S,\mathcal{H})$  is a randomized hypothesis  $p\in \Delta (\mathcal{H})$  that satisfies, for all  $S\in S$ ,  $\mathbb{E}[\ell (p,(x,y))\cdot 1[x\in S]]\leq \min_{h^{*}\in \mathcal{H}}\mathbb{E}[\ell (h^{*},(x,y))\cdot 1[x\in S]] + \varepsilon$ . We always assume such a hypothesis  $p$  exists in class  $\mathcal{H}$ .
A (near) optimal sample complexity for multi-group learning of  $O\left(\ln (|\mathcal{S}| |\mathcal{H}|) / \varepsilon^2\right)$  was attained by [39] using a reduction to sleeping experts. [39] also asked whether there exists a simpler optimal algorithm that does not rely on sleeping experts. We answer this affirmatively by designing an optimal
algorithm that just runs two Hedge algorithms. The following is a direct implication of Lemma 3.3 and the observation that multi-group learning directly reduces to learning an  $\varepsilon$ -competitive solution.
Theorem 5.3. Fix a set of groups  $\mathcal{S} \subseteq 2^{\mathcal{X}}$ , loss  $\ell : \mathcal{Y} \times \mathcal{Y} \to [0,1]$ , hypothesis class  $\mathcal{H} : \mathcal{X} \to \mathcal{Y}$  and distribution  $D$ . There is a no-regret no-regret algorithm that takes  $2T = O\left(\ln (|\mathcal{S}| |\mathcal{H}|) / \varepsilon^{2}\right)$  samples from  $D$  and returns an  $\varepsilon$ -optimal solution to the multi-group learning problem  $(\mathcal{S}, \mathcal{H})$ .
In this section, we study the empirical performance of multicalibration algorithms on the UCI Adult Income dataset [26], a real-world dataset for predicting individuals' incomes based on the US Census. We defer additional results, datasets, and methods to the Appendix G.
Experiment setup. This experiment, summarized in Table 2, aims to learn a predictor on the UCI dataset that predicts the 'income' label and is multicalibrated on the values of eight other labeled attributes including 'age'. We discretize with 0.1-width bins  $(\lambda = 10)$ , and perform random 80-20 train/test splits of the dataset. This results in approximately 24000 training samples, 6000 test samples, and 130 groups. Each multicalibration algorithm is evaluated on 10 seeds, each after 50 iterations, with performance measured by their average iterate's and last iterate's multicalibration violations.
We study four multicalibration algorithms based on no-regret best-response dynamics, which use an empirical risk minimizer as the adversary and implements either Hedge [14], Prod [28], Optimistic Hedge (OptHedge) [36], or Gradient Descent (GD) as the learner. We study two algorithms based on no-regret no-regret dynamics, which plays against itself either Hedge (Hedge-Hedge) or Optimistic Hedge (OptHedge-OptHedge). Learning rate decay is tuned on the training set by sweeping over  $[0.8, 0.85, 0.9, 0.95]$  for the learner and  $[0.9, 0.95, 0.98, 0.99]$  for the adversary.