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+# Leonardo Cella<sup>1</sup> Massimiliano Pontil<sup>12</sup> Claudio Gentile<sup>3</sup>
+
+# Abstract
+
+We introduce and analyze a best arm identification problem in the rested bandit setting, wherein arms are themselves learning algorithms whose expected losses decrease with the number of times the arm has been played. The shape of the expected loss functions is similar across arms, and is assumed to be available up to unknown parameters that have to be learned on the fly. We define a novel notion of regret for this problem, where we compare to the policy that always plays the arm having the smallest expected loss at the end of the game. We analyze an arm elimination algorithm whose regret vanishes as the time horizon increases. The actual rate of convergence depends in a detailed way on the postulated functional form of the expected losses. We complement our analysis with lower bounds, indicating strengths and limitations of the proposed solution.
+
+# 1. Introduction
+
+Multi-armed bandits are a mathematical framework of sequential decision problems that have played a fundamental role in machine learning and statistics (see e.g. Bubeck et al., 2012; Cesa-Bianchi, 2016; Lattimore & Szepesvári, 2020; Siegmund, 2003, and references therein). This framework consists of a sequence of  $T$  interactions (or rounds) between a learning agent and an unknown environment. During each round the learner picks an action from a set of options  $\mathcal{K}$ , usually referred to as arms, and the environment consequently generates a feedback (e.g., in the form of a loss value) associated with the chosen action/pulled arm. Multi-armed bandits have applications to a wide variety of domains including clinical trials, online advertising, and marketing.
+
+In the standard i.i.d. stochastic bandit setting (Auer et al.,
+
+$^{1}$ Italian Institute of Technology, Genoa, Italy  $^{2}$ University College London, United Kingdom  $^{3}$ Google Research, New York, USA. Correspondence to: Leonardo Cella <leonardocella@gmail.com>.
+
+Proceedings of the  $38^{th}$  International Conference on Machine Learning, PMLR 139, 2021. Copyright 2021 by the author(s).
+
+2002), the feedback generated when pulling an arm is modeled as a random variable sampled from a prescribed distribution associated with the selected arm. In contrast, in this paper we are interested in a non-stationary stochastic bandit setting called rested bandits (Allesiardo et al., 2017; Besbes et al., 2014; Cella & Cesa-Bianchi, 2020; Kleinberg & Immorlica, 2018; Levine et al., 2017; Seznec et al., 2018), whereby the feedback/losses received upon pulling arms are not i.i.d. anymore. That is, the distribution of losses changes as a function of the number of times each arm has been pulled so far. As a relevant example, assume the expected loss of action  $i \in \mathcal{K}$  at a given round is decreasing with time and takes the parametric form
+
+$$
+\frac {\alpha_ {i}}{\sqrt {\tau}} + \beta_ {i}, \tag {1}
+$$
+
+where  $\tau$  is the number of times arm  $i$  has been pull up to that round, and  $\alpha_{i}$  and  $\beta_{i}$  are unknown parameters.
+
+Considering decreasing expected losses is reasonable whenever the properties of the chosen arm improve as we allocate resources to them. For example, this is the case in scenarios where the goal is to find the best talent in a pool of candidates, say, the most valuable worker to train in an online labor platform having limited training time.
+
+Overall, we interpret this problem as an algorithm selection or selective training problem. In this scenario, an arm represents a learning device that satisfies Equation (1) (like a specific neural network architecture) and the goal is to keep training the learner that will be the best at the end of the game. The parameters  $\alpha_{i}$  and  $\beta_{i}$  in (1) may therefore quantify relevant properties of such models. For instance, in a standard statistical learning setting, parameter  $\alpha_{i}$  can quantify the complexity (which may or may not be known) of the  $i$ -th model class,  $\beta_{i}$  might encode the representational power of that class in the form of the statistical risk of the best-in-class hypothesis (which is typically unknown), while the dependence on  $1 / \sqrt{\tau}$  is meant to suggest a plausible behavior of the generalization error of the  $i$ -th algorithm as a function of the training set size  $\tau$ . Within this setting, an arm  $i \in \mathcal{K}$  with small  $\alpha_{i}$  and large  $\beta_{i}$  may represent an empirical risk minimizer (ERM) operating on a simple model class where the ERM has an estimation error getting small with few samples, but which only underfits the data without effectively minimizing the approximation error. Conversely, an arm  $i \in \mathcal{K}$  with large  $\alpha_{i}$  and small  $\beta_{i}$  may correspond
+
+to an ERM operating on a complex model class with large estimation error (where overfitting is likely to occur) and small approximation error.
+
+Given a budget of  $T$  training samples, our specific goal is to design a strategy for online selective training, whereby at each round we have to decide which algorithm the next training example has to be fed to. This problem is of fundamental importance since, in many practical situations, performing a batch model selection (or model training) might be too computationally demanding. Thus, the goal is to design a strategy (a learning policy) that interacts with different learning algorithms with the goal of spending the budget of  $T$  samples on the algorithm/model that is likely to perform best after training. Pulling an arm corresponds to feeding the current sample to the associated algorithm, while observing the feedback corresponds to being able to estimate in an approximate manner (e.g., on a separate test set) the generalization error of the trained algorithm for that arm, this error being a decreasing function of the number of samples the chosen algorithm has so far been trained over. Because the algorithms to select from may originate from different modeling assumptions, we also view this as a best model identification problem.
+
+Contributions. We first propose a novel notion of regret which is suited to the online learning problem we consider here. This regret criterion frames our problem as a best arm identification problem within a rested bandit scenario. We then characterize the structure of the problem by proving a non-asymptotic lower bound restricted to the 2-arm case. Finally, we propose two action elimination algorithms, and show for one of the two algorithms a regret upper bound that essentially matches the above-mentioned lower bound.
+
+Notation. For a integer  $N > 0$ , we abbreviate the set  $\{1, \dots, N\}$  by  $[N]$ . We use  $\mathbb{E}[\cdot]$  and  $\mathbb{P}[\cdot]$ , to denote expected value and probability measure, respectively. Moreover, for a given  $\sigma$ -algebra  $\mathcal{F}$ ,  $\mathbb{E}_{\mathcal{F}}[\cdot]$  and  $\mathbb{P}_{\mathcal{F}}[\cdot]$  denote their conditional counterparts.
+
+# 2. Related works
+
+Our problem can be seen as a (rested variant of) the best arm identification problem, in that our metric reminds the simple-regret that was previously designed for the best-arm identification problem in the standard (stationary) stochastic multi-armed bandit setting (e.g. Even-Dar et al., 2006; Audibert & Bubeck, 2010; Gabillon et al., 2012; Kaufmann et al., 2016). The best arm identification problem has been investigated from two slightly different viewpoints. In the so called fixed-confidence variant, the goal is to minimize the sample complexity (that is, the number of pulls) needed to guarantee that, with some fixed confidence level, the selected arm is the one with smallest expected loss. In the
+
+fixed-budget variant, the goal is to find the best-arm within a fixed number of rounds (budget), while minimizing the probability of error.
+
+In our case, we want the learning algorithm to single out with high probability (fixed confidence) the best base learner but, due to the non-stationary nature of the expected loss of base learners, we also want to do so with as few pulls as possible. Hence, we are in a sense combining the two criteria of fixed confidence and fixed budget.
+
+A problem which is similar in spirit to ours is that of online model selection in bandit settings (that is, the case where the base learners are themselves bandit algorithms). This has been investigated in a number of papers in recent years, e.g., (Agarwal et al., 2017; Foster et al., 2019; 2020; Pacchiano et al., 2020; Cutkosky et al., 2021). In particular Agarwal et al. (2017) consider a very general class of base learners which have to satisfy reasonable stability assumptions. In order to deal with bandit information, importance weighted feedback is given to the bandit learners. In (Foster et al., 2019; 2020) the emphasis is specifically on linear bandit model selection problems, where model selection operates on the input dimension (Foster et al., 2019) or the amount of misspecification (Foster et al., 2020). Similar to (Agarwal et al., 2017), in (Pacchiano et al., 2020) the authors investigate the problem of algorithm selection in contextual bandits where contexts are stochastic. In order to bypass the stability assumption in (Agarwal et al., 2017), an additional smoothed transformation is introduced. The positive side effect induced by this additional step is the ability to feed the base learners with the original feedback with no re-weighting.
+
+Unlike all the above works, we assume the expected loss of the base learners (which are not limited to bandit policies) depends in specific ways on the number of times each base learner is selected, this dependence being known up unknown parameters that have to be estimated. More importantly, we investigate a performance metric that is different from the standard cumulative regret incurred with respect to the best allocation policy, as studied in (Agarwal et al., 2017; Foster et al., 2020; 2019; Pacchiano et al., 2020). Further remarks on this comparison is given in the next section.
+
+A different stream of literature, which is loosely related to our work, is hyperparameter optimization; see (Li et al., 2017) for a representative example. The main difference with our setting is that, besides the standard exploration-exploitation trade-off, here we also have to deal with a trade-off induced by non-stationarity. Hyperparameter optimization algorithms like the one in (Li et al., 2017) adaptively searches in the space of hyperparameters, and the goal is akin to best arm identification. Yet, the feedback is assumed to be stationary, since hyperparameter values do not correspond to stateful objects (as in the case of our
+
+base learners), and hyperparameter configurations are usually evaluated on a separate validation set. An adversarial variant of hyperparameter optimization was considered in (Jamieson & Talwalkar, 2016), but their notion of regret is different from ours.
+
+In the bandits literature, there are two standard ways of modeling non-stationarity: restless (Whittle, 1988; Tekin & Liu, 2012; Ortner et al., 2014; Russac et al., 2019) and rested (Levine et al., 2017; Mintz et al., 2017; Kleinberg & Immoralica, 2018; Seznec et al., 2018; Pike-Burke & Grunewalder, 2019; Cella & Cesa-Bianchi, 2020; Kolobov et al., 2020) bandits. In the restless case, the non-stationary nature of the feedback is determined only by the environment, and the learning policies either try to detect changes in the payoff distribution in order to restart the learning model, or to apply a weight-decay scheme to the collected observations. On the contrary, in rested bandits, the non-stationarity depends on the learning policy itself. For instance, in (Cella & Cesa-Bianchi, 2020; Kleinberg & Immoralica, 2018; Kolobov et al., 2020; Pike-Burke & Grunewalder, 2019), an arm expected payoff distribution is parametrized by the elapsed time since that arm was last pulled. The main leverage given to the proposed solutions is the possibility of observing more unbiased samples corresponding to a fixed arm-delay pair. This simplifies the parameter estimation problem. Similar to the setting we are proposing, in (Levine et al., 2017; Seznec et al., 2018) the authors assume the expected loss of an arm to be monotonically increasing in the number of times the arm was pulled. The striking difference is that, in their variant, a simple greedy solution which at each round selects the currently-best arm is actually an optimal solution. Therefore, their learning problem reduces to estimating for each arm the expected loss corresponding to its next pull and always select the most promising one. In our setting (see Section 3 below), because expected losses are decreasing, a similar solution would be far from optimal, since our objective is to identify the arm minimizing the resulting loss at the end of the game.
+
+# 3. Learning setting
+
+We consider a set of  $K$  arms (or learning agents)  $\mathcal{K} = [K] = \{1,\dots ,K\}$ , whose average performance improves as we play them. At each round  $t\in [T]$ , the learner picks an arm  $I_{t}\in \mathcal{K}$  and observes the realization  $X_{I_t,t}$  of a loss random variable whose (conditional) expectation  $\mu_{I_t,t}$  is a decreasing function of the number of times arm  $I_{t}$  has been pulled so far. Specifically, for any  $i\in \mathcal{K}$  and  $t\in [T]$ , denote by  $\tau (i,t)$  the number of times arm  $i$  has been pulled up to time  $t$ , and by  $\mathcal{F}_t$  the  $\sigma$ -algebra generated by the past history of pulls and loss random variables  $I_1,X_{I_1,1},\ldots ,X_{I_{t - 1},t - 1}$ . Given a time horizon  $T$ , a learning policy  $\pi$  is a function that maps at each time  $t\in [T]$  the observed history
+
+$I_{1},X_{I_{1},1},\ldots ,I_{t - 1},X_{I_{t - 1},t - 1}$  to the next action  $I_{t}\in \mathcal{K}$  At the end of round  $T$  policy  $\pi$  has to commit to (or to output) a given action  $i_{\mathrm{out}}\in \mathcal{K}$  .We define
+
+$$
+\mu_ {i, t} \equiv \mathbb {E} _ {\mathcal {F} _ {t}} \left[ X _ {i, t} \right] = \frac {\alpha_ {i}}{\left(1 + \tau (i , t - 1)\right) ^ {\rho}} + \beta_ {i}, \tag {2}
+$$
+
+where exponent  $\rho \in (0,1]$  is a known parameter common to all arms while, for all arms  $i\in \mathcal{K}$ , scaling parameter  $\alpha_{i}$  and position parameter  $\beta_{i}$  are assumed to be non-negative but unknown to the learning algorithm. We assume  $\alpha_{i}\in [0,U]$  and  $\beta_{i}\in [0,1]$ , where the upper extreme  $U$  is a known quantity. Hence,  $\mu_{i,t}$  is the expected loss of arm  $i$  at round  $t$ , conditioned on the fact that  $i$  has already been played  $\tau (i,t - 1)$  times during the previous  $t - 1$  rounds.
+
+As a shorthand, from now on we will use  $\mu_i(\tau)$  to denote the expected loss of arm  $i\in \mathcal{K}$  if pulled so far  $\tau \in [T]$  times. Notice that when  $\alpha_{i} = 0$  for all  $i\in \mathcal{K}$  our setting reduces to the standard stochastic multi-armed bandit setting (e.g. (Auer et al., 2002)).<sup>1</sup> It is the decaying component  $\frac{\alpha_i}{(1 + \tau(i,t - 1))^{\rho}}$  that makes this setting an instance of the rested bandit setting, where the stochastic behavior of the arms depends on the actual policy  $I_{1},\ldots ,I_{t - 1}$  that has so far been deployed during the game.
+
+We compare a learning policy  $\pi$  to the optimal policy that knows all parameters  $\{\alpha_{i},\beta_{i}\}_{i\in \mathcal{K}}$  in advance, and pulls from beginning to end the arm  $i_T^*$  whose expected loss at time  $T$  is smallest, i.e.,
+
+$$
+i _ {T} ^ {*} = \arg \min  _ {i \in \mathcal {K}} \left(\frac {\alpha_ {i}}{T ^ {\rho}} + \beta_ {i}\right).
+$$
+
+We define the pseudo regret of  $\pi$  after  $T$  rounds as
+
+$$
+R _ {T} ^ {\pi} (\underline {{\mu}}) = \mu_ {i _ {\text {o u t}}} \left(\tau_ {\text {o u t}}\right) - \mu_ {i _ {T} ^ {*}} (T), \tag {3}
+$$
+
+where  $\tau_{\mathrm{out}} = \tau (i_{\mathrm{out}},T)$  is the random variable counting the number of pulls of arm  $i_{\mathrm{out}}\in \mathcal{K}$  after  $T$  rounds. In the above,  $\underline{\mu}\in \{\mu_i:[T]\to [0,1]\}_{i\in \mathcal{K}}$  collectively denotes the non-stationary environment generating the observed losses. Our goal is to bound pseudo-regret  $R_T^\pi (\underline{\mu})$  with high probability, where the probability is w.r.t. the random draw of variables  $X_{i,t}$  (and possibly the random choice of  $I_{1},\ldots ,I_{T}$  and  $i_{\mathrm{out}}$ ). Notice that  $R_T^\pi (\underline{\mu})$  is always non-negative.
+
+A closer inspection of Eq. (3) reveals that, unlike standard best-arm identification problems (e.g., (Even-Dar et al., 2006; Audibert & Bubeck, 2010; Gabillon et al., 2012; Kaufmann et al., 2016)), our objective here is not limited to predicting which arm is best at the end of the game, but also to pull it as much as we can, that is, to single it out as early as possible. This also entails that if the arm our policy  $\pi$  pulls the most throughout the  $T$  rounds is  $i \neq i_T^*$ ,
+
+then it may be better for  $\pi$  to output  $i_{\mathrm{out}} = i$  rather than  $i_T^*$  itself, even if  $\pi$  gets to know at some point the identity of  $i_T^*$  and starts pulling it from that time onward. This is because if, say, for some  $t_0$  close to  $T$  we have  $\tau (i,t_0) = t_0$  and  $\tau (i_T^*,t_0) = 0$ , then we may well have  $\mu_{i,t_0} < \mu_{i_T^*,T - t_0}$  so that (3) is smaller for  $i_{\mathrm{out}} = i$  than for  $i_{\mathrm{out}} = i_T^*$ . In order to gather further insights, it is also worth considering the simple policy  $\pi$  which selects all arms  $T / K$  times, and then outputs the best arm  $i_T^*$ . According to Eq. (3),  $\pi$  will still suffer significant regret, since it did not play  $i_T^*$  often enough throughout the  $T$  rounds (that is,  $\pi$  has explored "too much" on sub-optimal arms). We can thus claim that, thanks to the presence of the  $\tau_{\mathrm{out}}$  variable, our regret in (3) is only seemingly non-cumulative.
+
+Finally, observe that the average loss  $\mu_{i,t}$  in (2) can be expressed as the linear combination  $\mu_{i,t} = x_t^\top \theta_i^*$ , where  $\theta_i^* = [\alpha_i,\beta_i]^\top$  is the unknown vector associated with arm  $i$ , and  $x_{t} = [1 / \tau (i,t - 1)^{\rho},1]^\top$  is the "context" vector at time  $t$ . This might give the impression of some kind of linear contextual bandit (e.g., (Soare et al., 2014)) in the best arm identification regime. Yet, this impression is erroneous, since in our problem  $x_{t}$  is itself generated by the learning policy during its online functioning.
+
+In the sequel, we also adopt the notion of state  $\underline{\tau} = (\tau_1,\tau_2,\dots \tau_K)\in [T]^K$  to encode the case where, for all  $i\in \mathcal{K}$ , arm  $i$  has been pulled  $\tau_{i}$  times. Notice that when the learning policy is at state  $(\tau ,\ldots ,\tau)$ , keep sampling all arms in a round-robin fashion (exploring) entails observing  $K$  many samples with expected value  $\mu_1(\tau),\dots ,\mu_K(\tau)$  respectively, and ending up into state  $(\tau +1,\dots ,\tau +1)\in [T]^K$ . Conversely, when the learning policy is at state  $(\tau ,\dots ,\tau)$ , then keep pulling the same arm  $i\in \mathcal{K}$  for the remaining  $T - K\tau$  rounds (exploiting) corresponds to reaching the furthest still reachable state where arm  $i$  (which will then be the most pulled one) will have expected loss  $\mu_i(T - (K - 1)\tau)$ .
+
+On the comparison to Bandit Model Selection. The reader may wonder to what extent our task is similar to the bandit model selection problem (Agarwal et al., 2017; Foster et al., 2019; 2020; Pacchiano et al., 2020; Cutkosky et al., 2021) we alluded to in the related works section. At a high-level, this question is similar to the difference between best-arm identification (BAI) and regret minimization (RM). We can coarsely claim that the cited papers correspond to the generalization to stateful arms of RM, while here we generalize BAI to stateful arms having a specific parametric form of their losses. In fact, we cannot easily compare to bandit model selection, since our problem is substantially different. To see why, consider two base learners (say, two students) whose expected loss curves intersect just at time  $T - 1$ . The first one has  $\alpha_{1} = 0$  and  $\beta_{1} > 0$  (not improving over time but looking promising at the beginning), the
+
+second one has  $\alpha_{2} > 0$  and  $\beta_{2} = 0$  (lagging behind at the beginning, but able to ramp up faster over time). In our BAI setting, the optimal policy would stick to arm 2 since  $\mu_{2}(T) < \mu_{1}(T)$ , while an RM policy would clearly seek to play arm 1. This also helps elucidate the key role played by  $T$ : As  $T \to \infty$  arm 2 becomes more attractive even in the RM sense, while if we stop earlier, say  $T / 2$  (so that the two loss curves no longer intersect), then arm 1 becomes better also in our BAI setting. All in all, in our setting one has to depart from the general idea, quite common in RM, of studying regret "for large  $T$  (or even "for large  $K$ ), since the complex interplay among problem parameters makes these investigations less meaningful than in RM. Observe that, as  $T \to \infty$  our setting turns to standard stationary BAI, since  $\mu_{i,t} \rightarrow \beta_{i}$  in Eq. (2). Hence, what matters here is the case when  $T$  is not large.
+
+# 4. Main trade-offs and lower bound
+
+In this section we provide a distribution-dependent lower bound for the proposed setting. This will also give us the chance to comment on the specific features of our learning task in terms of the main trade-offs a learning policy has to face. We start by defining the class of arm-elimination policies as those which periodically remove sub-optimal arms and keep sampling in a round-robin fashion<sup>2</sup> the remaining arms across the rounds. The following simple fact holds.
+
+Fact 1. The regret incurred by an arbitrary policy  $\pi$  is invariant to permutations of the chronological order of its actions. In fact, (3) only depends on the arm  $i_{\mathrm{out}}$  selected at the end, and the number of times  $\tau_{\mathrm{out}}$  that arm has been chosen during the  $T$  rounds. Hence, for any  $\pi$ , there exists an arm-elimination policy  $\pi'$  whose regret is not worse (i.e., having the same pair  $i_{\mathrm{out}}, \tau_{\mathrm{out}}$ ).
+
+Proof. Let for simplicity  $K = 2$ : (i) Since expected losses (2) decrease with the number of pulls, it is easy to see that the best thing a policy can do so as to minimize regret (3) is to decide as early as possible which arm to commit to, pull that arm from that point on, and output it as  $i_{\mathrm{out}}$ . Any deviation from this results in higher regret. (ii) In the initial stage before commitment, the best thing a policy can do is to equalize the number of pulls of the two arms (hence the round-robin sampling of an arm-elimination policy). Any deviation from this equalization strategy can be penalized by the adversary generating parameters  $(\alpha_i, \beta_i)$ : if we pull arm 1 more than arm 2 the adversary may have made 2 the optimal arm at the time we commit, and vice versa.
+
+We can therefore restrict our lower bound investigation to arm-elimination policies. An advantage of this restriction is a more convenient characterization of the state space  $\{\underline{I}\}$
+
+![](images/8f129616029838c6f5c663a5beb29a28def4ccc2d213e6e2b6f3f482f4c3cd5b.jpg)  
+Figure 1. Expected losses associated with the arms in Example 1.
+
+associated with the learning problem. The size of the state space is clearly of the form  $T^K$ .
+
+Another relevant aspect of our problem is that, based on (2) and (3), for each given state  $(\tau_{1},\dots,\tau_{K})\in [T]^{K}$ , there are at most  $K$  many candidate optimal and still reachable states that any policy could end up to. These are specifically the  $K$  alternative states that the learning policy at hand would reach by committing to one of the  $K$  arms for all the remaining rounds. All other states (which are exponential many) can easily be seen to be sub-optimal.
+
+Before moving to the main result of this section (the regret lower bound), we would like to give an additional characterization of the considered class of policies. The missing component which gives a well-specified policy is the condition governing the arm elimination. Since expected losses (2) are non-increasing, and given the regret criterion (3), once a policy is confident that sticking to an arm would give a smaller expected loss than the one associated with the last pull, this policy might be tempted to eliminate all the other arms. In the next example we show that operating this way can be sub-optimal.
+
+Example 1. Let us consider the specific instance of our problem with  $K = 2$  arms whose expected losses are sketched in Figure 1. Whereas the first arm is stationary  $\mu_1(\tau) = \beta_1$ , the second is not,  $\mu_2(\tau) = \frac{\alpha_2}{\tau^\rho} + \beta_1$ . At state  $\underline{\tau}_1 = (\tau_1, \tau_1)$  it may occur that the  $\tau_1$  observations associated with arm 2 are enough to realize that  $\mu_2(T - \tau_1) < \mu_2(\tau_1)$ . Hence the learning policy knows that if it kept sampling arm 2 for the remaining  $T - 2\tau_1$  pulls it would achieve a smaller (expected) loss compared to  $\mu_2(\tau_1)$ . The same would not hold for the other arm, as it is stationary.
+
+Let us now denote by  $\tau_{2}$  the number of pulls it takes to figure out that  $\beta_{1} < \mu_{2}(T - \tau_{2})$ . It could be the case that  $\tau_{2} > \tau_{1}$  (that is, as in Figure 1, we have  $\mu_{2}(T - \tau_{2}) - \beta_{1} < \mu_{2}(\tau_{1}) - \mu_{2}(T - \tau_{1})$ ). In order to maximize  $\tau_{\mathrm{out}}$  (so as to minimize regret (3)) a naive policy might eliminate arm 1 after  $\tau_{1}$  observations. This would translate into choosing the wrong value of  $i_{\mathrm{out}}$ , hence clearly incurring a regret. Conversely, a smarter policy that keeps exploring up to state
+
+$(\tau_{2}, \tau_{2})$  would return  $i_{\mathrm{out}} = 1$  and yield  $\tau_{\mathrm{out}} = T - \tau_{2}$ . Notice that, thanks to the stationary nature of the optimal arm, the regret incurred by this policy is indeed zero.
+
+All in all, the above observations help better understand the structure of our problem, which will be useful in all technical proofs (see the appendix). We can now turn our attention to the lower bound. In doing so, we generalize the results in (Bubeck et al., 2013), which in turn adopts a hypothesis testing argument that hinges on a lower bound for the minimax risk of hypothesis testing (ee e.g. Tsybakov, 2008, Ch. 2). Notice that the classical lower bound result for stationary bandits (Lai & Robbins, 1985) cannot easily be adapted here since, being asymptotic in nature, that result tends to lose the non-stationary component of our expected losses (2), and thus the cumulated effect of this non-stationarity on the  $\tau_{\mathrm{out}}$  variable.
+
+As done in (Bubeck et al., 2013), for all arms and all possible number of pulls, we consider all families of loss distributions  $\{\mathbb{P}_{\mu}\}$ , indexed by their expected value  $\mu$ , and such that  $KL(\mathbb{P}_{\mu}, \mathbb{P}_{\mu'}) = C(\mu - \mu')^2$  for some absolute constant  $C > 0$  (e.g., in the case of normal distributions,  $KL(\mathcal{N}(\mu, \sigma), \mathcal{N}(\mu', \sigma)) = \frac{1}{2} (\mu - \mu')^2$ ). In the sequel, we use  $\tau_{\mathrm{sub}} = T - \tau_{\mathrm{out}}$  to denote the number of rounds spent by pulling all arms different from  $i_{\mathrm{out}}$ . Additionally, we denote by  $\mathbb{P}_{\underline{\mu}(\tau)} = \mathbb{P}_{\mu_1(\tau)} \otimes \ldots \otimes \mathbb{P}_{\mu_K(\tau)}$  the product distribution that generates the losses from  $\mathbb{P}_{\mu_i(\tau)}$  when pulling arm  $i \in \mathcal{K}$  for the  $\tau$ -th time. The result that follows restricts to the two arm case, $^3$  and delivers a bound on the regret that holds in expectation over the loss random draws.
+
+Theorem 1. Let  $\mathbb{P}_{\underline{\mu} (\tau)} = \mathbb{P}_{\mu_1(\tau)}\otimes \mathbb{P}_{\mu_2(\tau)}$  be defined by distributions whose expected values are  $\mu_{1}(\tau) = \frac{\alpha_{1}}{\tau^{\rho}} +\beta$  and  $\mu_{2}(\tau) = \frac{\alpha_{2}}{\tau^{\rho}} +\beta +\Delta$  , respectively, where  $\Delta >0$  is an unknown but fixed constant. Additionally, let  $\mathbb{P}_{\underline{\mu}'(\tau)} = \mathbb{P}_{\mu_1'(\tau)}\otimes \mathbb{P}_{\mu_2'(\tau)}$  be another product distribution, whose expected value components are  $\mu_1^\prime (\tau) = \mu_1(\tau)$  and  $\mu_2^\prime (\tau) = \mu_1(\tau) - \Delta$  . Then, for any policy  $\pi$  , and any horizon  $T\geq 1$  , the quantity max{E[Rr(μ)],E[Rr(μ')]} can be lower bounded by
+
+$$
+\left\{ \begin{array}{l l} \mu_ {2} (T - \tau_ {\mathrm {s u b}}) - \mu_ {i _ {T} ^ {*}} (T) & i f T - \tau_ {\mathrm {s u b}} \leq \lceil \overline {{\tau}} \rceil , \alpha_ {1} > \alpha_ {2} \\ \mu_ {1} (T - \tau_ {\mathrm {s u b}}) - \mu_ {i _ {T} ^ {*}} (T) & i f T - \tau_ {\mathrm {s u b}} \geq \lceil \overline {{\tau}} \rceil , \alpha_ {1} > \alpha_ {2} \\ \alpha_ {1} \left(\frac {1}{(T - \tau_ {\mathrm {s u b}}) ^ {\rho}} - \frac {1}{T ^ {\rho}}\right) & i f \alpha_ {1} \leq \alpha_ {2} , \end{array} \right.
+$$
+
+where  $\overline{\tau} = \left(\frac{\alpha_1 - \alpha_2}{\Delta}\right)^{\frac{1}{\rho}}$  satisfies  $\mu_{1}(\overline{\tau}) = \mu_{2}(\overline{\tau})$ . We also have  $i_T^* = \{1\}$  if  $\overline{\tau} < T$  and  $i_T^* = \{2\}$  otherwise. Finally,
+
+$\tau_{\mathrm{sub}}$  is the smallest  $\tau \in [T / 2]$  which is strictly larger than
+
+$$
+\begin{array}{l} \min  \left\{\frac {1}{8 C \Delta_ {\tau} ^ {2}} \log \left(C \Delta_ {\tau} ^ {2} \tau / 4\right), \frac {1}{8 C \widetilde {\Delta} _ {1 , \tau} ^ {2}} \log \left(C \widetilde {\Delta} _ {1, \tau} ^ {2} \tau / 4\right), \right. \\ \left. \frac {1}{8 C \widetilde {\Delta} _ {2 , \tau} ^ {2}} \log \left(C \widetilde {\Delta} _ {2, \tau} ^ {2} \tau / 4\right) \right\}, \\ \end{array}
+$$
+
+where  $\Delta_{\tau} = \mu_1(\tau) - \mu_2(\tau)$  and  $\widetilde{\Delta}_{i,\tau} = \mu_i(T - \tau - 1) - \mu_i(T - \tau)$ .
+
+The proof is given in the appendix. The main idea behind it is that under the assumptions of Theorem 1, we have two quantities characterizing the lower bound on the number of sub-optimal pulls  $\tau_{\mathrm{sub}}$ . The first one is associated with  $i_{\mathrm{out}}$ , and is of order  $1 / \Delta_{\tau}^{2}$ . The second one is induced by the objective of minimizing the incurred expected loss at  $\tau_{\mathrm{out}}$  and is of order  $1 / \widetilde{\Delta}_{i,\tau}^{2}$ . The main point here is that exploring towards arm  $i_{\mathrm{out}}$  is worthwhile only if it does not cause a higher incurred loss  $\mu_{i_{\mathrm{out}}}(\tau_{\mathrm{out}})$ . The second ingredient is specified by relation between  $\alpha_{1}$  and  $\alpha_{2}$ . In fact, if  $\alpha_{1} \leq \alpha_{2}$  we have that the second arm's expected loss  $\mu_{2}(\tau)$  is always worse than the one of the first, if considered at the same number of pulls  $\tau$ . On the other hand, if  $\alpha_{1} > \alpha_{2}$  their order relation depends on  $\overline{\tau}$ .
+
+We would like to emphasize that even when  $\alpha_{1} = \alpha_{2}$ , the horizon  $T$  is large enough, and  $\Delta \rightarrow 0$  (that is, the two arms are less and less statistically distinguishable), an optimal strategy for our regret minimization problem is by no means to pull both arms an equal  $(T / 2)$  number of times. Rather, an optimal strategy would commit to one of the two arms as soon as it is confident enough on which of them has the smaller loss (at any reachable state), unless trying to determine the best arm causes a bigger regret than immediately committing to any of the two. The quantity  $\tau_{\mathrm{sub}}$  (at least in the two-arm case) will play a key role in characterizing the statistical complexity of our learning problem.
+
+# 5. Estimation of parameters
+
+In order to minimize the regret  $R_T^\pi(\underline{\mu})$ , any reasonable policy  $\pi$  has to be able to estimate, for all arms  $i \in \mathcal{K}$ , the associated expected loss  $\mu_i(\cdot)$ , and it has to do so at any still reachable state where arm  $i$  will be pulled  $\tau_{\mathrm{out}}$  times. To this effect, we now introduce two statistically independent estimators. Upon pulling arm  $i \in \mathcal{K}$  for  $2\tau$  times, we define
+
+$$
+\widehat {X} _ {i, \tau} = \frac {1}{\tau} \sum_ {s = 1} ^ {\tau} X _ {i} (s), \quad \widetilde {X} _ {i, \tau} = \frac {1}{\tau} \sum_ {s = \tau + 1} ^ {2 \tau} X _ {i} (s), \quad (4)
+$$
+
+where  $X_{i,s}$  denotes the loss incurred by arm  $i$  after having pulled it  $s$  times  $(\mathbb{E}[X_{i,s}] = \mu_{i,s})$ . Notice, that due to the way we have defined  $\mu_{i,s}$ , the above estimators are empirical averages of independent but non-identically distributed
+
+random variables, the independence deriving from the fact that pulling one arm does not influence the distribution of the others. Moreover, because of the time decay, the expectation of  $\widehat{X}_{i,\tau}$  cannot be smaller than the one of  $\widetilde{X}_{i,\tau}$ .
+
+We combine these two estimators together with standard concentration inequalities to derive a joint estimator for  $(\alpha_{i},\beta_{i})$ . Since the two estimators are non-redundant, this allows us to come up with estimators for  $\alpha_{i}$  and  $\beta_{i}$  individually. Using Bernstein's inequality, we can derive confidence bounds around  $\widehat{X}_{i,\tau}$  and  $\widetilde{X}_{i,\tau}$  as functions of  $\beta_{i}$  and  $\alpha_{i}$ . Specifically, for each arm  $i\in \mathcal{K}$ , number of pulls  $2\tau \in [T]$ , the expectation  $\mathbb{E}[\widehat{X}_{i,\tau}]$  is contained with probability at least  $1 - \delta$  in the interval  $[\widehat{X}_{i,\tau} - \mathrm{CB}_{\widehat{X},\tau}(\delta),\widehat{X}_{i,\tau} + \mathrm{CB}_{\widehat{X},\tau}(\delta)]$  where
+
+$$
+\mathbf {C B} _ {\widehat {X}, \tau} (\delta) = \left(\sqrt {U} + 1\right) ^ {2} \sqrt {\frac {2}{\tau} \log \frac {1}{\delta}} + \frac {(U + 1) \log \frac {1}{\delta}}{\tau}.
+$$
+
+A similar argument follows for  $\mathbb{E}[\widetilde{X}_{i,\tau}]$  and  $\mathrm{CB}_{\widetilde{X},\tau}(\delta)$ . We defer to the appendix the details of the exact expression for the confidence intervals. Starting from these definitions we can build the following set of inequalities
+
+$$
+\mathbb {E} \left[ \widehat {X} _ {i, \tau} \right] - \mathbf {C B} _ {\widehat {X}, \tau} (\delta) \leq \widehat {X} _ {i, \tau} \leq \mathbb {E} \left[ \widehat {X} _ {i, \tau} \right] + \mathbf {C B} _ {\widehat {X}, \tau} (\delta)
+$$
+
+$$
+\mathbb {E} [ \widetilde {X} _ {i, \tau} ] - \mathbf {C B} _ {\widetilde {X}, \tau} (\delta) \leq \widetilde {X} _ {i, \tau} \leq \mathbb {E} [ \widetilde {X} _ {i, \tau} ] + \mathbf {C B} _ {\widetilde {X}, \tau} (\delta)
+$$
+
+which can be solved for  $\alpha_{i}$  and  $\beta_{i}$  individually. As shown in the appendix, this gives rise to the following confidence intervals for  $\alpha_{i}$ :
+
+$$
+\begin{array}{l} \alpha_ {i} \in \overbrace {\frac {\tau \Delta \widehat {X} _ {i , \tau}}{\sum_ {s = 1} ^ {\tau} \frac {1}{s ^ {\rho}} - \sum_ {s = \tau + 1} ^ {2 \tau} \frac {1}{s ^ {\rho}}}} ^ {\widehat {\alpha} _ {i, \tau}} \pm \\ \pm \frac {5 \tau^ {\rho} (\sqrt {U} + 1) ^ {2}}{\rho} \left[ \frac {\log 1 / \delta}{\tau} + \sqrt {\frac {1}{\tau} \log \frac {1}{\delta}} \right], \tag {5} \\ \end{array}
+$$
+
+where  $\Delta \widehat{X}_{i,\tau} = \widehat{X}_{i,\tau} - \widetilde{X}_{i,\tau}$ . For brevity, the confidence interval centroid will be denoted by  $\widehat{\alpha}_{i,\tau}$ . Similarly,  $\beta_{i}$  can be shown to satisfy
+
+$$
+\begin{array}{l} \beta_ {i} \in \overbrace {\widehat {X} _ {i , \tau} - \frac {\widehat {\alpha} _ {i , \tau}}{\tau} \sum_ {s = 1} ^ {\tau} \frac {1}{s ^ {\rho}}} ^ {\widehat {\beta} _ {i, \tau}} \pm \\ \frac {5 (\sqrt {U} + 1) ^ {2}}{(1 - \rho) \rho} \left[ \frac {\log 1 / \delta}{\tau} + \sqrt {\frac {1}{\tau} \log \frac {1}{\delta}} \right], \tag {6} \\ \end{array}
+$$
+
+where  $\widehat{\beta}_{i,\tau}$  denotes the centroid of confidence interval (6). Despite we have provided separate estimators for  $\alpha_{i}$  and  $\beta_{i}$ , it is important to stress that our interest here is not to estimate them separately. Combining these estimators gives
+
+$$
+\widehat {\mu} _ {i, \tau} (\tau_ {\mathrm {o u t}}) = \frac {\widehat {\alpha} _ {i , \tau}}{\tau_ {\mathrm {o u t}} ^ {\rho}} + \widehat {\beta} _ {i, \tau},
+$$
+
+an estimate of the expected loss incurred by arm  $i \in \mathcal{K}$  as if we had pulled it  $\tau_{\mathrm{out}}$  times after having observed only  $2\tau$  realizations of  $X_{i,t}$ . All the above can be summarized by the following theorem.
+
+Theorem 2. After observing  $X_{i,1},\ldots ,X_{i,2\tau}$  loss realizations of arm  $i\in \mathcal{K}$ , we can predict the expected loss  $\mu_{i,\tau_{\mathrm{out}}}$  of arm  $i$  as it were pulled  $\tau_{\mathrm{out}}$ -many times (with  $\tau_{\mathrm{out}} > \tau$ ). In particular, we have that with probability at least  $1 - \delta$  jointly over  $i\in \mathcal{K}$ ,  $\tau \in [T]$  and  $\tau_{\mathrm{out}}\in [T]$ ,
+
+$$
+\widehat {\mu} _ {i, \tau} (\tau_ {\mathrm {o u t}}) - C B _ {\mu , \tau} (\delta) \leq \mu_ {i, \tau_ {\mathrm {o u t}}} \leq \widehat {\mu} _ {i, \tau} (\tau_ {\mathrm {o u t}}) + C B _ {\mu , \tau} (\delta),
+$$
+
+where
+
+$$
+\mathrm {C B} _ {\mu , \tau} (\delta) = \frac {1 0 \left(\sqrt {U} + 1\right) ^ {2}}{(1 - \rho) \rho} \left[ \frac {\log \frac {\tau K T}{\delta}}{\tau} + \sqrt {\frac {1}{\tau} \log \frac {\tau K T}{\delta}} \right].
+$$
+
+Hence, the approach contained in Theorem 2 allows us to obtain confidence intervals for  $\mu_{i,\tau_{\mathrm{out}}}$  shrinking with  $\tau$  as  $\frac{1}{\sqrt{\tau}}$  up to a numerical constant depending on  $\rho$  and  $U$ .
+
+Finally, observe that these confidence intervals are non-vacuous only when  $\rho \in (0,1)$ , that is, excluding the extreme cases  $\rho = 0$  and  $\rho = 1$ . The case  $\rho = 0$  is indeed uninteresting, since it yields a stationary case which is equivalent to the one achieved by the setting  $\alpha_{i} = 0$  for all  $i$ . In fact, due to the specific nature of the empirical averages in (4), when  $\rho = 0$  the centroid  $\widehat{\alpha}_{i,\tau}$  occurring in (5) is not well defined, independent of the number of observed samples  $\tau$ . On the other hand, because our derivations rely on approximations of the form  $\sum_{s=1}^{\tau} \frac{1}{s^{\rho}} \approx \frac{s^{1 - \rho}}{1 - \rho}$ , which only hold for  $\rho \neq 1$ , the case  $\rho = 1$  should be treated separately via standard approximations of the form  $\sum_{s=1}^{\tau} \frac{1}{s} \approx \log \tau$ .
+
+The above estimators will be the building blocks of our learning algorithms, presented in the next section. In particular, the definition of  $CB_{\mu,\tau}(\delta)$  given in Theorem 2 above will be repeatedly used throughout the rest of the paper.
+
+# 6. Regret minimization
+
+In this section we present two learning policies. We first describe as a warm-up a simple explore-then-commit strategy, then we present a more sophisticated strategy inspired by the Successive Reject algorithm (Audibert & Bubeck, 2010). For both policies, we set the confidence parameter  $\delta$  to  $\frac{1}{T}$ .
+
+The first solution we propose is a rested bandit variant of the standard explore-then-commit (ETC) policy (e.g. Lattimore & Szepesvári, 2020, Ch. 6)). In its original formulation, ETC requires as input a parameter  $n \in [T]$  specifying the number of initial pulls associated with each arm. Once all the arms have been pulled  $n$  times, the exploratory phase finishes. The original ETC algorithm then sticks to the most
+
+# Algorithm 1 Explore-Then-Commit (ETC)
+
+Require: Confidence parameter  $\delta = 1 / T$
+
+1: for  $n \in 1, \ldots, \lfloor T / K \rfloor$  do  
+2: pull each arm once  
+3:  $\tau_{\mathrm{out}} = T - n(K - 1)$  
+4: if  $\exists i\in \mathcal{K}$  ..  $\widehat{\mu}_{i,n}(\tau_{\mathrm{out}}) <   \min_{j\in \mathcal{K}\setminus \{i\}}\widehat{\mu}_{j,n}(\tau_{\mathrm{out}}) -$ $2CB_{\mu ,n}(\delta)$  then  
+5:  $i_{\mathrm{out}} = \arg \min_{i\in \mathcal{K}}\widehat{\mu}_{i,n}(\tau_{\mathrm{out}})$  
+6: break; {The exploration phase terminates}  
+7: end if  
+8: end for  
+9: Play  $i_{\mathrm{out}}$  until round  $T$  {Commit}  
+10: Output  $i_{\mathrm{out}}$
+
+promising arm according to the estimates computed during the exploration. Hence the two phases of exploration and exploitation are kept separate. This strategy has a clear limitation. Since the exploration parameter  $n$  is an input to the algorithm, the original ETC algorithm does not adapt the length of the exploration phase to the actual samples, so that understanding how to best set  $n$  is not a simple task. One thing that is worth noticing is that in the 2-arm bandit case, this parameter  $n$  takes values in the range  $[T/2]$ . If  $\tau_{\mathrm{sub}}$  in our lower bound of Theorem 1 equals  $T/2$  (that is, when  $T/2$  is smaller than  $\log T / \Delta_{\tau}^{2}$ ,  $\log T / \widetilde{\Delta}_{1,\tau}^{2}$  and  $\log T / \widetilde{\Delta}_{2,\tau}^{2}$ ), we cannot commit to any specific arm, and the ETC algorithm results in a solo-exploration strategy which is indeed optimal in this case (up to the choice of  $i_{\mathrm{out}}$ ). Algorithm 1 describes a variant of the standard ETC policy adapted to our rested bandit scenario. At a generic round  $t = Kn$ , this algorithm starts committing to an arm  $i \in \mathcal{K}$  only when we are confident with probability at least  $1 - \delta$  that  $i$  is the arm with lowest expected loss if pulled for the remaining  $T - Kn$  times (Line 4). Hence, unlike the original ETC algorithm, this algorithm implicitly computes  $n$  on the fly based on the observed samples. Finally, upon committing to an arm, our algorithm does not reconsider its decision based on the newly collected samples (Line 9). We have the following result, that help elucidate the benefit of adaptively inferring  $n$ .
+
+Theorem 3. Consider the same two-arm setting  $\mathbb{P}_{\underline{\mu} (\tau)} = \mathbb{P}_{\mu_1(\tau)}\otimes \mathbb{P}_{\mu_2(\tau)}$  contained in Theorem 1 and the notation introduced therein. Running Algorithm 1 with  $T\geq 1$  yields
+
+$$
+R _ {T} ^ {E T C} (\underline {{\mu}}) \leq \mu_ {i _ {\mathrm {o u t}}} (T - n _ {0}) - \mu_ {i _ {T} ^ {*}} (T)
+$$
+
+with probability at least  $1 - \frac{1}{T}$ , where  $n_0 = \min \left\{\frac{T}{2}, \frac{c_\rho}{\Delta_{n_0}^2}\right\}$  and  $c_\rho = \frac{1600(\sqrt{U} + 1)^4}{\rho^2(1 - \rho)^2} \log (4n_0T^2)$ .
+
+This result is optimal up to a logarithmic factor (namely,  $i_{\mathrm{out}} \in \arg \min_{i \in \mathcal{K}} \mu_i(T - n_0)$  and  $n_0 = \tau_{\mathrm{sub}}$  up to a logarithmic factor) whenever  $n_0 = \frac{c_\rho}{\Delta_{n_0}^2}$  and  $\tau_{\mathrm{sub}} = \frac{1}{8C\Delta_\tau^2}\log \left(C\Delta_\tau^2\tau /4\right)$ . Conversely, when at least one of the
+
+above conditions is not met, the bound contains an additional  $\tilde{O}\left(1 / \sqrt{n_0}\right)$  term if compared to the result of Theorem 1, where  $\tilde{O} (\cdot)$  hides log  $T$  factors.
+
+Notice that finding the commitment condition for ETC cannot be readily obtained by available results in the bandit literature, as this requires a specific understanding of the interplay among the loss curves. Even in the two arm case of Theorem 3, it is not possible to avoid the unfriendly implicit form of the regret bound coming from the definition of  $n_0$  therein.
+
+Starting from ETC, in the next section we present our final learning policy, which will be analyzed both in the general  $K$ -armed case and in the specific setting contained in the lower bound of Section 4.
+
+# 6.1. Towards an Optimal Policy
+
+The first limitation of the ETC strategy in Algorithm 1 becomes clear when considering more than 2 arms. Let us consider an instance with  $K = 3$  arms where there exist two values  $n_2, n_3$  satisfying:
+
+$$
+\widehat {\mu} _ {1} \left(n ^ {\prime}\right) <   \widehat {\mu} _ {2} \left(n ^ {\prime}\right) - 2 C B _ {\mu , n _ {2}} \left(n _ {2}\right) \quad \forall n ^ {\prime} > n _ {2}
+$$
+
+$$
+\widehat {\mu} _ {1} \left(n ^ {\prime}\right) <   \widehat {\mu} _ {3} \left(n ^ {\prime}\right) - 2 C B _ {\mu , n _ {3}} \left(n _ {3}\right) \quad \forall n ^ {\prime} > n _ {3}.
+$$
+
+The ETC policy in Algorithm 1 has a single counter  $n$  that has to satisfy at the same time  $K - 1 = 2$  arm elimination conditions (line 4 of Algorithm 1). The best this algorithm can do in order not to commit to the wrong arm is to keep exploring up to  $n = \max \{n_2, n_3\}$ . The obvious drawback of this solution is that ETC would then waste  $|n_2 - n_3|$  pulls on the sub-optimal arms 2 and 3, rather than selecting  $i_{\mathrm{out}} = 1$ . We now present in Algorithm 2 the strategy REST-SURE (RESTed SUccessive REject), a rested version of the Successive Reject algorithm from (Even-Dar et al., 2006; Audibert & Bubeck, 2010). As for its stationary counterpart, REST-SURE keeps sampling all the active arms in a round-robin fashion, and then periodically removes arms once it is confident about their sub-optimality (line 12). The key adaptation to our rested bandit scenario is that one arm is deemed sub-optimal when there is a better arm in any of the still reachable states.
+
+Going into some details of the pseudocode, the stopping condition in lines 4 of Algorithm 2 is inspired by the same reasoning governing the commitment in the stationary bandit problem. This condition tells us that exploration has provided enough information to identify (with high probability) arm  $i_{\mathrm{out}} = \arg \min_{i\in \mathcal{K}}\mu_i(\tau_{\mathrm{out}})$  at the best reachable state. The second stopping condition (line 8) is due to the non-stationary component in the expected loss (2). This condition controls the trade-off between the estimation of  $i_{\mathrm{out}} = \arg \min_{i\in \mathcal{K}}\mu_i(\tau_{\mathrm{out}})$  and the minimization of the incurred expected loss, namely the impact on the value of
+
+Algorithm 2 REST-SURE  
+Require: Confidence parameter  $\delta = 1 / T$    
+1: Initialize:  $\mathcal{A}_0 = \mathcal{K},n = 0,\tau_{\mathrm{out}} = T$  , and  $t = 0$    
+2: for  $t = 1,\dots ,T$  do   
+3:  $\tau_{\mathrm{out}} = T - t + n$    
+4: if  $\exists i\in \mathcal{A}_n:\widehat{\mu}_i(\tau_{\mathrm{out}}) <   \min_{j\in \mathcal{A}_n\backslash \{i\}}\widehat{\mu}_j(\tau_{\mathrm{out}}) -$ $2CB_{\mu ,n}(\delta)$  then   
+5:  $i_{\mathrm{out}} = \arg \min_{i\in \mathcal{A}_n}\widehat{\mu}_i(\tau_{\mathrm{out}})$    
+6: break; {Found  $i^{*}_{\tau_{\mathrm{out}}}$  w.h.p.}   
+7: end if   
+8: if  $\min_{i\in \mathcal{A}_n}\widehat{\mu}_{i,n}(\tau_{\mathrm{out}} - |\mathcal{A}_n| + 1) - 2CB_{\mu ,n}(\delta)>$ $\min_{i\in \mathcal{A}_n}\widehat{\mu}_{i,n}(\tau_{\mathrm{out}})$  then   
+9:  $i_{\mathrm{out}}$  randomly chosen in  $\mathcal{A}_n$    
+10: break; {No advantage in learning  $i^{*}_{\tau_{\mathrm{out}}}$ }   
+11: end if   
+12:  $\mathcal{A}_{n + 1} = \mathcal{A}_n$    
+13: for each arm  $i\in \mathcal{A}_{n + 1}$  such that  $\forall m\in [n,\tau_{\mathrm{out}}]$  ..  $\exists j\in \mathcal{A}_{n + 1}:\widehat{\mu}_{i,n}(m) - \widehat{\mu}_{j,n}(m) > 2CB_{\mu ,n}(\delta)$  do   
+14:  $\mathcal{A}_{n + 1} = \mathcal{A}_{n + 1}\setminus \{i\}$  {Arm elimination}   
+15: end for   
+16: Pull once each active arm  $i\in \mathcal{A}_{n + 1}$    
+17:  $t = t + |\mathcal{A}_{n + 1}|;\quad n = n + 1$    
+18: end for   
+19: Play  $i_{\mathrm{out}}$  until round  $T$  {Commit}   
+20: Output  $i_{\mathrm{out}}$
+
+$\tau_{\mathrm{out}}$ . In particular, this condition stops the policy in its exploration towards the identity of  $i_{\mathrm{out}}$  as soon as this would cause an increased regret due to a reduced valued of  $\tau_{\mathrm{out}}$ .
+
+We need the following additional notation. We set for brevity  $\Delta_{j,i}(\tau) = \mu_j(\tau) - \mu_i(\tau)$  for any  $\tau \in [T]$ ,  $K_n = K - n$ , and  $\mu^{*}(\tau) = \min_{i\in \mathcal{K}}\mu (\tau)$  denotes the smallest expected loss over all arms after each one of them has been pulled  $\tau$  times. The following is the main result of this section.
+
+Theorem 4. For all  $K > 1$ , if REST-SURE is run on  $K$  arms having arbitrary non-stationary loss distributions  $\mathbb{P}_{\underline{\mu}(m)} = \mathbb{P}_{\mu_1(m)}, \otimes, \ldots, \otimes, \mathbb{P}_{\mu_K(m)}$  with support in  $[0,1]$  and expected value parameterized according to (2), then with probability at least  $1 - \frac{1}{T}$  the pseudo-regret of REST-SURE after  $T$  interactions satisfies
+
+$$
+R _ {T} ^ {\mathrm {R E S T - S U R E}} (\underline {{\mu}}) \leq \mu_ {i _ {\mathrm {o u t}}} (T - \overline {{n}}) - \mu_ {i _ {T} ^ {*}} (T),
+$$
+
+where  $\overline{n} = \sum_{s\in [K - 1]}n_{\sigma (s)}$ , and  $n_{\sigma (s)}$  is defined as the
+
+![](images/45818f3c9368e2faa89b45a8115a0724bb5fa5dbe2d34ba09f0fc3a859007057.jpg)  
+Figure 2. Expected losses associated with the arms in Example 2.
+
+smallest  $n\in [T]$  which is greater than
+
+$$
+\begin{array}{l} \min \left\{\frac {T - \sum_ {j = 1} ^ {s - 1} n _ {\sigma (j)}}{K _ {s - 1}}, \right. \\ \frac {c _ {\rho} \log (n K ^ {2} T ^ {2})}{\min _ {j \in \mathcal {A} _ {s - 1} ^ {*}, m \in [ n _ {\sigma (s) , \tau_ {\mathrm {o u t}} (s)} ]} \Delta_ {\sigma (s) , j} ^ {2} (m)}, \\ \frac {c _ {\rho} \log (n K ^ {2} T ^ {2})}{\left(\mu^ {*} \left(\tau_ {\mathrm {o u t}} (s) - K _ {s + 1}\right) - \mu^ {*} \left(\tau_ {\mathrm {o u t}} (s)\right)\right) ^ {2}}, \\ \left. \frac {c _ {\rho} \log (n K ^ {2} T ^ {2})}{\left(\min  _ {j \in \mathcal {K}} \left(\Delta_ {\sigma (s) , j} (\tau_ {\text {o u t}} (s))\right) ^ {2} \right.} \right\}. \\ \end{array}
+$$
+
+In the above,  $\mathcal{A}_s^* = \mathcal{K} \setminus \{\sigma(1), \ldots, \sigma(s-1)\}$ ,  $\tau_{\mathrm{out}}(n) = T - \sum_{s=1}^{n} K_{s+1} n_{\sigma(s)}$  and  $c_\rho = \frac{1600(\sqrt{U} + 1)^2}{\rho^2(1 - \rho)^2}$ . Notice that  $\sigma(s) = \arg \min_{j \in \mathcal{A}_{s-1}^*} n_{\sigma(j)}$ . Finally,  $i_{\mathrm{out}} \in \arg \min_{i \in \mathcal{K}} \mu_i(T - \overline{n})$  only if  $\min_{j \in \mathcal{A}_{s-1}^*} \Delta_{\sigma(s),j}(\tau_{\mathrm{out}}(s))$  is greater than  $\mu^*(\tau_{\mathrm{out}}(s) - K_{s+1}) - \mu^*(\tau_{\mathrm{out}}(s))$ . Conversely, when the latter condition is not met we can only guarantee that  $\mu_{i_{\mathrm{out}}} (T - \overline{n}) \leq \mu^*(T - \overline{n}) + \tilde{O}(1 / \sqrt{n_{i_{\mathrm{out}}}})$ .
+
+Notice that  $\overline{n}$  is solely a function of the problem parameters  $\{\alpha_{i},\beta_{i}\}_{i\in \mathcal{K}},\rho ,K,U$  and  $T$ . This is because so are the involved quantities  $\sigma (s)$  and  $n_{\sigma (s)}$ . The exact expression for  $\overline{n}$  might be somewhat hard to interpret. The first term in the min plays the same role as term  $T / 2$  in Theorem 1, and guarantees the total number of pulls is most  $T$ . The second term is obtained from the arm-elimination condition of line 12. The third term in the min is obtained by analyzing the condition at lines 7-8. Finally, the commitment to arm  $i_{\mathrm{out}}$  yields the fourth term. In order to clarify the heavy statement of Theorem 4, we now present a symbolic example.
+
+Example 2. For the sake of illustration, let us consider the specific instance of the above theorem with  $K = 3$  arms whose expected losses are sketched in Figure 2. In that figure,  $\tau_{1}$  is the number of times REST-SURE needs to pull each arm before eliminating arm 2. Hence, we have  $\sigma(1) = 2$  and  $n_{\sigma(1)} = \tau_{1}$ . Similarly,  $\tau_{2}$  is the number of pulls associated with the remaining arms  $A_{\tau_{2}} = \{1,3\}$  before committing to arm 3. Hence, with probability at least  $1 - 1 / T$ , we have  $\sigma(2) = 1$ ,  $i_{\mathrm{out}} = 3$  and  $\overline{n} = \tau_{1} + \tau_{2}$ .
+
+The proof of Theorem 4 is an extension of the proof of Theorem 3, and is given in the appendix. To conclude, we now show that the bound of REST-SURE matches the lower bound given in Theorem 1 up to logarithmic factors.
+
+Corollary 1. Let us consider the same two-arm setting  $\mathbb{P}_{\underline{\mu} (\tau)} = \mathbb{P}_{\mu_1(\tau)}\otimes \mathbb{P}_{\mu_2(\tau)}$  as in Theorem 1 and the notation introduced therein. Running Algorithm 2 with  $T\geq 1$  yields
+
+$$
+R _ {T} ^ {\mathrm {R E S T - S U R E}} (\underline {{\mu}}) \leq \mu_ {i _ {\mathrm {o u t}}} (T - n _ {0}) - \mu_ {i _ {T} ^ {*}} (T)
+$$
+
+with probability at least  $1 - \frac{1}{T}$ . In the above,  $n_0 = \min \left\{\frac{c_\rho}{\Delta_{n_0}^2}, \frac{c_\rho}{\overline{\Delta}_{n_0}^2}, \frac{T}{2}\right\}$  and  $c_\rho = 25600(\sqrt{U} + 1)^4 \log (4n_0 T^2)$ . This result is optimal up to a logarithmic factor (namely,  $i_{\mathrm{out}} \in \arg \min_{i \in \mathcal{K}} \mu_i(T - n_0)$  and  $n_0 = \tau_{\mathrm{sub}}$  up to a logarithmic factor) whenever  $n_0 = \frac{c_\rho}{\Delta_{n_0}^2}$  and  $\tau_{\mathrm{sub}} = \frac{1}{8C\Delta_{\tau}^2} \log \left(C\Delta_{\tau}^2\tau / 4\right)$ . Conversely, when at least one of the above conditions is not met, the bound contains an additional  $\tilde{O}\left(1 / \sqrt{n_0}\right)$  term if compared to the result of Theorem 1, where  $\tilde{O}(\cdot)$  hides  $\log T$  factors.
+
+Notice that, differently from Theorem 3, the definition of  $n_0$  now matches the one of  $\tau_{\mathrm{sub}}$  in Theorem 1 up to a logarithmic factor. Furthermore, the extra term  $\tilde{O}\left(\frac{1}{\sqrt{n_0}}\right)$  is always smaller than the one mentioned in Theorem 3.
+
+# 7. Conclusions and Ongoing Research
+
+In this work we have proposed an online algorithm selection problem formulated as a best arm identification within a specific rested bandit scenario. Here, each arm represents a candidate learning model and each pull corresponds to giving the associated learner more i.i.d training samples, thus allowing the learner to reduce its generalization error. We formulated an ad hoc notion of regret, provided a lower bound for the learning problem, and analyzed two alternative strategies, one of which we have shown to be optimal in the cases covered by the lower bound.
+
+We considered losses of the parametric form  $f(\tau; (\alpha, \beta)) = \frac{\alpha}{\tau^{\rho}} + \beta$  due to their relevance when considering the typical behavior of generalization error as a function of training set size. Yet, our analysis can be generalized to any parametric family of non-increasing functions  $f(\tau; \theta)$ , where  $\theta$  is a vector of parameters.
+
+At last we note that, while the focus of this paper has been theoretical, future work may be devoted to study the empirical performance of our methods and the underlying bounds. In Appendix D we included simple preliminary experiments on synthetic data that help corroborate our theoretical findings. A deeper investigation on real-world data is left to the future.
+
+# References
+
+Agarwal, A., Luo, H., Neyshabur, B., and Schapire, R. Corralling a band of bandit algorithms. In Conference on Learning Theory, pp. 12-38, 2017.  
+Allesiardo, R., Féraud, R., and Maillard, O.-A. The nonstationary stochastic multi-armed bandit problem. International Journal of Data Science and Analytics, 3(4): 267-283, 2017.  
+Audibert, J.-Y. and Bubeck, S. Best Arm Identification in Multi-Armed Bandits. In  $COLT - 23$ th Conference on Learning Theory - 2010, pp. 13 p., Haifa, Israel, June 2010. URL https://hal-enpc.archives-ouvertes.fr/hal-00654404.  
+Auer, P., Cesa-Bianchi, N., and Fischer, P. Finite-time analysis of the multiarmed bandit problem. Machine learning, 47(2-3):235-256, 2002.  
+Besbes, O., Gur, Y., and Zeevi, A. Stochastic multi-armed-bandit problem with non-stationary rewards. In Advances in neural information processing systems, pp. 199-207, 2014.  
+Bubeck, S., Cesa-Bianchi, N., et al. Regret analysis of stochastic and nonstochastic multi-armed bandit problems. Foundations and Trends in Machine Learning, 5 (1):1-122, 2012.  
+Bubeck, S., Perchet, V., and Rigollet, P. Bounded regret in stochastic multi-armed bandits. In Conference on Learning Theory, pp. 122-134, 2013.  
+Cella, L. and Cesa-Bianchi, N. Stochastic bandits with delay-dependent payoffs. In International Conference on Artificial Intelligence and Statistics, pp. 1168-1177, 2020.  
+Cesa-Bianchi, N. Multi-armed Bandit Problem. Springer New York, New York, NY, 2016. ISBN 978-1-4939-2864-4.  
+Cutkosky, A., Dann, C., Das, A., Gentile, C., Pacchiano, A., and Purohit, M. Dynamic balancing for model selection in bandits and r1. In ICML, 2021.  
+Even-Dar, E., Mannor, S., and Mansour, Y. Action elimination and stopping conditions for the multi-armed bandit and reinforcement learning problems. Journal of machine learning research, 7(Jun):1079-1105, 2006.  
+Foster, D., Gentile, C., Mohri, M., and Zimmert, J. Adapting to misspecification in linear contextual bandits and beyond. In Advances in Neural Information Processing Systems, 2020.
+
+Foster, D. J., Krishnamurthy, A., and Luo, H. Model selection for contextual bandits. In Advances in Neural Information Processing Systems, pp. 14714-14725, 2019.  
+Gabillon, V., Ghavamzadeh, M., and Lazaric, A. Best arm identification: A unified approach to fixed budget and fixed confidence. In Advances in Neural Information Processing Systems, pp. 3212-3220, 2012.  
+Jamieson, K. and Talwalkar, A. Non-stochastic best arm identification and hyperparameter optimization. In Artificial Intelligence and Statistics, pp. 240-248, 2016.  
+Kaufmann, E., Cappé, O., and Garivier, A. On the complexity of best-arm identification in multi-armed bandit models. The Journal of Machine Learning Research, 17 (1):1-42, 2016.  
+Kleinberg, R. and Immorlica, N. Recharging bandits. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pp. 309-319. IEEE, 2018.  
+Kolobov, A., Bubeck, S., and Zimmert, J. Online learning for active cache synchronization. arXiv preprint arXiv:2002.12014, 2020.  
+Lai, T. L. and Robbins, H. Asymptotically efficient adaptive allocation rules. Advances in applied mathematics, 6(1): 4-22, 1985.  
+Lattimore, T. and Szepesváři, C. Bandit algorithms. Cambridge University Press, 2020.  
+Levine, N., Crammer, K., and Mannor, S. Rotting bandits. In Advances in neural information processing systems, pp. 3074-3083, 2017.  
+Li, L., Jamieson, K., DeSalvo, G., Rostamizadeh, A., and Talwalkar, A. Hyperband: A novel bandit-based approach to hyperparameter optimization. The Journal of Machine Learning Research, 18(1):6765-6816, 2017.  
+Mintz, Y., Aswani, A., Kaminsky, P., Flowers, E., and Fukuoka, Y. Non-stationary bandits with habituation and recovery dynamics. arXiv preprint arXiv:1707.08423, 2017.  
+Ortner, R., Ryabko, D., Auer, P., and Munos, R. Regret bounds for restless markov bandits. Theoretical Computer Science, 558:62-76, Nov 2014. ISSN 0304-3975. doi: 10. 1016/j.tcs.2014.09.026. URL http://dx.doi.org/ 10.1016/j.tcs.2014.09.026.  
+Pacchiano, A., Phan, M., Abbasi-Yadkori, Y., Rao, A., Zimmert, J., Lattimore, T., and Szepesvari, C. Model selection in contextual stochastic bandit problems. arXiv preprint arXiv:2003.01704, 2020.
+
+Pike-Burke, C. and Grunewalder, S. Recovering bandits. In Advances in Neural Information Processing Systems, pp. 14122-14131, 2019.  
+Russac, Y., Vernade, C., and Cappé, O. Weighted linear bandits for non-stationary environments. In Advances in Neural Information Processing Systems, pp. 12017-12026, 2019.  
+Seznec, J., Locatelli, A., Carpentier, A., Lazaric, A., and Valko, M. Rotting bandits are no harder than stochastic ones. arXiv preprint arXiv:1811.11043, 2018.  
+Siegmund, D. Herbert Robbins and sequential analysis. Annals of statistics, pp. 349-365, 2003.  
+Soare, M., Lazaric, A., and Munos, R. Best-arm identification in linear bandits. In Advances in Neural Information Processing Systems, pp. 828-836, 2014.  
+Tekin, C. and Liu, M. Online learning of rested and restless bandits. IEEE Transactions on Information Theory, 58 (8):5588-5611, 2012.  
+Tsybakov, A. B. Introduction to nonparametric estimation. Springer Science & Business Media, 2008.  
+Whittle, P. Restless bandits: Activity allocation in a changing world. booktitle of applied probability, 25(A):287-298, 1988.

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+# Better Training using Weight-Constrained Stochastic Dynamics
+
+Benedict Leimkuhler 1 Tiffany Vlaar 1 Timothée Pouchon 1 Amos Storkey 2
+
+# Abstract
+
+We employ constraints to control the parameter space of deep neural networks throughout training. The use of customized, appropriately designed constraints can reduce the vanishing/exploding gradients problem, improve smoothness of classification boundaries, control weight magnitudes and stabilize deep neural networks, and thus enhance the robustness of training algorithms and the generalization capabilities of neural networks. We provide a general approach to efficiently incorporate constraints into a stochastic gradient Langevin framework, allowing enhanced exploration of the loss landscape. We also present specific examples of constrained training methods motivated by orthogonality preservation for weight matrices and explicit weight normalizations. Discretization schemes are provided both for the overdamped formulation of Langevin dynamics and the underdamped form, in which momenta further improve sampling efficiency. These optimization schemes can be used directly, without needing to adapt neural network architecture design choices or to modify the objective with regularization terms, and see performance improvements in classification tasks.
+
+# 1. Introduction
+
+We study stochastic training methods based on Langevin dynamics combined with algebraic constraints. Our general framework allows for incorporating constraints into standard training schemes and sampling methods for neural networks. Constraints provide direct control of the parameter space of a model and hence afford a means to improve its generalization performance. As applications, we consider magnitude control and orthogonality of neural network weights.
+
+$^{1}$ Department of Mathematics, University of Edinburgh, United Kingdom  $^{2}$ Department of Informatics, University of Edinburgh, United Kingdom. Correspondence to: Tiffany Vlaar <Tiffany.Vlaar@ed.ac.uk>.
+
+Proceedings of the  $38^{th}$  International Conference on Machine Learning, PMLR 139, 2021. Copyright 2021 by the author(s).
+
+Current approaches to enhance the generalization performance of overparameterized neural networks consist of both explicit and implicit regularization techniques (Neyshabur et al., 2015). Examples of the former are L1 (Williams, 1995; Tibshirani, 1996) and L2 (Hoerl & Kennard, 1970) regularization, which modify the loss by adding a parameter norm penalty term. Batch normalization (BatchNorm) (Ioffe & Szegedy, 2015) is a technique that causes an implicit regularization effect. BatchNorm can be viewed as tantamount to a constraint imposed on the network's parameters during training. Although BatchNorm is widely used, explanations for the method's success remain elusive (Santurkar et al., 2018; Yao et al., 2019). The reliance on increasingly complex strategies does little to enhance the explainability of neural networks, so robust simplification of all aspects of training is desirable. The constrained approach proposed in this paper provides a conceptually straightforward and interpretable framework that offers direct control of parameter spaces, without requiring modifications to the neural network architecture or objective. The transparency of this approach allows for drawing a direct connection between the use of weight constraints and the generalisation performance of the resulting neural network.
+
+In neural network (NN) training one aims to minimize the loss  $L_{X}(\theta)$  for parameters  $\theta \in \mathbb{R}^n$  and data  $X$ . Constraints can be seen as limiting cases of penalty-based regularization which replaces minimization of the loss  $L_{X}(\theta)$  by that of the augmented loss  $L_{X}^{c}(\theta) = L_{X}(\theta) + \frac{1}{\varepsilon^{2}} g(\theta)^{2}$ , where  $g(\cdot)$  is a suitable smooth function of the parameters. In the limit  $\varepsilon \to 0$ , these penalty terms introduce an undesirable stiffness and consequent stability restriction in gradient-based training, which limits the choice of step size (see Figure 5 for an illustration). It is therefore natural to relate the above system to a constrained optimization task subject to  $g(\theta) = 0$  (see Section 3).
+
+A popular NN training scheme is stochastic gradient descent (SGD). SGD may be improved by incorporating momenta (Sutskever et al., 2013) and additive noise (Welling & Teh, 2011; Wenzel et al., 2020), or more generally by embedding the loss gradient in a Langevin dynamics (LD) framework (Cheng et al., 2017). We will combine the resulting discretized stochastic differential equation (SDE) approach with constraints (Sec. 4). The benefit of using constrained SDEs for NN training is illustrated in Figure 1, where the
+
+![](images/20139cd9ae082ad378323deff884d3efb00f77d63258458086e220d661dbd58d.jpg)
+
+![](images/657f261f0d19c2f0e955d96536dacd61f64861fa96d70e9e41ca82cfee86d9f0.jpg)
+
+![](images/95be81c3b35bb85317e3a1a5d5b743e8266175f93aec42a1ab27875e73e2594c.jpg)
+
+![](images/27bdaa6d6ca234ba64b66e84407571e44a2f9710ce8da4d3a5fe51decc56a83a.jpg)
+
+![](images/0f3c82fca69f4273bb6ccbff15991c66fa771bd51c43fcbb1eceeb31585effe4.jpg)
+
+![](images/1d96c4546e4c81d88aedb3004b6cfd4c4204ae6045bc48701dcf5e31fd749bcb.jpg)
+
+![](images/ae79537308b1514ddaf34a7a66538ec30356399cd2b5c0751391c7dfd06e5133.jpg)
+
+![](images/9682358b48ab0fb966d9301aa1b62dcaf111b5f27081b10ae9d505127629a472.jpg)
+
+![](images/cd3f871f3f5432ff9149098dfda3e00f4f48bd88335e64b549a5229ebda53371.jpg)
+
+![](images/747b1e79b32df116048b932e3a5b908bfa814e3bae7b772fa10965bd42a40efc.jpg)
+
+![](images/c00ca8f5423fd18e76541bb1f07f3d95fecb1419c4418e4e702ef613d1f99d7a.jpg)  
+Figure 1. Classifiers obtained using different optimizers: SGD (1st column), SGD with weight decay (WD) (2nd col.), constrained SGD (C-SGD) (3rd col.), SGLD (Welling & Teh, 2011) (4th col.), constrained SGLD (5th col.) using a 500-node single hidden layer perceptron for a spiral binary classification problem. Top and middle row show training and test data points, respectively, and decision boundaries of the trained classifier. Bottom row shows loss curves. Hyperpar. settings: all:  $h = 0.05$ ,  $2\%$  subsampling; SGD with WD = 1e-4; C-SGD:  $r_0 = 1$ ,  $r_1 = 5$  (see Eq. (2)); SGLD and C-SGLD:  $\tau = 5\mathrm{e} - 5$  (see Eq. (7)). We observe that although the use of WD can stabilize the test loss, it does not improve test accuracy (2nd col.). In contrast, our constrained approach (3rd col.) maintains a stable test loss throughout training and improved generalization performance. The use of additive noise (or low temperature) in combination with the constraints (C-SGLD, 5th col.) strongly outperforms standard SGD:  $91.7\%$  vs.  $81\%$  test acc., resp., and obtains smoothened classification boundaries.
+
+![](images/53a20b2fac41b83b29ca0d6d5924d7a56e67731b79710bc91b4b166f998efbff.jpg)
+
+![](images/e72fb9cbb21a87dd2416a9c329e0c0ff35cdb3acb4fcb1895fa8c0211da0a977.jpg)
+
+![](images/ed093e3bb9170a37f5b2caadf08dd3da9a2e51e553c4e47935d7cf6cc4b5f091.jpg)
+
+![](images/16a080bca3b7465c7d089abe466c09bdd9d7b45fa636eca18b68906262fd06b4.jpg)
+
+combination of using additive noise and magnitude constraints (as defined by Eq. (2)) leads to smoother classification boundaries and significantly enhanced generalization performance (compare the 5th column, the constrained SDE approach, with column 1, standard SGD). These observations are maintained over 100 runs (see Fig. 2, Fig. 3, and Table 1). We distinguish between two different types of smoothness of the resulting classifiers: first, the curvature of the classification boundary and second, the sharpness of the transition between prediction regions belonging to different classes. As shown in Table 1 and Fig. 3 the use of magnitude constraints throughout training generates classifiers which exhibit both types of smoothness. The use of additive noise throughout training further reduces the curvature of the classification boundary. In contrast, the use of weight decay is not sufficient for SGD to obtain the same levels of smoothness. See Appendix D for further numerical details.
+
+Table 1. Accompanies Fig. 1 and 2, with same hyperparameter settings. We present estimates of the mean, standard deviation (std), and maximum (max) curvature of classifier boundaries obtained using different optimizers evaluated over 100 runs after training for a fixed number of 10,000 epochs. We computed our curvature estimates using the method described in Appx. D, which we suggest is indicative of the curvature of the locally smoothed classification boundary and allows us to compare the relative curvature estimates of classifiers trained using different optimizers. The combined use of constraints and additive noise (C-SGLD) obtains much lower curvatures compared to SGD with weight decay (WD).
+
+<table><tr><td rowspan="2">Optimizer</td><td colspan="3">Curvature Approximation</td></tr><tr><td>Mean</td><td>Std</td><td>Max</td></tr><tr><td>SGD</td><td>519</td><td>4.33 ·104</td><td>3.26·108</td></tr><tr><td>SGD with WD</td><td>51.1</td><td>3.80 ·103</td><td>1.14 ·107</td></tr><tr><td>C-SGD</td><td>9.38</td><td>317</td><td>5.58 ·105</td></tr><tr><td>SGLD</td><td>8.73</td><td>189</td><td>6.27 ·105</td></tr><tr><td>C-SGLD</td><td>6.08</td><td>40.8</td><td>1.43 ·105</td></tr></table>
+
+![](images/b7013810f68daad411498c79e4b9d840686cc5773293bfd453e859d261919718.jpg)  
+Figure 2. Same data and hyperparameter settings as for Fig. 1, but these results are averaged over 100 runs. Constrained approaches, C-SGD and C-SGLD (with additive noise), clearly outperform standard SGD and SGD with WD in terms of test loss and test accuracy.
+
+![](images/b867d411511a3cf2eb30c4ccc2f972d620fadffda9c97fd3d74fbee53048aa93.jpg)
+
+![](images/4787491b462fb0a5332bcf5adba5fb7ece846ad0003c9230d5c080b4b1a3fb81.jpg)  
+Figure 3. Same data and hyperparameter settings as in Fig. 1 and 2. We show gradients of the prediction along horizontal  $(y = 0)$  and vertical cross-sections  $(x = 0)$ . The results are averaged over 100 runs and evaluated at 10,000 epochs. Our constrained approach C-SGLD exhibits less sharp transitions between classes than SGD (middle/right plot). The size of the constraint directly controls this property (this is illustrated in the left plot for the input layer with constraint size  $r_0$  in Eq. (2) for all input layer weights).
+
+![](images/05cdf99b181bcf4e79e5d63733949ddb409f2ca0c81c9bea367817e89344b30b.jpg)
+
+![](images/a7e5c4bebc2c3c2f892931dda2d43a7fcbb6030172cbebbde575c62c3b1d5c23.jpg)
+
+Apart from such magnitude constraints, the general framework provided in this paper allows for straightforward incorporation of other constraints. Another specific example we consider is orthogonality of the weight matrix. We provide detailed algorithms for both of these purposes and for a general constraint in a Langevin dynamics setting (Sec. 4 and Appendix B) and show improved generalization performance on classification tasks (Sec. 5).
+
+Concretely, our contributions are:
+
+- We introduce the use of constrained stochastic differential equations for neural network training.  
+- We provide a general mathematical framework that allows for implementation of new constraints.  
+- We propose specific constraints, magnitude control and orthogonality of neural network weights, and provide algorithms to accompany these. The benefit of using these is illustrated for several networks and datasets and is shown to outperform soft constraints (such as weight decay or orthogonal regularization).
+
+- We provide PyTorch code to support our algorithms, which can be found on https://github.com/TiffanyVlaar/ConstrainedNNtraining
+
+# 2. Background and Related Work
+
+Neural network loss landscapes are notoriously difficult to characterize rigorously due to their high-dimensionality and non-convexity. Although they appear to contain multiple, roughly equivalent local minima which exhibit nearly zero training loss (Choromanska et al., 2015; Kawaguchi, 2016; Keskar et al., 2017), not all these minima obtain the same generalization performance (Chaudhari et al., 2017; Wu et al., 2017). The training of deep neural networks is hypersensitive to e.g., the choice of initialization (Sutskever et al., 2013), optimizer (Wilson et al., 2017), and hyperparameter settings (Jastrzewski et al., 2018), including learning rate scheduling (Loshchilov & Hutter, 2017; Smith, 2017). Without careful hyperparameter tuning, the loss landscape may not be explored sufficiently by the optimization scheme, thus resulting in a reduced generalization performance of the trained network (Zhang et al., 2015; Keskar et al., 2017).
+
+Sampling methods, which use small amounts of additive noise (Leimkuhler et al., 2019; Wenzel et al., 2020), have been found to enhance exploration and speed the approach to 'good' minima, which enhance their generalization to nearby data sets. Hence, we incorporate the flexibility to use additive noise to enhance exploration in our optimization schemes by taking a constrained SDE approach to neural network training. We propose a general mathematical framework for this purpose and consider the ergodic properties of the idealized SDEs associated with gradient schemes, which may help these methods to ensure robust exploration of a useful range of parameters (Sec. 4). We further propose specific constraints (Sec. 3) and show that the use of these leads to enhanced performance compared to soft constraints, such as weight decay or orthogonal regularization (Sec. 5).
+
+Magnitude control of neural network weights. In this work we consider a circle constraint, which limits the magnitude of the size of the weights (we typically leave the biases unconstrained). A corresponding soft constraint, which adds a penalty term to the loss, is weight decay or L2 regularization (Hoerl & Kennard, 1970). We also propose a sphere constraint, which is analogous to max-norm (Srebro & Shraibman, 2005; Srivastava et al., 2014) as used in some regularization procedures. However, applying this constraint in combination with additive noise does yield a distinctive training method.
+
+Orthogonality of the weight matrix. The concept of orthogonality has surfaced several times in the recent neural network literature. Orthogonal matrices have properties (norm preservation, unit singular values) which are thought to provide enhanced numerical stability (Zhou et al., 2006; Rodríguez et al., 2017). An orthogonal matrix  $Q \in \mathbb{R}^{r \times s}$  (i.e.,  $Q^T Q = I_s$ ) is an isometry:  $\| Qz \| = \| z \| \forall z \in \mathbb{R}^s$ . Orthogonal weight matrices were shown to mitigate the vanishing/exploding gradient problem in RNNs (Pascanu et al., 2013; Arjovsky et al., 2016; Vorontsov et al., 2017) and are developing a growing following in the CNN literature as well (Rodriguez et al., 2017; Bansal et al., 2018; Huang et al., 2018; Li et al., 2019). Orthogonal initialization is linked to achieving dynamical isometry (Saxe et al., 2013; Pennington et al., 2017; 2018), which can accelerate training. Xiao et al. (2018) were able to train 10,000 layer vanilla CNNs, without learning rate decay, BatchNorm or residual connections, by using initial orthogonal convolution kernels.
+
+Methods for enforcing orthogonality during training include the use of 'soft' constraints which add a restraint term to the loss (Brock et al., 2017; Xie et al., 2017; Bansal et al., 2018) and hard constraints based on optimization over Stiefel manifolds (Huang et al., 2018; Jia et al., 2019). The latter requires repeated singular value decomposition of high-dimensional matrices during training, which is costly. Int his work we propose a straightforward algorithm to incorporate orthonor
+
+mality constraints for rectangular matrices within our NN training framework, with manageable additional cost. We make no empirical claims over other manifold optimization methods, but rather provide a framework for network optimization that is theoretically sound, flexible enough to incorporate new constraints, and demonstrates good properties relative to standard SGD training or simple soft constraint approaches.
+
+Constrained SDEs. In this work we focus on optimization schemes for neural networks using constrained Langevin dynamics in both its overdamped and underdamped (with momentum) form. A discussion of the properties of unconstrained Langevin dynamics in its overdamped and underdamped forms was studied in Pavliotis (2014). We consider the specific issues associated to the extension of the standard framework to constrained SDEs. The ergodic properties of constrained Langevin (in the absence of gradient noise) were previously studied in Lelièvre et al. (2010) (overdamped) and Lelièvre et al. (2012) (underdamped). Exponential convergence to equilibrium for constrained overdamped Langevin is a consequence of a Poincaré inequality. Poincaré inequalities on manifolds and their use in the analysis of diffusion processes are presented in Bakry et al. (2013), Chapter 4. Finally, Langevin dynamics discretizations are studied in Faou & Lelièvre (2009); Lelièvre et al. (2010) ( overdamped) and Lelièvre et al. (2012); Leimkuhler & Matthews (2016) (underdamped).
+
+An alternative to the use of constrained SDEs are constrained Hamiltonian Monte Carlo (HMC) methods (Graham & Storkey, 2017; Zappa et al., 2018; Lelièvre et al., 2020). Although HMC schemes have nil sampling bias if fully converged, their acceptance rates depend on stepsize and system size (Beskos et al., 2013; Bou-Rabee & Sanz-Serna, 2018). In practice SDE-based methods are often preferred in many high-dimensional sampling calculations compared to HMC schemes as they are found to offer greater overall efficiency for a fixed computational budget.
+
+# 3. Neural Networks with Constraints
+
+Imposing good priors on neural networks is known to improve performance, e.g. convolutional neural networks (CNNs) suit image datasets better than overparameterized fully connected NNs, despite being a subset of the latter (d'Ascoli et al., 2019). Using constraints also arises naturally in the control of vanishing/exploding gradients. In Appendix C we illustrate this and also provide a connection between the magnitude of the weights and the smoothness of the interpolant. These observations suggest the use of constraints to control the magnitudes of individual weights and/or to limit the growth of gradients in deep networks. We present various approaches in this section.
+
+We consider a  $L$ -layer neural network, which has parameters  $\theta \in \mathbb{R}^n$ , with a weight matrix  $W^\ell \in \mathbb{R}^{d^\ell \times d^{\ell -1}}$  and bias vector  $b^\ell \in \mathbb{R}^{d^\ell}$  for each layer  $\ell$ . To allow for inequality constraints, we define slack variables vector  $\xi \in \mathbb{R}^{n^\xi}$  and consider variable  $q = (\theta ,\xi)\in \mathbb{R}^d$ , where  $d = n + n^{\xi}$ . The constraint manifold is
+
+$$
+\Sigma = \left\{q \in \mathbb {R} ^ {d} \mid g (q) = 0 \right\}, \quad g: \mathbb {R} ^ {d} \rightarrow \mathbb {R} ^ {m}. \tag {1}
+$$
+
+We partition  $\theta = (\theta^u,\theta^c)$  into unconstrained  $\theta^u\in \mathbb{R}^{n^u}$  and constrained  $\theta^c\in \mathbb{R}^{n^c}$  parameters. We typically only constrain the neural network weights, not the biases.
+
+Circle constraints: In a circle constraint, we restrict each parameter in  $\theta^c$  as  $|\theta_i^c |\leq r_i$ , where  $r_i > 0$  is given. We thus introduce  $m = n^{c} = n^{\xi}$  slack variables  $\xi_{i}$  and define
+
+$$
+g _ {i} (q) = \left| \theta_ {i} ^ {c} \right| ^ {2} + \left| \xi_ {i} \right| ^ {2} - r _ {i} ^ {2} \quad 1 \leq i \leq m. \tag {2}
+$$
+
+If  $q \in \Sigma$ , then the parameters in  $\theta^c$  are bounded as desired.
+
+Sphere constraints: In a similar way, we could opt to restrict the sums of squares of weights associated to the input channels of any node. For layer  $\ell$ , we denote the  $i$ -th row of the weight matrix  $W^{\ell}$  as  $\theta^{c,i}$ , set  $\theta^u = b^\ell$ , introduce  $m = d^\ell$  slack variables  $\xi_{i}$ , and define as sphere constraint:
+
+$$
+g _ {i} (q) = \left\| \theta^ {c, i} \right\| ^ {2} + \left| \xi_ {i} \right| ^ {2} - r _ {i} ^ {2}, 1 \leq i \leq m, \tag {3}
+$$
+
+where  $\|\cdot\|$  denotes the Euclidean norm. Sphere constraints are analogous to max-norm (Srebro & Shraibman, 2005; Srivastava et al., 2014), but have been unexplored in combination with additive noise. We leave this for future work.
+
+Orthogonality constraints: We set  $\theta^u = b^\ell$ , and define as orthogonality constraint for layer  $\ell$  with  $n^\ell$  parameters
+
+$$
+g (q) = \left\{ \begin{array}{l l} {\left(W ^ {\ell}\right) ^ {T} W ^ {\ell} - I _ {n ^ {\ell - 1}}} & {\text {i f} n ^ {\ell - 1} \leq n ^ {\ell},} \\ {W ^ {\ell} \left(W ^ {\ell}\right) ^ {T} - I _ {n ^ {\ell}}} & {\text {o t h e r w i s e}.} \end{array} \right. \tag {4}
+$$
+
+As the matrix equality  $g(q) = 0$  is symmetric, it corresponds to  $m = s(s + 1) / 2$  constraints, where  $s = \min \{n^{l - 1}, n^l\}$ .
+
+# 4. Constrained SDEs and their Discretization
+
+In this chapter we describe SDE-based methods for constrained neural network training. We first introduce standard (unconstrained) Langevin dynamics in Section 4.1. Then in Section 4.2 we discuss properties of constrained Langevin dynamics (LD), such as ergodicity and exponential convergence to equilibrium, which ensures the effectiveness of our schemes as training methods. In Section 4.3 we discuss the discretization of constrained Langevin dynamics in both the overdamped and the underdamped case, where the use of momenta allows us to accelerate the training process. The choice of discretization scheme will strongly affect the efficiency and robustness of the resulting training method.
+
+Hence, to allow for ease and efficacy of implementation of our methods, we describe the most appropriate discretization schemes in detail in Appendix B for both the general setting and for the specific constraints we consider in this paper, i.e., circle and orthogonality constraints.
+
+# 4.1. Langevin Dynamics
+
+Consider the unconstrained Langevin system of SDEs
+
+$$
+\mathrm {d} \theta_ {t} = p _ {t} \mathrm {d} t, \tag {5}
+$$
+
+$$
+\mathrm {d} p _ {t} = \left(- \nabla L \left(\theta_ {t}\right) - \gamma p _ {t}\right) \mathrm {d} t + \sqrt {2 \gamma \tau} \mathrm {d} \mathcal {W} _ {t},
+$$
+
+with momenta  $p$ , parameters  $\theta$ , loss  $L(\theta)$ , temperature hyperparameter  $\tau \geq 0$ , friction hyperpar.  $\gamma$ , and  $d$ -dim. Wiener process  $\mathcal{W}$  (Leimkuhler & Matthews, 2015). Under some mild assumptions, Langevin dynamics is provably ergodic, which means that its solutions sample the distribution:
+
+$$
+\rho \propto \exp [ - (L (\theta) + \| p \| ^ {2} / 2) / \tau ]. \tag {6}
+$$
+
+The temperature hyperparameter  $\tau$ , which controls the additive noise level, provides a direct connection between a pure optimization and sampling approach. The standard Bayes posterior is recovered for  $\tau = 1$ , whereas setting  $\tau = 0$  will provide maximum a posteriori (MAP) point estimates. The range of values in between corresponds to an artificially sharpened posterior, where as  $\tau \rightarrow 0$ , the posterior probability mass is confined closer and closer to the modes of the distribution. Using low temperatures (Leimkuhler et al., 2019; Wenzel et al., 2020), sampling methods have been found to enhance exploration and speed the approach to 'good' minima, which enhance their generalization to nearby data sets. In this work we therefore consider a constrained SDE approach to neural network training to allow for the incorporation of both constraints and additive noise.
+
+# 4.2. Constrained Langevin: Ergodicity and Central Limit Theorem.
+
+The neural network loss function naturally extends to the variable  $q = (\theta, \xi) \in \mathbb{R}^d$  taking the form  $V(q) = L(\theta)$  (note that in particular  $\nabla_{\xi} V = 0$ ). The first continuous training method we consider is the constrained overdamped Langevin $^2$  system
+
+$$
+\mathrm {d} q _ {t} = - \nabla V (q _ {t}) \mathrm {d} t + \sqrt {2 \tau} \mathrm {d} \mathcal {W} _ {t} - \nabla_ {q} g (q _ {t}) \mathrm {d} \lambda_ {t}, \tag {7}
+$$
+
+$$
+0 = g (q _ {t}),
+$$
+
+where  $\mathcal{W}$  is a  $d$ -dim. Wiener process,  $\tau \geq 0$  is the temperature hyperparameter, and  $\lambda_{t}$  is an  $\mathbb{R}^m$ -valued vector of Lagrange multipliers. Provided the initial configuration  $q_{0}$  satisfies the constraint, any trajectory  $q_{t}$  of (7) remains on the constraint manifold  $\Sigma$  defined in Eq. (1). For  $\beta^{-1} = \tau > 0$ , (7) is equivalent to an underlying ergodic (unconstrained) SDE (see Appx. A.1) with unique invariant measure
+
+$$
+\mathrm {d} \nu_ {\Sigma} = Z ^ {- 1} e ^ {- \beta V (q)} \mathrm {d} \sigma_ {\Sigma}, Z = \int_ {\Sigma} e ^ {- \beta V (q)} \mathrm {d} \sigma_ {\Sigma}, \tag {8}
+$$
+
+where  $\sigma_{\Sigma}$  is the surface measure on  $\Sigma$
+
+Ergodicity ensures that averages of observables with respect to  $\nu_{\Sigma}$  can be approximated by time averages of trajectories of (7). To ensure the practical use of (7) as a training method, we need the convergence to occur in a reasonable time. Thanks to the reversibility of the underlying SDE (see Appx. A.1), exponential convergence to equilibrium occurs as a consequence of a Poincaré inequality for  $\nu_{\Sigma}$  (see Appx. A.2, A.3 and Bakry et al. (2013)). We provide a summary of the results here and refer to Appx. A for more details.
+
+A Poincaré inequality holds under a curvature-dimension assumption: there exists  $\rho > 0$  such that
+
+$$
+C D (\rho , \infty): \quad \operatorname {R i c} _ {\mathfrak {g}} + \beta \nabla_ {\mathfrak {g}} ^ {2} V \geq \rho \mathfrak {g}, \tag {9}
+$$
+
+in the sense of symmetric matrices. The terms in (9) rely on the structure of  $\Sigma$  as a Riemannian manifold:  $\mathfrak{g}$  is the Riemannian metric,  $\mathrm{Ric}_{\mathfrak{g}}$  is the Ricci curvature tensor and  $\nabla_{\mathfrak{g}}^{2}V$  is the Hessian of  $V$  on the manifold. Under (9) we have the following result ((Bakry et al., 2013), Appx. A.2).
+
+Theorem 4.1 Assume that there exists  $\rho >0$  and  $N > n$  such that  $CD(\rho ,N)$  holds. Then  $\nu_{\Sigma}$  satisfies a Poincaré inequality: there exists a constant  $L > 0$  such that
+
+$$
+\int_ {\Sigma} \left| \phi (q) - \langle \phi \rangle_ {\nu_ {\Sigma}} \right| ^ {2} d \nu_ {\Sigma} (q) \leq \frac {1}{2 L} \int_ {\Sigma} \left| \Pi (q) \nabla \phi (q) \right| ^ {2} d \nu_ {\Sigma}
+$$
+
+$$
+\forall \phi \in H ^ {1} (\nu_ {\Sigma}), \tag {10}
+$$
+
+where  $\Pi (q)$  is the projection onto the cotangent space  $T_q^*\Sigma$  Eq. (A-8) and  $H^{1}(\nu_{\Sigma})$  is the space of functions with square  $\nu_{\Sigma}$ -integrable gradients Eq. (A-7).
+
+Consequences of Theorem 4.1 are the exponential convergence and a central limit theorem (CLT) for the convergence in Eq. (A-2) (see Appx. A.3).
+
+Corollary 4.2 If (9) holds then
+
+$$
+\int_ {\Sigma} \left| \mathbb {E} \left(\phi \left(q _ {t}\right) \mid q _ {0}\right) - \langle \phi \rangle_ {\nu_ {\Sigma}} \right| ^ {2} d \nu_ {\Sigma} (q _ {0}) \leq C (\phi) e ^ {- 2 L / \beta t}
+$$
+
+$$
+\forall \phi \in H ^ {1} \left(\nu_ {\Sigma}\right), \tag {11}
+$$
+
+where  $C(\phi)$  depends only on  $\phi$ . Furthermore we have the following convergence in law:
+
+$$
+\sqrt {T} \left(\langle \phi \rangle_ {T} - \langle \phi \rangle_ {\nu_ {\Sigma}}\right)\rightarrow \mathcal {N} (0, \sigma_ {\phi} ^ {2}) \quad a s T \rightarrow \infty ,
+$$
+
+where the asymptotic variance  $\sigma_{\phi}^{2}$  is bounded as
+
+$$
+\sigma_ {\phi} ^ {2} \leq \frac {\beta}{L} \int_ {\Sigma} \left| \phi - \langle \phi \rangle_ {\nu_ {\Sigma}} \right| ^ {2} d \nu_ {\Sigma}.
+$$
+
+In  $\mathbb{R}^n$  assumption (9) is equivalent to convexity of  $V$ , which is known to be too strong a requirement (a confining assumption is sufficient, see e.g. Lelièvre & Stoltz (2016)). Although (9) can certainly be weakened, the above results ensure that provided the curvature of the manifold is well behaved, sampling on  $\Sigma$  has similar properties as on a flat space.
+
+Introducing momenta  $p$  leads to constrained underdamped Langevin dynamics, the 2nd order counterpart of Eq. (7)
+
+$$
+\mathrm {d} q _ {t} = p _ {t} \mathrm {d} t, 0 = g (q _ {t}), \tag {12}
+$$
+
+$$
+\mathrm {d} p _ {t} = \left(- \nabla V \left(q _ {t}\right) - \gamma p _ {t}\right) \mathrm {d} t + \sqrt {2 \gamma \tau} \mathrm {d} \mathcal {W} _ {t} - \nabla g \left(q _ {t}\right) \mathrm {d} \lambda_ {t},
+$$
+
+where  $\gamma$  is the friction hyperparameter. The constraint induces a cotangency condition:  $p\in T_q^*\Sigma$ , where  $T_{q}^{*}\Sigma = \{p\in \mathbb{R}^{d}\mid \nabla^{T}g(q)p = 0\}$  is the cotangent space of the manifold  $\boldsymbol{\Sigma}$ . The corresponding phase space is the cotangent bundle  $T^{*}\Sigma = \{(q,p)\mid q\in \Sigma ,p\in T_{q}^{*}\Sigma \}$ . Given an initial pair  $(q,p)\in T^{*}\Sigma$ , any trajectory  $(q_{t},p_{t})$  of (12) stays on  $T^{*}\Sigma$  for all time.
+
+(12) is equivalent to an underlying ergodic SDE, whose invariant measure is  $\mathrm{d}\mu = e^{-\beta H(q,p)}\mathrm{d}\sigma_{T^{*}\Sigma}$ , with Hamiltonian  $H(q,p) = V(q) + \frac{1}{2} p^T p$  and Liouville measure of the cotangent bundle  $\sigma_{T^{*}\Sigma}$  (Lelièvre et al., 2012). Based on the result for the unconstrained case, we expect exponential convergence to equilibrium also to hold here, but will leave this technical proof (e.g. based on hypocoercivity (Villani, 2009; Lelièvre & Stoltz, 2016)) for future work.
+
+# 4.3. Discretization of Constrained Langevin Dynamics.
+
+The simplest iteration scheme  $q_{n} \in \Sigma \mapsto q_{n+1} \in \Sigma$  for constrained overdamped Langevin dynamics (7) consists of an Euler-Maruyama step followed by projection onto the constraint manifold  $\Sigma$ . The best choice for the projection is constraint-specific.
+
+For circle constraints we suggest orthogonal projection, which is both explicit and robust (we describe this in detail in Appx. B.3). For orthogonality constraints, we derive an efficient quasi-Newton scheme to solve the non-linear system for the projection step (Appx. B.5). We present the resulting training scheme in Algorithm 1, where we denote  $Q = W^{\ell}$  if  $n^{\ell} \leq n^{\ell - 1}$  and  $Q = (W^{\ell})^{T}$  otherwise, and present one training iteration  $Q_{n} \in \Sigma \mapsto Q_{n + 1} \in \Sigma$ . Further, we denote  $h$  as the stepsize,  $G(Q) = \nabla_{Q} V(Q)$  and  $\tilde{G}$  the gradient of the loss evaluated on a randomly subsampled partial data set.  $R_{n}$  is an independent standard random normal matrix of the same size as  $Q$ . The initialization must be done with care: the constrained parameters and the potential slack variable must satisfy the constraint initially.
+
+# Algorithm 1 Orthog. constraint overdamped Langevin
+
+# Every step:
+
+$$
+\begin{array}{l} Q ^ {(0)} = Q _ {n} - h \tilde {G} (Q _ {n}) + \sqrt {2 \tau h} R _ {n}, \\ \mathbf {f o r} k = 0 \mathbf {t o} K - 1 \mathbf {d o} \\ Q ^ {(k + 1)} = Q ^ {(k)} - \frac {1}{2} Q _ {n} \left(\left(Q ^ {(k)}\right) ^ {T} Q ^ {(k)} - I _ {s}\right), \\ e n d \mathrm {f o r} \\ Q _ {n + 1} = Q ^ {(K)}. \\ \end{array}
+$$
+
+For underdamped Langevin dynamics a common way of building discretization schemes is via the use of splitting methods (Leimkuhler & Matthews, 2016). For the constrained underdamped Langevin system (12) an ABO splitting strategy under  $0 = g(q_{t})$ ,  $0 = \nabla_{q}g(q_{t})p_{t}$  gives:
+
+A:  $\mathrm{d}q_{t} = p_{t}\mathrm{d}t$ $\mathrm{d}p_{t} = -\nabla_{q}g(q_{t})\mathrm{d}\lambda_{t},$
+
+B:  $\mathrm{d}q_{t} = 0$ $\mathrm{d}p_{t} = -\nabla_{q}V(q_{t})\mathrm{d}t - \nabla_{q}g(q_{t})\mathrm{d}\mu_{t},$  (13)
+
+O:  $\mathrm{d}q_{t} = 0$ ,  $\mathrm{d}p_{t} = -\gamma p_{t}\mathrm{d}t + \sqrt{2\gamma\tau}\mathrm{d}\mathcal{W}_{t} - \nabla g(q_{t})\mathrm{d}\nu_{t}$
+
+In the specific case  $\tau = 0$  and by re-scaling  $\mu = e^{-\gamma h} / h$  and  $\delta t = h^2$ , an OBA sequence is equivalent to the standard PyTorch form of SGD with momentum  $\mu$  and stepsize  $\delta t$  (Paszke et al., 2017; Leimkuhler et al., 2019). As alternative one could use a symmetric splitting method, e.g. BAOAB method (Leimkuhler et al., 2016), but this would lose its accuracy order advantage in the presence of gradient noise.
+
+In (13) the B and O components can be solved exactly (in law) while the A component can be approximated using a standard scheme for constrained ODEs (e.g. SHAKE or RATTLE (Leimkuhler & Reich, 2004)[Chap. 7]). Importantly, the A component does not involve the evaluation of the gradient. For circle constraints the A step can be solved explicitly and the corresponding algorithm is provided in detail in Appendix B.4. For orthogonality constraints all details are provided in Appendix B.6, but we will provide the algorithm here. For  $Q \in \Sigma$ , the projection onto the cotangent space  $T_{Q}^{*}\Sigma$  is defined as  $\Pi_{Q}: \mathbb{R}^{r \times s} \to \mathbb{R}^{r \times s}$ ,
+
+$$
+\bar {P} \mapsto \Pi_ {Q} \bar {P} = \bar {P} - \frac {1}{2} Q \left(\bar {P} ^ {T} Q + Q ^ {T} \bar {P}\right). \tag {14}
+$$
+
+We initialize the parameters and momenta (using projection (14)) to obey the constraint. Then the ABO steps  $(Q_{n},P_{n})\in T^{*}\Sigma \mapsto (Q_{n + 1},P_{n + 1})\in T^{*}\Sigma$  are given by Algorithm 2, where  $\tilde{G} (Q)$  is the gradient of the loss evaluated on a subset of the data. More details in Appx. B.
+
+# 5. Numerical Experiments
+
+The use of constraints can enhance generalization performance. We support this claim by comparing the performance of neural network architectures trained using the constrained approaches described in this paper to nets trained using unconstrained SGD. We typically set  $\tau = 0$  and use
+
+# Algorithm 2 Orthog. constraint underdamped Langevin
+
+# Every step:
+
+$$
+\left. \begin{array}{l} Q ^ {(0)} = Q _ {n} + h P _ {n}, \\ \text {f o r} k = 0 \text {t o} K - 1 \text {d o} \\ Q ^ {(k + 1)} = Q ^ {(k)} - \frac {1}{2} Q _ {n} \left(\left(Q ^ {(k)}\right) ^ {T} Q ^ {(k)} - I _ {s}\right), \\ \text {e n d f o r} \\ Q _ {n + 1} = Q ^ {(K)}, \bar {P} _ {n + 1} = P _ {n} + \frac {1}{h} \left(Q _ {n + 1} - Q ^ {(0)}\right), \\ P _ {n + 1} = \Pi_ {Q _ {n + 1}} \bar {P} _ {n + 1} \\ \bar {P} _ {n + 1} = P _ {n} - h \tilde {G} (Q _ {n}), \\ P _ {n + 1} = \Pi_ {Q _ {n}} \bar {P} _ {n + 1}, \\ P _ {n + 1} = e ^ {- \gamma h} P _ {n} + \sqrt {\tau \left(1 - e ^ {- 2 \gamma h}\right)} R _ {n}, \\ P _ {n + 1} = \Pi_ {Q _ {n}} \bar {P} _ {n + 1} \end{array} \right\} (\mathrm {A})
+$$
+
+equivalent learning rates to present a fair comparison between constrained and unconstrained approaches. We denote our circle and orthogonal Constrained overdamped Langevin Algorithms as c-CoLA-  $od$  and o-CoLA-  $od$ , respectively. We compare underdamped variants (CoLA-  $ud$ ) with SGD with momentum (SGD-m).
+
+# 5.1. Orthogonality Constraints
+
+In Fig. 4 we want to train a multi-layer perceptron (MLP) with  $p$  hidden layers on a tightly wound spiral binary classification problem (Fig. D1) and compare the performance of SGD with our orthogonality-preserving overdamped Langevin method o-CoLA-od. For SGD we show results for i) standard PyTorch initialization, ii) orthogonal initialization, and iii) orthogonal regularization ('soft constraint'), where a penalty term is added to the loss to encourage orthogonality of the NN weight matrices. Our o-CoLA-od method clearly outperforms all of these variants in terms of test accuracy for MLPs with more than 3 hidden layers. In Appx. D (Fig. D2) we show that the use of a small temperature perturbation can speed up training even further and slightly increase the test accuracy. The performance of the soft constraint approach can be somewhat improved by lowering the stepsize, yet cannot match the performance of o-CoLA-od (see Figure 5). This illustrates the undesirable stiffness introduced into the system by using penalty-based regularization. The use of o-CoLA-od also removes the need to tune an additional parameter (the penalty strength).
+
+For a ResNet-34 architecture with BatchNorm and learning rate (LR) decay on CIFAR-10 (Krizhevsky & Hinton, 2009) data our underdamped orthogonal constrained method, without weight decay (WD) significantly outperforms SGD-m without WD (Fig. 6). The overdamped case is presented in the supplement, Fig. D3. In future work we will explore the nuances of combining orthogonality constraints with BatchNorm, residual connections and LR decay.
+
+![](images/09e58417be545843233ad451a2954dd5271fd117bacc15809bcf7ebfa9fb4e96.jpg)  
+Figure 4. Test acc. of MLPs with  $p$ -number of 100-node hidden layers (HL), ReLU activation. The MLPs are trained on a 4-turn spiral dataset (Fig. D1) using SGD with standard initialization (1st column), SGD with orthogonal initialization (2nd col.), SGD with orthogonal regularization ('soft constraint') by adding a penalty term with strength  $\lambda$  to the loss (3rd col.), and o-CoLA-od with  $\tau = 0$  (4th col.). For the orthogonal regularization approach and o-CoLA-od we constrain weights in all layers, apart from input and output layers. We set stepsize  $h = 0.1$  for all methods and use  $5\%$  subsampling. We found the optimal penalty strength  $\lambda = 0.05$  for the orthogonal regularization method through line search. Results are averaged over 10 runs. We observe that our o-CoLA-od method significantly outperforms unconstrained SGD and SGD with a soft constraint for MLPs with more than 3 hidden layers.
+
+# 5.2. Circle Constraints
+
+We evaluate our circle constrained c-CoLA-ud method on the Fashion-MNIST data set (Xiao et al., 2017). We reduce the amount of training data to 10K samples and use the remaining 60K samples as test data. c-CoLA-ud clearly outperforms SGD-m in terms of both test accuracy and test loss for a 1000-node single hidden layer perceptron (see Fig. 7). The lower test loss of c-CoLA-ud is maintained during training and the method shows no signs of overfitting, thus eliminating the need for early stopping. Even with weight decay, SGD-m is outperformed by its constrained counterpart (for more detailed hyperparameter studies see Appx. D). We also show that a small transformer (Vaswani et al., 2017) with 2 encoder layers (each with 2-head self-attention and 200-node feed-forward network) trained using c-CoLA-ud achieves a lower validation loss on NLP datasets than its unconstrained counterpart, SGD-m (Table 2).
+
+# 6. Conclusion
+
+We provide a general framework that can be used to directly influence the parameter space of deep neural networks. The constrained SDE-based algorithms described in this paper allow for the use of additive noise to enhance exploration but can also be used directly in combination with standard SGD approaches. We provide a mathematical framework to study these regularized training methods as discretizations of constrained Langevin dynamics and provide detailed discretization schemes (see Appendix B). As specific examples of constraints we consider circle and orthogonality constraints, which obtain improved generalization performance on classification tasks compared to unconstrained SGD and soft constraint approaches. Further uses of our general framework are left for future work.
+
+Table 2. Minimum val. loss on Penn Treebank data (batchsize 1024) (Marcus et al., 1993) and Wikitext-2 (batchsize 128) (Merit et al., 2017) using a transformer trained using c-CoLA-ad or SGDm. Hyperpar. c-CoLA-ad:  $h = 0.4, r = 0.5, r_L = 0.1, r_N = 1, r_A = 1, \tau = 0, \gamma = 0.5$  (Treebank) and  $\gamma = 1$  (Wikitext-2), where the subscripts  $L, N, A$  represent the radii belonging to the linear, norm and self-attention layers respectively. The transformer trained using c-CoLA-ad obtains lower validation losses. Studies with weight decay are provided in the supplement.  
+
+<table><tr><td>Optimizer</td><td>Penn Treebank</td><td>Wikitext-2</td></tr><tr><td>c-CoLA-ud</td><td>4.81</td><td>5.09</td></tr><tr><td>SGD h = 0.1</td><td></td><td></td></tr><tr><td>mom = 0.7</td><td>4.87</td><td>5.13</td></tr><tr><td>mom = 0.8</td><td>4.83</td><td>5.13</td></tr><tr><td>mom = 0.9</td><td>4.84</td><td>5.13</td></tr><tr><td>SGD h = 0.2</td><td></td><td></td></tr><tr><td>mom = 0.7</td><td>4.83</td><td>5.13</td></tr><tr><td>mom = 0.8</td><td>4.83</td><td>5.14</td></tr></table>
+
+# Acknowledgements
+
+The authors wish to thank Gabriel Stoltz and Tony Lelièvre for helpful discussions on constrained SDEs. Benedict Leimkuhler is a fellow of the Alan Turing Institute which is supported by EPSRC grant EP/N510129/1. During the creation of this paper Timothee Pouchon was supported by the Swiss National Science Foundation, project P2ELP2_188037. Tiffany Vlaar is supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh.
+
+![](images/44e37c219b4642d08568cb158177a167d547430b3c9beef2c83a7eeab8a4c53b.jpg)  
+Figure 5. Same set-up as for Figure 4. MLPs with varying numbers of hidden layers (HL) were trained using o-CoLA-od with  $h = 0.1$  (right-most) and using SGD with a penalty term added to the loss (results are presented in the 1st three columns with varying stepsizes  $h$  and penalty strengths  $\lambda$ ). Results are averaged over 10 runs. We illustrate that the use of a penalty-based soft constraint introduces an undesirable stiffness into the system, needing the stepsize to be lowered to improve performance and to allow for the use of larger penalty strengths. The soft constraint approach is unable to reach the same performance as our o-CoLA-od method (right-most) and its performance is heavily dependent on the choice of penalty strength and step size.
+
+![](images/2e5c5aabe1f0ef6e4ca4ee5802273eaa3db34b1865ad143af4b499e290f85fd3.jpg)
+
+![](images/17d758b01fe7f8e73963025eec57b7e1007ec0faf79888def1cc37c773685cb5.jpg)
+
+![](images/56590481c6d8a9e59be9cd584a072744190e0ed8b5ad9af61af7f6e34960ed47.jpg)
+
+![](images/1565a5e3f97804d2517b443a05ec02e2bb20cde5da642fc7e5c0ea313f113fb0.jpg)  
+Figure 6. Train (left) & test (middle) loss and test accuracy (right) averaged over 5 runs of a ResNet-34 with BatchNorm trained using SGD-m vs. o-CoLA-ud with  $\tau = 0$  on CIFAR-10. For SGD we initially use  $h = 0.1$  and decay by a factor 10 every 50 epochs (indicated by the vertical black dotted lines). We set momentum  $= 0.9$  and present results with and without WD. o-CoLA-ud (with  $\gamma = 0.5$ ) did not use WD. Its learning rate was re-scaled to match the parameters of SGD-m and used the same LR schedule. The o-CoLA-ud method without weight decay strongly outperforms SGD-m without weight decay.
+
+![](images/f542d0955d694387706ddf05bf547a4ef623eb2c77c8310719ff5d781f1785a1.jpg)
+
+![](images/343822f027eb7bda38b53b35c7d021d3d88f139d12781950f09b85eb4ddee771.jpg)
+
+![](images/de06ec9d9e12a0cd05b5cb740c5111359803404b07bfb44a2bebc331a0c49c9c.jpg)  
+Figure 7. Test loss (left) and test accuracy (right) averaged over 5 runs of a 1000-node SHLP trained using SGD-m vs. c-CoLA-ad on Fashion-MNIST (batchsize 128, number of training data samples reduced to 10K). After a line search we chose the best performing hyperparameter setting for SGD, namely  $h = 0.1$ ,  $mom = 0.8$ , and varied the amount of weight decay (WD). Standard deviations are provided in the supplement. Hyperparameters c-CoLA-ad:  $h = 0.3$ ,  $\gamma = 1$ ,  $r_0 = 0.05$ ,  $r_1 = 0.1$ ,  $\tau = 0$ . Due to the small training dataset size both methods quickly reached  $100\%$  training accuracy, but c-CoLA-ad is superior in its test loss and test accuracy.
+
+![](images/22e55ed8b07ff37bab6aa9781886f397b6a999cc5834c9a06cd4246551d95cde.jpg)
+
+# References
+
+Arjovsky, M., Shah, A., and Bengio, Y. Unitary evolution recurrent neural networks. In International Conference on Machine Learning, pp. 1120-1128, 2016.  
+Bakry, D. and Emery, M. Diffusions hypercontractives. In Azéma, J. and Yor, M. (eds.), Séminaire de Probabilités XIX 1983/84, pp. 177-206, Berlin, Heidelberg, 1985. Springer Berlin Heidelberg. ISBN 978-3-540-39397-9.  
+Bakry, D., Gentil, I., and Ledoux, M. Analysis and geometry of Markov diffusion operators, volume 348. Springer Science & Business Media, 2013.  
+Bansal, N., Chen, X., and Wang, Z. Can we gain more from orthogonality regularizations in training deep CNNs? In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pp. 4266-4276, 2018.  
+Beskos, A., Pillai, N., Roberts, G., Sanz-Serna, J.-M., and Stuart, A. Optimal tuning of the hybrid Monte Carlo algorithm. Bernoulli, 19(5A):1501-1534, 2013.  
+Bhattacharya, R. N. On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 60(2):185-201, 1982.  
+Bou-Rabee, N. and Sanz-Serna, J. Geometric integrators and the Hamiltonian Monte Carlo method. Acta Numerica, 27:113-206, 2018.  
+Brock, A., Lim, T., Ritchie, J. M., and Weston, N. J. Neural photo editing with introspective adversarial networks. *ICLR*, 2017.  
+Chaudhari, P., Choromanska, A., Soatto, S., LeCun, Y., Baldassi, C., Borgs, C., Chayes, J., Sagun, L., and Zecchina, R. Entropy-SGD: Biasing gradient descent into wide valleys. *ICLR*, 2017.  
+Cheng, X., Chatterji, N. S., Bartlett, P. L., and Jordan, M. I. Underdamped Langevin MCMC: A non-asymptotic analysis. arXiv:1707.03663, 2017.  
+Choromanska, A., Henaff, M., Mathieu, M., Arous, G. B., and LeCun, Y. The loss surfaces of multilayer networks. AISTATS, 2015.  
+d'Ascoli, S., Sagun, L., Bruna, J., and Biroli, G. Finding the needle in the haystack with convolutions: on the benefits of architectural bias. NeurIPS, 2019.  
+Faou, E. and Lelievre, T. Conservative stochastic differential equations: Mathematical and numerical analysis. Mathematics of computation, 78(268):2047-2074, 2009.
+
+Graham, M. and Storkey, A. Asymptotically exact inference in differentiable generative models. In Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, volume 54, pp. 499-508, 2017.  
+He, K., Zhang, X., Ren, S., and Sun, J. Delving deep into rectifiers: Surpassing human-level performance on Imagenet classification. In Proceedings of the IEEE international conference on computer vision, pp. 1026-1034, 2015.  
+Hoerl, A. and Kennard, R. Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12: 55-67, 1970.  
+Huang, L., Liu, X., Lang, B., Wei Yu, A., and Li, B. Orthogonal weight normalization: Solution to optimization over multiple dependent stiefel manifolds in deep neural networks. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.  
+Ioffe, S. and Szegedy, C. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International Conference on Machine Learning, pp. 448-456, 2015.  
+Jastrzegski, S., Kenton, Z., Arpit, D., Ballas, N., Fischer, A., Bengio, Y., and Storkey, A. Three factors influencing minima in SGD. ICANN, 2018.  
+Jia, K., Li, S., Wen, Y., Liu, T., and Tao, D. Orthogonal deep neural networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2019.  
+Kawaguchi, K. Deep learning without poor local minima NeurIPS, 2016.  
+Keskar, N., Mudigere, D., Nocedal, J., and M. Smelyanskiy, P. T. On large-batch training for deep learning: Generalization gap and sharp minima. *ICLR*, 2017.  
+Kipnis, C. and Varadhan, S. R. S. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Communications in Mathematical Physics, 104(1):1-19, 1986.  
+Kirkpatrick, S., Gelatt, C., and Vecchi, M. Optimization by simulated annealing. Science, 220:671-680, 1983.  
+Krizhevsky, A. and Hinton, G. Learning multiple layers of features from tiny images. 2009.  
+Leimkuhler, B. and Matthews, C. Molecular Dynamics: With Deterministic and Stochastic Numerical Methods. Interdisciplinary Applied Mathematics. Springer, 2015.  
+Leimkuhler, B. and Matthews, C. Efficient molecular dynamics using geodesic integration and solvent-solute
+
+splitting. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2189): 20160138, 2016.  
+Leimkuhler, B. and Reich, S. Simulating Hamiltonian dynamics, volume 14. Cambridge university press, 2004.  
+Leimkuhler, B., Matthews, C., and Stoltz, G. The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics. *IMA Journal of Numerical Analysis*, 36(1):13-79, 2016.  
+Leimkuhler, B., Matthews, C., and Vlaar, T. Partitioned integrators for thermodynamic parameterization of neural networks. Foundations of Data Science, 1(4):457-489, 2019.  
+Lelièvre, T. and Stoltz, G. Partial differential equations and stochastic methods in molecular dynamics. Acta Numerica, 25:681-880, 2016.  
+Lelièvre, T., Stoltz, G., and Rousset, M. Free energy computations: A mathematical perspective. Imperial College Press, 2010. ISBN 9781848162488.  
+Lelièvre, T., Rousset, M., and Stoltz, G. Langevin dynamics with constraints and computation of free energy differences. Mathematics of computation, 81(280):2071-2125, 2012.  
+Lelièvre, T., Stoltz, G., and Zhang, W. Multiple projection MCMC algorithms on submanifolds. arXiv:2003.09402, 2020.  
+Li, Q., Haque, S., Anil, C., Lucas, J., Grosse, R., and Jacobsen, J. Preventing gradient attenuation in Lipschitz constrained convolutional networks. NeurIPS, 2019.  
+Loshchilov, I. and Hutter, F. Stochastic gradient descent with warm restarts. *ICLR*, 2017.  
+Marcus, M. P., Santorini, B., and Marcinkiewicz, M. A. Building a large annotated corpus of English: The Penn Treebank. Computational Linguistics, 19(2):313-330, 1993.  
+Marinari, E. and Parisi, G. Simulated tempering: a new Monte Carlo scheme. *Europhysics Letters*, 1992.  
+Merit, S., Xiong, C., Bradbury, J., and Socher, R. Pointer sentinel mixture models. *ICLR*, 2017.  
+Neyshabur, B., Tomioka, R., and Srebro, N. In search of the real inductive bias: On the role of implicit regularization in deep learning. In Bengio, Y. and LeCun, Y. (eds.), 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Workshop Track Proceedings, 2015.
+
+Pascanu, R., Mikolov, T., and Bengio, Y. On the difficulty of training recurrent neural networks. In International conference on machine learning, pp. 1310-1318, 2013.  
+Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E., DeVito, Z., Lin, Z., Desmaison, A., Antiga, L., and Lerer, A. Automatic differentiation in PyTorch. 2017.  
+Pavliotis, G. A. Stochastic processes and applications: diffusion processes, the Fokker-Planck and Langevin equations, volume 60. Springer, 2014.  
+Pennington, J., Schoenholz, S., and Ganguli, S. Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice. In Advances in Neural Information Processing Systems, pp. 4785-4795, 2017.  
+Pennington, J., Schoenholz, S., and Ganguli, S. The emergence of spectral universality in deep networks. In International Conference on Artificial Intelligence and Statistics, pp. 1924-1932, 2018.  
+Persson, P. The level set method. Lecture notes MIT 16.920J / 2.097J / 6.339J, Numerical Methods for Partial Differential Equations, October 2006.  
+Rodríguez, P., González, J., Cucurull, G., Gonfaus, J. M., and Roca, X. Regularizing CNNs with locally constrained decorrelations. *ICLR*, 2017.  
+Santurkar, S., Tsipras, D., Ilyas, A., and Madry, A. How does batch normalization help optimization? In Advances in Neural Information Processing Systems, pp. 2483-2493, 2018.  
+Saxe, A. M., McClelland, J. L., and Ganguli, S. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. arXiv:1312.6120, 2013.  
+Smith, L. N. Cyclic learning rates for training neural networks. Workshop on Application of Computer Vision, 2017.  
+Srebro, N. and Shraibman, A. Rank, trace-norm and maxnorm. In International Conference on Computational Learning Theory, pp. 545-560. Springer, 2005.  
+Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., and Salakhutdinov, R. Dropout: a simple way to prevent neural networks from overfitting. The journal of machine learning research, 15(1):1929-1958, 2014.  
+Sutskever, I., Martens, J., Dahl, G., and Hinton, G. On the importance of initialization and momentum in deep learning. ICML, 2013.  
+Tibshirani, R. Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society: Series B (Methodological), 58(1):267-288, 1996.
+
+Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., and Polosukhin, I. Attention is all you need. In Advances in Neural Information Processing Systems, pp. 5998-6008, 2017.  
+Villani, C. Hypocoercivity. Memoirs of the American Mathematical Society, 202(950), 2009.  
+Vorontsov, E., Trabelsi, C., Kadoury, S., and Pal, C. On orthogonality and learning recurrent networks with long term dependencies. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 3570-3578. JMLR.org, 2017.  
+Welling, M. and Teh, Y. W. Bayesian learning via stochastic gradient Langevin dynamics. In Proceedings of the 28th International Conference on Machine Learning (ICML-11), pp. 681-688, 2011.  
+Wenzel, F., Roth, K., Veeling, B. S., Swiatkowski, J., Tran, L., Mandt, S., Snoek, J., Salimans, T., Jenatton, R., and Nowozin, S. How good is the Bayes posterior in deep neural networks really? arXiv:2002.02405, 2020.  
+Williams, P. Bayesian regularization and pruning using a Laplace prior. Neural computation, 7(1):117-143, 1995.  
+Wilson, A., Roelofs, R., Stern, M., Srebro†, N., and Recht, B. The marginal value of adaptive gradient methods in machine learning. *NeurIPS*, 2017.  
+Wu, L., Zhu, Z., and E, W. Towards understanding generalization of deep learning: Perspective of loss landscapes. ICML, 2017.  
+Xiao, H., Rasul, K., and Vollgraf, R. Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. arXiv:1708.07747, 2017.  
+Xiao, L., Bahri, Y., Sohl-Dickstein, J., Schoenholz, S., and Pennington, J. Dynamical isometry and a mean field theory of CNNs: How to train 10,000-layer vanilla convolutional neural networks. In International Conference on Machine Learning, pp. 5393-5402, 2018.  
+Xie, D., Xiong, J., and Pu, S. All you need is beyond a good init: Exploring better solution for training extremely deep convolutional neural networks with orthonormality and modulation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 6176-6185, 2017.  
+Yao, Z., Gholami, A., Keutzer, K., and Mahoney, M. PyHessian: Neural networks through the lens of the Hessian. arXiv:1912.07145, 2019.  
+Zappa, E., Holmes-Cerfon, M., and Goodman, J. Monte Carlo on manifolds: Sampling densities and integrating
+
+functions. Communications on Pure and Applied Mathematics, 71(12):2609-2647, 2018.  
+Zhang, S., Choromanska, A., and LeCun, Y. Deep learning with elastic averaging SGD. NeurIPS, 2015.  
+Zhou, J., Do, M., and Kovacevic, J. Special paraunitary matrices, Cayley transform, and multidimensional orthogonal filter banks. IEEE Transactions on Image Processing, 15(2):511-519, 2006.

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+# Beyond  $\log^2 (T)$  Regret for Decentralized Bandits in Matching Markets
+
+Soumya Basu $^{1}$  Karthik Abinav Sankararaman $^{2}$  Abishek Sankararaman $^{3}$
+
+# Abstract
+
+We design decentralized algorithms for regret minimization in two-sided matching markets with one-sided bandit feedback that significantly improves upon the prior works (Liu et al., 2020a; Sankararaman et al., 2021; Liu et al., 2020b). First, for general markets, for any  $\varepsilon > 0$ , we design an algorithm that achieves a  $O(\log^{1 + \varepsilon}(T))$  regret to the agent-optimal stable matching, with unknown time horizon  $T$ , improving upon the  $O(\log^2(T))$  regret achieved in (Liu et al., 2020b). Second, we provide the optimal  $\Theta(\log(T))$  regret for markets satisfying uniqueness consistency – markets where leaving participants don't alter the original stable matching. Previously,  $\Theta(\log(T))$  regret was achievable (Sankararaman et al., 2021; Liu et al., 2020b) in the much restricted serial dictatorship setting, when all arms have the same preference over the agents. We propose a phase based algorithm, where in each phase, besides deleting the globally communicated dominated arms, the agents locally delete arms with which they collide often. This local deletion is pivotal in breaking deadlocks arising from rank heterogeneity of agents across arms. We further demonstrate superiority of our algorithm over existing works through simulations.
+
+# 1. Introduction
+
+Decentralized decision making by competing agents under uncertainty, each one motivated by one's own objective, is a key feature in online market places, e.g. TaskRabbit, UpWork, DoorDash. An emerging line of research (Aridor et al., 2020; Johari et al., 2021; Liu et al., 2020a; Sankararaman et al., 2021; Liu et al., 2020b) in the field of multi-agent
+
+<sup>1</sup>Google, Mountain View, CA, USA <sup>2</sup>Facebook, Menlo Park, CA, USA <sup>3</sup>Amazon, Palo Alto, CA, USA. Correspondence to: Soumya Basu <basusoumya@utexas.edu>, Karthik Abinav Sankararaman <karthikabinavs@gmail.com>, Abishek Sankararaman <abishek.90@gmail.com>.
+
+Proceedings of the  $38^{th}$  International Conference on Machine Learning, PMLR 139, 2021. Copyright 2021 by the author(s).
+
+bandits is dedicated to understanding algorithmic principles in the interplay of competition, learning and regret minimization. The two-sided matching market (Gale & Shapley, 1962) is one such thread, where regret minimization is first studied in (Liu et al., 2020a) with a centralized arbiter, and in (Sankararaman et al., 2021; Liu et al., 2020b) at different levels of decentralization.
+
+We study the fully decentralized two-sided matching market, comprising of  $N$  demand-side entities (a.k.a. agents) and  $K$  supply-side entities (a.k.a. arms). Each entity has a separate preference ranking of the opposite side agents, i.e. each agent over  $K$  arms, and each arm over  $N$  agents, and aims to match with the most preferred entity. In the bandit setting, the agents lack prior knowledge of their respective preferences, thus need to learn the preferences only through their own past interactions, while the arms know their preferences. In each round, every agent simultaneously chooses an arm of their choice, and are either matched to their arm of choice and receive a stochastic reward (reward defines agents' preference); or are blocked, are notified of this, and receive no reward.
+
+A matching between agents and arms is stable if there exist no pair of agents and arms that are not matched with each other under the matching, prefer each other over their current partners. When each participant knows its preference and the agents propose to the arms across multiple rounds, the system admits any stable matching as a Nash equilibrium. Further, the agent-optimal stable matching is the one, which yields the highest reward among multiple possible stable matchings, to all agents (Gale & Shapley, 1962). Our objective is to design a uniform protocol for the agents, which allows each agent to quickly find and match with its agent-optimal arm, thus maximizing their cumulative reward.
+
+Under a restrictive special case, known as serial dictatorship, where all arms have the same preference (Sankararaman et al., 2021) shows it is possible for each agent to attain  $O(NK\log (T))$  cumulative regret (the gap between optimal and achieved reward) in  $T$  rounds. They rely on an Upper confidence bound (UCB) based algorithm, called UCB-D3, that uses strategic deletion of arms done through global communication alongside UCB based explore-exploit. When the decentralization is relaxed, and each agent observes the
+
+response of all the arms per round, (Liu et al., 2020b) designs Collision Avoidance UCB (CA-UCB), which avoids collision by arm deletion and reduced switching of arms. This provides  $O(\exp(N^4) \log^2(T))$  regret for each agent with respect to an agent-pessimial stable matching. Under agent-pessimial stable matching, all the agents obtain the minimum reward among all stable matching, thus it is less desirable than the agent-optimal stable matching. Moreover, the relaxation of decentralization comes at the cost of privacy, and the possible lack of truthfulness of agents which can hurt the performance, and discourage participation.
+
+Our first contribution is to show through a simple phase based explore then commit protocol each agent can achieve  $O(K\log^{1 + \varepsilon}(T))$  regret for any  $\varepsilon > 0$ , for sufficiently large time horizon  $T$ . We further focus on two-sided markets that satisfy uniqueness consistency (UnqC) (Karpov, 2019), where the stable match is robust —namely if any subset of matched pairs of a stable match leave, the remaining matches are still stable. This also relaxes the stringent requirement of homogeneous preferences imposed by the serial dictatorship model. Arguably, robustness to matches departing the system and heterogeneity of preferences is a desirable property in large dynamic markets such as TaskRabbit, InstaCart etc. Under UnqC, where UCB-D3 fails due to heterogeneous arm ranking, we design UCB-D4 (UCB-D3 augmented with local deletions) that allows each agent to achieve  $O(NK\log(T))$  regret. In local deletion, an agent aggressively and locally removes the arms to which it collides above a well designed threshold.
+
+# 1.1. Main Contributions
+
+1. General markets. The best existing regret bound for general stable matching due to (Liu et al., 2020b) obtains a  $O(\exp(N^4)\log^2(T))$  agent-pessimial regret under partial decentralized feedback in time horizon  $T$ . The CA-UCB algorithm is used where agents switch with very low frequency to the arm with highest UCB index in a carefully chosen subset. The slow switching allows for collision avoidance but at a cost of high regret. Therefore, we design, phased ETC, a simple phase based algorithm, used uniformly by agents, which sets up a protocol that allows collision free exploration at the beginning of each phase. Then using the Gale-Shapley (Gale & Shapley, 1962) algorithm commits to a stable matching with their individual estimated preference lists. We prove this algorithm achieves a  $O(K\log^{1+\varepsilon}(T))$  agent-optimal regret under fully decentralized feedback for any  $\varepsilon > 0$ , by setting the exploration duration based on  $\varepsilon$ . Although our proposed algorithm beats the state-of-the-art CA-UCB guarantees in many dimensions - fully vs partially decentralized, agent optimal vs pessimal regret,  $O(K\log^{1+\varepsilon}(T))$  vs  $O(\exp(N^4)\log^2(T))$  - it suffers from cold-start, i.e. it works for  $T = \Omega(\exp(N/\Delta^2))$ , where  $\Delta$  is the minimum reward gap across all arms and
+
+agents. This leaves open the quest for an optimal  $O(\log (T))$  regret without the curse of cold-start.
+
+2. Markets with uniqueness consistency. We next focus on markets with uniqueness consistency (or UnqC in short), where there is a unique and robust stable matching. In this setting, the best known result (excluding serial dictatorship) is  $O(\exp(N^4) \log^2(T))$ , achieved by CA-UCB. For serial dictatorship, a special case of UnqC where all arms have the same preference order, the best result,  $O((j - 1)K \log(T))$  for agent ranked  $j$  for all arms, is obtained by UCB-D3. UCB-D3, a phase based algorithm however, cannot incur sub-linear regret when the preferences are heterogeneous (empirically shown in the Section 6). We introduce UCB-D4, a generalization of UCB-D3 that handles heterogeneous arm preferences under uniqueness consistency, and achieves the coveted  $O(\log(T))$  regret for any  $T = \Omega(N / \Delta^2)$ , i.e. without the cold-start problem. Our key algorithmic and theoretical insights behind this result are as follows.
+
+Algorithmic. UCB-D4 augments UCB-D3 with a local deletion, where an agent in each phase deletes an arm locally if it experiences collision more than a  $(\beta \times$  phase length) times, for an appropriate  $\beta > 0$ . The local deletion plays an important role in eliminating deadlocks that can be created under UCB-D3. In particular, consider the case when uniqueness consistency holds. Here arms have heterogeneous preferences, and agent  $a$  and agent  $b$  can block each other from exploring their non-stable matched arms. Interestingly, we discover that such deadlocks do not occur when playing the stable matched arms. Due to the specific nature of the deadlock, if an agent 'frequently' collides with an arm it is safe to delete that arm locally. The key is to carefully set the threshold of local deletion. A small threshold can remove the stable matched arm with constant probability due to the stochastic feedback, thus incur linear regret. Whereas, a large threshold deter the agent from getting out of the deadlock fast enough to achieve  $O(\log (T))$  regret. UCB-D4 with  $\beta < 1 / K$  strikes the correct balance.
+
+Theoretical. We introduce a dual induction proof-technique linked with  $\alpha$ -condition. The  $\alpha$ -condition bestows two orders: one among agents - left order, and one among arms - right order. In the left order, the agents have their stable matched arm as best arm once the stable matched arms for higher order agents are removed. For the right order, same holds with the roles of agents and arms swapped. Our dual induction uses these two orders. We show local deletion ensures the arms inductively, following the right order, become 'available' for their respective stable matched agents. This allows the agents to inductively, following the left order, identify and broadcast their respective stable matched arm. The second induction, depends on first, and is driven by global deletion of dominated arms, alongside deadlock resolution due to local deletion.
+
+3. Empirical. We compare the performance of our algorithms against CA-UCB, an algorithm for matching markets under partial decentralization: the response of all arms per round is visible. Our proposed phased ETC and UCB-D4 both work under full decentralization: only sees response of the proposed arm. Extensive simulations show, despite the restrictive feedback, phased ETC outperforms CA-UCB in general instances, while UCB-D4 does so under uniqueness consistency. We also empirically validate that UCB-D3 produces linear regret under uniqueness consistency.
+
+# 1.2. Related Work
+
+The field of bandit learning has a vast literature with multiple textbooks on this subject (Lattimore & Szeptsvári, 2020; Bubeck & Cesa-Bianchi, 2012; Slivkins, 2019). Our work falls in the multi-agent bandits setting, which is effective in modelling applications like wireless networks (Avner & Mannor, 2016; Darak & Hanawal, 2019), online advertising (Hillel et al., 2013), data-centers and search scheduling (Sankararaman et al., 2019) where multiple decision makers interact causing an interesting interplay of competition, learning and consensus (Boursier & Perchet, 2020; Brânzei & Peres, 2019). The study of multi-agent bandits in two-sided matching markets is initiated by (Liu et al., 2020a). They study the centralized setting, when the agents pass on their estimated preference to a single decision maker who then proposes to the arms. In this setting, the authors design a UCB based protocol that completely avoids collision (due to centralized proposals) and attains a  $O(NK\log(T))$  regret with respect to the agent-pessimial stable matching. The papers of (Sankararaman et al., 2021) and (Liu et al., 2020b) are the closest to our work as both of them studies decentralized (the latter only partially) two-sided matching markets. We have already mentioned their relation to this work, and how we improve upon them. Two-sided markets under full information, commonly known as the stable marriage problem was introduced in the seminal work of (Gale & Shapley, 1962) where they established the notion of stability and provided the optimal algorithm to obtain a stable matching. Our results rely on the recent combinatorial characterization, namely  $\alpha$ -condition, of UnqC in stable marriage problem by (Karpov, 2019). Further, in the economics literature, uncertainty in two-sided matching have been studied recently in (Johari et al., 2021; Ashlagi et al., 2019) in directions tangential to our work. We provide a detailed related work in Appendix B.
+
+# 2. System Model
+
+Agents and arms. We have  $N$  agents and  $K \geq N$  arms. When agent  $j \in [N]$  is matched to arm  $k \in [K]$  it receives a reward sampled (independent of everything) from a latent distribution with support  $[0,1]$ , and mean  $\mu_{jk} \in [0,1]$ . We
+
+assume that  $\{\mu_{jk}\}_{j,k}$  are all distinct. For each agent, the arm means impose a preference order over the arms, with higher means preferred over lower means. Similarly, every arm  $k\in [K]$  has a total preference order over the arms  $>\operatorname {arm}(k)$ ; for  $j,j^{\prime}\in [N]$ , if  $\operatorname {agent}(j) > _{\operatorname {arm}(k)}$  agent  $(j^{\prime})$  then arm  $k$  prefers agent  $j$  over  $j^{\prime}$ . When context is clear, we use  $j$  for agent  $(j)$ , and  $k$  for  $\operatorname {arm}(k)$ . For any subset of agents and arms, the preference profile is the preference order of the agents restricted to the given set of arms, and vice versa. The game proceeds in  $T$  rounds (value of  $T$  unknown to the agents) where every agent simultaneously plays one arm, and is either matched to that arm or is notified that it was not matched. In every round, each arm is matched to the most preferred agent playing that arm in that round (if any).
+
+Stable Matching. Given the preference order for agents and arms, consider a matching denoted by the set  $\mathbf{k}^*,\mathbf{j}^*$ , with  $k_{j}^{*}$  denoting the matched arm for agent  $j$ , and  $j_{k}^{*}$  denoting the matched agent for arm  $k$ . Under a stable-matching there exist no two pairs  $(j,k)$  and  $(j',k')$  such that  $k = k_{j}^{*}$  and  $k' = k_{j'}^*$ , but  $\mu_{jk} < \mu_{jk'}$  and  $\mathrm{agent}(j) > _{\mathrm{arm}(k')} \mathrm{agent}(j')$  if agent  $j$  matches with  $k'$  both improve their position. An agent-optimal stable match is unique and is one where all agents obtain their respective best possible match among all possible stable matchings (see, (Gale & Shapley, 1962)).
+
+Decentralized bandit with no information. All agents have common information, that we have  $N$  total agents and  $K$  arms each labeled 1 through  $K$ . In each round, every agent observes only the outcome of its action and cannot observe the other agents' play/outcome. Specifically, when it is rejected by the arm it observes the collision signal, otherwise it observes the reward obtained from the arm it proposed to. Thus, our feedback structure is fully decentralized where in each round, an agent's decision to play an arm to play can depend only on common information before the start, its past actions and outcomes. Agents can however agree to a common protocol that map their observed outcomes to arms in every round.
+
+Agent optimal regret. Total reward obtained by any agent is compared against that obtained by playing the agent optimal stable match in all rounds. Let  $P_{j}(t) \in [K]$  be the arm played by agent  $j$  in round  $t$  and  $M_{j}(t) \in \{0,1\}$  be the indicator random variable denoting whether agent  $j$  was matched to  $P_{j}(t)$ . Let  $k_{j}^{*}$  be the agent-optimal stable match of agent  $j$ . The  $T$ -round individual regret for an agent is
+
+$$
+R _ {j} (T) = T \mu_ {j k _ {j} ^ {*}} - \mathbb {E} \left[ \sum_ {t = 1} ^ {T} M _ {j} (t) \mu_ {j P _ {j} (t)} \right].
+$$
+
+The goal is to design a protocol that all agents follow to minimize their individual regret.
+
+We make a few important remarks on the model.
+
+1. Feedback structure. Our feedback structure is same
+
+as (Sankararaman et al., 2021), and more restrictive than the one proposed very recently in (Liu et al., 2020b) called decentralized bandit with partial information (see also, Section 6). In the latter, all the agents can observe the matching between arms and agents in each round, alongside it's own reward or collision information. Further, the knowledge of preference order of the arms is explicitly assumed.
+
+2. Why agent optimal regret? Our rationale for this is to compare against an oracle in which the arm-preferences of the agents are common information known to all agents, and each agent plays to maximize its individual total reward. This corresponds to a repeated game, in which, the set of pure strategy Nash equilibria corresponds to all agents playing a particular stable match in all rounds. The optimal equilibria that maximizes the reward for all agents simultaneously, corresponds to the agent-optimal stable matching.
+
+3. Model generalizations. We focus on  $N \leq K$ , as for  $N > K$  some agents will remain unmatched under full information, and have 0 regret by definition. Our algorithms, can be modified easily to handle such cases. Further, we can admit any sub-Gaussian reward distribution with finite mean in place of rewards supported on [0, 1], with minor changes in our proof.
+
+# 3. Achieving  $O(\log^{(1 + \varepsilon)}(T))$  Regret in Matching Markets for  $\varepsilon > 0$
+
+In this section, we give a simple Phased Explore then Commit (ETC) style algorithm that obtains an asymptotic per-agent regret of  $O(\log^{1 + \varepsilon}(T))$  for any given  $\varepsilon > 0$ . The algorithm proceeds in phases of exponentially growing lengths, with phase  $i$  lasting  $2^i$  rounds. In each phase  $i \geq 0$ , all agents explore for the first  $i^\varepsilon$  rounds, and then subsequently exploit by converging to a stable matching through the Gale-Shapley strategy. The explorations are organized so that the agents avoid collision, which can be done in a simple decentralized fashion, where each agent gets assigned an unique id in the range  $\{1, \dots, N\}$  (Algorithm 3 in Appendix A). The pseudo-code for phased ETC is given in Algorithm 1.
+
+The minimum reward gap across all arms and agents is  $\Delta = \min \{|\mu_{jk} - \mu_{jk'}|:j\in [N],k,k'\in [K]\}$
+
+Theorem 1. If every agent runs Algorithm 1 with input parameter  $\varepsilon > 0$ , then the regret of any agent  $j$  after  $T$  rounds satisfies
+
+$$
+\begin{array}{l} R _ {T} ^ {(j)} \leq \frac {K (\log_ {2} (T) + 2) ^ {1 + \varepsilon}}{1 + \varepsilon} + (N ^ {2} + K) (\log_ {2} (T) + 2) \\ + 2 ^ {\left(\left(8 / \Delta^ {2}\right) ^ {1 / \varepsilon} 4 ^ {(1 + \varepsilon) / \varepsilon} + 1\right)} + \frac {e}{e - 2}. \\ \end{array}
+$$
+
+The proof is given in Appendix C. Several remarks are in order now on the regret upper bound of phased-ETC.
+
+Algorithm 1 Phased ETC Algorithm  
+1: Index  $\leftarrow$  INDEX-ESTIMATION()  
+2: for  $N + 1 \leq t \leq T$  do  
+3:  $i \gets \lfloor \log_2(t - 1) \rfloor$   
+4: if  $t - 2^i + 1 \leq K \lfloor i^\varepsilon \rfloor$  then  
+5: Play arm  $(t + \text{Index} - 2^i + 1) \mod K$   
+6: else  
+7: Play GALE-SHAPLEY-MATCHING (Gale & Shapley, 1962) of Algorithm 5 with arm-preference ordered by empirical means computed using all the explore samples thus far.  
+8: end if  
+9: end for
+
+1. Comparison with CA-UCB. From an asymptotic viewpoint (when  $T \to \infty$ ), our result improves upon the recent result of (Liu et al., 2020b). Their proposed algorithm CA-UCB, achieves a regret of  $O(\log^2(T))$  with respect to the agent-pessimal stable matching. Theorem 1 shows that, even under our (restrictive) decentralized bandit model, the phased ETC algorithm achieves  $O(\log^{1 + \varepsilon}(T))$  regret with respect to the agent-optimal stable matching.  
+2. Exponential dependence on gap. The constant in the regret bound has an exponential dependence on  $\Delta^{-2}$ , which limits its applicability to practical values of  $T$ . We note that, the CA-UCB algorithm proposed in (Liu et al., 2020b), has an exponential dependence on  $N^4$  that is multiplied with the  $\log^2(T)$  term in their regret bound. It is an interesting open problem, to obtain an algorithm, that obtains  $O(\log^{1 + \varepsilon}(T))$  regret for the general matching bandit case, without any exponential dependence.  
+3. Comparison with Single Phase ETC. An ETC algorithm was proposed in (Liu et al., 2020a), which requires the knowledge of minimum reward gap  $(\Delta)$  and the time horizon  $(T)$  to compute the necessary exploration. We remove the dependence on the reward gap and time horizon by our interleaved exploration and exploitation in the phased-ETC.
+
+In the following section, we present UCB-D4 — UCB-D3 with Local Deletion, a decentralized algorithm that achieves regret scaling as  $O\left(\frac{\log(T)}{\Delta^2}\right)$  without any exponential dependence on problem parameters, whenever the underlying system satisfies uniqueness consistency.
+
+# 4. Achieving  $O(\log (T))$  Regret under Uniqueness Consistency
+
+We first introduce the uniqueness consistency in stable matching systems, and provide known combinatorial characterization of such systems, before presenting our algorithm.
+
+Uniqueness Consistency. The uniqueness consistency is an
+
+important subclass of graphs with unique stable-matching, and is defined as below after (Karpov, 2019).
+
+Definition 1 ((Karpov, 2019)). A preference profile satisfies uniqueness consistency iff (i) there exists a unique stable matching  $(\mathbf{k}^{*},\mathbf{j}^{*})$ ; (ii) for any subset of arms or agents, the restriction of the preference profile on this subset with their stable-matched pair according to  $(\mathbf{k}^{*},\mathbf{j}^{*})$  has a unique stable matching.
+
+The uniqueness consistency implies that even if an arbitrary subset of agent leaves the system with their respective stable matched arms, we are left with a system with unique stable matching among the rest of agents and arms. This allows for any algorithm, which is able to identify at least one stable pair in a unique stable-matching system, to iteratively lead the system to the global unique stable matching.
+
+Combinatorial characterization. In (Karpov, 2019), a necessary and sufficient condition for the uniqueness consistency of the stable matching system is established in the form of  $\alpha$ -condition (defined shortly). However, in order to connect  $\alpha$ -condition with the serial dictatorship studied in (Sankararaman et al., 2021), we find it instructive to first present the sequential preference condition (SPC) (Eeckhout, 2000), a generalization of serial dictatorship, before defining the  $\alpha$ -condition in full generality. The definition of SPC (Eeckhout, 2000) (which is stated in terms of men and women preferences) restated in our system where we have agents and arms is as follows.
+
+Let  $\sigma$  denote a pair of permutations of  $[N]$  and  $[K]$ . Then  $[K]_{\sigma} = \{a_1^{(\sigma)}, \ldots, a_K^{(\sigma)}\}$  and  $[N]_{\sigma} = \{A_1^{(\sigma)}, \ldots, A_N^{(\sigma)}\}$  denote permutations of the ordered sets  $[K]$  and  $[N]$ , respectively. The  $k$ -th arm in  $[K]_{\sigma}$  is the  $a_k^{(\sigma)}$ -th arm in  $[K]$ , and the  $j$ -th agent in  $[N]_{\sigma}$  is the  $A_j^{(\sigma)}$ -th agent in  $[N]$ .
+
+Definition 2. Sequential preference condition (SPC) is satisfied iff there is an order of agents and arms so that
+
+$$
+\begin{array}{l} \forall j \in [ N ] _ {\mathrm {s p c}}, \forall k \in [ K ] _ {\mathrm {s p c}}, k > j,: \mu_ {j j} > \mu_ {j k}, a n d \\ \forall k \leq N, \forall j \in [ N ] _ {\text {s p c}}, j > k: \operatorname {a g e n t} j _ {k} ^ {*} > _ {\operatorname {a r m} k} \operatorname {a g e n t} j. \\ \end{array}
+$$
+
+We next introduce a generalization of the SPC condition known as  $\alpha$ -condition. This was first introduced in (Karpov, 2019) recently, and shown to be the weakest sufficient condition for a market to admit an unique stable matching. We again restate the definition from (Karpov, 2019) in our scenario.
+
+Definition 3. The  $\alpha$ -condition is satisfied iff there is a stable matching  $(\mathbf{k}^*, \mathbf{j}^*)$ , a left-order of agents and arms
+
+$$
+s. t. \forall j \in [ N ] _ {l}, \forall k > j, k \in [ K ] _ {l}: \mu_ {j k _ {j} ^ {*}} > \mu_ {j k},
+$$
+
+and a (possibly different) right-order of agents and arms
+
+$$
+\begin{array}{l} s. t. \forall k <   j \leq N, a _ {k} \in [ K ] _ {r}, A _ {j} \in [ N ] _ {r}: \\ \operatorname {a g e n t} A _ {j _ {a _ {k}} ^ {*}} > _ {\operatorname {a r m} a _ {k}} \operatorname {a g e n t} A _ {j}. \\ \end{array}
+$$
+
+The following theorem in (Karpov, 2019) provides the characterization of uniqueness consistency, where unacceptable mates are absent (an unacceptable pair  $(j,k)$  means agent  $j$  does not accept arm  $k$ , or the vice versa).<sup>1</sup>
+
+Theorem 2 ((Karpov, 2019)). If there is no unacceptable mates, then the  $\alpha$ -condition is a necessary and sufficient condition for the uniqueness consistency.
+
+Without loss of generality, henceforth we consider the SPC order to be identity, i.e.  $[N]_{\mathrm{spc}} = [N]$  and  $[K]_{\mathrm{spc}} = [K]$ . Again, without loss of generality we consider the left order in the  $\alpha$ -condition to be identity, i.e.  $[N]_l = [N]$  and  $[K]_l = [K]$ . Therefore, for the rest of the paper we only deal with arm order  $a_{k}^{(r)} = a_{k}$  and agent order  $A_{j}^{(r)} = A_{j}$ , for arm  $k \in [K]$  and agent  $j \in [N]$ .
+
+We end with a few remarks on these three systems.
+
+1. Serial Dictatorship, SPC, and  $\alpha$ -condition. As mentioned earlier, the  $\alpha$ -condition generalizes SPC. SPC is satisfied when the left order is identical to the right order in  $\alpha$ -condition. Further, SPC generalizes serial dictatorship, as the unique rank of the agents in the latter and their respective stable matched arms creates an SPC order. We present examples of the three systems.
+
+1. This system is Serial dictatorship, SPC, and  $\alpha$ -condition
+
+$$
+\operatorname {a g e n t}: a: 1 > 2 > 3, b: 1 > 2 > 3, c: 2 > 1 > 3,
+$$
+
+$$
+\operatorname {a r m}: 1: a > b > c, 2: a > b > c, 3: a > b > c.
+$$
+
+2. This system is not Serial dictatorship as arms do not have a unique rank. But it satisfies SPC and  $\alpha$ -condition, with a valid SPC order  $\{(a,1),(b,2),(c,3)\}$ .
+
+$$
+\operatorname {a g e n t}: a: 1 > 2 > 3, b: 1 > 2 > 3, c: 2 > 1 > 3,
+$$
+
+$$
+\operatorname {a r m}: 1: a > b > c, 2: a > b > c, 3: a > c > b.
+$$
+
+3. The third system is not Serial dictatorship or SPC as there is no SPC order. But it satisfies  $\alpha$ -condition as a valid left order is  $\{(a,1), (b,2), (c,3)\}$ , and the corresponding right order is  $\{(b,2), (c,3), (1,a)\}$ .
+
+$$
+\operatorname {a g e n t}: a: 1 > 2 > 3, b: 1 > 2 > 3, c: 2 > 1 > 3,
+$$
+
+$$
+\operatorname {a r m}: 1: b > c > a, 2: b > a > c, 3: b > c > a.
+$$
+
+Currently, the  $\alpha$ -condition is the weakest sufficient condition for uniqueness of stable matching. Necessary condition for the uniqueness of stable matching remains elusive and is a long standing open problem in combinatorics.
+
+2. Stable matching under SPC and  $\alpha$ -condition. The definition of SPC implies that the unique stable matching is
+
+obtained when, under the SPC order, for every  $i \leq N$ , the agent  $i$  is matched to the arm  $i$ . Furthermore, under  $\alpha$ -condition for any  $j \in [N]$ , the agent  $j$  is matched with arm  $j$ , and the agent  $A_{j}$  is matched with the arm  $a_{j}$  in the stable matching. We present a proof in the Appendix E.
+
+# 4.1. UCB-D4: UCB Decentralized Dominate-Delete with Local Deletions
+
+In this section, we describe our main algorithm. The algorithm applies a novel technique called local deletions that interleaves with the phased global deletion strategy of UCB-D3 algorithm proposed in (Sankaraman et al., 2021). Algorithm 2 describes this algorithm in detail.
+
+Algorithm 2 UCB-D4 algorithm (for an agent  $j$ )  
+1: Input: Parameters  $\beta \in (0,1 / K)$ , and  $\gamma > 1$   
+2: Set Index  $(j) \gets$  INDEX-ESTIMATION()  
+3: Global deletion  $G_{j}[0] = \phi$ ,  $\forall j \in [N]$   
+4: for phase  $i = 1,2,\ldots$  do  
+5: Reset the collision counters  $C_{jk}[i] = 0, \forall k \in [K]$   
+6: Delete dominated arms,  $\mathcal{A}_{j}[i] \gets [K] \setminus G_{j}[i - 1]$   
+7: if  $t < 2^{i} + N + NK(i - 1)$  then  
+8: Local deletion  $L_{j}[i] \gets \{k : C_{jk}[i] \geq |\beta 2^{i}|\}$   
+9: Play an arm  $P_{j}(t) \in \arg \max_{k \in \mathcal{A}_{j}[i] \setminus L_{j}[i]} (\widehat{\mu}_{k,j}(t - 1) + \sqrt{\frac{2\gamma \log(t)}{N_{k,j}(t - 1)}})$   
+10: if Arm  $k = P_{j}(t)$  is matched then  
+11: Update estimate  $\widehat{\mu}_{k,j}$ , and matching count  $N_{k,j}$   
+12: else  
+13: Increase collision counter  $C_{jk}[i] \gets C_{jk}[i] + 1$   
+14: end if  
+15: else if  $t = 2^{i} + N + NK(i - 1)$  then  
+16:  $\mathcal{O}_{j}[i] \gets$  most matched arm so far in phase  $i$   
+17:  $G_{j}[i] \gets$  COMMUNICATION  $(i, \mathcal{O}_{j}[i])$   
+18: end if  
+19: end for
+
+The index of each agent  $j$ , namely  $\mathrm{Index}(j)$ , is set as the rank of this agent for the arm 1, which can be learned accurately in  $N$  rounds and in a distributed way (details in Algorithm 3 in Appendix A).
+
+At a high-level, the algorithm proceeds in phases as follows. At each phase  $i$ , every agent  $j$  first updates its set of active arms for the phase given by  $\mathcal{A}_j[i]$  by removing all the arms in the global dominated set  $G_j[i - 1]$ . Then for the next  $2^i$  time-steps, every agent plays the arm with the highest UCB from its active set  $\mathcal{A}_j[i]$ . Here,  $\hat{\mu}_{jk}(t)$  denotes the estimated mean reward for, and  $N_{jk}(t)$  the number of matches with, arm  $k$  by agent  $j$  at the end of round  $t$ . Whenever the agent collides with an arm at least  $\lceil \beta 2^i \rceil$  times (tracked by collision counters  $C_{jk}[i]$ ), it removes this arm from the active set.
+
+Finally, in the last  $NK$  steps of this phase, each agent participates in a communication protocol (Algorithm 4 in Appendix A) to update the global dominated sets  $G_{j}[i]$ . In the rounds from  $(K \times (\mathrm{Index}(j) - 1))$  to  $(K \times \mathrm{Index}(j) - 1)$ , agent  $j$  proposes to the  $K$  arms in round robin order, and adds the rejected arms to its globally dominated set  $G_{j}[i]$ . In the remaining rounds, it proposes to its most played arm in phase  $i$ , i.e., arm  $\mathcal{O}_j[i]$ . We show that the globally dominated set  $G_{j}[i] = \{\mathcal{O}_{j^{\prime}}[i]: j^{\prime} > \mathcal{O}_{j^{\prime}}[i] j\}$ .
+
+# 5. Main Results
+
+We now present our main result in Theorem 3 which show that UCB-D4 attains near optimal logarithmic regret when the stable matching satisfies uniqueness consistency.
+
+System parameters. We introduce the following definitions first that will be used in the regret upper bound:
+
+1. The blocking agents for agent  $j \in [N]$  and arm  $k \in [K]$ ,  $\mathcal{B}_{jk} := \{j' : \mathrm{agent}(j') >_{\mathrm{arm}(k)} \mathrm{agent}(j)\}$ .  
+2. The dominated arms for agent  $j \in [N]$ ,  $\mathcal{D}_j := \{k_{j'}^* : j' \leq j - 1\}$ .  
+3. The blocked non-dominated arms for agent  $j \in [N]$ ,  $\mathcal{H}_j = \{k : \exists j' \in \mathcal{B}_{jk} : k \notin \mathcal{D}_j \cup \mathcal{D}_{j'}\}$ .  
+4. The max blocking agent for agent  $j \in [N] J_{\max}(j) = \max (j + 1, \{j' : \exists k \in \mathcal{H}_j, j' \in \mathcal{B}_{jk}\})$ .  
+5. The right-order mapping for  $\alpha$ -condition for agent  $j \in [N]$  is  $lr(j)$  so that  $A_{lr(j)} = j$  with  $A_j$  as defined in Definition 3. We define the  $lr_{\max}(j) = \max \{lr(j'): j' \leq j\}$ .  
+6. The gaps of each agent  $j$ , is given as  $\Delta_{jk} = (\mu_{jk_j^*} - \mu_{jk})$  which can be negative for some arms  $k$ . We define  $\Delta_{\min} = \min \{\Delta_{jk}: j \in [N], k \in [K], \Delta_{jk} > 0\}$  as the minimum positive gap across arms and agents.
+
+Some comments are due on some of the above definitions. The dominated arms is defined similar to (Sankararaman et al., 2021). These arms may have higher mean, hence higher long term UCB, than the agent's stable matched arm. If not removed, the agent will incur linear regret due to collision from these arms which are played by blocking agents in steady state. The blocked non-dominated arms are absent in serial dictatorship, but in the SPC and  $\alpha$ -condition they emerge due to heterogeneity of arm preference orders. These arms are not necessarily removed during the global deletion (i.e. through the set  $G_{j}[i]$  in UCB-D4), and may create a deadlock for an agent where the agent keeps playing these arms without exploring them due to collisions.
+
+Regret bounds. We present the  $O(\log (T))$  regret bound for the systems following uniqueness consistency in Theorem 3.
+
+Theorem 3. For a stable matching instance satisfying  $\alpha$ -condition (Definition 3), suppose each agent follows UCB-D4 (Algorithm 2) with  $\gamma > 1$  and  $\beta \in (0,1 / K)$ .
+
+Further, let
+
+$$
+\begin{array}{l} f _ {\alpha} (j) = l r _ {\max } (j) + j, i ^ {*} = \max  \{8, i _ {1}, i _ {2} \}, \\ i _ {1} = \min \{i: (N - 1) \frac {1 0 \gamma i}{\Delta_ {m i n} ^ {2}} <   \beta 2 ^ {(i - 1)} \}, \\ a n d i _ {2} = \min  \{i: (N - 1 + N K (i - 1)) \leq 2 ^ {i + 1} \}. \\ \end{array}
+$$
+
+Then the regret for an agent  $j \in [N]$  is upper bounded by
+
+$$
+\begin{array}{l} \mathbb{E}[R_{j}(T)]\leq \underbrace{\sum_{k\notin\mathcal{D}_{j}\cup k_{j}^{*}}\frac{8\gamma}{\Delta_{jk}}\left(\log(T) + \sqrt{\frac{\pi}{\gamma}\log(T)}\right)}_{sub - optimal match} \\ + \underbrace {\sum_ {k \notin \mathcal {D} _ {j}} \sum_ {j ^ {\prime} \in \mathcal {B} _ {j k}: k \notin \mathcal {D} _ {j ^ {\prime}}} \frac {8 \gamma \mu_ {k _ {j} ^ {*}}}{\Delta_ {j ^ {\prime} k} ^ {2}} \left(\log (T) + \sqrt {\frac {\pi}{\gamma} \log (T)}\right)} _ {\text {c o l l i s i o n}} \\ + \underbrace {(K - 1 + | \mathcal {B} _ {j k _ {j} ^ {*}} |) \log_ {2} (T)} _ {\text {c o m m u n i c a t i o n}} \\ + \underbrace {O \left(\frac {N ^ {2} K ^ {2}}{\Delta_ {\operatorname* {m i n}} ^ {2}} + (\beta | \mathcal {H} _ {j} | f _ {\alpha} (J _ {\max } (j)) + f _ {\alpha} (j) - 2) 2 ^ {i ^ {*}}\right)} _ {\text {t r a n s i e n t p h a s e , i n d e p e n d e n t o f T}}. \\ \end{array}
+$$
+
+Furthermore, if the system satisfies SPC (Definition 2) the  $T$  independent term in the above regret bound can be replaced by  $O\left(\frac{N^2K^2}{\Delta_{\min}^2} + (\beta |\mathcal{H}_j|J_{\max}(j) + j - 1)2^{i^*}\right)$ .
+
+We conclude with some remarks about our main result.
+
+Scaling of regret bounds. Our regret bounds in Theorem 3 are desirable for the following reasons.
+
+1. Dependence on the arm gaps and horizon. Our regret upper bound is near optimal in the arm gaps and horizon, as it matches with the regret lower bound of  $\Omega\left(\frac{\log(T)}{\Delta_{\min}^2}\right)$  in Corollary 6 of (Sankararaman et al., 2021).  
+2. Polynomial dependence. The constant term associated with the regret bound is polynomial in all the system parameters, including the minimum gap, as
+
+$$
+2 ^ {i ^ {*}} = O \left(\max  \left\{\frac {N}{\beta \Delta_ {\text {m i n}} ^ {2}} \log (\frac {N}{\beta \Delta_ {\text {m i n}} ^ {2}}), N K \log (N K) \right\}\right).
+$$
+
+Hence, UCB-D4 regret bounds, under uniqueness consistency, has no exponential dependence on system parameters.
+
+3. Dependence on preference profile. The preference profile influences the regret mainly in three ways. First, each agent deletes the dominated arms  $\mathcal{D}_j$  so incurs no long term regret for those arms. The first term in the regret bound scales as  $(K - j)$ . Second, for any agent and any non-dominated arms (which are not globally deleted) the blocking agents during exploration creates collision leading to the second term scaling as at most  $|\mathcal{H}_j\setminus \mathcal{D}_j|$ . Finally, the constant in the regret bound scales linearly with the order of the agent and all of its blocking agents  $(j + J_{\max}(j))$  in SPC. For the uniqueness consistency, the constant depends on both the
+
+left and right order  $(f_{\alpha}(j) = j + lr_{\max}(j))$  of the agent and all of its blocking agents  $(f_{\alpha}(j) + f_{\alpha}(J_{\max}(j)))$
+
+Local deletion threshold. The local deletion threshold is set as  $\beta \times$  phase length  $+\Theta (1)$  with  $\beta < 1 / K$ . Increasing the threshold leads to higher regret until local deletion vanishes. This happens as more collision is allowed until an arm is deleted. But higher threshold allows for quick detection of the stable matched arm. However, decreasing the threshold leads to a more aggressive deletion leading to lower regret from collision per phase, at a cost of longer detection time for the stable matched arm. In particular, if instead we set local deletion threshold as  $\Theta (\mathrm{polylog}(\mathrm{phase~length}))$  then the constant  $2^{i^*}$  becomes exponential, i.e.  $\exp (N / \Delta_{\mathrm{min}}^2)$ . At one extreme, for  $\beta \geq 1 / K$ , the global deletion may stop freezing as the stable matched arm for an agent is not guaranteed to emerge as the most matched arm. In the other extreme, for a threshold of  $\Theta (\log (\mathrm{phase~length}))$  with large enough probability the stable matched arm may get locally deleted. In both extremes, the UCB-D4 ceases to work.
+
+# 5.1. Key Insights into the Proof of Theorem 3
+
+The full proof of Theorem 3 is presented in the Appendix E. We first present why UCB-D3 (Sankararaman et al., 2021) that works for the serial dictatorship setting fails under SPC.
+
+Deadlocks beyond serial dictatorship. Under SPC while running UCB-D3 (Sankararaman et al., 2021) the blocked non-dominated arms cause deadlock where two agents are unable to sample their respective best stable matched arm leading to linear regret. This is best explained by an example. Let us consider the 3 agent  $(a, b, \text{and } c)$ , and 3 arm  $(1, 2, \text{and } 3)$  system with the preference lists given as
+
+$$
+\begin{array}{l} \text {a g e n t}: a: 1 > 2 > 3, b: 2 > 1 > 3, c: 3 > 1 > 2, \\ \operatorname {a r m}: 1: a > b > c, 2: a > b > c, 3: a > c > b. \\ \end{array}
+$$
+
+This system satisfies SPC with the agent optimal stable matching  $\{(a,1),(b,2),(c,3)\}$ , but it is not a serial dictatorship. If we run the UCB-D3 algorithm in (Sankararaman et al., 2021) then the agent  $a$  matches with arm 1, and agent  $b$  and  $c$  both delete arm 1 using global deletion. However, without additional coordination, with non-negligible probability, agent  $b$  may not match with arm 3 (collision with agent  $c$ ), and agent  $c$  may not match with arm 2 (collision with agent  $b$ ). No further global deletion is guaranteed resulting in a deadlock, hence linear regret (see, Section 6).
+
+We first focus on the proof for SPC condition before tackling  $\alpha$ -condition, as the latter builds on the former.
+
+Proof Sketch for SPC. There are two main components of the proof, inductive freezing of stable matching pairs, and vanishing of local deletion, both in expected constant time.
+
+Inductive freezing of stable matching pairs. Local deletion
+
+of arms with more than  $(\beta \times$  phase length) collisions, for  $\beta < 1 / K$ , ensures the agent ranked 1 (under SPC) removes all the blocked non-dominated arms, thus it never gets stuck in a deadlock. This gives agent 1 opportunity to detect arm 1 as its most matched arm in expected constant time, as in the presence of arm 1, agent 1 matches with any non-deleted sub-optimal arm  $O(\log (\mathrm{phase~length}))$  times with high probability under UCB dynamics. Note that agent 1 never deletes arm 1 (either globally or locally) due to the SPC condition. Agent 2 next deletes dominated arm 1, and for similar reasons as above matches most with arm 2. The proof is completed using an induction over the agents following the SPC order, where in agent  $j$  freezing happens in  $O(ji^{*})$  phases.
+
+Constant time vanishing of local deletion. We establish that the local deletion vanishes in expected constant time. Indeed, when all the agents settle to their respective stable matched arms, the sub-optimal play is limited to  $O(\log (\mathrm{phase~length}))$ , hence total collision to blocked non-dominated arm is also logarithmic in phase length. This leads to the vanishing of local deletion, which requires  $(\beta \times$  phase length) collisions, with an extra  $O(\beta |\mathcal{H}_j|J_{\max}(j)2^{i^*})$  regret, where  $J_{\max}(j)$  is the maximum blocking agent.
+
+Stable regime. The terms that grow with  $T$  as  $O(\log (T))$  are accounted for by keeping count of expected (1) number of sub-optimal matches, (2) number of collisions from blocking agents, and (3) the regret due to communication sub-phases.
+
+Uniqueness consistency - A tale of two orders. When moving from SPC instances to uniqueness consistency (equivalently  $\alpha$ -condition due to Theorem 2) we no longer have the simple inductive structure that we leverage in the proof sketch of SPC. Under  $\alpha$ -condition we have two orders instead. The left order states when arm 1 to arm  $(j - 1)$  are removed agent  $j$  has arm  $j$  as the most preferred arm. Whereas, the right order states when agent  $A_{1}$  to agent  $A_{(k - 1)}$  (recall  $\{A_1,\ldots ,A_N\}$  is a separate permutation of the agents) are removed arm  $a_{k}$  prefers agent  $A_{k}$ . Global deletion here must follow left order. However, unlike SPC the set of blocking agents for agent 1 and arm 1 (i.e.  $\mathcal{B}_{11}$ ) is nonempty. Thus, agent 1 cannot get matched with arm 1 majority of time unless the agents in  $\mathcal{B}_{11}$  stop playing arm 1. We next show how this is resolved.
+
+Proof Sketch of  $\alpha$ -condition. We show due to local deletion an inductive warmup of stable matching pairs precedes the two phases mentioned in the proof sketch of SPC.
+
+Inductive warmup of stable matching pairs. We begin with an observation that holds for every stable matching: for any two stable matching pair  $(j,k)$ ,  $(j',k')$ , the arm  $k$  is either sub-optimal (has mean reward lesser than  $k'$ ) for agent  $j'$ ,
+
+or arm  $k$  prefers agent  $j$  over agent  $j'$ . Hence, in our system, for any  $2 \leq j \leq N$ , either (a) arm  $a_j$  (stable matched arm for agent  $A_j$ ) is suboptimal for agent  $A_1$ , or (b) arm  $a_j$  prefers  $A_j$  over  $A_1$ . In case (b) arm  $A_1$  never causes collision in  $a_j$  for agent  $A_j$ . Now suppose case (a) holds. Due to  $\alpha$ -condition, the agent  $A_1$  is the most preferred for arm  $a_1$ . Therefore, arm  $a_1$  is always available to agent  $A_1$ , and  $a_1$  has higher mean than  $a_j$ . This implies the UCB algorithm plays arm  $a_j$  only  $O(\log (\text{phase length}))$  times with high probability in any phase. Once this happens, arm  $a_2$  is almost always available to  $A_2$ , and the induction sets in. This is used to prove  $A_j$  is warmed up in  $O(ji^*)$  phases, or agent  $j$  is warmed up in  $O(lr(j)i^*)$  phases. Once agent 1 is warmed up the inductive freezing starts as it matches with arm 1. The rest closely follows the proof of SPC.
+
+# 6. Numerical Simulations
+
+In this section, we present our numerical simulations with 5 agents and 6 arms. Additional results with larger instances are deferred to the Appendix F.
+
+Baselines. We use two baselines with their own feedback. 1. Centralized UCB (UCB-C) is proposed in (Liu et al., 2020a) for the centralized feedback setting, where each agent in every round submits its preference order (based on UCB indices) to a centralized agent who assigns the agent optimal matching under this preference, and the rewards are observed locally.  
+2. Collision Avoidance UCB (CA-UCB) is proposed in (Liu et al., 2020b) for the decentralized with partial information setting where at each round all agents observe the player matched to each arm, but rewards are observed locally.
+
+The centralized setting is the most relaxed where no collision happens, followed by the partial decentralized setting where each round the matching of arms can be learned without any collision. Both are relaxed compared to our decentralized with no information setting where any global information can be obtained only through collisions.
+
+Results. We generate random instances to compare the performance of UCB-C, CA-UCB and UCB-D4 in their respective settings. Since (Liu et al., 2020b) does not mention how to set their hyper-parameter  $\lambda$ , we report the best result by running a grid search over  $\lambda$ . For UCB-D4, we use  $\beta = 1/2K$  and  $\gamma = 2$ . We simulate all the algorithms on the same sample paths, for a total 50 sample paths and report mean,  $75\%$  and  $25\%$  agent-optimal regret.
+
+Figure 1 shows that in a general instance phased ETC outperforms CA-UCB even with a restricted feedback, whereas, as expected, UCB-C outperforms phased ETC. Figure 2 shows that when uniqueness condition holds UCB-D4, despite the restricted feedback, outperforms CA-UCB, while it is comparable to the centralized UCB-C.
+
+Linear Regret of UCB-D3 in SPC. Figure 3 shows that even when the SPC condition holds, UCB-D3 may result in linear regret, emphasizing the importance of UCB-D4.
+
+![](images/fd4e1c173f825b01e7848873445fe53ca2aba186e50d549dc1749741a4ee6f11.jpg)  
+Figure 1. A general instance with  $N = 5$ , and  $K = 6$ . Regret: UCB-C (blue) < Phased-ETC (orange) < CA-UCB (green).
+
+![](images/fa435f568f68d14328494d91e2ab9e57b6fe5c003570ac034dccd751861a2658.jpg)  
+Figure 2. An  $\alpha$ -condition instance with  $N = 5$ , and  $K = 6$ . Regret: UCB-C (blue) < UCB-D4 (green) < CA-UCB (orange).
+
+![](images/47c166cc56f93c8eca426171d69425ef4aa153524b67c9582fc36bdfd7652b8d.jpg)  
+Figure 3. Linear regret of UCB-D3 for an SPC instance.
+
+# References
+
+Aridor, G., Mansour, Y., Slivkins, A., and Wu, Z. S. Competing bandits: The perils of exploration under competition. arXiv preprint arXiv:2007.10144, 2020.  
+Ashlagi, I., Krishnaswamy, A. K., Makhijani, R., Saban, D., and Shiragur, K. Assortment planning for two-sided sequential matching markets. arXiv preprint arXiv:1907.04485, 2019.  
+Avner, O. and Mannor, S. Multi-user lax communications: a multi-armed bandit approach. In IEEE INFOCOM 2016-The 35th Annual IEEE International Conference on Computer Communications, pp. 1-9. IEEE, 2016.  
+Bistritz, I. and Leshem, A. Game of thrones: Fully distributed learning for multiplayer bandits. Mathematics of Operations Research, 2020.  
+Boursier, E. and Perchet, V. SIC-MMAB: synchronisation involves communication in multiplayer multi-armed bandits. In Wallach, H. M., Larochelle, H., Beygelzimer, A., d'Alché-Buc, F., Fox, E. B., and Garnett, R. (eds.), Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pp. 12048-12057, 2019. URL https://proceedings.neurips.cc/paper/2019/ hash/c4127b9194fe8562c64dc0f5bf2c93bc Abstract.html.  
+Boursier, E. and Perchet, V. Selfish robustness and equilibria in multi-player bandits. In Abernethy, J. D. and Agarwal, S. (eds.), Conference on Learning Theory, COLT 2020, 9-12 July 2020, Virtual Event [Graz, Austria], volume 125 of Proceedings of Machine Learning Research, pp. 530-581. PMLR, 2020. URL http://proceedings.mlr.press/v125/boursier20a.html.  
+Bränzei, S. and Peres, Y. Multiplayer bandit learning, from competition to cooperation. arXiv preprint arXiv:1908.01135, 2019.  
+Bubeck, S. and Cesa-Bianchi, N. Regret analysis of stochastic and nonstochastic multi-armed bandit problems. Foundations & Trends in Machine Learning, 2012.  
+Buccapatnam, S., Tan, J., and Zhang, L. Information sharing in distributed stochastic bandits. In 2015 IEEE Conference on Computer Communications (INFOCOM), pp. 2605-2613. IEEE, 2015.  
+Chawla, R., Sankararaman, A., Ganesh, A., and Shakkottai, S. The gossiping insert-eliminate algorithm for multiagent bandits. In Chiappa, S. and Calandra, R. (eds.), The
+
+23rd International Conference on Artificial Intelligence and Statistics, AISTATS 2020, 26-28 August 2020, Online [Palermo, Sicily, Italy], volume 108 of Proceedings of Machine Learning Research, pp. 3471-3481. PMLR, 2020. URL http://proceedings.mlr.press/v108/chawla20a.html.  
+Dai, X. and Jordan, M. I. Learning strategies in decentralized matching markets under uncertain preferences. arXiv preprint arXiv:2011.00159, 2020.  
+Darak, S. J. and Hanawal, M. K. Multi-player multi-armed bandits for stable allocation in heterogeneous ad-hoc networks. IEEE Journal on Selected Areas in Communications, 37(10):2350-2363, 2019.  
+Dubey, A. and Pentland, A. S. Cooperative multi-agent bandits with heavy tails. In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pp. 2730-2739. PMLR, 2020. URL http://proceedings.mlr.press/v119/dubey20a.html.  
+Eeckhout, J. On the uniqueness of stable marriage matchings. Economics Letters, 69(1):1-8, 2000.  
+Gale, D. and Shapley, L. S. College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9-15, 1962.  
+Hillel, E., Karnin, Z. S., Koren, T., Lempel, R., and Somekh, O. Distributed exploration in multi-armed bandits. In Burges, C. J. C., Bottou, L., Ghahramani, Z., and Weinberger, K. Q. (eds.), Advances in Neural Information Processing Systems 26: 27th Annual Conference on Neural Information Processing Systems 2013. Proceedings of a meeting held December 5-8, 2013, Lake Tahoe, Nevada, United States, pp. 854-862, 2013. URL https://proceedings.neurips.cc/paper/2013/hash/598b3e71ec378bd83e0a727608b5db01 Abstract.html.  
+Johari, R., Kamble, V., and Kanoria, Y. Matching while learning. Operations Research, 2021.  
+Kalathil, D., Nayyar, N., and Jain, R. Decentralized learning for multiplayer multiarmed bandits. IEEE Transactions on Information Theory, 60(4):2331-2345, 2014.  
+Karpov, A. A necessary and sufficient condition for uniqueness consistency in the stable marriage matching problem. *Economics Letters*, 178:63–65, 2019.  
+Kolla, R. K., Jagannathan, K., and Gopalan, A. Collaborative learning of stochastic bandits over a social network. IEEE/ACM Transactions on Networking, 26(4): 1782-1795, 2018.
+
+Landgren, P., Srivastava, V., and Leonard, N. E. Distributed cooperative decision making in multi-agent multi-armed bandits. Automatica, 125:109445, 2021.  
+Larrnaaga, M., Ayesta, U., and Verloop, I. M. Dynamic control of birth-and-death restless bandits: Application to resource-allocation problems. IEEE/ACM Transactions on Networking, 24(6):3812-3825, 2016.  
+Lattimore, T. and Szepesvári, C. Bandit algorithms. Cambridge University Press, 2020.  
+Liu, L. T., Mania, H., and Jordan, M. I. Competing bandits in matching markets. In Chiappa, S. and Calandra, R. (eds.), The 23rd International Conference on Artificial Intelligence and Statistics, AISTATS 2020, 26-28 August 2020, Online [Palermo, Sicily, Italy], volume 108 of Proceedings of Machine Learning Research, pp. 1618-1628. PMLR, 2020a. URL http://proceedings.mlr.press/v108/liu20c.html.  
+Liu, L. T., Ruan, F., Mania, H., and Jordan, M. I. Bandit learning in decentralized matching markets. arXiv preprint arXiv:2012.07348, 2020b.  
+Mehrabian, A., Boursier, E., Kaufmann, E., and Perchet, V. A practical algorithm for multiplayer bandits when arm means vary among players. In Chiappa, S. and Calandra, R. (eds.), The 23rd International Conference on Artificial Intelligence and Statistics, AISTATS 2020, 26-28 August 2020, Online [Palermo, Sicily, Italy], volume 108 of Proceedings of Machine Learning Research, pp. 1211-1221. PMLR, 2020. URL http://proceedings.mlr.press/v108/mehrabian20a.html.  
+Rosenski, J., Shamir, O., and Szlak, L. Multi-player bandits - a musical chairs approach. In Balcan, M. and Weinberger, K. Q. (eds.), Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, volume 48 of JMLR Workshop and Conference Proceedings, pp. 155-163. JMLR.org, 2016. URL http://proceedings.mlr.press/v48/rosenski16.html.  
+Sankararaman, A., Ganesh, A., and Shakkottai, S. Social learning in multi agent multi armed bandits. Proceedings of the ACM on Measurement and Analysis of Computing Systems, 3(3):1-35, 2019.  
+Sankararaman, A., Basu, S., and Sankararaman, K. A. Dominate or delete: Decentralized competing bandits in serial dictatorship. In Banerjee, A. and Fukumizu, K. (eds.), The 24th International Conference on Artificial Intelligence and Statistics, AISTATS 2021, April 13-15, 2021, Virtual Event, volume 130 of Proceedings
+
+of Machine Learning Research, pp. 1252-1260. PMLR, 2021. URL http://proceedings.mlr.press/v130/sankararaman21a.html.  
+Slivkins, A. Introduction to multi-armed bandits. Foundations and Trends® in Machine Learning, 12(1-2):1-286, 2019.

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+# Beyond the Pareto Efficient Frontier: Constraint Active Search for Multiobjective Experimental Design
+
+Gustavo Malkomes<sup>1</sup> Bolong Cheng<sup>1</sup> Eric Hans Lee<sup>1</sup> Michael McCourt<sup>1</sup>
+
+# Abstract
+
+Many problems in engineering design and simulation require balancing competing objectives under the presence of uncertainty. Sample-efficient multiobjective optimization methods focus on the objective function values in metric space and ignore the sampling behavior of the design configurations in parameter space. Consequently, they may provide little actionable insight on how to choose designs in the presence of metric uncertainty or limited precision when implementing a chosen design. We propose a new formulation that accounts for the importance of the parameter space and is thus more suitable for multiobjective design problems; instead of searching for the Paretoefficient frontier, we solicit the desired minimum performance thresholds on all objectives to define regions of satisfaction. We introduce an active search algorithm called Expected Coverage Improvement (ECI) to efficiently discover the region of satisfaction and simultaneously sample diverse acceptable configurations. We demonstrate our algorithm on several design and simulation domains: mechanical design, additive manufacturing, medical monitoring, and plasma physics.
+
+# 1. Introduction
+
+Accelerated design and optimization via machine learning is receiving increasing interest in multiple science and engineering disciplines, such as materials design, drug discovery, and chemical engineering (Forrester et al., 2008; Negoescu et al., 2011; Molesky et al., 2018). Using numerical simulation to study the impact of design decisions prior to manufacturing has become a common strategy to reduce the number of fabrications, thus reducing the substantial cost of development cycles.
+
+In recent years, active learning has gained popularity in scientific communities. These adaptive methods more efficiently identify promising candidates compared to brute-force or evolutionary algorithms (Song et al., 2018; Attia et al., 2020; Haghanifar et al., 2020; Duris et al., 2020). Bayesian optimization (BO) is arguably the most prominent example of active learning, where the goal is to seek the optimizer  $\arg \max_{\mathbf{x}} f(\mathbf{x})$  of an expensive-to-evaluate black-box objective  $f$ .
+
+However, in most real world applications, multiple competing objectives need to be investigated (Koziel et al., 2014; Singh et al., 2016). Optimality of multiobjective problems is usually defined by the concept of Pareto efficiency, where no individual objective can be improved without loss in at least one other objective. Most work on optimizing multiple objectives focuses on searching for the Pareto efficient frontier (and the corresponding parameters). In general, for an  $m$ -objective problem, the (nearly) Pareto efficient solutions are assumed to be a dense set of points on an  $(m - 1)$ -dimensional manifold (Konakovic-Lukovic et al., 2020). Unfortunately, using the Pareto frontier as the measurement of success is limiting in engineering and design applications; candidates near the Pareto frontier, unaccounted for in most optimization literature, are still of scientific interest in the real world (del Rosario et al., 2020).
+
+In practice, design objectives can also be formulated as inequality constraints, where the decision makers seek to find design choices that satisfy known performance thresholds on each objective. This formulation is related to the topic of level set estimation (LSE) in the single objective setting. Its goal is to determine the decision boundary of parameter space that separates it above and below the threshold on the objective (Gotovos et al., 2013). However, the notion of level set does not naturally extend to the multiobjective settings; a naive approach of identifying the level set for each objective and taking the intersection is not computationally efficient. In addition, most LSE algorithms focus on searching for the decision boundary; in the multiobjective setting, this behavior cannot reveal the tradeoff among objectives, since there are not enough samples within the region of interest.
+
+Figure 1 illustrates the limitations of multiobjective BO and
+
+![](images/6ed65d2199d9b00b77e4bde5742f5840e89e5eb59bf9da9b2b4f9eb6086f5d11.jpg)
+
+![](images/7efc5071668330baa8484020418efc1703b038d67f5b5c7c7e4c5731b840ec44.jpg)
+
+![](images/fcc2781d025bf7cc41ac513fc55bd05614849faf74ebeda921fd829cd52acddf.jpg)
+
+![](images/8e4a8c1589dd850fe0b1e0cc3be2ff59dddf9d3b149223650fb14e1d153c799a.jpg)  
+Figure 1. Comparison of BO, LSE, constraint active search, and random search behaviors on the same multiobjective problem. The top row shows the observed samples in the parameter space; the bottom row, in the metric space. The shaded region represents the satisfactory regions of interest, where both objectives exceed the known thresholds (labeled with dashed black lines in the metric space).
+
+![](images/146070f434493296c8784e9cb4af0eb887bc2fd87c249e9676f81ed4b1fa91b4.jpg)
+
+![](images/85023c14eb6fc35637466314b59f24654f5a7a73d098554cb9f32199f1dd7206.jpg)
+
+LSE respectively, and highlights constraint active search as a new perspective on multiobjective problems. BO focuses on searching for the Pareto efficient parameters. LSE focuses on searching for the boundary of the region satisfying the thresholds. Constraint active search attempts to sample from that region of interest.
+
+Our paper makes the following contributions:
+
+- We propose an alternative formulation for the multi-objective design problem as an active search problem. In this setting, the objectives are treated as constraints with known threshold values; the goal is to find diverse samples that satisfy all the thresholds. We call this formulation constraint active search (CAS).  
+- We develop an algorithm called expected coverage improvement (ECI) for constraint active search that preserves the sample efficiency of active search whilst also diversely sampling from the regions of interest. We provide theoretical analysis on the sample diversity of ECI; specifically, it guarantees a constant approximation ratio to the optimal sample diversity.  
+- We compare ECI to various baselines on a suite of synthetic multiobjective design benchmarks as well as real-world multiobjective design and simulation applications in materials science, medical monitoring, and plasma physics.
+
+# 2. Related Work
+
+# 2.1. Multiobjective Bayesian Optimization
+
+Bayesian optimization (BO) is a popular method for optimizing expensive, black-box objective functions. BO consists of two components: a probabilistic model, to model the objective function  $f$ , and an acquisition function, to determine which parameters  $x$  to sample next. In this section, we focus on reviewing the literature of BO in the multiobjective setting. We refer the readers to Frazier (2018) for a comprehensive review of BO in the single objective setting.
+
+There are several strategies at adapting BO to multiobjective optimization. The simplest approach is linear scalarization, where single objective BO strategies optimize a linear combination of the objectives. Knowles (2006) first proposes using a random scalarization of the objective functions to search for the entire frontier. This method is further improved in Paria et al. (2020), which models each individual objective with a separate Gaussian process (GP).
+
+A related approach involves reformulating the problem as a constrained optimization problem, commonly known as the  $\varepsilon$ -constraint method (Haghanifar et al., 2020). In particular, this formulation leverages prior works from constrained Bayesian optimization literature (Gelbart, 2015; Gardner et al., 2014). In constrained BO, a probabilistic model is formed for each constraint and used as a multiplicative penalty term in the acquisition function (Letham et al., 2019). Recent works on constrained BO borrow ideas
+
+from the mathematical optimization community such as the trust region (Eriksson & Poloczek, 2021) and Alternating Directions Methods of Multiplier (Ariafar et al., 2019).
+
+A separate line of research in multiobjective BO improves the hypervolume of the Pareto efficient frontier directly (Emmerich et al., 2011; Konakovic-Lukovic et al., 2020; Daulton et al., 2020). Belakaria et al. (2019) adapts the entropy search acquisition function for the multiobjective setting, by maximizing the entropy of the Pareto frontier. Belakaria et al. (2020) and Fernandez-Sanchez et al. (2021) build upon this work to incorporate additional black-box inequality constraints. Almost all multiobjective BO literature uses the hypervolume of the Pareto frontier as their performance criterion. While hypervolume provides a single numerical value for the ease of comparison, using it as the sole criterion of success misses the goal of multi-objective optimization — understanding tradeoffs between parameters and objectives.
+
+# 2.2. Level Set Estimation
+
+The goal of level set estimation is to determine the regions where the objective function lies above or below a known threshold level. Bryan et al. (2005) first uses a GP-based active learning approach to tackle this problem through the straddle heuristic, defined as
+
+$$
+\operatorname {S T R A D D L E} (\mathbf {x}) = 1. 9 6 \sqrt {v (\mathbf {x})} - | f (\mathbf {x}) - \tau |,
+$$
+
+where  $\tau$  is the threshold of the level set and  $v$  is the model uncertainty. Gotovos et al. (2013) generalizes the straddle heuristic and provides theoretical guarantees on the classification quality. Zanette et al. (2019) maximizes the expected increase in the volume of the predicted level set. Iwazaki et al. (2020) considers the LSE problem with input uncertainty, where the input parameter values are subject to perturbation from a known Normal distribution. This work is related to research in safe BO (Sui et al., 2015; 2018), where only configurations exceeding the threshold with high probability are sampled.
+
+Almost all LSE methods focus on identifying the decision boundary. To our best knowledge, Wang et al. (2018) is the closest example of learning and diversely sampling from a region of interest (in the single objective setting). It divides the sampling procedure into two disjoint algorithms: one determines the level set and the other samples diversely from the interior of the learned level set.
+
+The notion of a level set does not naturally extend beyond more than one objective. Bryan & Schneider (2008) only considers the multiobjective problem in a composite setting, where a single threshold is used for a linear combination of the objective functions. One naive and inefficient approach to extend existing LSE algorithms to multiobjective problems is to estimate the level set of each objective separately
+
+and then intersect each region of interest.
+
+# 2.3. Active Search
+
+Our proposed work is also related to the topic of active search (Garnett et al., 2012). Active search can be seen as a special case of Bayesian optimization, where one has binary observations and cumulative reward. The goal of active search is to sequentially discover members of a rare, desired class. Inspecting any element is assumed to be expensive, representing, for instance, the cost of performing a real world laboratory experiment.
+
+Garnett et al. (2012) explores this problem using Bayesian decision theory, introduces a natural utility function for active search, and derives the optimal policy for this framework. In general, only myopic (one-step) approximations to the optimal policy are computationally tractable. More recently, Jiang et al. (2017) proposes an nonmyopic and efficient approximation to the optimal policy, which is later extended to the batch setting in Jiang et al. (2018). Active search, however, is not designed to solve multiobjective continuous problems—a problem that we address next.
+
+# 3. Constraint Active Search
+
+Scientists, engineers, and decision makers seldom rely on single-point observations to make final design decisions. Instead, they seek a diverse set of design configurations to enrich their understanding of each objective function's impact and its respective tradeoffs. As Figure 1 illustrates, neither BO nor LSE is well-suited to address these experimental design concerns. The former clusters around optima, and the latter clusters around the level set boundary.
+
+We propose an alternative formulation to the multiobjective design problem better suited to gather a diverse spectrum of design configurations. The goal of this formulation is to efficiently cover an unknown, possibly disconnected, set  $S$  defined by the objective thresholds  $\tau$  in a sequential fashion. We call this problem constraint active search. In Section 5, we illustrate how CAS may enable a practitioner to make better decisions than those made by exclusively identifying the Pareto frontier on an additive manufacturing problem.
+
+This concept is related to del Rosario et al. (2020) which advocates that the Pareto frontier itself is insufficient for scientific and engineering settings. That work introduces the notion of Pareto shells, the set of near Pareto optimal solutions, as the desired outcome of multiobjective optimization. We propose that, when thresholds  $\tau$  are available, studying the full region of constraint satisfaction<sup>1</sup> is preferable.
+
+![](images/4566d4519126b3b101cb9ff21d3cfd551761947d900d3613bbadf02266cd8578.jpg)  
+Figure 2. An illustration of our utility function when constraint active search is run on the 2D sinusoids problem in (Bryan et al., 2005). Each of the four subfigures above plots a single realization of the underlying stochastic model and outlines the decision boundary of the satisfactory region in magenta. The observations within the boundary are marked with blue crosses, and a dotted blue ball of radius  $r = 0.2$  is drawn around each to indicate coverage. A ball is also drawn around the candidate point, itself marked with a black star. The improvement of the utility is the volume of the blue region, and expected coverage improvement is the average value of all realizations.
+
+![](images/efe100f3d98d91e6d273d127ff2d995a679c69d0c90153af9292e1f7d965ec47.jpg)
+
+# 3.1. Problem Statement
+
+Suppose we want to search for design configurations in a compact search space  $\mathcal{X}$ . We may judge the quality of a design  $\mathbf{x} \in \mathcal{X}$  by evaluating  $m$  expensive black-box objective functions  $f_{1}, f_{2}, \ldots, f_{m}$ , each mapping  $\mathcal{X}$  to  $\mathbb{R}$ . We seek designs  $\mathbf{x}$  that yield acceptable performance, defined by threshold values  $\boldsymbol{\tau} = [\tau_{1}, \tau_{2}, \dots, \tau_{m}]^{\top}$ . Specifically, we wish to sequentially select configurations from the set:
+
+$$
+\mathcal {S} = \{\mathbf {x} \mid f (\mathbf {x}) \succeq \tau \},
+$$
+
+where  $f(\mathbf{x})\succeq \pmb{\tau}:= f_{i}(\mathbf{x})\geq \tau_{i},i = 1,\ldots ,m.$  At iteration  $t$  we assume we have evaluated  $t$  design configurations  $\mathbf{X}_t = \{\mathbf{x}_j\}_{j = 1}^t$  and their respective outputs  $\mathbf{Y}_t = \{\mathbf{y}_j\}_{j = 1}^t$  where  $\mathbf{y}_j = [y_{j_1},y_{j_2},\dots,y_{j_m}]$  is the potentially noisy observation of the objective functions  $f_{1},f_{2},\ldots f_{m}$  evaluated at  $\mathbf{x}_j$  . These observations form the dataset  $\mathcal{D}_t = (\mathbf{X}_t,\mathbf{Y}_t)$
+
+We assume we have  $m$  independent probabilistic models that capture our prior beliefs about observations  $y_{i} = f_{i}(\mathbf{x}_{i}) + \varepsilon_{i}$  for  $i = 1,2,\ldots ,m$ , as a probability distribution over  $p(\mathbf{y})$ , where  $\varepsilon_{i}$  is additive Gaussian noise. We also assume  $p(\mathbf{y})$  allows posterior updates when  $\mathcal{D}$  is observed, i.e.,  $p(\mathbf{y} \mid \mathcal{D})$ . Let  $Z$  be an indicator variable that tells us if a point  $\mathbf{x}$  will satisfy the thresholds,  $Z(\mathbf{x}) = \mathbb{1}\left[\mathbf{y}(\mathbf{x}) \succeq \boldsymbol{\tau}\right]$ . Our model of  $\mathbf{y}$  allows us to compute the probability of any point  $\mathbf{x}$  belonging to the satisfactory region  $\mathcal{S}$  after observing  $\mathcal{D}_t$ , which we denote by  $p(Z(\mathbf{x}) = 1 \mid \mathcal{D}_t)$ . We delay discussing the statistical model's details until Section 4, as they are immaterial to decision-making.
+
+# 3.2. Expected Coverage Improvement
+
+Our approach to constraint active search considers an additional resolution parameter  $r$ ; in real-world problems one can often define  $r$  in terms of the robustness of the design to perturbations. Typically, most experimental design problems possess a sense of known resolution, such as simulation
+
+accuracy or manufacturing precision/tolerance. This presumes that any design configuration within distance  $r$  of another does not convey extra information about  $S$ .
+
+Definition 3.1 (Coverage neighborhood). The coverage neighborhood of any  $\mathbf{x}$  is defined as
+
+$$
+\mathbb {N} _ {r} (\mathbf {x}) = \left\{\mathbf {x} ^ {\prime}: d (\mathbf {x}, \mathbf {x} ^ {\prime}) <   r \right\},
+$$
+
+for an a priori fixed  $r\in \mathbb{R}^+$  and an appropriate distance function  $d:\mathcal{X}\times \mathcal{X}\mapsto \mathbb{R}^{+}$ . The coverage neighborhood of a set of points  $\mathbf{X}$  is defined as
+
+$$
+\mathbb {N} _ {r} (\mathbf {X}) = \bigcup_ {\mathbf {x} \in \mathbf {X}} \mathbb {N} _ {r} (\mathbf {x}).
+$$
+
+For Euclidean distance,  $\mathbb{N}_r(\mathbf{x})$  is simply a ball of radius  $r$ .
+
+Constraint active search prioritizes datasets  $\mathcal{D}$  that cover as much volume of the satisfactory region  $S$  as possible. We express this desire with the following utility function:
+
+$$
+u (\mathcal {D}) = \operatorname {V o l} \left(\mathbb {N} _ {r} (\mathbf {X}) \cap \mathcal {S}\right).
+$$
+
+Intuitively,  $u(\mathcal{D})$  measures the total volume of  $S$  that is covered by neighborhood  $\mathbb{N}_r(\mathbf{x})$  for all  $\mathbf{x} \in \mathbf{X}$ . At iteration  $t$ , the expected improvement of this utility function for sampling a candidate configuration  $^2 \mathbf{x}$  is:
+
+$$
+\begin{array}{l} \alpha (\mathbf {x} \mid \mathcal {D} _ {t}) = \mathbb {E} _ {\mathbf {y}} \left[ u \left(\mathcal {D} _ {t} \cup (\mathbf {x}, \mathbf {y})\right) - u (\mathcal {D} _ {t}) \right] \\ = \mathbb {E} _ {Z} \left[ \operatorname {V o l} \left(\left\{\mathbb {N} _ {r} (\mathbf {x}) \cap \mathcal {S} _ {Z} \right\} \backslash \mathbb {N} _ {r} \left(\mathbf {X} _ {\mathbf {t}}\right)\right) \right]. \tag {1} \\ \end{array}
+$$
+
+We can also refer to (1) as an acquisition function. At iteration  $t$ , our policy selects the location  $\mathbf{x}^*$  that maximizes the acquisition function, i.e.,  $\mathbf{x}^* = \arg \max_{\mathbf{x} \in \mathcal{X}} \alpha(\mathbf{x} \mid \mathcal{D}_t)$ . We visualize the intuition behind our policy for different
+
+realizations of  $Z$  in Figure 2. Our acquisition function considers the volume of  $S_Z$  that a new candidate location is expected to cover under our current belief about  $Z$ .
+
+A common practice in the LSE and active search literature is to work on discrete search spaces:  $\mathcal{X}$  could be discretized by sampling a pool of  $n$  points. In this discretized setting, (1) simplifies to
+
+$$
+\begin{array}{l} \alpha (\mathbf {x} \mid \mathcal {D} _ {t}) = \mathbb {E} _ {Z} \left[ \sum_ {\mathbf {x} ^ {\prime} \in \mathbb {N} (\mathbf {x}) \backslash \mathbb {N} _ {r} (\mathbf {X})} Z (\mathbf {x} ^ {\prime}) \right] \\ = \sum_ {\mathbf {x} ^ {\prime} \in \mathbb {N} (\mathbf {x}) \backslash \mathbb {N} _ {r} (\mathbf {X})} \mathbb {E} _ {Z} \left[ Z \left(\mathbf {x} ^ {\prime}\right) \right] \\ = \sum_ {\mathbf {x} ^ {\prime} \in \mathbb {N} (\mathbf {x}) \backslash \mathbb {N} _ {r} (\mathbf {X})} p (Z (\mathbf {x} ^ {\prime}) = 1 \mid \mathcal {D} _ {t}). \quad (2) \\ \end{array}
+$$
+
+In practice, the argmax of  $\alpha (\mathbf{x}\mid \mathcal{D}_t)$  is often nonunique, especially at the start of search. This reflects the intuition that when only a few observations sparsely cover  $S$ , multiple—and possibly many—candidates will contribute equally to the expected coverage. Crucially, if this occurs, we break ties among the multiple optimum points by selecting the one furthest from all observations in  $\mathbf{X}_t$ , which encourages selecting a diverse set of points. We call this policy expected coverage improvement (ECI).
+
+# 3.3. Theoretical Analysis of ECI
+
+We present three theoretical results for ECI (proved in the supplement). All assume that  $S$  is fixed a priori — that  $S$  does not change as iteration proceeds.
+
+Theorem 1. If we have a budget of one iteration left, then ECI is one-step Bayes-optimal among all feasible policies.
+
+This result is not particularly surprising. The next pair of less trivial theorems concern the convergence of CAS using ECI. Because CAS seeks to sample a diverse set of points, we measure this diversity with fill distance. Fill distance is the standard measure of point diversity in the quasi-Monte Carlo (Joy et al., 1996), experimental design (Pronzato & Müller, 2012), and meshfree approximation (Fasshauer & McCourt, 2015) communities. The convergence of the fill distance of ECI thus describes the limiting behavior of its sample diversity.
+
+Given a set of sample points  $\mathbf{X}$ , the fill distance is formally defined as the following:
+
+$$
+\operatorname {F I L L} (\mathbf {X}, \mathcal {S}) = \sup  _ {\mathbf {x} \in \mathcal {S}} \min  _ {\mathbf {x} _ {j} \in \mathbf {X}} d (\mathbf {x} _ {j}, \mathbf {x}). \tag {3}
+$$
+
+In Euclidean space,  $\mathrm{FILL}(\mathbf{X},\mathcal{S})$  is the radius of the largest empty ball one can fit in  $\mathcal{S}$ , and measures the spacing of  $\mathbf{X}$  in  $\mathcal{S}$ . The smaller a set's fill, the better distributed it is within  $\mathcal{S}$ . Minimal fill is defined in the following sense:
+
+$$
+\rho = \min  _ {\mathbf {X} \in \mathcal {X} ^ {n}} \operatorname {F I L L} (\mathbf {X}, \mathcal {S}).
+$$
+
+Low-discrepancy sequences and Latin hypercubes achieve low fill in simple domains. However, computing the minimal fill is generically NP-hard (Pronzato & Müller, 2012).
+
+Having defined fill distance, we can now frame convergence in terms of the limiting behavior of  $\mathrm{FILL}(\mathbf{X},\mathcal{S})$  as  $n$  approaches infinity. This is formalized as follows.
+
+Theorem 2. If  $S$  is dense, ECI produces a dense sequence of evaluation points in  $S$ . Equivalently, the limit as  $n$  approaches infinity of  $\mathrm{FILL}(\mathbf{X}, S)$  is zero.
+
+A more interesting remark is that we are able to bound the rate at which ECI fills the set  $S$ . We can guarantee that ECI produces a fill bounded above by  $4\rho$ , and that for a sufficiently large iteration budget, ECI produces a fill that is no more than  $2\rho$ .
+
+Theorem 3. For a fixed and known  $S$ , let  $\rho = \min_{\mathbf{X}} \mathrm{FILL}(\mathbf{X}, S)$ . For any  $n$ , ECI produces a fill bounded above by  $4\rho$ . Furthermore, there exists a finite  $\mathbf{n}^*$  for which ECI produces a 2-approximation ratio of  $\rho$  for all  $n \geq \mathbf{n}^*$ .
+
+# 4. Experiments
+
+In our experiments, we build an independent Gaussian process (GP) to model each objective (Rasmussen & Williams, 2005). In particular, for a GP model and a single objective,
+
+$$
+p (Z (\mathbf {x}) = 1 \mid \mathcal {D} _ {t}) = \int_ {\tau} ^ {\infty} \phi (y \mid \mu (\mathbf {x}), v (\mathbf {x})) d y, \tag {4}
+$$
+
+where  $\phi$  is the Normal PDF, and  $\mu (\mathbf{x})$  and  $v(\mathbf{x})$  are the GP's posterior mean and variance, respectively, evaluated at  $\mathbf{x}$ . For  $m$  objectives modeled by  $m$  independent GPs,  $p(Z(\mathbf{x}) = 1\mid \mathcal{D}_t)$  is the product of probabilities associated with each model. Our GPs use a zero mean function and a  $C^4$  Matérn covariance kernel with length scales fixed  $a$  priori (more information is provided in the supplement).
+
+# 4.1. Baseline Methodologies
+
+In addition to random search (RND), we compare our algorithm to several (modified) baseline algorithms aforementioned in Section 2:  $\varepsilon$ -constraint BO (Haghanifar et al., 2020), STRADDLE (Bryan et al., 2005), and one-step active search (Garnett et al., 2012). We extend STRADDLE to the multiobjective setting by alternating the objective functions at each iteration. For one-step active search, we maximize the point with highest probability  $p(Z(\mathbf{x}) = 1 \mid \mathcal{D}_t)$  at each iteration, which we call ONE-S. In addition, we develop two information theoretic baselines:
+
+Mutual information (EZ): We consider the mutual information between  $\mathbf{y}$  and  $Z$ . Assuming noiseless observations  $f_{i}(\mathbf{x}) = \mathbf{y}_{i}, \forall i$ , this simplifies to the entropy of  $Z$ :
+
+$$
+\operatorname {M I} (\mathbf {y}; Z) = H (\mathbf {y}) - \mathbb {E} _ {Z} [ H (\mathbf {y} \mid Z) ] = H (Z),
+$$
+
+Figure 3. Median metric values across 20 repetitions for the additive manufacturing application. We plot the fill distance, number of positive samples, hypervolume, and coverage recall respectively. Shaded region corresponds to values between lower and upper quartiles.  
+![](images/f967955cf37be0e91984a8a53146387945f4f4f3df0949479278a06341050642.jpg)  
+ECI BO EZ EISR STRADDLE ONE-S RND
+
+![](images/1ff814dd5a9f2a181289e7b1df3fcda57f38542c66cf2690200a5845c81918de.jpg)
+
+![](images/6227e21093d87455d869fce494bc357f0fb29a3436300886156b3b78c1085b4e.jpg)
+
+![](images/264ad642ee4621595643521bbba81baa4ad397f2c3246f2c67229e2dca46d61f.jpg)
+
+where  $H(\cdot)$  is the entropy of a random variable. This would be the one-step optimal strategy that recovers the most information about the boundaries of  $S$ .
+
+Entropy inside  $S$  (EISR): We consider the utility:
+
+$$
+u _ {\mathrm {E I S R}} (\mathbf {x}) = \left\{ \begin{array}{l l} 0 & Z = 0, \\ H (f \mid \mathbf {x}) & Z = 1. \end{array} \right.
+$$
+
+$u_{\mathrm{EISR}}$  rewards sampling configurations that are inside  $S$  and have high entropy. The corresponding acquisition function can be written as
+
+$$
+\alpha_ {\mathrm {E I S R}} = \mathbb {E} _ {Z} \left[ u _ {\mathrm {E I S R}} (\mathbf {x}) \right] = p (Z (\mathbf {x}) = 1) H (\mathbf {y}).
+$$
+
+# 4.2. Performance Criteria
+
+We emphasize that no single criterion can sufficiently convey the full strength of any methodology; this is especially true in the multiobjective setting, in which different performance criteria already exist to quantify different algorithmic goals. We hope to impart a nuanced comparison of ECI to existing baselines. To that end, we consider the following four criteria.
+
+Fill distance: We want an algorithm that can effectively explore the entire satisfactory region. This is quantified by fill distance, which is defined in Equation 3.
+
+Positive samples: The number of satisfactory sampled points is commonly used in active search literature to judge the quality of the algorithm. In our experiments, these are the observed samples inside of  $S$ .
+
+Hypervolume: We measure the hypervolume of region in metric space bounded by the Pareto frontier and the defined thresholds. In particular, we conjecture that algorithms which excel at maximizing the hypervolume may underperform on other criteria, and vice versa.
+
+Coverage recall: Finally, we want to combine the idea of maximizing positive samples and minimizing fill distance
+
+to measure sample diversity. We propose a recall metric that considers coverage. Given a known  $S$  and samples  $\mathbf{X}$ , we define the coverage recall as
+
+$$
+\text {C o v e r a g e} = | \mathrm {N} _ {r} (\mathbf {X}) \cap \mathcal {S} | / | \mathcal {S} |.
+$$
+
+Note that the numerator is exactly the utility function that we have proposed in Section 3.1. We will simply refer to this as recall in the following text.
+
+# 4.3. Multiobjective Design Problem Suite
+
+We adopt eight multiobjective engineering design problems presented in the REPROBLEM suite (Tanabe & Ishibuchi, 2020). These synthetic functions are specifically designed to mimic real-world applications and avoid having unrealistic properties of traditional benchmark functions. We follow the naming convention from the original text and detail the problems we selected in the supplement.
+
+# 4.4. Additive Manufacturing
+
+Haghanifar et al. (2020) studies a numerical simulation to help design an additive manufacturing strategy for minimizing the reflection of light at multiple angles of incidence. We conduct the same search for different nanostructures which balance a desire for minimizing normal reflection with reflection at an oblique angle.
+
+The simulation, conducted using the Numerical FDTD simulator (Lumerical Inc., 2020), has a level of inaccuracy associated with the numerical methods. In addition, there is a level of imprecision associated with the actual manufacturing process which means that the desired design parameters are only realized to limited precision during fabrication. Our goal for this reflection minimization problem is to find nanostructured glass that satisfying  $R_{\mathrm{normal}}(\mathbf{x}) \leq \tau_{\mathrm{normal}}$  and  $R_{\mathrm{oblique}}(\mathbf{x}) \leq \tau_{\mathrm{oblique}}$ . The design parameters are the dimension of the nanostructures (a cone): maximum cone width, bottom diameter, top diameter, and height of the structure.
+
+Table 1. Selected experimental results, consisting of the mean over 20 independent trials. Hypervolume has been rescaled for ease of viewing:  $\mathrm{RE33\times 10^{-5}}$ , Additive  $\times 10^{-2}$ , EEG  $\times 10^{2}$ , Plasma  $\times 10^{0}$ . We denote with bold text the best mean for each problem and each metric. Please refer to the supplementary material for the complete set of experimental results and information regarding the variation around the mean (standard error). For each problem, ECI has the best recall performance.  
+
+<table><tr><td>Function</td><td>m</td><td>d</td><td>r</td><td>Methods</td><td>Fill distance ↓</td><td># Positive ↑</td><td>Coverage Recall ↑</td><td>Hypervolume ↑</td></tr><tr><td rowspan="7">RE33</td><td rowspan="7">3</td><td rowspan="7">4</td><td rowspan="7">0.08</td><td>RND</td><td>0.34</td><td>23.80</td><td>0.14</td><td>5.53</td></tr><tr><td>ONE-S</td><td>0.60</td><td>101.55</td><td>0.18</td><td>7.03</td></tr><tr><td>EZ</td><td>0.26</td><td>34.05</td><td>0.25</td><td>2.77</td></tr><tr><td>STRADDLE</td><td>0.35</td><td>13.70</td><td>0.10</td><td>2.92</td></tr><tr><td>BO</td><td>0.42</td><td>36.55</td><td>0.11</td><td>8.65</td></tr><tr><td>EISR</td><td>0.24</td><td>100.05</td><td>0.23</td><td>7.46</td></tr><tr><td>ECI</td><td>0.27</td><td>100.95</td><td>0.73</td><td>7.91</td></tr><tr><td rowspan="7">Additive</td><td rowspan="7">2</td><td rowspan="7">4</td><td rowspan="7">35.0</td><td>RND</td><td>108.22</td><td>22.40</td><td>0.42</td><td>7.94</td></tr><tr><td>ONE-S</td><td>211.86</td><td>156.75</td><td>0.60</td><td>8.77</td></tr><tr><td>EZ</td><td>92.37</td><td>106.05</td><td>0.89</td><td>8.52</td></tr><tr><td>STRADDLE</td><td>79.30</td><td>21.10</td><td>0.43</td><td>7.78</td></tr><tr><td>BO</td><td>107.84</td><td>121.95</td><td>0.82</td><td>9.07</td></tr><tr><td>EISR</td><td>102.01</td><td>121.40</td><td>0.98</td><td>8.56</td></tr><tr><td>ECI</td><td>34.92</td><td>56.00</td><td>1.00</td><td>8.07</td></tr><tr><td rowspan="7">EEG</td><td rowspan="7">2</td><td rowspan="7">6</td><td rowspan="7">0.25</td><td>RND</td><td>0.38</td><td>4.40</td><td>0.58</td><td>3.17</td></tr><tr><td>ONE-S</td><td>0.48</td><td>31.20</td><td>0.50</td><td>3.45</td></tr><tr><td>EZ</td><td>0.44</td><td>17.95</td><td>0.58</td><td>3.45</td></tr><tr><td>STRADDLE</td><td>0.41</td><td>2.00</td><td>0.38</td><td>2.28</td></tr><tr><td>BO</td><td>0.52</td><td>4.30</td><td>0.35</td><td>2.40</td></tr><tr><td>EISR</td><td>0.45</td><td>22.85</td><td>0.52</td><td>3.55</td></tr><tr><td>ECI</td><td>0.38</td><td>8.30</td><td>0.79</td><td>3.65</td></tr><tr><td rowspan="7">Plasma³</td><td rowspan="7">1</td><td rowspan="7">9</td><td rowspan="7">0.50</td><td>RND</td><td>1.02</td><td>10.85</td><td>0.19</td><td>5.63</td></tr><tr><td>ONE-S</td><td>1.01</td><td>12.80</td><td>0.20</td><td>5.88</td></tr><tr><td>EZ</td><td>1.07</td><td>10.70</td><td>0.17</td><td>5.48</td></tr><tr><td>STRADDLE</td><td>1.07</td><td>10.70</td><td>0.17</td><td>5.48</td></tr><tr><td>BO</td><td>1.04</td><td>12.60</td><td>0.15</td><td>5.74</td></tr><tr><td>EISR</td><td>1.03</td><td>10.55</td><td>0.14</td><td>6.08</td></tr><tr><td>ECI</td><td>0.90</td><td>31.25</td><td>0.65</td><td>6.64</td></tr></table>
+
+# 4.5. EEG for Brain Activity Reconstruction
+
+Electroencephalography (EEG) is a non-invasive strategy for reconstructing cerebral electric currents from their resulting electrostatic potential measured on a human's scalp (Ala et al., 2017). This process consists of solving an inverse problem, requiring localization of the electric dipoles' locations and moments, subject to the associated model of dipole/potential interaction (Sorrentino & Piana, 2017). The limited spatial resolution of EEG is a well-known problem with its practical viability (Samuelsson et al., 2021), making constraint active search a potentially interesting tool for identifying regions of brain activity.
+
+We use a meshfree solver to generate a scalp potential from a proposed dipole location through a quasi-stationary approximation to Maxwell's equations (Ala et al., 2015). We then
+
+apply our strategy to identify the proposed dipole which most closely matches the true potential created by the true dipole, which is fixed at the start but hidden. We measure scalp potential at 28 locations and try to minimize the 25th and 75th percentiles of absolute deviation between the proposed and true potential; this has the effect of pushing both ends of the interquartile range as low as possible.
+
+# 4.6. Plasma Physics
+
+A stellarator is device that uses a set of magnetic coils to confine a plasma hot enough to sustain nuclear fusion (Spitzer Jr, 1958). Stellarator coils lack rotational symmetry,
+
+![](images/3eb3a37fbc55de1a77838e50137859bf891e628c5e1336439b2790c5ddeaf788.jpg)  
+Figure 4. Left two panels: difference from the expected design (in CAD) and the fabricated sample of a nanostructure. Right two panels: comparison of acceptable configurations in metric space and in the parameter space. Small variations in the Pareto optimal points can lead to significant discrepancies in the design space. CAS points provide more information about satisfactory design configuration.
+
+![](images/1658e3c9ed9082383c941e94e50b8b9874d0ac1ca17de7abc0427e50248de784.jpg)
+
+![](images/d892ebf90cd246d309275dc80922d76c52dd4e39cca005f59dd2b2d0113c3e19.jpg)
+
+![](images/827cf488d0af1df8098968472ea8203911f2a93f6dc8c3c251c039b1b3abb0d2.jpg)
+
+and possess notably warped shapes due to the complex, quasi-symmetric magnetic field they must produce.
+
+Recently, (Giuliani et al., 2020) developed a stellarator simulator, PyPlasmaOpt, which significantly streamlines the simulation procedure of previous packages such as StellOpt (Strickler et al., 2003). PyPlasmaOpt paramaterizes each coil as a curve in 3D Cartesian coordinates  $\Gamma (\theta) = (x(\theta),y(\theta),z(\theta))$ , where each coordinate admits the following Fourier expansion, e.g., in the case of  $x$ :
+
+$$
+x (\theta) = c _ {0} + \sum_ {k = 1} ^ {n _ {\text {o r d e r}}} s _ {k} \sin (k \theta) + c _ {k} \cos (k \theta).
+$$
+
+For each coil, the parameters  $c_k$  and  $s_k$  represent the search space that we must search over. We consider a relatively simple, order one coil which has nine parameters total.
+
+We compute the objective function to be minimized by simulating the stellarator using the given coil shapes (which solves a certain first order, nonlinear ordinary differential equation). The resulting summary information from the simulation is our objective  $f$  and contains three terms:
+
+$$
+f (\mathbf {x}) = R _ {\text {m a g n e t i c}} (\mathbf {x}) + R _ {\text {t r a n s f o r m}} (\mathbf {x}) + R _ {\text {s h a p e}} (\mathbf {x}).
+$$
+
+The first term quantifies the quasi-symmetry of the magnetic field—the smaller the first term, the more desirable the resulting field. The second term locks the solution into a target rotational transform. The third term penalizes overly complex coils too impractical to manufacture in real life.
+
+# 4.7. Discussion
+
+We exhaustively tested ECI and other baselines on eleven total problems, but due to limited space, we present a representative subset of our experimental results in this section. Our analysis holds true for the seven additional problems, whose results are in the supplement along with accompanying experimental setups for reproducibility purposes.
+
+Specifically, Table 1 shows the mean value over 20 trials for
+
+each performance metric at termination for the RE33 function from the design problem suite and all our real-world applications. These results demonstrate that no algorithm can be the best in all performance metrics. In fact, it highlights each method's strengths and weaknesses, confirming our thesis that evaluating the performance of multiobjective problems is itself multiobjective.
+
+As expected, BO typically outperforms all methods on the hypervolume criterion, which only judges the Pareto frontier. However, it is consistently worse than its peers when judged by diversity (fill distance) and coverage (recall). Our proposed method ECI, on the other hand, is effective at diverse sampling inside the satisfactory region  $S$ ; it is almost always the best performing method for coverage recall and fill distance. Unsurprisingly, ONE-S excels at accumulating satisfactory samples since it greedily maximizes the number of positive samples by design. These samples are highly clustered, as revealed by the high fill distance and low coverage recall values.
+
+Each policy's objective is evident when we consider the sample progression. Figure 3 shows the variation on performance as a function of the number of observed samples for the additive manufacturing problem. For instance, ECI maximizes the coverage recall and also minimizes the fill distance but stagnates at maximizing hypervolume and number of positives.
+
+Our investigation reveals that different multiobjective algorithms prioritize different performance criteria. Therefore, one can tailor a strategy (perhaps an ensemble of methods) according to each application's need. If metric optimization is of ultimate relevance, we recommend multiobjective Bayesian optimization. If the decision maker's goal is to identify rare elements regardless of diversity, then active search is the best choice. However, we argue that selecting a diverse set of satisfactory configurations is the most suited objective for design problems. We provide an example to support this claim in the following section.
+
+# 5. Constraint Active Search in Practice
+
+This research was motivated, in part, by a collaboration with materials scientists using additive manufacturing to fabricate nanostructured glass with multiple desirable optical and physical characteristics. Multiobjective problems are intrinsically subjective — if we could quantify the tradeoffs between each metric precisely a priori, we would solve a single-objective problem instead. Therefore, multiobjective methodologies provide the decision makers with a set of points at termination. Traditional multiobjective optimization offers a Pareto optimal set of points; our alternative gives a set of diverse points in parameter space that meet the user constraints. The decision maker, a domain expert, must then decide which one or more of these points to promote to physical manufacturing, often considering two key complications:
+
+- Computational and manufacturing imprecision prevent an exact physical realization of the results. See Figure 4 (left two panels) for an example of the CAD model and the associated fabricated component, which suffered from limited manufacturing precision.  
+- Some metrics are unavailable during optimization - maybe they can only be computed after manufacturing (such as anti-fogging or durability) or require costly expert analysis (such as the supervisor stating expert opinions on a manufacturing scale).
+
+Both of these complications are better addressed through sample diversity in parameter space (not in metric space) — the primary motivation of CAS.
+
+Inaccuracies arising from the transfer of computational results to manufactured outcomes are best dealt with by choosing points most likely to achieve acceptable performance after fabrication. Because CAS explores the parameter space, it has a greater ability to select results far (e.g., in Euclidean sense) from unacceptable outcomes than multiobjective optimization. When only studying the Pareto frontier, no energy is spent exploring the safety of those points from undesirable results. Figure 4 (right) shows the numerical simulation results for the additive manufacturing example; interior points in parameter space are more likely to be selected for physical manufacturing.
+
+For metrics only computable after fabrication, users benefit from a diverse set of designs in parameter space so they may manufacture a select few and still hope to achieve good results on these previously unavailable metrics.
+
+# 6. Conclusion and Future Work
+
+We introduce constraint active search as a novel methodology to explore multiobjective experimental design options while satisfying minimum performance criteria. One key
+
+differentiation between constraint active search and existing multiobjective optimization literature is the emphasis on the design parameters: constraint active search values sample diversity in the satisfactory region.
+
+We propose a new search policy, expected coverage improvement, designed to efficiently improve the coverage of sampled designs over the satisfactory region; theoretical properties of this strategy include a constant approximation ratio to the optimal sample diversity. We define and study several performance criteria when judging the performance of proposed algorithms in multiobjective settings, and we explain the implications of prioritizing each. Using these criteria, we compare our new policy against existing strategies such as Bayesian optimization, level set estimation, and active search on synthetic and real world problems.
+
+There are many open problems in the topic of constraint active search for design and simulation; here we prioritize a few topics for future investigation.
+
+Distinctness/Precision: A key element of ECI (as well as the active search baseline) is the choice of the resolution  $r$ , through which the design precision is defined. This is acceptable, but limited: categorical parameters may exist; different parts of the domain may have different manufacturing precision; some parameters may have different levels of precision than others. Eventually, it would be beneficial to allow for a more abstract definition of design distinctness to support more complicated design circumstances.
+
+Unknown precision: In most design/manufacturing settings, the resolution  $r$  is known a priori. It is possible, however, that this will need to be discovered over the course of the search process (or, maybe, after the search process). For this situation, our proposed policy would benefit from an adaptive  $r$  definition, which might start at a very large value e.g.,  $10\%$  of the maximum distance between points in the domain, and then scale down as the search progresses.
+
+Parallelism: ECI is limited to the sequential setting; extending the search policy to the batch setting and finding a computationally tractable policy is non-trivial.
+
+Judging performance: Some of our proposed performance criteria are only computable for benchmark problems where the true satisfactory region can be estimated. In practice, judging the results of a search algorithm is less straightforward. We believe creating more practical performance criteria can accelerate our understanding of this problem.
+
+# Acknowledgements
+
+We want to thank Sajad Haghanifar and Paul Leu for providing the scanning electron microscope images and motivating the research presented in this paper.
+
+# References
+
+Ala, G., Fasshauer, G., Francomano, E., Ganci, S., and McCourt, M. J. The method of fundamental solutions in solving coupled boundary value problems for M/EEG. SIAM Journal on Scientific Computing, 37(4): B570-B590, 2015.  
+Ala, G., Fasshauer, G. E., Francomano, E., Ganci, S., and McCourt, M. J. An augmented MFS approach for brain activity reconstruction. Mathematics and Computers in Simulation, 141:3-15, 2017.  
+Ariafar, S., Coll-Font, J., Brooks, D., and Dy, J. ADMMBO: Bayesian optimization with unknown constraints using ADMM. Journal of Machine Learning Research, 20(123): 1-26, 2019.  
+Attia, P. M., Grover, A., Jin, N., Severson, K. A., Markov, T. M., Liao, Y.-H., Chen, M. H., Cheong, B., Perkins, N., Yang, Z., Herring, P. K., Aykol, M., Harris, S. J., Braatz, R. D., Ermon, S., and Chueh, W. C. Closed-loop optimization of fast-charging protocols for batteries with machine learning. Nature, 578(7795):397-402, 2020. ISSN 1476-4687. doi: 10.1038/s41586-020-1994-5.  
+Belakaria, S., Deshwal, A., and Doppa, J. R. Max-value entropy search for multi-objective Bayesian optimization. In Wallach, H. M., Larochelle, H., Beygelzimer, A., d'Alché-Buc, F., Fox, E. B., and Garnett, R. (eds.), Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pp. 7823-7833, 2019.  
+Belakaria, S., Deshwal, A., and Doppa, J. R. Max-value entropy search for multi-objective Bayesian optimization with constraints. Third Workshop on Machine Learning and the Physical Sciences (NeurIPS 2020), 2020.  
+Bryan, B. and Schneider, J. G. Actively learning level-sets of composite functions. In Cohen, W. W., McCallum, A., and Roweis, S. T. (eds.), Machine Learning, Proceedings of the Twenty-Fifth International Conference (ICML 2008), Helsinki, Finland, June 5-9, 2008, volume 307 of ACM International Conference Proceeding Series, pp. 80-87. ACM, 2008. doi: 10.1145/1390156.1390167.  
+Bryan, B., Schneider, J. G., Nichol, R., Miller, C. J., Genovese, C. R., and Wasserman, L. A. Active learning for identifying function threshold boundaries. In Advances in Neural Information Processing Systems 18 [Neural Information Processing Systems, NIPS 2005, December 5-8, 2005, Vancouver, British Columbia, Canada], pp. 163-170, 2005.
+
+Daulton, S., Balandat, M., and Bakshy, E. Differentiable expected hypervolume improvement for parallel multi-objective Bayesian optimization. In Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., and Lin, H. (eds.), Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.  
+del Rosario, Z., Rupp, M., Kim, Y., Antonio, E., and Ling, J. Assessing the frontier: Active learning, model accuracy, and multi-objective candidate discovery and optimization. The Journal of Chemical Physics, 153(2):024112, 2020. doi: 10.1063/5.0006124.  
+Duris, J., Kennedy, D., Hanuka, A., Shtalenkova, J., Edelen, A., Baxevanis, P., Egger, A., Cope, T., McIntire, M., Ermon, S., and Ratner, D. Bayesian optimization of a free-electron laser. Phys. Rev. Lett., 124:124801, 2020. doi: 10.1103/PhysRevLett.124.124801.  
+Emmerich, M. T. M., Deutz, A. H., and Klinkenberg, J. W. Hypervolume-based expected improvement: Monotonicity properties and exact computation. In 2011 IEEE Congress of Evolutionary Computation (CEC), pp. 2147-2154, 2011. doi: 10.1109/CEC.2011.5949880.  
+Eriksson, D. and Poloczek, M. Scalable constrained Bayesian optimization. In Banerjee, A. and Fukumizu, K. (eds.), The 24th International Conference on Artificial Intelligence and Statistics, AISTATS 2021, April 13-15, 2021, Virtual Event, volume 130 of Proceedings of Machine Learning Research, pp. 730-738. PMLR, 2021.  
+Fasshauer, G. E. and McCourt, M. J. Kernel-based Approximation Methods Using Matlab. World Scientific, 2015. doi: 10.1142/9335.  
+Fernández-Sánchez, D., Garrido-Merchán, E. C., and Hernández-Lobato, D. Improved max-value entropy search for multi-objective Bayesian optimization with constraints, 2021. URL https://arxiv.org/abs/2011.01150.  
+Forrester, A., Sobester, A., and Keane, A. Engineering design via surrogate modelling: a practical guide. John Wiley & Sons, 2008.  
+Frazier, P. I. Bayesian optimization. In Gel, E. and Ntaimo, L. (eds.), *Recent Advances in Optimization and Modeling of Contemporary Problems*, pp. 255–278. INFORMS, 2018.  
+Gardner, J. R., Kusner, M. J., Xu, Z. E., Weinberger, K. Q., and Cunningham, J. P. Bayesian optimization with inequality constraints. In Proceedings of the 31th International Conference on Machine Learning, ICML 2014,
+
+Beijing, China, 21-26 June 2014, volume 32 of JMLR Workshop and Conference Proceedings, pp. 937-945. JMLR.org, 2014.  
+Garnett, R., Krishnamurthy, Y., Xiong, X., Schneider, J., and Mann, R. Bayesian optimal active search and surveying. In Proceedings of the 29th International Coference on International Conference on Machine Learning, ICML'12, pp. 843-850, Madison, WI, USA, 2012. Omnipress. ISBN 9781450312851.  
+Gelbart, M. A. Constrained Bayesian optimization and applications. PhD thesis, Harvard, 2015.  
+Giuliani, A., Wechsung, F., Cerfon, A., Stadler, G., and Landreman, M. Single-stage gradient-based stellarator coil design: Optimization for near-axis quasi-symmetry, 2020. URL https://arxiv.org/abs/2010.02033.  
+Gotovos, A., Casati, N., Hitz, G., and Krause, A. Active learning for level set estimation. In Rossi, F. (ed.), *IJCAI* 2013, Proceedings of the 23rd International Joint Conference on Artificial Intelligence, Beijing, China, August 3-9, 2013, pp. 1344-1350. *IJCAI/AAAI*, 2013.  
+Haghanifar, S., McCourt, M., Cheng, B., Wuenschell, J., Ohodnicki, P., and Leu, P. W. Discovering high-performance broadband and broad angle antireflection surfaces by machine learning. Optica, 7(7):784-789, 2020. doi: 10.1364/OPTICA.387938.  
+Iwazaki, S., Inatsu, Y., and Takeuchi, I. Bayesian experimental design for finding reliable level set under input uncertainty. IEEE Access, 8:203982-203993, 2020. doi: 10.1109/ACCESS.2020.3036863.  
+Jiang, S., Malkomes, G., Converse, G., Shofner, A., Moseley, B., and Garnett, R. Efficient nonmyopic active search. In Precup, D. and Teh, Y. W. (eds.), Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017, volume 70 of Proceedings of Machine Learning Research, pp. 1714-1723. PMLR, 2017.  
+Jiang, S., Malkomes, G., Abbott, M., Moseley, B., and Garnett, R. Efficient nonmyopic batch active search. In Bengio, S., Wallach, H. M., Larochelle, H., Grauman, K., Cesa-Bianchi, N., and Garnett, R. (eds.), Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, pp. 1107-1117, 2018.  
+Joy, C., Boyle, P. P., and Tan, K. S. Quasi-Monte Carlo methods in numerical finance. Management Science, 42 (6):926-938, 1996.
+
+Knowles, J. ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Transactions on Evolutionary Computation, 10(1):50-66, 2006. doi: 10.1109/TEVC.2005.851274.  
+Konakovic-Lukovic, M., Tian, Y., and Matusik, W. Diversity-guided multi-objective Bayesian optimization with batch evaluations. In Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., and Lin, H. (eds.), Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.  
+Koziel, S., Bekasiewicz, A., Couckuyt, I., and Dhaene, T. Efficient multi-objective simulation-driven antenna design using co-Kriging. IEEE Transactions on Antennas and Propagation, 62(11):5900-5905, 2014.  
+Letham, B., Karrer, B., Ottoni, G., and Bakshy, E. Constrained Bayesian optimization with noisy experiments. Bayesian Anal., 14(2):495-519, 2019. doi: 10.1214/18-BA1110.  
+Lumerical Inc. Lumerical device multiphysics simulation suite, FDTD solver, 2020. URL https://www.lumerical.com/products/.  
+Molesky, S., Lin, Z., Piggott, A. Y., Jin, W., Vuckovic, J., and Rodriguez, A. W. Inverse design in nanophotonics. Nature Photonics, 12(11):659-670, 2018.  
+Negoescu, D. M., Frazier, P. I., and Powell, W. B. The knowledge-gradient algorithm for sequencing experiments in drug discovery. INFORMS Journal on Computing, 23(3):346-363, 2011.  
+Paria, B., Kandasamy, K., and Poczos, B. A flexible framework for multi-objective Bayesian optimization using random scalarizations. In Adams, R. P. and Gogate, V. (eds.), Proceedings of The 35th Uncertainty in Artificial Intelligence Conference, volume 115 of Proceedings of Machine Learning Research, pp. 766-776, Tel Aviv, Israel, 2020. PMLR.  
+Pronzato, L. and Müller, W. G. Design of computer experiments: space filling and beyond. Statistics and Computing, 22(3):681-701, 2012.  
+Rasmussen, C. E. and Williams, C. K. I. Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning). The MIT Press, 2005.  
+Samuelsson, J. G., Peled, N., Mamashli, F., Ahveninen, J., and Hämäläinen, M. S. Spatial fidelity of MEG/EEG source estimates: A general evaluation approach. NeuroImage, 224:117430, 2021.
+
+Singh, P., Couckuyt, I., Elsayed, K., Deschrijver, D., and Dhaene, T. Shape optimization of a cyclone separator using multi-objective surrogate-based optimization. Applied Mathematical Modelling, 40(5-6):4248-4259, 2016.  
+Song, J., Tokpanov, Y. S., Chen, Y., Fleischman, D., Fountaine, K. T., Atwater, H. A., and Yue, Y. Optimizing photonic nanostructures via multi-fidelity Gaussian processes. NeurIPS Workshop on Machine Learning for Molecules and Materials, 2018.  
+Sorrentino, A. and Piana, M. Inverse modeling for MEG/EEG data. In Mathematical and Theoretical Neuroscience, pp. 239-253. Springer, 2017.  
+Spitzer Jr, L. The stellarator concept. The Physics of Fluids, 1(4):253-264, 1958.  
+Strickler, D., Berry, L., and Hirshman, S. Integrated plasma and coil optimization for compact stellarators. Technical report, International Atomic Energy Agency (IAEA), 2003.  
+Sui, Y., Gotovos, A., Burdick, J. W., and Krause, A. Safe exploration for optimization with Gaussian processes. In Bach, F. R. and Blei, D. M. (eds.), Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, Lille, France, 6-11 July 2015, volume 37 of JMLR Workshop and Conference Proceedings, pp. 997-1005. JMLR.org, 2015.  
+Sui, Y., Zhuang, V., Burdick, J. W., and Yue, Y. Stagewise safe bayesian optimization with Gaussian processes. In Dy, J. G. and Krause, A. (eds.), Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholm, Sweden, July 10-15, 2018, volume 80 of Proceedings of Machine Learning Research, pp. 4788-4796. PMLR, 2018.  
+Tanabe, R. and Ishibuchi, H. An easy-to-use real-world multi-objective optimization problem suite. Applied Soft Computing, 89:106078, 2020. ISSN 1568-4946. doi: https://doi.org/10.1016/j.asoc.2020.106078.  
+Wang, Z., Garrett, C. R., Kaelbling, L. P., and Lozano-Perez, T. Active model learning and diverse action sampling for task and motion planning. In International Conference on Intelligent Robots and Systems (IROS), 2018.  
+Zanette, A., Zhang, J., and Kochenderfer, M. J. Robust super-level set estimation using Gaussian processes. In Berlingerio, M., Bonchi, Francesco and Gartner, T., Hurley, N., and Ifrim, G. (eds.), Machine Learning and Knowledge Discovery in Databases, pp. 276-291, Cham, 2019. Springer International Publishing. ISBN 978-3-030-10928-8.

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+# Beyond Variance Reduction: Understanding the True Impact of Baselines on Policy Optimization
+
+Wesley Chung  $^{*1}$  Valentin Thomas  $^{*2}$  Marlos C. Machado  $^{3,4,5}$  Nicolas Le Roux  $^{6,12}$
+
+# Abstract
+
+Bandit and reinforcement learning (RL) problems can often be framed as optimization problems where the goal is to maximize average performance while having access only to stochastic estimates of the true gradient. Traditionally, stochastic optimization theory predicts that learning dynamics are governed by the curvature of the loss function and the noise of the gradient estimates. In this paper we demonstrate that the standard view is too limited for bandit and RL problems. To allow our analysis to be interpreted in light of multi-step MDPs, we focus on techniques derived from stochastic optimization principles (e.g., natural policy gradient and EXP3) and we show that some standard assumptions from optimization theory are violated in these problems. We present theoretical results showing that, at least for bandit problems, curvature and noise are not sufficient to explain the learning dynamics and that seemingly innocuous choices like the baseline can determine whether an algorithm converges. These theoretical findings match our empirical evaluation, which we extend to multi-state MDPs.
+
+# 1. Introduction
+
+In the standard multi-arm bandit setting (Robbins, 1952), an agent needs to choose, at each timestep  $t$ , an arm  $a_{t} \in \{1,\dots,n\}$  to play, receiving a potentially stochastic reward  $r_t$  with mean  $\mu_{a_t}$ . The goal of the agent is usually to maximize the total sum of rewards,  $\sum_{i=1}^{T} r_i$ , or to maximize the average performance at time  $T$ ,  $\mathbb{E}_{i \sim \pi} \mu_i$  with  $\pi$  being the probability of the agent of drawing each arm (Bubeck
+
+*Equal contribution ¹Mila, McGill University ²Mila, University of Montreal ³DeepMind ⁴Amii, University of Alberta ⁵Work partially done at Google Research ⁶Google Research, Brain Team. Correspondence to: <wesley.chung2@gmail.com, vltn.thomas@gmail.com, marlosm@deepmind.com, nicolas@leroux.name>.
+
+Proceedings of the  $38^{th}$  International Conference on Machine Learning, PMLR 139, 2021. Copyright 2021 by the author(s).
+
+& Cesa-Bianchi, 2012). While the former measure is often used in the context of bandits, $^{1}$ $\mathbb{E}_{i\sim \pi}\mu_{i}$  is more common in the context of Markov Decision Processes (MDPs), which have multi-arm bandits as a special case.
+
+In this paper we focus on techniques derived from stochastic optimization principles, such as EXP3 (Auer et al., 2002; Seldin et al., 2013). In particular, we study policy gradient methods, a family of algorithms useful in the more general MDP setting which have seen empirical success in recent times (Schulman et al., 2017).
+
+We analyze the problem of learning to maximize the average reward,  $J$ , by gradient ascent:
+
+$$
+\theta^ {*} = \arg \max  _ {\theta} J (\theta) = \arg \max  _ {\theta} \sum_ {a} \pi_ {\theta} (a) \mu_ {a}, \tag {1}
+$$
+
+with  $\mu_{a}$  being the average reward of arm  $a$ . In this case, we are mainly interested in outputting an effective policy at the end of the optimization process, without explicitly considering the performance of intermediary policies.
+
+Optimization theory predicts that the convergence speed of stochastic gradient methods will be affected by the variance of the gradient estimates and by the geometry of the function  $J$ , represented by its curvature. Roughly speaking, the geometry dictates how effective true gradient ascent is at optimizing  $J(\theta)$  while the variance can be viewed as a penalty, capturing how much slower the optimization process is by using noisy versions of this true gradient. More concretely, doing one gradient step with stepsize  $\alpha$ , using a stochastic estimate  $g_{t}$  of the gradient, leads to (Bottou et al., 2018):
+
+$$
+\mathbb {E} \left[ J \left(\theta_ {t + 1}\right) \right] - J \left(\theta_ {t}\right) \geq \left(\alpha - \frac {L \alpha^ {2}}{2}\right) \| \mathbb {E} [ g _ {t} ] \| _ {2} ^ {2} - \frac {L \alpha^ {2}}{2} \operatorname {V a r} [ g _ {t} ],
+$$
+
+when  $J$  is  $L$ -smooth, i.e. its gradients are  $L$ -Lipschitz.
+
+As large variance has been identified as an issue for policy gradient (PG) methods, many works have focused on reducing the noise of the updates. One common technique is the use of control variates (Greensmith et al., 2004; Hofmann et al., 2015), referred to as baselines in the context of RL. These baselines  $b$  are subtracted from the observed returns to obtain shifted returns,  $r(a_{i}) - b$ , and do not change the
+
+expectation of the gradient. In MDPs, they are typically state-dependent. While the value function is a common choice, previous work showed that the minimum-variance baseline for the REINFORCE (Williams, 1992) estimator is different and involves the norm of the gradient (Peters & Schaal, 2008). Reducing variance has been the main motivation for many previous works on baselines (e.g., Gu et al., 2016; Liu et al., 2017; Grathwohl et al., 2017; Wu et al., 2018; Cheng et al., 2020), but the influence of baselines on other aspects of the optimization process has hardly been studied. We take a deeper look at baselines and their effects on optimization.
+
+# CONTRIBUTIONS
+
+We show that baselines can impact the optimization process beyond variance reduction and lead to qualitatively different learning curves, even when the variance of the gradients is the same. For instance, given two baselines with the same variance, the more negative baseline promotes committal behaviour where a policy quickly tends towards a deterministic one, while the more positive baseline leads to non-committal behaviour, where the policy retains higher entropy for a longer period.
+
+Furthermore, we show that the choice of baseline can even impact the convergence of natural policy gradient (NPG), something variance cannot explain. In particular, we construct a three-armed bandit where using the baseline minimizing the variance can lead to convergence to a deterministic, sub-optimal policy for any positive stepsize, while another baseline, with larger variance, guarantees convergence to the optimal policy. As such a behaviour is impossible under the standard assumptions in optimization, this result shows how these assumptions may be violated in practice. It also provides a counterexample to the convergence of NPG algorithms in general, a popular variant with much faster convergence rates than vanilla PG when using the true gradient in tabular MDPs (Agarwal et al., 2019).
+
+Further, we identify on-policy sampling as a key factor to these convergence issues as it induces a vicious cycle where making bad updates can lead to worse policies, in turn leading to worse updates. A natural solution is to break the dependency between the sampling distribution and the updates through off-policy sampling. We show that ensuring all actions are sampled with sufficiently large probability at each step is enough to guarantee convergence in probability. Note that this form of convergence is stronger than convergence of the expected iterates, a more common type of result (e.g., Mei et al., 2020b; Agarwal et al., 2019).
+
+We also perform an empirical evaluation on multi-step MDPs, showing that baselines have a similar impact in that setting. We observe a significant impact on the empirical performance of agents when using two different sets of
+
+basielines yielding the same variance, once again suggesting that learning dynamics in MDPs are governed by more than the curvature of the loss and the variance of the gradients.
+
+# 2. Baselines, learning dynamics & exploration
+
+The problem defined in Eq. 1 can be solved by gradient ascent. Given access only to samples, the true gradient cannot generally be computed and the true update is replaced with a stochastic one, resulting in the following update:
+
+$$
+\theta_ {t + 1} = \theta_ {t} + \frac {\alpha}{N} \sum_ {i} r \left(a _ {i}\right) \nabla_ {\theta} \log \pi_ {\theta} \left(a _ {i}\right), \tag {2}
+$$
+
+where  $a_i$  are actions drawn according to the agent's current policy  $\pi_{\theta}$ ,  $\alpha$  is the stepsize, and  $N$ , which can be 1, is the number of samples used to compute the update. To reduce the variance of this estimate without introducing bias, we can introduce a baseline  $b$ , resulting in the gradient estimate  $(r(a_i) - b)\nabla_{\theta}\log \pi_{\theta}(a_i)$ .
+
+While the choice of baseline is known to affect the variance, we show that baselines can also lead to qualitatively different behaviour of the optimization process, even when the variance is the same. This difference cannot be explained by the expectation or variance, quantities which govern the usual bounds for convergence rates (Bottou et al., 2018).
+
+# 2.1. Committal and non-committal behaviours
+
+To provide a complete picture of the optimization process, we analyze the evolution of the policy during optimization. We start in a simple setting, a deterministic three-armed bandit, where it is easier to produce informative visualizations.
+
+To eliminate variance as a potential confounding factor, we consider different baselines with the same variance. We start by computing the baseline leading to the minimum-variance of the gradients for the algorithm we use. For vanilla policy gradient, we have  $b_{\theta}^{*} = \frac{\mathbb{E}[r(a_{i})\|\nabla\log\pi_{\theta}(a_{i})\|_{2}^{2}]}{\mathbb{E}[\|\nabla\log\pi_{\theta}(a_{i})\|_{2}^{2}]}$  (Peters & Schaal, 2008; Greensmith et al., 2004) (see Appendix D.1 for details and the NPG version). Note that this baseline depends on the current policy and changes throughout the optimization. As the variance is a quadratic function of the baseline, the two baselines  $b_{\theta}^{+} = b_{\theta}^{*} + \epsilon$  and  $b_{\theta}^{-} = b_{\theta}^{*} - \epsilon$  result in gradients with the same variance (see Appendix D.4 for details). Thus, we use these two perturbed baselines to demonstrate that there are phenomena in the optimization process that variance cannot explain.
+
+Fig. 1 presents fifteen learning curves on the probability simplex representing the space of possible policies for the three-arm bandit, when using NPG and a softmax parameterization. We choose  $\epsilon = 1/2$  to obtain two baselines with the same variance:  $b_{\theta}^{+} = b_{\theta}^{*} + 1/2$  and  $b_{\theta}^{-} = b_{\theta}^{*} - 1/2$ .
+
+Inspecting the plots, the learning curves for  $\epsilon = -1 / 2$  and
+
+![](images/79c8424993c8cd07acbcb486e6803fec87083030aefa543c5c90060d164797f3.jpg)  
+(a)  $b_{\theta}^{-} = b_{\theta}^{*} - 1 / 2$
+
+![](images/ebaa7ba3b2dfff14a3f86fc2923daf4e13cfd0d13f58b03b98b51cada8f2215f.jpg)  
+(b)  $b_{\theta} = b_{\theta}^{*}$
+
+![](images/6eb2d179cd5fd058f560c77a32e8ff64374cae9fc2ae6989de1a4e013f5266cb.jpg)  
+(c)  $b_{\theta}^{+} = b_{\theta}^{*} + 1 / 2$  
+Figure 1: We plot 15 different trajectories of natural policy gradient with softmax parameterization, when using various baselines, on a 3-arm bandit problem with rewards  $(1,0.7,0)$  and stepsize  $\alpha = 0.025$  and  $\theta_0 = (0,3,5)$ . The black dot is the initial policy and colors represent time, from purple to yellow. The dashed black line is the trajectory when following the true gradient (which is unaffected by the baseline). Different values of  $\epsilon$  denote different perturbations to the minimum-variance baseline. We see some cases of convergence to a suboptimal policy for both  $\epsilon = -1/2$  and  $\epsilon = 0$ . This does not happen for the larger baseline  $\epsilon = 1/2$  or the value function as baseline. Figure made with Ternary (Harper & Weinstein, 2015).
+
+![](images/a394ffa671eb381a59c10e5979fc515ca84f0c11103ee46419c5130326b8db2d.jpg)  
+(d)  $b_{\theta} = V^{\pi_{\theta}}$
+
+$\epsilon = 1/2$  are qualitatively different, even though the gradient estimates have the same variance. For  $\epsilon = -1/2$ , the policies quickly reach a deterministic policy (i.e., a neighborhood of a corner of the probability simplex), which can be suboptimal, as indicated by the curves ending up at the policy choosing action 2. On the other hand, for  $\epsilon = 1/2$ , every learning curve ends up at the optimal policy, although the convergence might be slower. The learning curves also do not deviate much from the curve for the true gradient. Again, these differences cannot be explained by the variance since the baselines result in identical variances.
+
+Additionally, for  $b_{\theta} = b_{\theta}^{*}$ , the learning curves spread out further. Compared to  $\epsilon = 1/2$ , some get closer to the top corner of the simplex, leading to convergence to a suboptimal solution, suggesting that the minimum-variance baseline may be worse than other, larger baselines. In the next section, we theoretically substantiate this and show that, for NPG, it is possible to converge to a suboptimal policy with the minimum-variance baseline; but there are larger baselines that guarantee convergence to an optimal policy.
+
+We look at the update rules to explain these different behaviours. When using a baseline  $b$  with NPG, sampling  $a_{i}$  results in the update
+
+$$
+\begin{array}{l} \theta_ {t + 1} = \theta_ {t} + \alpha \left[ r \left(a _ {i}\right) - b \right] F _ {\theta} ^ {- 1} \nabla_ {\theta} \log \pi_ {\theta} \left(a _ {i}\right) \\ = \theta_ {t} + \alpha \frac {r (a _ {i}) - b}{\pi_ {\theta} (a _ {i})} \mathbf {1} _ {a _ {i}} + \alpha \lambda e \\ \end{array}
+$$
+
+where  $F_{\theta}^{-1} = \mathbb{E}_{a\sim \pi}[\nabla \log \pi_{\theta}(a)\nabla \log \pi_{\theta}(a)^{\top}],\mathbf{1}_{a_i}$  is a one-hot vector with 1 at index  $i$  , and  $\lambda e$  is a vector containing  $\lambda$  in each entry. The second line follows for the softmax policy (see Appendix D.2) and  $\lambda$  is arbitrary since shifting  $\theta$  by a constant does not change the policy.
+
+Thus, supposing we sample action  $a_{i}$ , if  $r(a_{i}) - b$  is positive,
+
+which happens more often when the baseline  $b$  is small (more negative), the update rule will increase the probability  $\pi_{\theta}(a_i)$ . This leads to an increase in the probability of taking the actions the agent took before, regardless of their quality (see Fig.1a for  $\epsilon = -1/2$ ). Because the agent is likely to choose the same actions again, we call this committal behaviour.
+
+While a smaller baseline leads to committal behaviour, a larger (more positive) baseline makes the agent second-guess itself. If  $r(a_i) - b$  is negative, which happens more often when  $b$  is large, the parameter update decreases the probability  $\pi_{\theta}(a_i)$  of the sampled action  $a_i$ , reducing the probability the agent will re-take the actions it just took, while increasing the probability of other actions. This might slow down convergence but it also makes it harder for the agent to get stuck. This is reflected in the  $\epsilon = 1/2$  case (Fig.1c), as all the learning curves end up at the optimal policy. We call this non-committal behaviour.
+
+While the previous experiments used perturbed variants of the minimum-variance baseline to control for the variance, this baseline would usually be infeasible to compute in more complex MDPs. Instead, a more typical choice of baseline would be the value function (Sutton & Barto, 2018, Ch. 13), which we evaluate in Fig. 1d. Choosing the value function as a baseline generated trajectories converging to the optimal policy, even though their convergence may be slow, despite it not being the minimum variance baseline. The reason becomes clearer when we write the value function as  $V^{\pi} = b_{\theta}^{*} - \frac{\mathrm{Cov}(r,\|\nabla\log\pi\|^{2})}{\mathbb{E}[\|\nabla\log\pi\|^{2}]}$  (see Appendix D.3). The term  $\mathrm{Cov}(r,\|\nabla\log\pi\|^{2})$  typically becomes negative as the gradient becomes smaller on actions with high rewards during the optimization process, leading to the value function being a noncommittal baseline, justifying a choice often made by practitioners.
+
+Additional empirical results can be found in Appendix A.1 for natural policy gradient and vanilla policy gradient for the softmax parameterization. Furthermore, we explore the use of different parameterizations: First, we testprojected stochastic gradient ascent and directly optimizing the policy probabilities  $\pi_{\theta}(a)$ . Next, we try the escort transform (Mei et al., 2020a), which was designed to improve the curvature of the objective. We find qualitatively similar results in all cases; baselines can induce committal and non-committal behaviour.
+
+# 3. Convergence to suboptimal policies with natural policy gradient (NPG)
+
+We empirically showed that PG algorithms can reach suboptimal policies and that the choice of baseline can affect the likelihood of this occurring. In this section, we provide theoretical results proving that it is indeed possible to converge to a suboptimal policy when using NPG. We discuss how this finding fits with existing convergence results and why standard assumptions are not satisfied in this setting.
+
+# 3.1. A simple example
+
+Standard convergence results assume access to the true gradient (e.g., Agarwal et al., 2019) or, in the stochastic case, assume that the variance of the updates is uniformly bounded for all parameter values (e.g., Bottou et al., 2018). These assumptions are in fact quite strong and are violated in a simple two-arm bandit problem with fixed rewards. Pulling the optimal arm gives a reward of  $r_1 = +1$ , while pulling the suboptimal arm leads to a reward of  $r_0 = 0$ . We use the sigmoid parameterization and call  $p_t = \sigma(\theta_t)$  the probability of sampling the optimal arm at time  $t$ .
+
+Our stochastic estimator of the natural gradient is
+
+$$
+g _ {t} = \left\{ \begin{array}{l} \frac {1 - b}{p _ {t}}, \text {w i t h p r o b a b i l i t y} p _ {t} \\ \frac {b}{1 - p _ {t}}, \text {w i t h p r o b a b i l i t y} 1 - p _ {t}, \end{array} \right.
+$$
+
+where  $b$  is a baseline that does not depend on the action sampled at time  $t$  but may depend on  $\theta_t$ . By computing the variance of the updates,  $\mathrm{Var}[g_t] = \frac{(1 - p_t - b)^2}{p_t(1 - p_t)}$ , we notice it is unbounded when the policy becomes deterministic, i.e.  $p_t \to 0$  or  $p_t \to 1$ , violating the assumption of uniformly bounded variance, unless  $b = 1 - p_t$ , which is the optimal baseline. Note that using vanilla (non-natural) PG would, on the contrary, yield a bounded variance. In fact, we prove a convergence result in its favour in Appendix B (Prop. 4).
+
+For NPG, the proposition below establishes potential convergence to a suboptimal arm and we demonstrate this empirically in Fig. 2.
+
+Proposition 1. Consider a two-arm bandit with rewards 1 and 0 for the optimal and suboptimal arms, respectively.
+
+![](images/7f5b62f0ece6d902927421c46529c15ff5ed7de8205fe7fe7f1145b25de21d48.jpg)  
+(a)  $\alpha = 0.05$
+
+![](images/9ec80a6256b463db054fbcd803cec59688b45b135e227c001d463565d689f8f0.jpg)  
+(b)  $\alpha = 0.1$  
+Figure 2: Learning curves for 100 runs of 200 steps, on the two-arm bandit, with baseline  $b = -1$  for three different step sizes  $\alpha$ . Blue: Curves converging to the optimal policy. Red: Curves converging to a suboptimal policy. Black: Avg. performance. The number of runs that converged to the suboptimal solution are  $5\%$ ,  $14\%$  and  $22\%$  for the three  $\alpha$ 's. Larger  $\alpha$ 's are more prone to getting stuck at a suboptimal solution but settle on a deterministic policy more quickly.
+
+![](images/6b5f401dfe04c620652548f9f9265e38720fad6adb10f5a578317b00fe447e15.jpg)  
+(c)  $\alpha = 0.15$
+
+Suppose we use natural policy gradient starting from  $\theta_0$ , with a fixed baseline  $b < 0$ , and fixed stepsize  $\alpha > 0$ . If the policy samples the optimal action with probability  $\sigma(\theta)$ , then the probability of picking the suboptimal action forever and having  $\theta_t$  go to  $-\infty$  is strictly positive. Additionally, if  $\theta_0 \leq 0$ , we have
+
+$P(\text{suboptimal action forever}) \geq (1 - e^{\theta_0})(1 - e^{\theta_0 + \alpha b})^{-\frac{1}{\alpha b}}.$
+
+Proof. All the proofs may be found in the appendix.  $\square$
+
+The updates provide some intuition as to why there is convergence to suboptimal policies. The issue is the committal nature of the baseline. Choosing an action leads to an increase of that action's probability, even if it is a poor choice. Choosing the suboptimal arm leads to a decrease in  $\theta$  by  $\frac{\alpha b}{1 - p_t}$ , thus increasing the probability the same arm is drawn again and further decreasing  $\theta$ . By checking the probability of this occurring forever,  $P(\text{suboptimal arm forever}) = \prod_{t=1}^{\infty} (1 - p_t)$ , we show that  $1 - p_t$  converges quickly enough to 1 that the infinite product is nonzero, showing it is possible to get trapped choosing the wrong arm forever (Prop. 1), and  $\theta_t \to -\infty$  as  $t$  grows.
+
+This issue could be solved by picking a baseline with lower variance. For instance, the minimum-variance baseline  $b = 1 - p_t$  leads to 0 variance and both possible updates are equal to  $+\alpha$ , guaranteeing that  $\theta \to +\infty$ , thus convergence. In fact, any baseline  $b \in (0,1)$  suffices since both updates are positive and greater than  $\alpha \min(b, 1 - b)$ . However, this is not always the case, as we show in the next section.
+
+To decouple the impact of the variance with that of the committal nature of the baseline, Prop. 2 analyzes the learning dynamics in the two-arm bandit case for perturbations of the optimal baseline, i.e. we study baselines of the form  $b = b^{*} + \epsilon$  and show how  $\epsilon$ , and particularly its sign, affects learning. Note that, because the variance is a quadratic func
+
+tion with its minimum in  $b^{*}$ , both  $+\epsilon$  and  $-\epsilon$  have the same variance. Our findings can be summarized as follows:
+
+Proposition 2. For the two-armed bandit defined in Prop. 1, when using a perturbed min-variance baseline  $b = b^{*} + \epsilon$ , the value of  $\epsilon$  determines the learning dynamics as follows:
+
+- For  $\epsilon < -1$ , there is a positive probability of converging to the suboptimal arm.  
+- For  $\epsilon \in (-1, 1)$ , we have convergence in probability to the optimal policy.  
+- For  $\epsilon \geq 1$ , the supremum of the iterates goes to  $+\infty$  in probability.
+
+While the proofs can be found in Appendix B.2, we provide here some intuition behind these results.
+
+For  $\epsilon < -1$ , we reuse the same argument as for  $b < 0$  in Prop. 1. The probability of drawing the correct arm can decrease quickly enough to lead to convergence to the suboptimal arm.
+
+For  $\epsilon \in (-1, 1)$ , the probability of drawing the correct arm cannot decrease too fast. Hence, although the updates, as well as the variance of the gradient estimate, are potentially unbounded, we still have convergence to the optimal solution in probability.
+
+Finally, for  $\epsilon \geq 1$ , we can reuse an intermediate argument from the  $\epsilon \in (0,1)$  case to argue that for any threshold  $C$ , the parameter will eventually exceed that threshold. For  $\epsilon \in (0,1)$ , once a certain threshold is crossed, the policy is guaranteed to improve at each step. However, with a large positive perturbation, updates are larger and we lose this additional guarantee, leading to the weaker result.
+
+We want to emphasize that not only we get provably different dynamics for  $\epsilon < -1$  and  $\epsilon \geq 1$ , showing the importance of the sign of the perturbation, but that there also is a sharp transition around  $|\epsilon| = 1$ , which cannot be captured solely by the variance.
+
+The above analysis was specific to these updates. To predict committal vs. non-committal behaviour more generally, it may be possible to utilize higher order moments or other distributional properties, even when the mean and variance is the same. Unfortunately, it is difficult to utilize higher-moment information in theoretical bounds in a general manner as Markov-type inequalities do not take into account the sign of the higher moment, which we think is where the committal vs. non-committal distinction would appear.
+
+# 3.2. Reducing variance with baselines can be detrimental
+
+As we saw with the two-armed bandit, the direction of the updates is important in assessing convergence. More specifically, problems can arise when the choice of baseline
+
+induces committal behaviour. We now show a different bandit setting where committal behaviour happens even when using the minimum-variance baseline, thus leading to convergence to a suboptimal policy. Furthermore, we design a better baseline which ensures all updates move the parameters towards the optimal policy. This cements the idea that the quality of parameter updates must not be analyzed in terms of variance but rather in terms of the probability of going in a bad direction, since a baseline that induces higher variance leads to convergence while the minimum-variance baseline does not. The following theorem summarizes this.
+
+Theorem 1. There exists a three-arm bandit where using the stochastic natural gradient on a softmax-parameterized policy with the minimum-variance baseline can lead to convergence to a suboptimal policy with probability  $\rho > 0$ , and there is a different baseline (with larger variance) which results in convergence to the optimal policy with probability 1.
+
+The bandit used in this theorem is the one we used for the experiments depicted in Fig. 1. The key is that the minimum-variance baseline can be lower than the second best reward; so pulling the second arm will increase its probability and induce committal behaviour. This can cause the agent to prematurely commit to the second arm and converge to the wrong policy. On the other hand, using any baseline whose value is between the optimal reward and the second best reward, which we term a gap baseline, will always increase the probability of the optimal action at every step, no matter which arm is drawn. Since the updates are sufficiently large at every step, this is enough to ensure convergence with probability 1, despite the higher variance compared to the minimum variance baseline. The key is that whether a baseline underestimates or overestimates the second best reward can affect the algorithm convergence and this is more critical than the resulting variance of the gradient estimates.
+
+As such, more than lower variance, good baselines are those that can assign positive effective returns to the good trajectories and negative effective returns to the others. These results cast doubt on whether finding baselines which minimize variance is a meaningful goal to pursue. The baseline can affect optimization in subtle ways, beyond variance, and further study is needed to identify the true causes of some improved empirical results observed in previous works. This importance of the sign of the returns, rather than their exact value, echoes with the cross-entropy method (De Boer et al., 2005), which maximizes the probability of the trajectories with the largest returns, regardless of their actual value.
+
+# 4. Off-policy sampling
+
+So far, we have seen that committal behaviour can be problematic as it can cause convergence to a suboptimal policy.
+
+This can be especially problematic when the agent follows a near-deterministic policy as it is unlikely to receive different samples which would move the policy away from the closest deterministic one, regardless of the quality of that policy.
+
+Up to this point, we assumed that actions were sampled according to the current policy, a setting known as on-policy. This setting couples the updates and the policy and is a root cause of the committal behaviour: the update at the current step changes the policy, which affects the distribution of rewards obtained and hence the next updates. However, we know from the optimization literature that bounding the variance of the updates will lead to convergence (Bottou et al., 2018). As the variance becomes unbounded when the probability of drawing some actions goes to 0, a natural solution to avoid these issues is to sample actions from a behaviour policy that selects every action with sufficiently high probability. Such a policy would make it impossible to choose the same, suboptimal action forever.
+
+# 4.1. Convergence guarantees with IS
+
+Because the behaviour policy changed, we introduce importance sampling (IS) corrections to preserve the unbiased updates (Kahn & Harris, 1951; Precup, 2000). These changes are sufficient to guarantee convergence for any baseline:
+
+Proposition 3. Consider a  $n$ -armed bandit with stochastic rewards with bounded support and a unique optimal action. The behaviour policy  $\mu_t$  selects action  $i$  with probability  $\mu_t(i)$  and let  $\epsilon_t = \min_i \mu_t(i)$ . When using NPG with importance sampling and a bounded baseline  $b$ , if  $\lim_{t \to \infty} t \epsilon_t^2 = +\infty$ , then the target policy  $\pi_t$  converges to the optimal policy in probability.
+
+Proof. (Sketch) Using Azuma-Hoeffding's inequality, we can show that for well chosen constants  $\Delta_i$ ,  $\delta$  and  $C > 0$ ,
+
+$$
+\mathbb {P} \left(\theta_ {t} ^ {1} \geq \theta_ {0} ^ {1} + \alpha \delta \Delta_ {1} t\right) \geq 1 - \exp \left(- \frac {\delta^ {2} \Delta_ {1} ^ {2}}{2 C ^ {2}} t \epsilon_ {t} ^ {2}\right)
+$$
+
+where  $\theta^1$  is the parameter associated to the optimal arm. Thus if  $\lim_{t\to \infty}t\epsilon_t^2 = +\infty$ , the RHS goes to 1. In a similar manner, we can upper bound  $\mathbb{P}\left(\theta_t^i\geq \theta_0^i +\alpha \delta \Delta_it\right)$  for all suboptimal arms, and applying an union bound, we get the desired result.
+
+The condition on  $\mu_t$  imposes a cap on how fast the behaviour policy can become deterministic: no faster than  $t^{-1/2}$ . Intuitively, this ensures each action is sampled sufficiently often and prevents premature convergence to a suboptimal policy. The condition is satisfied for any sequence of behaviour policies which assign at least  $\epsilon_t$  probability to each action at each step, such as  $\epsilon$ -greedy policies. It also holds if  $\epsilon_t$  decreases over time at a sufficiently slow rate. By choosing as behaviour policy  $\mu$  a linear interpolation between  $\pi$  and
+
+![](images/433881678f48ccedc43b64bfa594758f7c7bf03ae0df4d34687f179a3f40302c.jpg)  
+(a)  $b = 0 / \mathrm{IS}$
+
+![](images/3822516772b7d17aa65f32722e3fa3056385ac66a85b79e499a48cc78f2dd4dc.jpg)  
+(b)  $b = 0 / b^{*}$  
+Figure 3: Comparison between the variance of different methods on a 3-arm bandit. Each plot depicts the log of the ratio between the variance of two approaches. For example, Fig. (a) depicts  $\log \frac{\operatorname{Var}[g_{b = 0}]}{\operatorname{Var}[g_{\mathrm{IS}}]}$ , the log of the ratio between the variance of the gradients of PG without a baseline and PG with IS. The triangle represents the probability simplex with each corner representing a deterministic policy on a specific arm. The method written in blue (resp. red) in each figure has lower variance in blue (resp. red) regions of the simplex. The sampling policy  $\mu$ , used in the PG method with IS, is a linear interpolation between  $\pi$  and the uniform distribution,  $\mu(a) = \frac{1}{2}\pi(a) + \frac{1}{6}$ . Note that this is not the min. variance sampling distribution and it leads to higher variance than PG without a baseline in some parts of the simplex.
+
+![](images/10d7e08b9f3fd8d41ea9e3b1942896ef1ad79f2b1cffb82b2af7f385ad3092fb.jpg)  
+(c)  $b^{*}$  /IS.
+
+the uniform policy,  $\mu (a) = (1 - \gamma)\pi (a) + \frac{\gamma}{K},\gamma \in (0,1]$ , where  $K$  is the number of arms, we recover the classic EXP3 algorithm (Auer et al., 2002; Seldin et al., 2012).
+
+We can also confirm that this condition is not satisfied for the simple example we presented when discussing convergence to suboptimal policies. There,  $p_t$  could decrease exponentially fast since the tails of the sigmoid function decay exponentially and the parameters move by at least a constant at every step. In this case,  $\epsilon_t = \Omega(e^{-t})$ , resulting in  $\lim_{t \to \infty} te^{-2t} = 0$ , so Proposition 3 does not apply.
+
+# 4.2. Importance sampling, baselines & variance
+
+As we have seen, using a separate behaviour policy that samples all actions sufficiently often may lead to stronger convergence guarantees, even if it increases the variance of the gradient estimates in most of the space, as what matters is what happens in the high variance regions, which are usually close to the boundaries. Fig. 3 shows the ratios of gradient variances between on-policy PG without baseline, on-policy PG with the minimum variance baseline, and off-policy PG using importance sampling (IS) where the sampling distribution is  $\mu(a) = \frac{1}{2}\pi(a) + \frac{1}{6}$ , i.e., a mixture of the current policy  $\pi$  and the uniform distribution. While using the minimum variance baseline decreases the variance on the entire space compared to not using a baseline, IS actually increases the variance when the current policy is close to uniform. However, IS does a much better job at reducing the variance close to the boundaries of the simplex, where it actually matters to guarantee convergence.
+
+This suggests that convergence of PG methods is not so much governed by the variance of the gradient estimates in general, but by the variance in the worst regions, usually near the boundary. While baselines can reduce the variance, they generally cannot prevent the variance in those regions from exploding, leading to the policy getting stuck. Thus, good baselines are not the ones reducing the variance across the space but rather those that can prevent the learning from reaching these regions altogether. Large values of  $b$ , such that  $r(a_i) - b$  is negative for most actions, achieve precisely that. On the other hand, due to the increased flexibility of sampling distributions, IS can limit the nefariousness of these critical regions, offering better convergence guarantees despite not reducing variance everywhere.
+
+Importantly, although IS is usually used in RL to correct for the distribution of past samples (e.g., Munos et al., 2016), we advocate here for expanding the research on designing appropriate sampling distributions as done by Hanna et al. (2017; 2018) and Parmas & Sugiyama (2019). This line of work has a long history in statistics (c.f., Liu, 2008).
+
+# 4.3. Other mitigating strategies
+
+We conclude this section by discussing alternative strategies to mitigate the convergence issues. While they might be effective, and some are indeed used in practice, they are not without pitfalls.
+
+First, one could consider reducing the stepsizes, with the hope that the policy would not converge as quickly towards a suboptimal deterministic policy and would eventually leave that bad region. Indeed, if we are to use vanilla PG in the two-arm bandit example, instead of NPG, this effectively reduces the stepsize by a factor of  $\sigma(\theta)(1 - \sigma(\theta))$  (the Fisher information). In this case, we are able to show convergence in probability to the optimal policy. See Proposition 4 in Appendix B.
+
+Empirically, we find that, when using vanilla PG, the policy may still remain stuck near a suboptimal policy when using a negative baseline, similar to Fig. 2. While the previous proposition guarantees convergence eventually, the rate may be very slow, which remains problematic in practice. There is theoretical evidence that following even the true vanilla PG may result in slow convergence (Schaul et al., 2019), suggesting that the problem is not necessarily due to noise.
+
+An alternative solution would be to add entropy regularization to the objective. By doing so, the policy would be prevented from getting too close to deterministic policies. While this might prevent convergence to a suboptimal policy, it would also exclude the possibility of fully converging to the optimal policy, though the policy may remain near it.
+
+In bandits, EXP3 has been found not to enjoy high-probability guarantees on its regret so variants have been
+
+developed to address this deficiency (c.f. Lattimore & Szepesvári, 2020). For example, by introducing bias in the updates, their variance can be reduced significantly (Auer et al., 2002; Neu, 2015). Finally, other works have also developed provably convergent policy gradient algorithms using different mechanisms, such as exploration bonuses or ensembles of policies (Cai et al., 2019; Efroni et al., 2020; Agarwal et al., 2020).
+
+# 5. Extension to multi-step MDPs
+
+We focused our theoretical analyses on multi-arm bandits so far. However, we are also interested in more general environments where gradient-based methods are commonplace. We now turn our attention to the Markov Decision Process (MDP) framework (Puterman, 2014). An MDP is a set  $\{S, A, P, r, \gamma, \rho\}$  where  $S$  and  $A$  are the set of states and actions,  $P$  is the environment transition function,  $r$  is the reward function,  $\gamma \in [0,1)$  the discount factor, and  $\rho$  is the initial state distribution. The goal of RL algorithms is to find a policy  $\pi_{\theta}$ , parameterized by  $\theta$ , which maximizes the (discounted) expected return; i.e. Eq. 1 becomes
+
+$$
+\arg \max _ {\theta} J (\theta) = \arg \max _ {\theta} \sum_ {s} d _ {\gamma} ^ {\pi_ {\theta}} (s) \sum_ {a} \pi_ {\theta} (a | s) r (s, a),
+$$
+
+where there is now a discounted distribution over states induced by  $\pi_{\theta}$ . Although that distribution depends on  $\pi_{\theta}$  in a potentially complex way, the parameter updates are similar to Eq. 2:
+
+$$
+\theta_ {t + 1} = \theta_ {t} + \frac {\alpha}{N} \sum_ {i} [ Q (s _ {i}, a _ {i}) - b (s _ {i}) ] \nabla_ {\theta} \log \pi_ {\theta} (a _ {i} | s _ {i}),
+$$
+
+where  $(a_{i},s_{i})$  pairs are drawn according to the discounted state-visitation distribution induced by  $\pi_{\theta}$  and  $Q$  is the state-action value function induced by  $\pi_{\theta}$  (c.f. Sutton & Barto, 2018). To match the bandit setting and common practice, we made the baseline state dependent.
+
+Although our theoretical analyses do not easily extend to multi-step MDPs, we empirically investigated if the similarity between these formulations leads to similar differences in learning dynamics when changing the baseline. We consider a  $10 \times 10$  gridworld consisting of 4 rooms as depicted on Fig. 4a. We use a discount factor  $\gamma = 0.99$ . The agent starts in the upper left room and two adjacent rooms contain a goal state of value 0.6 or 0.3. The best goal (even discounted), with a value of 1, lies in the furthest room, so that the agent must learn to cross the sub-optimal rooms and reach the furthest one.
+
+Similar to the bandit setting, for a state  $s$ , we can derive the minimum-variance baseline  $b^{*}(s)$  assuming access to state-action values  $Q(s,a)$  for  $\pi_{\theta}$  and consider perturbations to it. Again, we use baselines  $b(s) = b^{*}(s) + \epsilon$  and  $b(s) = b^{*}(s) -$
+
+![](images/6a55f629b8fea33d51ad8fd8f351b020536c22790911acbd144ae52c0d114790.jpg)  
+(a) MDP
+
+![](images/62926f127fd7390676c3f49c1c95414bbcfef45b794b81a10b9147f71e1dfc9b.jpg)  
+(b) Returns
+
+![](images/95a6335fa55453661c1e8f62ed6ef4bba67e3ce0ec10198e2b2fc6aaabfb7678.jpg)  
+(c) Entropy (A)  
+Figure 4: We plot the discounted returns, the entropy of the policy over the states visited in each trajectory, and the entropy of the state visitation distribution, averaged over 50 runs, for multiple baselines. The baselines are of the form  $b(s) = b^{*}(s) + \epsilon$ , perturbations of the minimum-variance baseline, with  $\epsilon$  indicated in the legend. The shaded regions denote one standard error. Note that the policy entropy of lower baselines tends to decay faster than for larger baselines. Also, smaller baselines tend to get stuck on suboptimal policies, as indicated by the returns plot. See text for additional details.
+
+![](images/1562d06c86336075c40d39a597c226821c16f4f0967a2b4fdfb4e2a09ed2de5b.jpg)  
+(d) Entropy (S)
+
+$\epsilon$ , since they result in identical variances (this would not be the case if we used standard REINFORCE). We use a natural policy gradient estimate, which substitutes  $\nabla \log \pi(a_i | s_i)$  by  $F_{s_i}^{-1} \nabla \log \pi(a_i | s_i)$  in the update rule, where  $F_{s_i}$  is the Fisher information matrix for state  $s_i$  and solve for the exact  $Q(s, a)$  values using dynamic programming for all updates (see Appendix D.6 for details).
+
+In order to identify the committal vs. non-committal behaviour of the agent depending on the baseline, we monitor the entropy of the policy and the entropy of the stationary state distribution over time. Fig.4b shows the average returns over time and Fig.4c and 4d show the entropy of the policy in two ways. The first is the average entropy of the action distribution along the states visited in each trajectory, and the second is the entropy of the distribution of the number of times each state is visited up to that point in training.
+
+The action entropy for smaller baselines tends to decay faster compared to larger ones, indicating convergence to a deterministic policy. This quick convergence is premature in some cases since the returns are not as high for the lower baselines. In fact for  $\epsilon = -1$ , we see that the agent gets stuck on a policy that is unable to reach any goal within the time limit, as indicated by the returns of 0. On the other hand, the larger baselines tend to achieve larger returns with larger entropy policies, but do not fully converge to the optimal policy as evidenced by the gap in the returns plot.
+
+Since committal and non-committal behaviour can be directly inferred from the PG and the sign of the effective rewards  $R(\tau) - b$ , we posit that these effects extend to all MDPs. In particular, in complex MDPs, the first trajectories explored are likely to be suboptimal and a low baseline will increase their probability of being sampled again, requiring the use of techniques such as entropy regularization to prevent the policy from getting stuck too quickly. In some preliminary experiments with a deep RL policy gradient al
+
+girthm, PPO (Schulman et al., 2017), where we perturb the baseline by a fixed constant, seem to indicate that negative perturbations perform slightly worse than positive perturbations. The results are not conclusive though and there are many confounding factors in this setting which could affect the outcome, including clipping due to PPO, neural network generalization, and adaptive optimizers. It is likely that a more careful strategy to perturb the baseline is needed to gain benefits, similar to using exploration bonuses.
+
+# 6. Conclusion
+
+We presented results that dispute common beliefs about baselines, variance, and policy gradient methods in general. As opposed to the common belief that baselines only provide benefits through variance reduction, we showed that they can significantly affect the optimization process in ways that cannot be explained by the variance and that lower variance can even sometimes be detrimental.
+
+Different baselines can give rise to very different learning dynamics, even when they reduce the variance of the gradients equally. They do that by either making a policy quickly tend towards a deterministic one (committal behaviour) or by maintaining high-entropy for a longer period of time (non-committal behaviour). We showed that committal behaviour can be problematic and lead to convergence to a suboptimal policy. Specifically, we showed that stochastic natural policy gradient does not always converge to the optimal solution due to the unusual situation in which the iterates converge to the optimal policy in expectation but not almost surely. Moreover, we showed that baselines that lead to lower-variance can sometimes be detrimental to optimization, highlighting the limitations of using variance to analyze the convergence properties of these methods. We also showed that standard convergence guarantees for PG methods do not apply to some settings because the assumption of bounded variance of the updates is violated.
+
+The aforementioned convergence issues are also caused by the problematic coupling between the algorithm's updates and its sampling distribution since one directly impacts the other. As a potential solution, we showed that off-policy sampling can sidestep these difficulties by ensuring we use a sampling distribution that is different than the one induced by the agent's current policy. This supports the hypothesis that on-policy learning can be problematic, as observed in previous work (Schaul et al., 2019; Hennes et al., 2020). Nevertheless, importance sampling in RL is generally seen as problematic (van Hasselt et al., 2018) due to instabilities it introduces to the learning process. Moving from an imposed policy, using past trajectories, to a chosen sampling policy reduces the variance of the gradients for near-deterministic policies and can lead to much better behaviour. In general, other variance-reduction strategies may also be more effective (Xu et al., 2019).
+
+More broadly, this work suggests that treating bandit and reinforcement learning problems as a black-box optimization of a function  $J(\theta)$  may be insufficient to perform well. As we have seen, the current parameter value can affect all future parameter values by influencing the data collection process and thus the updates performed. Theoretically, relying on immediately available quantities such as the gradient variance and ignoring the sequential nature of the optimization problem is not enough to discriminate between certain optimization algorithms. In essence, to design highly-effective policy optimization algorithms, it may be necessary to develop a better understanding of how the optimization process evolves over many steps.
+
+# Acknowledgements
+
+We would like to thank Kris de Asis, Alan Chan, Ofir Nachum, Doina Precup, Dale Schuurmans, and Ahmed Touati for helpful discussions. We also thank Courtney Paquette, Vincent Liu, Scott Fujimoto and Csaba Szepesvári for reviewing an earlier version of this paper. Marlos C. Machado and Nicolas Le Roux are supported by a Canada CIFAR AI Chair.
+
+# References
+
+Agarwal, A., Kakade, S. M., Lee, J. D., and Mahajan, G. Optimality and approximation with policy gradient methods in markov decision processes. arXiv preprint arXiv:1908.00261, 2019.  
+Agarwal, A., Henaff, M., Kakade, S., and Sun, W. Pc-pg: Policy cover directed exploration for provable policy gradient learning. arXiv preprint arXiv:2007.08459, 2020.  
+Asadi, K. and Littman, M. L. An alternative softmax operator for reinforcement learning. In International Conference on Machine Learning, pp. 243-252. PMLR, 2017.  
+Auer, P., Cesa-Bianchi, N., Freund, Y., and Schapire, R. E. The nonstochastic multiarmed bandit problem. SIAM journal on computing, 32(1):48-77, 2002.  
+Bottou, L., Curtis, F. E., and Nocedal, J. Optimization methods for large-scale machine learning. Siam Review, 60(2):223-311, 2018.  
+Bubeck, S. and Cesa-Bianchi, N. Regret analysis of stochastic and nonstochastic multi-armed bandit problems. arXiv preprint arXiv:1204.5721, 2012.  
+Cai, Q., Yang, Z., Jin, C., and Wang, Z. Provably efficient exploration in policy optimization. arXiv preprint arXiv:1912.05830, 2019.  
+Cheng, C.-A., Yan, X., and Boots, B. Trajectory-wise control variates for variance reduction in policy gradient methods. In Conference on Robot Learning, pp. 1379-1394, 2020.  
+De Boer, P.-T., Kroese, D. P., Mannor, S., and Rubinstein, R. Y. A tutorial on the cross-entropy method. Annals of operations research, 134(1):19-67, 2005.  
+Efroni, Y., Shani, L., Rosenberg, A., and Mannor, S. Optimistic policy optimization with bandit feedback. arXiv preprint arXiv:2002.08243, 2020.  
+Grathwohl, W., Choi, D., Wu, Y., Roeder, G., and Duvenaud, D. Backpropagation through the void: Optimizing control variates for black-box gradient estimation. arXiv preprint arXiv:1711.00123, 2017.  
+Greensmith, E., Bartlett, P. L., and Baxter, J. Variance reduction techniques for gradient estimates in reinforcement learning. Journal of Machine Learning Research, 5(Nov): 1471-1530, 2004.  
+Gu, S., Lillicrap, T., Ghahramani, Z., Turner, R. E., and Levine, S. Q-prop: Sample-efficient policy gradient with an off-policy critic. arXiv preprint arXiv:1611.02247, 2016.
+
+Hanna, J. P., Thomas, P. S., Stone, P., and Niekum, S. Data-efficient policy evaluation through behavior policy search. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1394–1403. JMLR.org, 2017.  
+Hanna, J. P., Niekum, S., and Stone, P. Importance sampling policy evaluation with an estimated behavior policy. arXiv preprint arXiv:1806.01347, 2018.  
+Harper, M. and Weinstein, B. python-ternary: Ternary plots in python. Zenodo 10.5281/zenodo.594435, 2015. doi: 10.5281/zenodo.594435. URL https://github.com/marcharper/python-ternary.  
+Hennes, D., Morrill, D., Omidshafiei, S., Munos, R., Perolat, J., Lanctot, M., Gruslys, A., Lespiau, J.-B., Parmas, P., Duñez-Guzmán, E., et al. Neural replicator dynamics: Multiagent learning via hedging policy gradients. In Proceedings of the 19th International Conference on Autonomous Agents and MultiAgent Systems, pp. 492-501, 2020.  
+Hofmann, T., Lucchi, A., Lacoste-Julien, S., and McWilliams, B. Variance reduced stochastic gradient descent with neighbors. In Advances in Neural Information Processing Systems, pp. 2305-2313, 2015.  
+Kahn, H. and Harris, T. E. Estimation of particle transmission by random sampling. National Bureau of Standards applied mathematics series, 12:27-30, 1951.  
+Lattimore, T. and Szepesvári, C. Bandit Algorithms. Cambridge University Press, 2020. doi: 10.1017/9781108571401.  
+Liu, H., Feng, Y., Mao, Y., Zhou, D., Peng, J., and Liu, Q. Action-dependent control variates for policy optimization via stein's identity. arXiv preprint arXiv:1710.11198, 2017.  
+Liu, J. S. Monte Carlo strategies in scientific computing. Springer Science & Business Media, 2008.  
+Mei, J., Xiao, C., Dai, B., Li, L., Szepesvári, C., and Schuurmans, D. Escaping the gravitational pull of softmax. Advances in Neural Information Processing Systems, 33, 2020a.  
+Mei, J., Xiao, C., Szepesvari, C., and Schuurmans, D. On the global convergence rates of softmax policy gradient methods. In International Conference on Machine Learning, pp. 6820-6829. PMLR, 2020b.  
+Munos, R., Stepleton, T., Harutyunyan, A., and Bellemare, M. G. Safe and efficient off-policy reinforcement learning. In Advances in Neural Information Processing Systems, pp. 1046-1054, 2016.
+
+Neu, G. Explore no more: Improved high-probability regret bounds for non-stochastic bandits. arXiv preprint arXiv:1506.03271, 2015.  
+Parmas, P. and Sugiyama, M. A unified view of likelihood ratio and reparameterization gradients and an optimal importance sampling scheme. arXiv preprint arXiv:1910.06419, 2019.  
+Peters, J. and Schaal, S. Reinforcement learning of motor skills with policy gradients. *Neural networks*, 21(4):682-697, 2008.  
+Precup, D. Eligibility traces for off-policy policy evaluation. Computer Science Department Faculty Publication Series, pp. 80, 2000.  
+Puterman, M. L. Markov decision processes: discrete stochastic dynamic programming. John Wiley & Sons, 2014.  
+Robbins, H. Some aspects of the sequential design of experiments. Bulletin of the American Mathematical Society, 58(5):527-535, 1952.  
+Schaul, T., Borsa, D., Modayil, J., and Pascanu, R. Ray interference: a source of plateaus in deep reinforcement learning. arXiv preprint arXiv:1904.11455, 2019.  
+Schulman, J., Wolski, F., Dhariwal, P., Radford, A., and Klimov, O. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.  
+Seldin, Y., Szepesvári, C., Auer, P., and Abbasi-Yadkori, Y. Evaluation and analysis of the performance of the exp3 algorithm in stochastic environments. In EWRL, pp. 103-116, 2012.  
+Seldin, Y., Szepesvári, C., Auer, P., and Abbasi-Yadkori, Y. Evaluation and analysis of the performance of the exp3 algorithm in stochastic environments. In European Workshop on Reinforcement Learning, pp. 103-116. PMLR, 2013.  
+Sutton, R. S. and Barto, A. G. Reinforcement Learning: An Introduction. MIT Press, 2 edition, 2018.  
+van Hasselt, H., Doron, Y., Strub, F., Hessel, M., Sonnerat, N., and Modayil, J. Deep reinforcement learning and the deadly triad. CoRR, abs/1812.02648, 2018.  
+Williams, R. J. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229-256, 1992.  
+Wu, C., Rajeswaran, A., Duan, Y., Kumar, V., Bayen, A. M., Kakade, S., Mordatch, I., and Abbeel, P. Variance reduction for policy gradient with action-dependent factorized baselines. arXiv preprint arXiv:1803.07246, 2018.
+
+Xu, P., Gao, F., and Gu, Q. Sample efficient policy gradient methods with recursive variance reduction. arXiv preprint arXiv:1909.08610, 2019.

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+Andres Potapczynski  $^{*1}$  Luhuan Wu  $^{*2}$  Dan Biderman  $^{*1}$  Geoff Pleiss  $^{1}$  John P. Cunningham  $^{1,2}$
+
+# Abstract
+
+Scalable Gaussian Process methods are computationally attractive, yet introduce modeling biases that require rigorous study. This paper analyzes two common techniques: early truncated conjugate gradients (CG) and random Fourier features (RFF). We find that both methods introduce a systematic bias on the learned hyperparameters: CG tends to underfit while RFF tends to overfit. We address these issues using randomized truncation estimators that eliminate bias in exchange for increased variance. In the case of RFF, we show that the bias-to-variance conversion is indeed a trade-off: the additional variance proves detrimental to optimization. However, in the case of CG, our unbiased learning procedure meaningfully outperforms its biased counterpart with minimal additional computation. Our code is available at https://github.com/cunningham-lab/RTGPS.
+
+# 1. Introduction
+
+Gaussian Processes (GP) are popular and expressive nonparametric models, and considerable effort has gone into alleviating their cubic runtime complexity. Notable successes include inducing point methods (e.g. Snelson & Ghahramani, 2006; Titsias, 2009; Hensman et al., 2013), finite-basis expansions (e.g. Rahimi & Recht, 2008; Mutny & Krause, 2018; Wilson et al., 2020; Loper et al., 2020), nearest neighbor truncations (e.g. Datta et al., 2016; Katzfuss et al., 2021), and iterative numerical methods (e.g. Cunningham et al., 2008; Cutajar et al., 2016; Gardner et al., 2018). Common to these techniques is the classic speed-bias tradeoff: coarser GP approximations afford faster but more biased solutions that in turn affect both the model's predictions and learned hyperparameters. While a few papers analyze the
+
+*Equal contribution (randomly ordered). 1Zuckerman Institute, Columbia University 2Statistics Department, Columbia University. Correspondence to: Andres Potapczynski <ap3635@columbia.edu>, Luhuan Wu <lw2827@columbia.edu>, Dan Biderman <db3236@columbia.edu>.
+
+Proceedings of the  $38^{th}$  International Conference on Machine Learning, PMLR 139, 2021. Copyright 2021 by the author(s).
+
+bias of inducing point methods (Bauer et al., 2016; Burt et al., 2019), the biases of other approximation techniques, and their subsequent impact on learned GP models, have not been rigorously studied.
+
+Here we scrutinize the biases of two popular techniques – random Fourier features (RFF) (Rahimi & Recht, 2008) and conjugate gradients (CG) (e.g. Cunningham et al., 2008; Cutajar et al., 2016; Gardner et al., 2018). These methods are notable due to their popularity and because they allow dynamic control of the speed-bias tradeoff: at any model evaluation, the user can adjust the number of CG iterations or RFF features to a desired level of approximation accuracy. In practice, it is common to truncate these methods to a fixed number of iterations/features that is deemed adequate. However, such truncation will stop short of an exact (machine precision) solution and potentially lead to biased optimization outcomes.
+
+We provide a novel theoretical analysis of the biases resulting from RFF and CG on the GP log marginal likelihood objective. Specifically, we prove that CG is biased towards hyperparameters that underfit the data, while RFF is biased towards overfitting. In addition to yielding suboptimal hyperparameters, these biases hurt posterior predictions, regardless of the inference method used at test-time. Perhaps surprisingly, this effect is not subtle, as we will demonstrate. Our analysis suggests there is value in debiasing GP learning with CG and RFF. To do so, we turn to recent work that shows the merits of exchanging the speed-bias tradeoff for a speed-variance tradeoff (Beatson & Adams, 2019; Chen et al., 2019; Luo et al., 2020; Oktay et al., 2020). These works all introduce a randomization procedure that reweights elements of a fast truncated estimator, eliminating its bias at the cost of increasing its variance.
+
+We thus develop bias-free versions of GP learning with CG and RFF using randomized truncation estimators. In short, we randomly truncate the number of CG iterations and RFF features, while reweighting intermediate solutions to maintain unbiasedness. Our variant of CG uses the Russian Roulette estimator (Kahn, 1955), while our variant of RFF uses the Single Sample estimator of Lyne et al. (2015). We believe our RR-CG and SS-RFF methods to be the first to produce unbiased estimators of the GP log marginal likelihood with  $< \mathcal{O}(N^3)$  computation.
+
+Finally, through extensive empirical evaluation, we find our methods and their biased counterparts indeed constitute a bias-variance tradeoff. Both RR-CG and SS-RFF are unbiased, recovering nearly the same optimum as the exact GP method, while GP trained with CG and RFF often converge to solutions with worse likelihood. We note that bias elimination is not always practical. For SS-RFF, the optimization is slow, due to the large auxiliary variance needed to counteract the slowly decaying bias of RFF. On the other hand, RR-CG incurs a minimal variance penalty, likely due to the favorable convergence properties of CG. In a wide range of benchmark datasets, RR-CG demonstrates similar or better predictive performance compared to CG using the same expected computational time.
+
+To summarize, this work offers three main contributions:
+
+- theoretical analysis of the bias of CG- and RFF-based GP approximation methods (§3)  
+- RR-CG and SS-RFF: bias-free versions of these popular GP approximation methods (§4)  
+- results demonstrating the value of our RR-CG and SS-RFF methods ( $\S 3$  and 5).
+
+# 2. Background
+
+We consider observed data  $\mathcal{D} = \{(\mathbf{x}_i, y_i)\}_{i=1}^N$  for  $\mathbf{x}_i \in \mathbb{R}^d$  and  $y_i \in \mathbb{R}$ , and the standard GP model:
+
+$$
+\begin{array}{l} f (\cdot) \sim \mathcal {G P} (\mu (\cdot), k (\cdot , \cdot)), \\ \mathbf {y} _ {i} = f (\mathbf {x} _ {i}) + \epsilon_ {i}, \quad \epsilon_ {i} \sim \mathcal {N} \left(0, \sigma^ {2}\right) \\ \end{array}
+$$
+
+where  $k(\cdot, \cdot)$  is the covariance kernel,  $\mu(\cdot)$  is set to zero without loss of generality, and hyperparameters are collected into the vector  $\theta$ , which is optimized as:
+
+$$
+\begin{array}{l} \boldsymbol {\theta} ^ {*} = \arg \min  _ {\boldsymbol {\theta}} \mathcal {L} (\boldsymbol {\theta}) \\ \mathcal {L} (\boldsymbol {\theta}) = - \log p (\mathbf {y} \mid \mathbf {X}; \boldsymbol {\theta}) (1) \\ = \frac {1}{2} \left(\underbrace {\log | \widehat {\mathbf {K}} _ {\mathbf {X X}} |} _ {\text {m o d e l c o m p l e x i t y}} + \underbrace {\mathbf {y} ^ {\top} \widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \mathbf {y}} _ {\text {d a t a f i t}} + N \log 2 \pi\right) (1) \\ \end{array}
+$$
+
+where  $\widehat{\mathbf{K}}_{\mathbf{X}\mathbf{X}}\in \mathbb{R}^{N\times N}$  is the Gram matrix of all data points with diagonal observational noise:
+
+$$
+\widehat {\mathbf {K}} _ {\mathbf {X X}} [ i, j ] = k (\mathbf {x} _ {i}, \mathbf {x} _ {j}) + \sigma^ {2} \mathbb {I} _ {i = j}.
+$$
+
+Following standard practice, we optimize  $\theta$  with gradients:
+
+$$
+\frac {\partial \mathcal {L}}{\partial \boldsymbol {\theta}} = \frac {1}{2} \left(\operatorname {t r} \left\{\widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \frac {\partial \widehat {\mathbf {K}} _ {\mathbf {X X}}}{\partial \boldsymbol {\theta}} \right\} - \mathbf {y} ^ {\top} \widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \frac {\partial \widehat {\mathbf {K}} _ {\mathbf {X X}}}{\partial \boldsymbol {\theta}} \widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \mathbf {y}\right). \tag {2}
+$$
+
+Three terms thus dominate the computational complexity:  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y},\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  , and  $\operatorname {tr}\{\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\frac{\partial\widehat{\mathbf{K}}_{\mathbf{XX}}}{\partial\theta}\}$  . The common
+
+approach to computing this triad is the Cholesky factorization, requiring  $\mathcal{O}(N^3)$  time and  $\mathcal{O}(N^2)$  space.
+
+Extensive literature has accelerated the inference and hyperparameter learning of GP. Two very popular strategies are using conjugate gradients (Cunningham et al., 2008; Cutajar et al., 2016; Gardner et al., 2018; Wang et al., 2019) to approximate the linear solves in Eq. (2), and random Fourier features (Rahimi & Recht, 2008), which constructs a randomized finite-basis approximation of the kernel.
+
+# 2.1. Conjugate Gradients
+
+To apply conjugate gradients to GP learning, we begin by replacing the gradient in Eq. (2) with a stochastic estimate (Cutajar et al., 2016; Gardner et al., 2018):
+
+$$
+\frac {\partial \mathcal {L}}{\partial \boldsymbol {\theta}} \approx \frac {1}{2} \left(\mathbf {z} ^ {\top} \widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \frac {\partial \widehat {\mathbf {K}} _ {\mathbf {X X}}}{\partial \boldsymbol {\theta}} \mathbf {z} - \mathbf {y} ^ {\top} \widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \frac {\partial \widehat {\mathbf {K}} _ {\mathbf {X X}}}{\partial \boldsymbol {\theta}} \widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \mathbf {y}\right), \tag {3}
+$$
+
+where  $\mathbf{z}$  is a random variable such that  $\mathbb{E}[\mathbf{z}] = 0$  and  $\mathbb{E}[\mathbf{z}\mathbf{z}^T] = \mathbf{I}$ . Note that the first term constitutes a stochastic estimate of the trace term (Hutchinson, 1989). Thus, stochastic optimization of Eq. (1) can be reduced to computing the linear solves  $\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  and  $\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{z}$ .
+
+Conjugate gradients (CG) (Hestenes et al., 1952) is an iterative algorithm for solving positive definite linear systems  $\mathbf{A}^{-1}\mathbf{b}$ . It consists of a three-term recurrence, where each new term requires only a matrix-vector multiplication with  $\mathbf{A}$ . More formally, each CG iteration computes a new term of the following summation:
+
+$$
+\mathbf {A} ^ {- 1} \mathbf {b} = \sum_ {i = 1} ^ {N} \gamma_ {i} \mathbf {d} _ {i}, \tag {4}
+$$
+
+where the  $\gamma_{i}$  are coefficients and the  $\mathbf{d}_i$  are conjugate search directions (Golub & Van Loan, 2012).  $N$  iterations of CG produce all  $N$  summation terms and recover the exact solution. In practice, exact convergence may require more than  $N$  iterations due to inaccuracies of floating point arithmetic. However, the summation converges exponentially, and so  $J \ll N$  iterations may suffice to achieve high accuracy.
+
+CG is an appealing method for computing  $\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  and  $\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{z}$  due to its computational complexity and its potential for GPU-accelerated matrix products.  $J$  iterations takes at most  $\mathcal{O}(JN^2)$  time and  $\mathcal{O}(N)$  space if the matrix-vector products are performed in a map-reduce fashion (Wang et al., 2019). However, ill-conditioned kernel matrices hinder the convergence rate (Cutajar et al., 2016), and so the  $J^{\mathrm{th}}$  CG iteration may yield an inaccurate approximation of Eq. (3).
+
+# 2.2. Random Fourier Features
+
+Rahimi & Recht (2008) introduce a randomized finite-basis approximation to stationary kernels:
+
+$$
+k \left(\mathbf {x}, \mathbf {x} ^ {\prime}\right) = k \left(\mathbf {x} - \mathbf {x} ^ {\prime}\right) \approx \phi (\mathbf {x}) ^ {\top} \phi \left(\mathbf {x} ^ {\prime}\right) \tag {5}
+$$
+
+where  $\phi (\mathbf{x})\in \mathbb{R}^{J}$  and  $J\ll N$  . The RFF approximation relies on Bochner's theorem (Bochner et al., 1959): letting  $\tau = \mathbf{x} - \mathbf{x}'$  , all stationary kernels  $k(\tau)$  on  $\mathbb{R}^d$  can be exactly expressed as the Fourier dual of a nonnegative measure  $\mathbb{P}(\omega)\colon k(\tau) = \int \mathbb{P}(\omega)\exp (i\omega \tau)d\omega$  . A Monte Carlo approximation of this Fourier transform yields:
+
+$$
+k (\tau) \approx \frac {2}{J} \sum_ {j = 1} ^ {J / 2} \exp (i \omega_ {j} \tau), \quad \omega_ {j} \sim \mathbb {P} (\boldsymbol {\omega}),
+$$
+
+which simplifies to a finite-basis approximation:
+
+$$
+\mathbf {K} _ {\mathbf {X X}} \approx [ \phi (\mathbf {x} _ {1}) \dots \phi (\mathbf {x} _ {n}) ] [ \phi (\mathbf {x} _ {1}) \dots \phi (\mathbf {x} _ {n}) ] ^ {\top},
+$$
+
+$$
+\boldsymbol {\phi} (\mathbf {x}) = [ \cos (\boldsymbol {\omega} _ {i} ^ {\top} \mathbf {x}), \sin (\boldsymbol {\omega} _ {i} ^ {\top} \mathbf {x}) ] _ {i = 1} ^ {J / 2}, \quad \boldsymbol {\omega} _ {i} \sim \mathbb {P} (\boldsymbol {\omega}).
+$$
+
+For many common kernels,  $\mathbb{P}(\omega)$  can be computed in closed-form (e.g. RBF kernels have zero-mean Gaussian spectral densities). The approximated log likelihood can be computed in  $\mathcal{O}(J^3 + N)$  time and  $\mathcal{O}(JN)$  space using the Woodbury inversion lemma and the matrix determinant lemma, respectively. The number of random features  $J/2$  is a user choice, with typical values between 100-1000. More features lead to more accurate kernel approximations.
+
+# 2.3. Unbiased Randomized Truncation
+
+We will now briefly introduce Randomized Truncation Estimators, which are the primary tool we use to unbias the CG and RFF log marginal likelihood estimates. At a high level, assume that we wish to estimate some quantity  $\psi$  that can be expressed as a (potentially-infinite) series:
+
+$$
+\psi = \sum_ {j = 1} ^ {H} \Delta_ {j}, \quad H \in \mathbb {N} \cup \{\infty \}.
+$$
+
+Here and in the following sections,  $\Delta_j$  can either be random or deterministic. To avoid the expensive evaluation of the full summation, a randomized truncation estimator chooses a random term  $J \in \{1, \dots, H\}$  with probability mass function  $\mathbb{P}(J) = \mathbb{P}(\mathcal{I} = J)$  after which to truncate computation. In the following, we introduce two means of deriving unbiased estimators by weighting the summation terms.
+
+The Russian Roulette estimator (Kahn, 1955) obtains an unbiased estimator  $\bar{\psi}_J$  by truncating the sum after  $J \sim \mathbb{P}(J)$  terms and dividing the surviving terms by their survival probabilities:
+
+$$
+\bar {\psi} _ {J} = \sum_ {j = 1} ^ {J} \frac {\Delta_ {j}}{\mathbb {P} (\mathcal {J} \geq j)} = \sum_ {j = 1} ^ {H} \left(\frac {\mathbb {I} _ {J \geq j}}{\mathbb {P} (\mathcal {J} \geq j)}\right) \Delta_ {j}, \tag {6}
+$$
+
+and,  $\mathbb{E}[\bar{\psi}_J] = \sum_{i=1}^{H} \Delta_j = \psi$ . (See appendix for further derivation.) The choice of  $\mathbb{P}(J)$  determines both the computational efficiency and the variance of  $\bar{\psi}_J$ . A thin-tailed  $\mathbb{P}(J)$  will often truncate sums after only a few terms ( $J \ll H$ ).
+
+However, tail events  $(J\approx H)$  are upweighted inversely to their low survival probability, and so thin-tailed truncation distributions may lead to high variance.
+
+The Single Sample estimator (Lyne et al., 2015) implements an alternative reweighting scheme. After drawing  $J \sim \mathbb{P}(J)$ , it computes a single summation term  $\Delta_J$ , which it upweights by  $1 / \mathbb{P}(J)$ :
+
+$$
+\bar {\psi} _ {J} = \frac {\Delta_ {J}}{\mathbb {P} (J)} = \sum_ {j = 1} ^ {H} \left(\frac {\mathbb {I} _ {J = j}}{\mathbb {P} (\mathcal {I} = j)}\right) \Delta_ {j}. \tag {7}
+$$
+
+This procedure is unbiased, and it amounts to estimating  $\psi$  using a single importance-weighted sample from  $\mathbb{P}(J)$  (see appendix). Again,  $\mathbb{P}(J)$  controls the speed/variance trade-off. We refer the reader to (Beatson & Adams, 2019) for a detailed comparison of these two estimators. We emphasize that both estimators remain unbiased even if  $\Delta_j$  is a random variable, as long as it is independent from the random truncation integer  $J$ .
+
+# 3. GP Learning with CG and RFF is Biased
+
+Here we prove that early truncated CG and RFF provide biased approximations to the terms comprising the GP log marginal likelihood (Eq. 1). We also derive the bias decay rates for each method. We then empirically demonstrate these biases and show they affect the hyperparameters learned through optimization. Remarkably, we find that the above biases are highly systematic: CG-based GP learning favors underfitting hyperparameters while RFF-based learning favors overfitting hyperparameters.
+
+# 3.1. CG Biases GP Towards Underfitting
+
+In the GP literature, CG has often been considered an "exact" method for computing the log marginal likelihood (Cunningham et al., 2008; Cutajar et al., 2016), as the iterations are only truncated after reaching a pre-specified residual error threshold (e.g.  $10^{-10}$ ). However, as CG is applied to ever-larger kernel matrices it is common to truncate the CG iterations before reaching this convergence (Wang et al., 2019). While this accelerates the hyperparameter learning process, the resulting solves and gradients can no longer be considered "exact." In what follows, we show that the early-truncated CG optimization objective is not only approximate but also systematically biased towards underfitting.
+
+To analyze the early-truncation bias, we adopt the analysis of Gardner et al. (2018) that recovers the GP log marginal likelihood (Eq. 1) from the stochastic gradient's CG estimates of  $\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  and  $\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{z}$  (Eq. 3). Recall the two terms in the log marginal likelihood are the "data fit" term  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  and the "model complexity" term  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$ . The first term falls directly out of the CG estimate of  $\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$ , while a
+
+![](images/0143de43fb602592c899a0160ce8af6446173cff05534122875306913105d3e9.jpg)  
+Figure 1. CG (left) systematically overestimates log  $|\widehat{\mathbf{K}}_{\mathbf{X}\mathbf{X}}|$  and underestimates  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{X}\mathbf{X}}^{-1}\mathbf{y}$  whereas RFF (right) does the opposite. The dashed orange line shows the exact values computed by Cholesky. Our unbiased methods, RR-CG (left) and SS-RFF (right), recover the true log  $|\widehat{\mathbf{K}}_{\mathbf{X}\mathbf{X}}|$  and  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{X}\mathbf{X}}^{-1}\mathbf{y}$  values. For these two methods, the  $x$ -axis indicates the expected number of iterations/features.
+
+![](images/0d1fb09db399153f017bad5c70d68ef299a13b0938b9cbc4edab4a3b45f93089.jpg)
+
+![](images/456be6156675b672f167df15534c6faf6145667e680ff03656a01ac36bbec8d8.jpg)  
+Figure 2. Kernel lengthscale values learned by optimizing (biased) CG and RFF log marginal likelihood approximations. CG overestimates the optimal kernel lengthscale whereas RFF underestimates it. We plot the divergence (in log-ratio scale) between the learned and true lengthscales as a function of the number of CG iterations (left) and of the number of RFF samples (right).
+
+stochastic estimate of  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  can be obtained through the byproducts of CG's computation for  $\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{z}$ . Gardner et al. (2018) show that the CG coefficients in Eq. (4) can be manipulated to produce a partial tridiagonalization:  $\mathbf{T}_{\mathbf{z}}^{(J)} = \mathbf{Q}_{\mathbf{z}}^{(J)\top}\widehat{\mathbf{K}}_{\mathbf{XX}}\mathbf{Q}_{\mathbf{z}}^{(J)}$ , where  $\mathbf{T}_{\mathbf{z}}^{(J)} \in \mathbb{R}^{J \times J}$  is tridiagonal.  $\mathbf{T}_{\mathbf{z}}^{(J)}$  can compute the Stochastic Lanczos Quadrature estimate of  $\widehat{\mathbf{K}}_{\mathbf{XX}}$  (Ubaru et al., 2017; Dong et al., 2017):
+
+$$
+\begin{array}{l} \log | \widehat {\mathbf {K}} _ {\mathbf {X X}} | = \mathbb {E} \left[ \mathbf {z} ^ {T} (\log \widehat {\mathbf {K}} _ {\mathbf {X X}}) \mathbf {z} \right] \\ \approx \left\| \mathbf {z} \right\| ^ {2} \mathbf {e} _ {1} ^ {\top} \left(\log \mathbf {T} _ {\mathbf {z}} ^ {(J)}\right) \mathbf {e} _ {1}, \tag {8} \\ \end{array}
+$$
+
+where  $\log (\cdot)$  is the matrix logarithm and  $\mathbf{e}_1$  is the first row of the identity matrix. The following theorem analyzes the bias of these  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{X}\mathbf{X}}^{-1}\mathbf{y}$  and  $\log |\widehat{\mathbf{K}}_{\mathbf{X}\mathbf{X}}|$  estimates:
+
+Theorem 1. Let  $u_{J}$  and  $v_{J}$  be the estimates of  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{X}\mathbf{X}}^{-1}\mathbf{y}$  and  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  respectively after  $J$  iterations of CG; i.e.:
+
+$$
+u _ {J} = \mathbf {y} ^ {\top} \left(\sum_ {i = 1} ^ {J} \gamma_ {i} \mathbf {d} _ {i}\right), v _ {J} = \| \mathbf {z} \| ^ {2} \mathbf {e} _ {1} ^ {\top} \left(\log \mathbf {T} _ {\mathbf {z}} ^ {(J)}\right) \mathbf {e} _ {1}.
+$$
+
+If  $J < N$ , CG underestimates the inverse quadratic term and overestimates the log determinant in expectation:
+
+$$
+u _ {J} \leq \mathbf {y} ^ {\top} \widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \mathbf {y}, \quad \mathbb {E} _ {\mathbf {z}} [ v _ {J} ] \geq \log | \widehat {\mathbf {K}} _ {\mathbf {X X}} |. \tag {9}
+$$
+
+The biases of both terms decay at a rate of  $\mathcal{O}(C^{-2J})$ , where  $C$  is a constant that depends on the conditioning of  $\widehat{\mathbf{K}}_{\mathbf{XX}}$ .
+
+Proof sketch. The direction of the biases can be proved using a connection between CG and numeric quadrature.  $u_{J}$  and  $v_{J}$  are exactly equal to the  $J$ -point Gauss quadrature approximation of  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  and  $\| \mathbf{z}\|^{2}\mathbf{e}_{1}^{\top}(\log \mathbf{T}_{\mathbf{z}}^{(N)})\mathbf{e}_{1}$  represented as Riemann-Stieltjes integrals. The sign of the CG approximation bias follows from the standard Gauss quadrature error bound, which is negative for  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  and positive for  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$ . The convergence rates are from standard bounds on CG (Golub & Van Loan, 2012) and the analysis of Ubaru et al. (2017). See appendix for a full proof.
+
+Fig. 1 confirms our theoretical analysis and demonstrates the systematic biases of CG. We plot the log marginal likelihood terms for a subset of the PoleTele UCI dataset, varying the number of CG iterations  $(J)$  used to produce the estimates. Compared against the exact terms (computed with Cholesky), we see an overestimation of  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  and an underestimation of  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$ . These biases are most prominent when using few CG iterations.
+
+We turn to study the effect of CG learning on hyperparameters. Since the log marginal likelihood is a nonconvex function of  $\theta$ , it is not possible to directly prove how the bias affects  $\theta$ . Nevertheless, we know intuitively that underestimating  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{X}\mathbf{X}}^{-1}\mathbf{y}$  de-prioritizes model fit while overestimating  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  over-penalizes model complexity. Thus, the learned hyperparameters will likely underfit the data. Underfitting may manifest in an overestimation of the learned lengthscale  $\ell$ , as low values of  $\ell$  increase the flexibility and the complexity of the model. This hypothesis is empirically confirmed in Fig. 2 (left panel). We train a GP regression model on a toy dataset:  $y = x\sin (5\pi x) + \varepsilon$
+
+and  $\varepsilon \sim \mathcal{N}(0,0.01)$ . We fix all hyperparameters other than the lengthscale, which is learned using both CG-based optimization and (exact) Cholesky-based optimization. The overestimation of  $\ell$  decays with the number of CG iterations.
+
+# 3.2. RFF Biases GP Towards Overfitting
+
+Previous work has studied the accuracy of RFF's approximation to the entries of the Gram matrix  $k(\mathbf{x},\mathbf{x}^{\prime}) \approx \phi (\mathbf{x})^{\top}\phi (\mathbf{x}^{\prime})$  (Rahimi & Recht, 2008; Sutherland & Schneider, 2015). However, to the best of our knowledge there has been little analysis of nonlinear functions of this approximate Gram matrix, such as  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  and  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  appearing in the GP objective. Interestingly, we find that RFF systematically biases these terms:
+
+Theorem 2. Let  $\widetilde{\mathbf{K}}_J$  be the RFF approximation with  $J/2$  random features. In expectation,  $\widetilde{\mathbf{K}}_J$  overestimates the inverse quadratic and underestimates the log determinant:
+
+$$
+\mathbb {E} _ {\mathbb {P} (\omega)} \left[ \mathbf {y} ^ {\top} \widetilde {\mathbf {K}} _ {J} ^ {- 1} \mathbf {y} \right] \geq \mathbf {y} ^ {\top} \widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \mathbf {y} \tag {10}
+$$
+
+$$
+\mathbb {E} _ {\mathbb {P} (\boldsymbol {\omega})} \left[ \log | \widetilde {\mathbf {K}} _ {J} | \right] \leq \log | \widehat {\mathbf {K}} _ {\mathbf {X X}} |. \tag {11}
+$$
+
+The biases of both terms decay at a rate of  $\mathcal{O}(1 / J)$
+
+Proof sketch. The direction of the biases is a straightforward application of Jensen's inequality, noting that  $\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}$  is a convex function and  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  is a concave function. The magnitude of the bias is derived from a second-order Taylor expansion that closely resembles the analysis of Nowozin (2018). See appendix for full proof.
+
+Again, Fig. 1 confirms the systematic biases of RFF, which decay at a rate proportional to the number of features, as predicted by Thm. 2. Hence, RFF should affect the learned hyperparameters in a manner opposite to CG. Overestimating  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  emphasizes data fitting while underestimating  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  reduces the model complexity penalty, overall resulting in overfitting behavior. Following the intuition presented in Sec. 3.1, we expect the lengthscale to be underestimated, as empirically confirmed by Fig. 2 (right panel). The figure also illustrates the slow decay of the RFF bias.
+
+# 4. Bias-free Scalable Gaussian Processes
+
+We debias the estimates of both the GP training objective in Eq. (1) and its gradient in Eq. (2) (as approximated by CG and RFF) using unbiased randomized truncation estimators. To see how such estimators apply to GP hyperparameter learning, we note that both CG and RFF recover the true log marginal likelihood (or an unbiased estimate thereof) in their limits:
+
+Observation 1. CG recovers the exact log marginal likeli
+
+hood in expectation in at most  $N$  iterations:
+
+$$
+\mathbf {y} ^ {\top} \widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \mathbf {y} = \mathbf {y} ^ {\top} \left(\sum_ {j = 1} ^ {N} \gamma_ {j} \mathbf {d} _ {j}\right), \tag {12}
+$$
+
+$$
+\log | \widehat {\mathbf {K}} _ {\mathbf {X X}} | = \mathbb {E} _ {\mathbf {z}} \left[ \| \mathbf {z} \| ^ {2} \mathbf {e} _ {1} ^ {\top} \left(\log \mathbf {T} _ {\mathbf {z}} ^ {(N)}\right) \mathbf {e} _ {1} \right]. \tag {13}
+$$
+
+By the law of large numbers, RFF converges almost surely to the exact log marginal likelihood as the number of random features goes to infinity:
+
+$$
+\mathbf {y} ^ {\top} \widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \mathbf {y} = \lim  _ {J \rightarrow \infty} \mathbf {y} ^ {\top} \widetilde {\mathbf {K}} _ {J} ^ {- 1} \mathbf {y}, \tag {14}
+$$
+
+$$
+\log | \widehat {\mathbf {K}} _ {\mathbf {X X}} | = \lim  _ {J \rightarrow \infty} \log | \widetilde {\mathbf {K}} _ {J} |. \tag {15}
+$$
+
+To maintain the scalability of CG and RFFs while eliminating bias, we express the log marginal likelihood terms in Eqs. (12) to (15) as summations amenable to randomized truncation. We then apply the Russian Roulette and Single Sample estimators of Sec. 2.3 to avoid computing all summation terms while obtaining the same result in expectation.
+
+# 4.1. Russian Jouette-Truncated CG (RR-CG)
+
+The stochastic gradient in Eq. (3) requires performing two solves:  $\widehat{\mathbf{K}}_{\mathbf{X}\mathbf{X}}^{-1}\mathbf{y}$  and  $\widehat{\mathbf{K}}_{\mathbf{X}\mathbf{X}}^{-1}\mathbf{z}$ . Using the summation formulation of CG (Eq. 4), we can write these two solves as series:
+
+$$
+\widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \mathbf {y} = \sum_ {i = 1} ^ {N} \gamma_ {i} \mathbf {d} _ {i}, \quad \widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \mathbf {z} = \sum_ {i = 1} ^ {N} \gamma_ {i} ^ {\prime} \mathbf {d} _ {i} ^ {\prime},
+$$
+
+where each CG iteration computes a new term of the summation. By applying the Russian Roulette estimator from Eq. (6), we obtain the following unbiased estimates:
+
+$$
+\widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \mathbf {y} \approx \sum_ {j = 1} ^ {J} \left(\gamma_ {j} \mathbf {d} _ {j}\right) / \mathbb {P} (\mathcal {J} \geq j), \quad J \sim \mathbb {P} (J) \tag {16}
+$$
+
+$$
+\widehat {\mathbf {K}} _ {\mathbf {X X}} ^ {- 1} \mathbf {z} \approx \sum_ {j = 1} ^ {J ^ {\prime}} (\gamma_ {j} ^ {\prime} \mathbf {d} _ {j} ^ {\prime}) / \mathbb {P} (\mathcal {J} \geq j). J ^ {\prime} \sim \mathbb {P} (J ^ {\prime}),
+$$
+
+These unbiased solves produce an unbiased optimization gradient in Eq. (3); we refer to this approach as Russian Roulette CG (RR-CG). With the appropriate truncation distribution  $\mathbb{P}(J)$ , this estimate affords the same computational complexity of standard CG without its bias.
+
+We must compute two independent estimates of  $\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  with different  $J\sim \mathbb{P}(J)$  in order for the  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\frac{\partial\widehat{\mathbf{K}}_{\mathbf{XX}}}{\partial\theta}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  term in Eq. (3) to be unbiased. Thus, the unbiased gradient requires 3 calls to RR-CG, as opposed to the 2 CG calls needed for the biased gradient. Nevertheless, RR-CG has the same  $\mathcal{O}(JN^2)$  complexity as standard CG - and the additional solve can be computed in parallel with the others.
+
+We can also use the Russian Roulette estimator to compute the log marginal likelihood itself, though this is not strictly necessary for gradient-based optimization. (See appendix.)
+
+Choosing the truncation distribution. Since the Russian Roulette estimator is unbiased for any choice of  $\mathbb{P}(J)$
+
+we wish to choose a truncation distribution that balances computational cost and variance<sup>1</sup>. Beatson & Adams (2019) propose the relative optimization efficiency (ROE) metric, which is the ratio of the expected improvement of taking an optimization step with our gradient estimate to its computational cost. A critical requirement of the ROE analysis is the expected rate of decay of our approximations in terms of the number of CG iterations  $J$ . We summarize our estimates and choices of distribution as follows:
+
+Theorem 3. The approximation to  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  and to  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  using RR-CG decays at a rate of  $\mathcal{O}(C^{-2J})$ . Therefore the truncation distribution that maximizes the ROE is  $\mathbb{P}^{*}(J)\propto C^{-2J}$ , where  $C$  is a constant that depends on the conditioning of  $\widehat{\mathbf{K}}_{\mathbf{XX}}$ . The expected computation and variance of  $\mathbb{P}^{*}(J)$  is finite.
+
+Proof sketch. Beatson & Adams (2019) show that the truncation distribution that maximizes the ROE is proportional to the rate of decay of our approximation divided by its computational cost. The error of CG decays as  $\mathcal{O}(C^{-2J})$ , and the cost of each summation term is constant with respect to  $J$ . In practice, we vary the exponential decaying rate to control the expectation and variance of  $\mathbb{P}(J)$ . To further reduce the variance of our estimator, we set a minimum number of CG iterations to be computed, as in (Luo et al., 2020). See appendix for full proof.
+
+In practice, however, we do not have access to  $C$  since computing the conditioning of  $\widehat{\mathbf{K}}_{\mathbf{XX}}$  is impractical. Yet, we can change the base of the exponential to  $e$  and add a temperature parameter  $\lambda$  to rescale the function and control the rate of decay of the truncation distribution as a sensible alternative. Thus, we follow a more general exponential decay distribution:
+
+$$
+\mathbb {P} (J) \propto e ^ {- \lambda J}, \quad J = J _ {\min }, \dots , N \tag {17}
+$$
+
+where  $J_{\mathrm{min}}$  is the minimum truncation number. By varying the values of  $\lambda$  and  $J_{\mathrm{min}}$ , we can control the expectation and standard deviation of  $\mathbb{P}(J)$ . In practice we found that having the standard deviation value between 10 and 20 achieves stable GP learning process, which can be obtained by tuning  $\lambda$  between 0.05 and 0.1 for sufficiently large datasets (e.g.  $N \geq 500$ ). We also noticed that the method is not sensitive to these choices of hyperparameters and that they work well across all the experiments. The expected truncation number can be further tuned by varying  $J_{\mathrm{min}}$ . We emphasize that these choices impact the speed-variance tradeoff. By setting a larger  $J_{\mathrm{min}}$  we decrease the speed by requiring more baseline computations but also decrease the variance (since the minimum approximations have the largest deviations from the ground truth).
+
+Toy problem. In Fig. 1 we plot the empirical mean of the RR-CG estimator using  $10^{4}$  samples from an exponential truncation distribution. We find that RR-CG produces unbiased estimates of the  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  and  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  terms that are indistinguishable from the exact values computed with Cholesky. Reducing the expected truncation iteration  $\mathbb{E}(J)$  (x-axis) increases the standard error of empirical means, demonstrating the speed-variance trade-off.
+
+# 4.2. Single Sample-Truncated RFF (SS-RFF)
+
+Denoting  $\widetilde{\mathbf{K}}_j$  as the kernel matrix estimated by  $j$  random Fourier features, we can write  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  as the following telescoping series:
+
+$$
+\begin{array}{l} \log | \widehat {\mathbf {K}} _ {\mathbf {X X}} | = \log | \widetilde {\mathbf {K}} _ {1} | + \sum_ {j = 2} ^ {N / 2 - 1} \left(\log | \widetilde {\mathbf {K}} _ {j} | - \log | \widetilde {\mathbf {K}} _ {j - 1} |\right) (18) \\ + \log | \widehat {\mathbf {K}} _ {\mathbf {X X}} | - \log | \widetilde {\mathbf {K}} _ {N / 2 - 1} | \\ = \log | \widetilde {\mathbf {K}} _ {1} | + \sum_ {j = 2} ^ {N / 2} \Delta_ {j}, (19) \\ \end{array}
+$$
+
+where  $\Delta_{j}$  is defined as  $\log |\widetilde{\mathbf{K}}_j| - \log |\widetilde{\mathbf{K}}_{j - 1}|$  for all  $j < N / 2$ , and  $\Delta_{N / 2}$  is defined as  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}| - \log |\widetilde{\mathbf{K}}_{N / 2 - 1}|$ . Note that each  $\Delta_{j}$  is now a random variable, since it depends on the random Fourier frequencies  $\omega$ . Crucially, we only include  $N / 2$  terms in the series so that no term requires more than  $\mathcal{O}(N^3)$  computation in expectation. (For any  $j > N / 2$ ,  $\widetilde{\mathbf{K}}_j$  is a full-rank matrix and thus is as computationally expensive as the true  $\widehat{\mathbf{K}}_{\mathbf{XX}}$  matrix.) We construct a similar telescoping series for  $\mathbf{y}^\top \widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$ .
+
+As with Eq. (16), we approximate the series in Eq. (19) with a randomized truncation estimator, this time using the Single Sample estimator (7):
+
+$$
+\log | \widehat {\mathbf {K}} _ {\mathbf {X X}} | \approx \log | \widetilde {\mathbf {K}} _ {1} | + \Delta_ {J} / \mathbb {P} (J). \tag {20}
+$$
+
+where  $J$  is drawn from the truncation distribution  $\mathbb{P}(J)$  with support over  $\{2,3,\dots N / 2\}$ . Note that the Single Sample estimate requires computing 3 log determinants (log  $|\widetilde{\mathbf{K}}_1|$ ,  $\log |\widetilde{\mathbf{K}}_{J - 1}|$ , and  $\log |\widetilde{\mathbf{K}}_J|)$  for a total of  $\mathcal{O}(J^3 +NJ^2)$  computations and  $\mathcal{O}(NJ)$  memory. This is asymptotically the same requirement as standard RFF. The Russian Roulette estimator, on the other hand, incurs a computational cost of  $\mathcal{O}(NJ^3 +J^4)$  as it requires computing ( $\log |\widetilde{\mathbf{K}}_1|$  through  $\log |\widetilde{\mathbf{K}}_J|$ ) which quickly becomes impractical for large  $J$ .
+
+A similar Single Sample estimator constructs an unbiased estimate of  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$ . Backpropagating through these Single Sample estimates produces unbiased estimates of the log marginal likelihood gradient in Eq. (2).
+
+Choosing the truncation distribution. For the Single Sample estimator we do not have to optimize the ROE since
+
+minimizing the variance of this estimator does not result in a degenerate distribution.
+
+Theorem 4. The truncation distribution that minimizes the variance of the SS-RFF estimators for  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  and  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  is  $\mathbb{P}^{*}(J)\propto 1 / J$ . The expected variance and computation of  $\mathbb{P}^{*}(J)$  is finite.
+
+Proof sketch. The minimum variance distribution can be found by solving a constrained optimization problem. In practice, we can further decrease the variance of our estimator by fixing a minimum value of RFF features to be used in Eq. (19) and by increasing the step size  $(c\in \mathbb{N})$  between the elements at each  $\Delta_j = \log |\widetilde{\mathbf{K}}_{cJ}| - \log |\widetilde{\mathbf{K}}_{c(J - 1)}|$ . See appendix for full proof.
+
+For the experiments we started with 500 features and also tried various step sizes  $c \in \{1, 10, 100\}$ . The variance of the estimator decreases as we increase  $c$  since the probability weights will decrease in magnitude. Yet, despite of using the optimal truncation distribution, setting a high number of features and taking long steps  $c = 100$ , the variance of the estimator still requires taking several steps before converging, making SS-RFF computationally impractical.
+
+Toy problem. Similar to RR-CG, in Fig. 1 we plot the empirical mean of the SS-RFF estimator using  $10^{4}$  samples. We find that SS-RFF produces unbiased estimates of the  $\mathbf{y}^{\top}\widehat{\mathbf{K}}_{\mathbf{XX}}^{-1}\mathbf{y}$  and  $\log |\widehat{\mathbf{K}}_{\mathbf{XX}}|$  terms. However, these estimates have a higher variance when compared to the estimates of RR-CG. Reducing the expected truncation iteration  $\mathbb{E}(J)$  (x-axis) increases the standard error of the empirical means, demonstrating the speed-variance trade-off.
+
+# 4.3. Analysis of the Bias-free Methods
+
+Randomized truncations and conjugate gradients have existed for many decades (Hestenes et al., 1952; Kahn, 1955), but have rarely been used in conjunction. Filippone & Engler (2015) proposed a method closely related to our RRCG which performs randomized early-truncation of CG iterations to obtain unbiased posterior samples of the GP covariance parameters. We differ by tackling the GP hyperparameter learning problem: we provide the first theoretical analysis of the biases incurred by both CG and RFF, and proceed to tailor unbiased estimators for each method.
+
+To some extent, randomized truncation methods are antithetical to the original intention of CG: producing deterministic and nearly exact solves. For large-scale applications, where early truncation is necessary for computational tractability, the ability to trade bias for variance is beneficial. This fact is especially true in the context of GP learning, where the bias of early truncation is systematic and cannot be simply explained away as numerical imprecision.
+
+Randomized truncation estimates are often used to estimate
+
+![](images/495420eb0e1ebd18b3860233d98c70be461c456f67a546cd3aab9b44acb261bc.jpg)  
+Figure 3. Optimization landscape of a GP with two hyperparameters. The SS-RFF and RR-CG models converge to similar hyperparameter values that are nearly optimal, while the RFF and CG models converge to suboptimal solutions. In addition, the stochastic effect of the randomized truncation is visible in the trajectories of RR-CG and SS-RFF. Moreover, CG (and RR-CG) models truncate after 20 iterations (in expectation); RFF (and SS-RFF) models use 700 features (in expectation).
+
+infinite series, where it is challenging to design truncation distributions with finite expected computation and/or variance. We avoid such issues since CG and the telescoping RFF summations are both finite.
+
+# 5. Results
+
+First, we show that our bias-free methods recover nearly the same hyperparameters as exact methods (i.e. Cholesky-based optimization), whereas models that use CG and RFF converge to suboptimal hyperparameters. Since RR-CG and SS-RFF eliminate bias at the cost of increased variance, we then demonstrate the optimization convergence rate and draw conclusions on our methods' applicability. Finally, we compare models optimized with RR-CG against a host of approximate GP methods across a wide range of UCI datasets (Asuncion & Newman, 2007). All experiments are implemented in GPyTorch (Gardner et al., 2018)
+
+Optimization trajectories of bias-free GP. Fig. 3 displays the optimization landscape – the log marginal likelihood of the PoleTele dataset – as a function of (RBF) kernel lengthscale  $\ell$  and noise  $\sigma^2$ . As expected, an exact GP (optimized using Cholesky, Fig. 3 upper left) recovers the optimum. Notably, the GP trained with standard CG and RFF converges to suboptimal hyperparameters (upper right/ lower right). RR-CG and SS-RFF models (trained with 20 iterations and 700 features in expectation, respectively) successfully eliminate this bias, and recover nearly
+
+![](images/dcd08e7d969e8e426549f2ba1059524140bdb3052de3b257fc1793709a6344ac.jpg)  
+Figure 4. The GP optimization objective for models trained with RFF and SS-RFF. (PoleTele dataset, RBF kernel, Adam optimizer.) RFF models converge to sub-optimal log marginal likelihoods. SS-RFF models converge to (near) optimum values, yet require more than  $100 \times$  as many optimization steps.
+
+![](images/4e96164922f83c763403b80bec0a93cf8d9026cb7a85c2957e37581ddcd4a1e2.jpg)  
+Figure 5. The GP optimization objective for models trained with CG and RR-CG. (Bike dataset, RBF kernel, Adam optimizer.) RR-CG models converge to optimal solutions, while the (biased) CG models diverge. Increasing the expected truncation of RR-CG only slightly improves optimization convergence; models converge in  $< 100$  steps of Adam.
+
+the same parameters as the exact model (upper center/ lower left). These plots also show the speed-variance tradeoff of randomized truncation. SS-RFF and RR-CG have noisy optimization trajectories due to auxiliary truncation variance.
+
+Convergence of GP hyperparameter optimization. Figs. 4 and 5 plot the exact GP log marginal likelihood of the parameters learned by each method during optimization. Each trajectory corresponds to a RBF-kernel GP trained on the PoleTele dataset (Fig. 4) and the Bike dataset (Fig. 5).
+
+Fig. 4 shows that RFF models converge to solutions with worse log likelihoods, and more RFF features slow the rate of optimization. Additionally, we see the cost of the auxiliary variance needed to debias SS-RFF: while SS-RFF models achieve better optima than their biased counterparts, they take 2-3 orders of magnitude longer to converge, despite using a truncation distribution that minimizes variance. We thus conclude that SS-RFF has too much variance to be practical for GP hyperparameter learning.
+
+Fig. 5 on the other hand shows that RR-CG is minimally affected by its auxiliary variance. The GP trained with RR-CG converges in roughly 100 iterations, nearly matching Cholesky-based optimization. Decreasing the expected truncation value from  $\mathbb{E}[J] = 40$  to 20 slightly slows this convergence. We note that the bias induced by standard CG can be especially detrimental to GP learning. On this dataset, the biased models deviate from their Cholesky counterparts and eventually diverge away from the optimum.
+
+Predictive performance of bias-free GP. Lastly, we compare the predictive performance of GPs that use RR-CG, CG, and Cholesky for hyperparameter optimization. We emphasize that the RR-CG and CG methods only make use of early truncation approximations during training. At test time, we compute the predictive posterior by running CG to a tolerance of  $\leq 10^{-4}$ , which we believe can be considered "exact" for practical intents and purposes. Additionally, we include four other (biased) scalable GP approximations methods as baselines: RFF, Stochastic Variational Gaussian Processes (SVGP) (Hensman et al., 2013), generalized Product of Expert Gaussian Processes (POE) (Cao & Fleet, 2014; Deisenroth & Ng, 2015), and stochastic gradient-based Gaussian Processes (sgGP) (Chen et al., 2020). We note that the RFF, SVGP, and sgGP methods introduce both bias and variance, as these methods rely on randomization and approximation.
+
+We use CG with  $J = 100$  iterations, and RR-CG with  $\mathbb{E}[J] = 100$  expected iterations; both methods use the preconditioner of Gardner et al. (2018). All RFF models use 700 random features. For SVGP, we use 1,024 inducing points and minibatches of size 1,024 as in (Wang et al., 2019). The POE models are comprised of GP experts that are each trained on 1,024 data point subsets. For sgGP, the subsampled datasets are constructed by selecting a random point  $x, y$  and its 15 nearest neighbors as in (Chen et al., 2020). Each dataset is randomly split to  $64\%$  training,  $16\%$  validation and  $20\%$  testing sets. All kernels are RBF with a separate lengthscale per dimension. See appendix for more details.
+
+We report prediction accuracy (RMSE) and negative log likelihood (NLL) in Fig. 6 (see appendix for full tables on predictive performance and training time). We make two key observations: (i) RR-CG meaningfully debiases CG. When the bias of CG is not detrimental to optimization (e.g. CG with 100 iterations is close to convergence for the Elevators dataset), RR-CG has similar performance. However, when the CG bias is more significant (e.g. the KEGG dataset), the bias-free RR-CG improves the GP predictive RMSE and NLL. We also include a figure displaying the predictive performance of RR-CG and CG with increasing number of (expected) CG iterations in appendix. (ii) RR-CG recovers the same optimum as the "ground-truth" method
+
+![](images/fe3140defce32f95a96df48d017812ef4a61f27a7d073467b03ff6f9307c440e.jpg)  
+Figure 6. Root-mean-square-error (RMSE) and negative log likelihood (NLL) of GP trained with CG (light purple), RR-CG (dark purple) and various approximate methods (grey). Dashed red lines indicate Cholesky-based GP performance (when applicable). Results are averaged over 3 dataset splits. Missing RFF and sgGP results correspond to (very high) outlier NLL / RMSE values. In almost all experiments, GP learning with RR-CG achieves similar or better performance compared to that with CG at the same computational cost.
+
+(i.e. Cholesky) does, as indicated by the red-dashed line in Fig. 6. This result provides additional evidence that RR-CG achieves unbiased optimization.
+
+While RR-CG obtains the lowest RMSE on all but 2 datasets, we note that the (biased) GP approximations sometimes achieve lower NLL. For example, SVGP has a lower NLL than that of RR-CG on the Bike dataset, despite having a higher RMSE. We emphasize that this is not a failing of RR-CG inference. The SVGP NLL is even better than that of the exact (Cholesky) GP, suggesting a potential model misspecification for this particular dataset. Since SVGP overestimates the observational noise  $\sigma^2$  (Bauer et al., 2016), it may obtain a better NLL when outliers are abundant. Though we cannot compare against the Cholesky posterior on larger datasets, we hypothesize that the NLL/RMSE discrepancy on these datasets is due to a similar modeling issue.
+
+# 6. Conclusion
+
+We prove that CG and RFF introduce systematic biases to the GP log marginal likelihood objective: CG-based training will favor underfitting models, while RFF-based training will promote overfitting. Modifying these methods with randomized truncation converts these biases into variance, enabling unbiased stochastic optimization. Our results show that this bias-to-variance exchange indeed constitutes a trade-off. The convergence of SS-RFF is impractically slow, likely due to the truncation variance needed to eliminate RFF's slowly-decaying bias. However, for CG-based
+
+training, we find that variance is almost always preferable to bias. Models trained with RR-CG achieve better performance than those trained with standard CG, and tend to recover the hyperparameters learned with exact methods. Though models trained with CG do not always exhibit noticeable bias, RR-CG's negligible computational overhead is justifiable to counteract cases where the bias is significant.
+
+We reported experiments with at most 300K observations for our methods and baselines, which is substantial for GPs. We emphasize that RR-CG can be extended to datasets with over one million data points as in Wang et al. (2019). However, the computational cost is much higher, requiring multiple GPUs for training and testing.
+
+We note that the RR-CG algorithm is not limited to GP applications. Future work should explore applying RR-CG to other optimization problems with large-scale solves.
+
+# Acknowledgements
+
+This work was supported by the Simons Foundation, McKnight Foundation, the Grossman Center, and the Gatsby Charitable Trust.
+
+# References
+
+Asuncion, A. and Newman, D. Uci machine learning repository, 2007.
+
+Bauer, M., van der Wilk, M., and Rasmussen, C. E. Understanding probabilistic sparse gaussian process approxi
+
+mations. In Advances in Neural Information Processing Systems, 2016.  
+Beatson, A. and Adams, R. P. Efficient optimization of loops and limits with randomized telescoping sums. In International Conference on Machine Learning, 2019.  
+Bochner, S. et al. Lectures on Fourier integrals, volume 42. Princeton University Press, 1959.  
+Burt, D., Rasmussen, C. E., and Van Der Wilk, M. Rates of convergence for sparse variational gaussian process regression. In International Conference on Machine Learning, pp. 862-871, 2019.  
+Cao, Y. and Fleet, D. J. Generalized product of experts for automatic and principled fusion of gaussian process predictions. arXiv preprint arXiv:1410.7827, 2014.  
+Chen, H., Zheng, L., Al Kontar, R., and Raskutti, G. Stochastic gradient descent in correlated settings: A study on gaussian processes. Advances in Neural Information Processing Systems, 33, 2020.  
+Chen, R. T. Q., Behrmann, J., Duvenaud, D., and Jacobsen, J.-H. Residual flows for invertible generative modeling. Advances in Neural Information Processing Systems, 2019.  
+Cunningham, J. P., Shenoy, K. V., and Sahani, M. Fast gaussian process methods for point process intensity estimation. In International Conference on Machine learning, pp. 192-199, 2008.  
+Cutajar, K., Osborne, M. A., Cunningham, J. P., and Filippone, M. Preconditioning kernel matrices. In International Conference on Machine Learning, 2016.  
+Datta, A., Banerjee, S., Finley, A. O., and Gelfand, A. E. Hierarchical nearest-neighbor gaussian process models for large geostatistical datasets. Journal of the American Statistical Association, 111(514):800-812, 2016.  
+Deisenroth, M. and Ng, J. W. Distributed gaussian processes. In International Conference on Machine Learning, pp. 1481-1490. PMLR, 2015.  
+Dong, K., Eriksson, D., Nickisch, H., Bindel, D., and Wilson, A. G. Scalable log determinants for gaussian process kernel learning. In Advances in Neural Information Processing Systems, pp. 6327-6337, 2017.  
+Filippone, M. and Engler, R. Enabling scalable stochastic gradient-based inference for gaussian processes by employing the unbiased linear system solver (ulisse). In International Conference on Machine Learning, pp. 1015-1024. PMLR, 2015.
+
+Gardner, J., Pleiss, G., Weinberger, K. Q., Bindel, D., and Wilson, A. G. Gpytorch: Blackbox matrix-matrix gaussian process inference withgpu acceleration. In Advances in Neural Information Processing Systems, pp. 7576-7586, 2018.  
+Golub, G. H. and Van Loan, C. F. Matrix Computations. The Johns Hopkins University Press, 4th Edition, 2012.  
+Hensman, J., Fusi, N., and Lawrence, N. D. Gaussian processes for big data. In Uncertainty in Artificial Intelligence, 2013.  
+Hestenes, M. R., Stiefel, E., et al. Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49(1), 1952.  
+Hutchinson, M. F. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Communications in Statistics-Simulation and Computation, 18(3):1059-1076, 1989.  
+Kahn, H. Use of different monte carlo sampling techniques. Rand Corporation, 1955.  
+Katzfuss, M., Guinness, J., et al. A general framework for vecchia approximations of gaussian processes. Statistical Science, 36(1):124-141, 2021.  
+Loper, J., Blei, D., Cunningham, J. P., and Paninski, L. General linear-time inference for gaussian processes on one dimension. arXiv preprint arXiv:2003.05554, 2020.  
+Luo, Y., Beatson, A., Norouzi, M., Zhu, J., Duvenaud, D., Adams, R. P., and Chen, R. T. Q. Sumo: Unbiased estimation of log marginal probability for latent variable models. In International Conference on Learned Representations, 2020.  
+Lyne, A.-M., Girolami, M., Atchade, Y., Strathmann, H., Simpson, D., et al. On russian roulette estimates for bayesian inference with doubly-intractable likelihoods. Statistical science, 30(4):443-467, 2015.  
+Mutny, M. and Krause, A. Efficient high dimensional bayesian optimization with additivity and quadrature fourier features. Advances in Neural Information Processing Systems, pp. 9005-9016, 2018.  
+Nowozin, S. Debiasing evidence approximations: On importance-weighted autoencoders and jackknife variational inference. In International Conference on Learned Representations, 2018.  
+Oktay, D., McGreivy, N., Aduol, J., Beatson, A., and Adams, R. P. Randomized automatic differentiation. arXiv preprint arXiv:2007.10412, 2020.
+
+Rahimi, A. and Recht, B. Random features for large-scale kernel machines. In Advances in Neural Information Processing Systems, pp. 1177-1184, 2008.  
+Snelson, E. and Ghahramani, Z. Sparse Gaussian processes using pseudo-inputs. In Advances in Neural Information Processing Systems, 2006.  
+Sutherland, D. J. and Schneider, J. G. On the error of random fourier features. Conference on Uncertainty in Artificial Intelligence, 2015.  
+Titsias, M. K. Variational learning of inducing variables in sparse Gaussian processes. In International Conference on Artificial Intelligence and Statistics, pp. 567-574, 2009.  
+Ubaru, S., Chen, J., and Saad, Y. Fast estimation of tr(f(a)) via stochastic lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4):1075-1099, 2017.  
+Wang, K., Pleiss, G., Gardner, J., Tyree, S., Weinberger, K. Q., and Wilson, A. G. Exact gaussian processes on a million data points. In Advances in Neural Information Processing Systems, pp. 14648-14659, 2019.  
+Wilson, J., Borovitskiy, V., Terenin, A., Mostowsky, P., and Deisenroth, M. Efficiently sampling functions from gaussian process posteriors. In International Conference on Machine Learning, pp. 10292-10302, 2020.

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+# Bias-Robust Bayesian Optimization via Dueling Bandits
+
+Johannes Kirschner<sup>1</sup> Andreas Krause<sup>1</sup>
+
+# Abstract
+
+We consider Bayesian optimization in settings where observations can be adversarially biased, for example by an uncontrolled hidden confounder. Our first contribution is a reduction of the confounded setting to the dueling bandit model. Then we propose a novel approach for dueling bandits based on information-directed sampling (IDS). Thereby, we obtain the first efficient kernelized algorithm for dueling bandits that comes with cumulative regret guarantees. Our analysis further generalizes a previously proposed semiparametric linear bandit model to non-linear reward functions, and uncovers interesting links to doubly-robust estimation.
+
+# 1. Introduction
+
+Bayesian optimization (Mockus, 1982) is a model-based approach for zero-order global optimization with noisy feedback. It has been successfully applied to many applications such as hyper-parameter tuning of machine learning models, robotics and chemical design. Some variants such as Expected-Improvement (Bull, 2011) or the GP-UCB algorithm (Srinivas et al., 2010) come with theoretical guarantees, ensuring convergence to the global optimum in finite time under suitable regularity assumptions. Closely related is the field of bandit algorithms (Lattimore & Szepesvári, 2020), in particular the linear bandit model (Abe & Long, 1999; Dani et al., 2008; Abbasi-Yadtori et al., 2011).
+
+Most linear bandit algorithms and Bayesian optimization approaches require that the true function is realized in a known reproducing kernel Hilbert space (RKHS) and rely on unbiased evaluations of the objective. The regularity assumptions raise questions of robustness to miss-specification and adversarial attacks, and addressing these limitations has been the content of several recent works (Lykouris et al., 2018; Bogunovic et al., 2020b;a).
+
+We study a setting where the learner's objective is to maximize an unknown function  $f: \mathcal{X} \to \mathbb{R}$  with additive confounded feedback,
+
+$$
+y _ {t} = f \left(x _ {t}\right) + b _ {t} + \epsilon_ {t}, \tag {1}
+$$
+
+where  $x_{t}$  is the evaluation point chosen by the learner at time  $t$ ,  $\epsilon_{t}$  is  $\sigma^2$ -sub-Gaussian (zero-mean) observation noise, and  $b_{t}$  is an additive confounding term. We assume that  $b_{t}$  is chosen by an adversary, but does not depend on the input  $x_{t}$ . The bias term allows to model the influence of an unobserved and uncontrolled covariate, or a perturbation of the feedback signal imposed by an adversary. One can also interpret (1) as a contextual model, where  $b_{t}$  captures the effect of a changing (unobserved) context on the reward. We discuss further examples and applications in Section 1.1 below.
+
+The proposed feedback model (1) generalizes the semi-parametric contextual bandit model studied by Krishnamurthy et al. (2018), where  $f(x_{t}) = \langle x_{t},\theta \rangle$  is a linear function defined by a parameter  $\theta \in \mathbb{R}^d$ . They show that a doubly-robust least-squares estimator allows to recover reward differences  $\langle x - x',\theta \rangle$  for inputs  $x,x^{\prime}\in \mathbb{R}^{d}$  despite the confounding. They further propose an elimination-style algorithm, bandit orthogonalized semiparametric estimation (BOSE), which is based on sampling actions from a distribution that minimizes the variance of the estimator. However, finding low variance distributions requires solving a convex-quadratic feasibility problem, which is computationally demanding and in general leads to sampling distributions with support that spans  $\mathbb{R}^d$ .
+
+Contributions Our first contribution is two reductions of the confounded feedback model (1) to the dueling bandit setting. This allows us to leverage existing algorithms for dueling bandits in the confounded observation setting. We then propose the first efficient algorithm for kernelized dueling bandits that comes with theoretical guarantees on the cumulative regret. The approach is based on information-directed sampling (IDS), which was recently studied in the context of linear partial monitoring by Kirschner et al. (2020b). In particular, we propose an efficient approximation of IDS, that reduces the computation complexity from  $\mathcal{O}(|\mathcal{X}|^4)$  to  $\mathcal{O}(|\mathcal{X}|)$  on finite action sets  $\mathcal{X}$ . For continuous action sets, the proposed algorithm requires to optimize a (non-convex) acquisition function over the input space, akin to standard Bayesian optimization.
+
+# 1.1. Motivating Examples
+
+We start by motivating our problem through applications.
+
+Range-Adjusting Measurement Devices In many real-world optimization tasks, the observed feedback arises from a physical sensing device. Such measurement devices can be subject to calibration errors or might automatically adjust the output range for better sensitivity. For example, in optimization of free electron lasers (Kirschner et al., 2019; Duris et al., 2020), the target signal is measured with a gas-detector, which exploits a physical law to amplify the signal. An input voltage is used to control the amplification factor, which requires re-adjustment<sup>1</sup> with increasing target signal such that the physical relationship between the target and the measured output stays approximately linear<sup>2</sup> (Sorokin et al., 2019; Juranic et al., 2018). Our feedback model allows optimization that is robust to absolute changes in the target signal occurring at any time.
+
+Distributed Optimization of Additive Functions In high-dimensional settings, previous work on Bayesian optimization often uses structural assumptions to reduce the dependence of the sample complexity on the dimension. One popular choice is additivity (Kandasamy et al., 2015), for example coordinate-wise  $f(x) = \sum_{i\in [d]}f_i(x^i)$ . To optimize additive functions, we can apply  $d$  individual learners to optimize each 1-dimensional component  $f_{i}$  separately. Note that the learners cannot directly evaluate  $f_{i}(x_{i})$ , but only obtain the global noisy feedback  $f(x) + \epsilon$ , which depends on the choices  $x = (x^{1},\ldots ,x^{d})$  of all learners. When the learners act in parallel and there is no communication possible, the feedback of each learner is confounded as in (1) by the other learner's choices. Our robust approach guarantees that the learners are able to optimize each component successfully, despite the confounding.
+
+Adversarial Attacks Robustness to adversarial attacks was studied recently in the context of bandit algorithms (Lykouris et al., 2018; Bogunovic et al., 2020b) and Bayesian optimization (Bogunovic et al., 2020a). In all previous work that we are aware of, the corruption of the feedback is allowed to depend on the actions, with varying assumptions of whether or not the adversary observes the action choice of the learner. Note that our feedback model is more stringent, as it does not allow for action dependence. On the other hand, our model allows for sublinear regret even with a constant corruption in every round, whereas in previous work, the regret scales with the total amount of corruption.
+
+1Translating the range-adjusted signal into an absolute value is possible, but not straightforward and comes with other limitations. 2More precisely, the relationship depends on the photon energy, pulse intensity, and physical properties of the gas involved.
+
+# 1.2. Related Work
+
+There is a vast amount of literature on bandit algorithms (Lattimore & Szepesvári, 2020) and Bayesian optimization (Mockus, 1982; Srinivas et al., 2010; Shahriari et al., 2016; Frazier, 2018). Our feedback model is a generalization of the semi-parametric linear bandit setting proposed by Krishnamurthy et al. (2018) with applications for example in mobile health (Tewari & Murphy, 2017). A variant of Thompson sampling was analyzed in the same setting by Kim & Paik (2019), which they show outperforms the BOSE approach by Krishnamurthy et al. (2018), but the frequentist regret bound they derive has an extra factor in the dimension. The Thompson sampling variant is also computationally more efficient, but requires to explicitly compute the probabilities that each action is optimal under the posterior distribution. For Bayesian optimization, robust variants have been considered recently, for example with adversarial perturbations of the input (Bogunovic et al., 2018), corruption of the output (Bogunovic et al., 2020a) and distributionally robust optimization (Kirschner et al., 2020a). In the context of adversarial attacks, there is an increasing body of work (Lykouris et al., 2018; Li et al., 2019; Liu & Lai, 2020; Gupta et al., 2019; Bogunovic et al., 2020a;b), however the feedback model differs from ours, see also the discussion in Section 1.1.
+
+Of particular relevance to our work is the (stochastic)ueling bandit setting (Yue et al., 2012; Sui et al., 2018; Bengs et al., 2021) and kernelized variants (Sui et al., 2017a;b). Early work by Yue & Joachims (2009) applied theueling bandit model to the optimization setting with continuous action sets and concave reward functions, and established a connection to gradient-based optimization. However, to the best of our knowledge, none of the previous works provide bounds on the cumulative regret in the kernelized (nonconcave) setting. A kernelized algorithm with theoretical guarantees that requires point evaluations andueling feedback is by Xu et al. (2020). Closely related is also the work by Saha & Gopalan (2020) on linearueling bandits with possibly infinite input spaces. They establish a connection to the generalized linear bandit model (GLM) and propose an algorithm which, similar to ours, relies on finding an informative action pair with low regret. However, gap-dependent bounds and a kernelized variant was not provided, and for finite action sets  $\mathcal{X}$ , their algorithm requires  $\mathcal{O}(|\mathcal{X}|^2)$  computation steps per round. Recent work by Agarwal et al. (2021) considers the finite-armedueling bandit setting with adversarial corruptions of the feedback. Lastly, we remark that most previous work onueling bandits considers binary feedback, whereas here we are interested in quantitative feedback on the reward-difference between the chosen action pair. Formally, we consider sub-Gaussianueling feedback which includes the Bernoulli likelihood, but does not exploit the heteroscedasticity of binary observations.
+
+# 2. Setting
+
+Let  $\mathcal{X} \subset \mathbb{R}^d$  be a compact input space and  $f: \mathcal{X} \to \mathbb{R}$  a fixed and unknown objective function. In each round  $t = 1, \ldots, n$ , the learner chooses an action  $x_t \in \mathcal{X}$  and observes the confounded outcome
+
+$$
+y _ {t} = f (x _ {t}) + b _ {t} + \epsilon_ {t},
+$$
+
+where  $\epsilon_{t}$  is  $\sigma^2$ -sub-Gaussian, conditionally independent noise, and  $b_{t} \in \mathbb{R}^{d}$  is an unobserved and possibly time-dependent confounding term. We assume that  $b_{t}$  does not depend on the current action  $x_{t}$  chosen by the learner, and satisfies one of the following assumptions:
+
+a) The bias  $b_{t}$  is bounded,  $|b_{t}| \leq C_{max}$  and fixed at the beginning of round  $t$ , but can otherwise arbitrarily depend on  $(x_{s}, y_{s})_{s=1}^{t-1}$ .  
+b) The difference between two consecutive bias terms is bounded,  $|b_{t} - b_{t-1}| \leq D_{max}$  and  $b_{t}$  is fixed at the beginning of round  $t-1$ , but can otherwise arbitrarily depend on  $(x_{s}, y_{s})_{s=1}^{t-2}$ .
+
+Which assumption is used is specified in the relevant context. Let  $x^{*} \in \arg \max_{x \in \mathcal{X}} f(x)$  be the optimal action. The suboptimality gap is  $\Delta(x) = f(x^{*}) - f(x)$ . The learner's objective is to maximize the cumulative reward  $\sum_{t=1}^{n} f(x_{t})$ , or equivalently minimize the regret,
+
+$$
+R _ {n} = \sum_ {t = 1} ^ {n} f (x ^ {*}) - f (x _ {t}) = \sum_ {t = 1} ^ {n} \Delta (x _ {t}).
+$$
+
+For the analysis, we assume that the function  $f$  is in a known reproducing kernel Hilbert space (RKHS)  $\mathcal{H}$  with associated kernel  $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$  and bounded Hilbert norm  $\| f \|_{\mathcal{H}} \leq B$ . This is a standard assumption in Bayesian optimization (Srinivas et al., 2010; Chowdhury & Gopalan, 2017) and justifies the use of kernelized least-squares regression, formally introduced in Section 4. We further require that the kernel function is bounded,  $k(x, x) \leq 1$ .
+
+# 3. Reduction to Dueling Bandits
+
+It is clear from the observation model (1) that any additive shift of the objective, i.e.  $\tilde{f}(x) = f(x) + c$  for  $c \in \mathbb{R}$ , can be absorbed in the unobserved confounding terms  $b_{t}$ , hence rendering the observation sequences for  $f(x)$  and  $\tilde{f}(x)$  indistinguishable. In particular, the learner can only hope to recover the true function up to an additive constant. Fortunately, to determine the best action  $x^{*} = \arg \max_{x \in \mathcal{X}} f(x)$ , it suffices to estimate reward differences  $f(x^{1}) - f(x^{2})$  for actions  $x^{1}, x^{2} \in \mathcal{X}$ , which is indeed possible. To do so, previous work in the linear setting relies on doubly-robust estimation (Krishnamurthy et al., 2018; Kim & Paik, 2019). Here
+
+we take a different approach, and propose a generic reduction of the feedback model (1) to the dueling bandit model (Yue et al., 2012; Sui et al., 2018). The reduction has the advantage that we can leverage existing algorithms for dueling bandits, which also eliminates the need to find low-variance sampling distributions for doubly-robust estimation.
+
+In dueling bandits, the learner chooses two actions  $x_{t}^{1}, x_{t}^{2} \in \mathcal{X}$  and obtains (noisy) feedback on which of the two actions has higher reward. Meanwhile, the learner suffers regret for both actions, but the reward of each action is not observed. While most previous work on dueling bandits uses a binary feedback model, here, we are concerned with a quantitative version of the same setting, which is a special case of linear partial monitoring (Lin et al., 2014). Specifically, for a reward function  $f: \mathcal{X} \to \mathbb{R}$  and actions  $x_{t}^{1}, x_{t}^{2} \in \mathcal{X}$ , we define quantitative dueling feedback as follows:
+
+$$
+d _ {t} = f \left(x _ {t} ^ {1}\right) - f \left(x _ {t} ^ {2}\right) + \xi_ {t}, \tag {2}
+$$
+
+where  $\xi_{t}$  is  $\rho^2$ -sub-Gaussian observation noise and  $\rho$  is known to the learner. The distribution of  $\xi_{t}$  is allowed to depend on  $(x_{t}^{1}, x_{t}^{2})$  as long as the sub-Gaussian tail assumption is satisfied uniformly over all actions. Note that this includes binary feedback typically used forueling bandits, and, more generally, bounded noise distributions. Next, we present two reductions schemes to generate theueling bandit feedback (2) from confounded observations (1).
+
+Two-Point Reduction The first scheme uses two confounded observations to construct the dueling bandit feedback. Given two inputs  $x_{t}^{1}, x_{t}^{2} \in \mathcal{X}$  in round  $t$ , we evaluate both points according to (1), where the order of evaluation is uniformly randomized. The two observations are
+
+$$
+\begin{array}{l} y _ {t} ^ {1} = f \left(x _ {t} ^ {1}\right) + b _ {2 t + i _ {t}} + \epsilon_ {2 t + i _ {t}}, \\ y _ {t} ^ {2} = f (x _ {t} ^ {2}) + b _ {2 t + 1 - i _ {t}} + \epsilon_ {2 t + 1 - i _ {t}}, \\ \end{array}
+$$
+
+where  $i_t \sim \text{Bernoulli}(0.5)$ . We then define
+
+$$
+d _ {t} = y _ {t} ^ {1} - y _ {t} ^ {2}. \tag {3}
+$$
+
+Assuming that  $b_{2t}$  and  $b_{2t+1}$  are fixed before either of  $x_t^1, x_t^2$  is chosen by the learner and using that the observation noise  $\epsilon_t$  is zero-mean, one easily confirms that  $\mathbb{E}[d_t] = f(x_t^1) - f(x_t^2)$ . We further use the following properties of sub-Gaussian random variables. Any random variable  $X$  such that  $X \in [-B, B]$  is  $B^2$ -sub-Gaussian. For two independent random variables  $X_1, X_2$  that are  $B_1^2$ - and  $B_2^2$ -sub-Gaussian respectively,  $X_1 + X_2$  is  $(B_1^2 + B_2^2)$ -sub-Gaussian. Hence if  $|b_{2t} - b_{2t+1}| \leq D_{\max}$ , it follows that
+
+$$
+\begin{array}{l} \xi_ {t} = d _ {t} - \mathbb {E} [ d _ {t} ] \\ = b _ {2 t + i _ {t}} + \epsilon_ {2 t + i _ {t}} - \left(b _ {2 t + 1 - i _ {t}} + \epsilon_ {2 t + 1 - i _ {t}}\right) \\ \end{array}
+$$
+
+is  $(D_{\mathrm{max}}^2 + 2\sigma^2)$ -sub-Gaussian.
+
+One-Point Reduction Perhaps surprisingly, one can also construct the dueling bandit feedback from a single observation using randomization. For given inputs  $x_{t}^{1}, x_{t}^{2} \in \mathcal{X}$  we choose one point uniformly at random and evaluate the confounded function (1) to obtain a single observation
+
+$$
+y _ {t} = f \left(x _ {t} ^ {(1 + i _ {t})}\right) + b _ {t} + \epsilon_ {t},
+$$
+
+where  $i_t \sim \text{Bernoulli}(0.5)$ . The dueling bandit feedback is
+
+$$
+d _ {t} = (- 1) ^ {i _ {t}} 2 y _ {t}. \tag {4}
+$$
+
+Again, we get an unbiased observation of the reward difference,  $\mathbb{E}[d_t] = f(x_t^1) - f(x_t^2)$ . Further, if  $|b_{t}|\leq C_{\mathrm{max}}$ , then  $\xi_{t} = d_{t} - \mathbb{E}[d_{t}]$  is  $4(C_{\mathrm{max}}^{2} + \sigma^{2})$ -sub-Gaussian. Compared to the two-point reduction, here the sub-Gaussian variance  $\rho$  depends on the absolute value  $|b_{t}|$  of the confounding term instead of the difference  $|b_{t} - b_{t + 1}|$ . On the other hand, the one-point sampling scheme only requires  $b_{t}$  to be fixed before the choice of  $x_{t}$ , but may depend on all previous actions and observations.
+
+# 4. Information-Directed Sampling
+
+With the reduction to dueling feedback, we are set to readily apply any dueling bandit algorithm in the confounded setting. Furthermore, dueling bandits (as defined in (2)) are a special case of partial monitoring, for which also several algorithms exist. In the following, we adapt the information-directed sampling (IDS) approach (Russo & Van Roy, 2014), more specifically, the version proposed by Kirschner et al. (2020b) for linear partial monitoring. The main reason for this choice is that IDS works with quantitative dueling feedback, whereas most other work on dueling bandits focuses on settings with Bernoulli likelihood. Also, IDS can be formulated as a kernelized algorithm, and comes with theoretical guarantees on the regret. However, a direct adaptation of IDS to the dueling setting as proposed by Kirschner et al. (2020b) requires  $\mathcal{O}(|\mathcal{X}|^4)$  computational steps per round for finite action sets. In the following, we introduce an approximation of IDS, which obtains the same theoretical guarantees and only requires to optimize a simple score function over the action set. The resulting efficient kernelized dueling bandit algorithm may be of independent interest. On a high level, IDS samples actions from a distribution that minimizes a trade-off between an estimate of the regret and an information gain, as we elaborate below.
+
+Kernel Regression for Dueling Feedback The first step is to set up kernelized least-squares regression for the Dueling bandit feedback. Recall that we assume that  $\mathcal{H}$  is a RKHS with kernel function  $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$  and  $f \in \mathcal{H}$  with  $\|f\|_{\mathcal{H}} \leq B$ . In round  $t$ , the learner has already collected data  $\mathcal{D}_t = \{(x_s^1, x_s^2, d_s)\}_{s=1}^{t-1}$ , where  $x_s^1, x_s^2 \in \mathcal{X}$  is the input pair chosen at step  $s$ , and  $d_s$  is quantitativeueling
+
+bandit feedback defined in Eq. (2). The kernel least-squares estimator with regularizer  $\lambda > 0$  is
+
+$$
+\hat {f} _ {t} = \underset {f \in \mathcal {H}} {\arg \min } \sum_ {s = 1} ^ {t - 1} \left(f \left(x _ {s} ^ {1}\right) - f \left(x _ {s} ^ {2}\right) - d _ {s}\right) ^ {2} + \lambda \| f \| _ {\mathcal {H}} ^ {2} \tag {5}
+$$
+
+The solution corresponds to the posterior mean of the Gaussian process model with kernel  $k$  and prior variance  $\lambda^{-1}$  and can be computed in closed form. Let  $\mathbf{d}_t = [d_1, \ldots, d_{t-1}]^\top$  be the vector which collects the observations and define  $K_t \in \mathbb{R}^{t-1 \times t-1}$  and  $k_t(x) \in \mathbb{R}^{t-1}$  for  $x \in \mathcal{X}$  as follows:
+
+$$
+\left[ K _ {t} \right] _ {i j} \triangleq k \left(x _ {i} ^ {1}, x _ {j} ^ {1}\right) - k \left(x _ {i} ^ {1}, x _ {j} ^ {2}\right) - k \left(x _ {i} ^ {2}, x _ {j} ^ {1}\right) + k \left(x _ {i} ^ {2}, x _ {j} ^ {2}\right)
+$$
+
+$$
+[ k _ {t} (x) ] _ {i} \triangleq k (x, x _ {i} ^ {1}) - k (x, x _ {i} ^ {2})
+$$
+
+The least-squares solution  $\hat{f}_t$  evaluated at any  $x\in \mathcal{X}$  is  $\hat{f}_t(x) = k_t(x)^\top (K_t + \lambda \mathbf{1})^{-1}\mathbf{d}_t$ , where  $\mathbf{1}$  is the identity matrix in the appropriate dimension. To compute uncertainty estimates, we further define
+
+$$
+k _ {t} (x, y) \triangleq k (x, y) - k _ {t} (x) ^ {\top} \left(K _ {t} + \lambda \mathbf {1}\right) ^ {- 1} k _ {t} (y),
+$$
+
+$$
+\psi_ {t} (x, z) \triangleq k _ {t} (x, x) ^ {2} + k _ {t} (z, z) ^ {2} - 2 k _ {t} (x, z). \tag {6}
+$$
+
+For any  $t \geq 1$ ,  $x_1, x_2 \in \mathcal{X}$ , the estimate  $\hat{f}_t$  satisfies with probability at least  $1 - \delta$ ,
+
+$$
+\begin{array}{l} \left| \hat {f} _ {t} \left(x ^ {1}\right) - \hat {f} _ {t} \left(x ^ {2}\right) - \left(f \left(x ^ {1}\right) - f \left(x ^ {2}\right)\right) \right| ^ {2} \\ \leq \beta_ {t, \delta} \psi_ {t} \left(x ^ {1}, x ^ {2}\right), \tag {7} \\ \end{array}
+$$
+
+where the confidence coefficient is chosen as follows,
+
+$$
+\beta_ {t, \delta} ^ {1 / 2} \triangleq \rho \sqrt {\log \det (\mathbf {1} + \lambda^ {- 1} K _ {t}) + 2 \log \frac {1}{\delta}} + \sqrt {\lambda} B.
+$$
+
+The confidence bound is by Abbasi-Yadkori (2012, Theorem 3.11), which improves upon earlier results by Srinivas et al. (2010).
+
+Gap Estimate We use  $\hat{f}_t$  to define an estimate  $\hat{\Delta}_t(x)$  of the suboptimality gaps  $\Delta (x) = f(x^{*}) - f(x)$  for all  $x\in \mathcal{X}$ . Let  $\hat{x}_t^* = \arg \max_{x\in \mathcal{X}}\hat{f}_t(x)$  be the empirical estimate of the maximizer. Note that  $\hat{x}_t^*$  is always defined, despite the fact that in general we can determine  $\hat{f}_t$  only up to a constant shift (which does not affect the maximizer). Define
+
+$$
+\delta_ {t} \triangleq \max _ {z \in \mathcal {X}} \hat {f} _ {t} (z) - \hat {f} _ {t} (\hat {x} _ {t} ^ {*}) + \beta_ {t, \delta} ^ {1 / 2} \psi_ {t} (\hat {x} _ {t} ^ {*}, z) ^ {1 / 2},
+$$
+
+where  $\psi_t(x,z)$  is defined in (6). Intuitively,  $\delta_t$  is the largest plausible regret that the learner can occur from playing  $\hat{x}_t^*$  given the confidence estimates. The gap estimate is defined for any  $x\in \mathcal{X}$  as follows:
+
+$$
+\hat {\Delta} _ {t} (x) \triangleq \delta_ {t} + \hat {f} _ {t} \left(\hat {x} _ {t} ^ {*}\right) - \hat {f} _ {t} (x). \tag {8}
+$$
+
+The gap estimate satisfies an upper bound on the true gaps, provided that (7) holds, as summarized in the next lemma.
+
+Lemma 1. With probability  $1 - \delta$ , for all  $x \in \mathcal{X}$  and  $t \geq 1$ ,
+
+$$
+\Delta (x) \leq 2 \hat {\Delta} _ {t} (x).
+$$
+
+Proof. With probability at least  $1 - \delta$
+
+$$
+\begin{array}{l} \Delta (x) = f \left(x ^ {*}\right) - f (x) \\ = \max  _ {z \in \mathcal {X}} f (z) - f \left(\hat {x} _ {t} ^ {*}\right) + f \left(\hat {x} _ {t} ^ {*}\right) - f (x) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \stackrel {(i)} {\leq} \max _ {z \in \mathcal {X}} \hat {f} _ {t} (z) - \hat {f} _ {t} (\hat {x} _ {t} ^ {*}) + \beta_ {t, \delta} ^ {1 / 2} \psi_ {t} (z, \hat {x} _ {t} ^ {*}) ^ {1 / 2} \\ + \hat {f} _ {t} (\hat {x} _ {t} ^ {*}) - \hat {f} _ {t} (x) + \beta_ {t, \delta} ^ {1 / 2} \psi_ {t} (\hat {x} _ {t} ^ {*}, x) ^ {1 / 2} \\ \end{array}
+$$
+
+$$
+\stackrel {(i i)} {=} \hat {\Delta} _ {t} (x) + \beta_ {t, \delta} ^ {1 / 2} \psi_ {t} (\hat {x} _ {t} ^ {*}, x) ^ {1 / 2}.
+$$
+
+Here,  $(i)$  uses (7) twice and  $(ii)$  is by definition of  $\hat{\Delta}_t(x)$ . On the other hand,
+
+$$
+\begin{array}{l} \hat {\Delta} _ {t} (x) = \max  _ {z \in \mathcal {X}} \hat {f} _ {t} (z) - \hat {f} _ {t} (x) + \beta_ {t, \delta} ^ {1 / 2} \psi_ {t} (z, \hat {x} _ {t} ^ {*}) ^ {1 / 2} \\ \geq \beta_ {t, \delta} ^ {1 / 2} \psi_ {t} (\hat {x} _ {t} ^ {*}, x) ^ {1 / 2}. \\ \end{array}
+$$
+
+We used that  $\psi_t(\hat{x}_t^*, x) = \psi_t(x, \hat{x}_t^*)$  in the last step. The claim follows with the last two displays combined.
+
+Note that this gap estimate is different from the choice proposed by Kirschner et al. (2020b), and importantly, allows us to reduce computational complexity.
+
+Information Gain IDS further requires an information gain function  $I_{t}(x^{1}, x^{2})$  defined for each action, which in our case consists of an input pair,  $x^{1}, x^{2} \in \mathcal{X}$ . Several choices were proposed by Kirschner et al. (2020b). Here we use the differential of the log-determinant potential,
+
+$$
+\begin{array}{l} I _ {t} \left(x ^ {1}, x ^ {2}\right) \triangleq \log \left(1 + \psi_ {t} \left(x ^ {1}, x ^ {2}\right)\right), \tag {9} \\ = \log \frac {\det  (\mathbf {1} + K _ {t + 1})}{\det  (\mathbf {1} + K _ {t})} \\ \end{array}
+$$
+
+This choice of information gain corresponds to the Bayesian mutual information  $\mathbb{I}_t\big(d_t;f|(x_t^1,x_t^2) = (x^1,x^2)\big)$  in the Gaussian process model. In particular, the total information gain for (9) resembles the Gaussian entropy
+
+$$
+\gamma_ {n} = \sum_ {t = 1} ^ {n} I _ {t} \left(x _ {t} ^ {1}, x _ {t} ^ {2}\right) = \log \det  \left(\mathbf {1} _ {n} + K _ {n + 1}\right). \tag {10}
+$$
+
+The log-determinant depends on the kernel function and upper bounds are known for many popular choices. For example, the linear kernel,  $k(x,x^{\prime}) = \langle x,x^{\prime}\rangle$ , satisfies  $\gamma_{n} \leq \mathcal{O}(d\log (n))$  and the RBF kernel,  $k(x,x^{\prime}) = \exp (-\| x - x^{\prime}\|_{2}^{2} / 2)$ , satisfies  $\gamma_{n} \leq \log (n)^{d + 1}$  (Srinivas et al., 2010, Theorem 5).
+
+Algorithm 1: Approx. IDS for Dueling Feedback  
+Input: Action set  $\mathcal{X}$ , confidence coefficient  $\beta_{t,\delta}(\delta)$ , DuelingFeedback  $(x_t^1, x_t^2)$ , e.g. via (3) or (4)  
+1 for  $t = 1, 2, 3, \ldots$  do  
+2  $\hat{x}_t^* \gets \arg \max_{x \in \mathcal{X}} \hat{f}_t(x)$   
+3  $\hat{\Delta}_t(x) \gets \delta_t + \hat{f}_t(\hat{x}_t^*) - \hat{f}_t(x) \quad // Eq. (8)$   
+4  $I_t(x) \gets \log(1 + \psi_t(\hat{x}_t^*, x)) \quad // Eq. (9)$   
+5  $x_t, p \gets \arg \min_{x \in \mathcal{X}, p \in [0,1]} \frac{\left( (1 - p) \delta_t + p \hat{\Delta}_t(x) \right)^2}{p I_t(x)}$   
+6  $B_t \sim Bernoulli(p)$   
+7 if  $B_t == 1$ , then  
+8  $(x_t^1, x_t^2) \gets (\hat{x}_t^*, x_t)$   
+9  $d_t \gets$  DuelingFeedback  $(x_t^1, x_t^2)$   
+10 else  
+11  $(x_t^1, x_t^2) \gets (\hat{x}_t^*, \hat{x}_t^*) \quad // No feedback$
+
+(Approximate) Information-Directed Sampling Given the gap estimate  $\hat{\Delta}_t(x)$  and information gain  $I_t(x^1, x^2)$ , we optimize the following trade-off jointly over  $\mathcal{X} \times [0,1]$ ,
+
+$$
+z _ {t}, p _ {t} \triangleq \underset {z \in \mathcal {X}, p \in [ 0, 1 ]} {\arg \min } \frac {\left((1 - p) \delta_ {t} + p \hat {\Delta} _ {t} (z)\right) ^ {2}}{p I _ {t} \left(\hat {x} _ {t} ^ {*} , z\right)}. \tag {11}
+$$
+
+Note that for fixed  $z$ , the optimal trade-off probability is  $p(z) = \min \left(\frac{\delta_t}{\hat{\Delta}_t(z) - \delta_t}, 1\right)$ . Consequently, IDS samples the pair  $(\hat{x}_t^*, z_t)$  with probability  $p_t$ , and otherwise, with probability  $1 - p_t$ , the greedy pair  $(\hat{x}_t^*, \hat{x}_t^*)$ . Note that choosing the same action for dueling feedback provides no information, and in particular the least-squares estimate remains unchanged. The approach is summarized in Algorithm 1.
+
+Computational Complexity As usual, computing the kernel estimates requires to invert the kernel matrix  $K_{t}$ . With incremental updates, the exact kernel estimate and all related quantities can be computed in  $\mathcal{O}(d^{2}n^{3})$  steps in total over  $n$  rounds. Note that  $\hat{x}_t^*$  and  $\delta_t$  can be computed in  $\mathcal{O}(|\mathcal{X}|)$  steps assuming that all kernel quantities have been pre-computed. The trade-off (11) can be computed in  $\mathcal{O}(|\mathcal{X}|)$  steps, since it only requires to evaluate the gap estimates  $\hat{\Delta}(x)$  and information gain  $I_{t}(\hat{x}_{t}^{*}, x)$  for all  $x \in \mathcal{X}$ . Therefore the overall complexity is  $\mathcal{O}(d^{2}n^{3}|\mathcal{X}|)$ . Of course, Algorithm 1 can also be applied in the linear setting without kernelization, in which case the overall complexity is  $\mathcal{O}(d^{2}n|\mathcal{X}|)$ .
+
+We remark that (11) is an approximation of the IDS trade-off proposed by Kirschner et al. (2020b), which requires to optimize a similar quantity over distributions  $\mathcal{P}(\mathcal{X}\times \mathcal{X})$ . This is also possible, but the direct implementation suggested in (Kirschner et al., 2020b) requires  $\mathcal{O}(|\mathcal{X}|^4)$  compute steps per round to calculate the gap estimates and to find the IDS distribution.
+
+# 4.1. Regret Bounds
+
+In the language of linear partial monitoring, theueling feedback (2) is a so-called locally observable game, which informally means that any reward difference  $f(x) - f(x^{\prime})$  can be estimated from playing actions which have no more regret than playing either  $x$  or  $x^{\prime}$  alone (Kirschner et al., 2020b, Appendix C.5). Forueling bandits, IDS (without the approximation and sampling scheme that we introduce here) has regret at most  $R_{n}\leq \mathcal{O}(\sqrt{n\beta_{n}\gamma_{n}})$ , see Kirschner et al. (2020b, Corollary 18). Here we show that Algorithm 1 satisfies a similar result. Note that the regret guarantee applies generally to settings with quantitativeueling feedback, where we define regret as follows:
+
+$$
+R _ {n} ^ {\text {d u e l}} = \sum_ {t = 1} ^ {n} \Delta \left(x _ {t} ^ {1}\right) + \Delta \left(x _ {t} ^ {2}\right).
+$$
+
+Theorem 1. For  $\rho^2$ -sub-Gaussian dueling bandit feedback (2), Algorithm 1 satisfies with probability at least  $1 - \delta$
+
+$$
+R _ {n} ^ {\text {d u e l}} \leq \mathcal {O} \left(\sqrt {n \beta_ {n , \delta} \left(\gamma_ {n} + \log \frac {1}{\delta}\right)}\right).
+$$
+
+Note that the learner requires knowledge of the sub-Gaussian variance  $\rho^2$  and the Hilbert norm bound  $\|f\|_{\mathcal{H}} \leq B$ , which appear in the definition of the confidence coefficient  $\beta_{n,\delta}$ . The regret guarantee has the same scaling as the best known bound for GP-UCB in standard Bayesian optimization (Srinivas et al., 2010; Chowdhury & Gopalan, 2017). Note that  $\beta_{n,\delta} = \mathcal{O}\left(\gamma_n + \log \frac{1}{\delta}\right)$  hence combined with bounds for the total information gain  $\gamma_n$ , we can derive bounds for specific choices of the kernel function. For example in the linear setting, we get  $R_n^{dual} \leq \mathcal{O}\left(\rho \sqrt{n}(d\log(n) + \log\frac{1}{\delta})\right)$ , which is the same as for LinUCB (Abbasi-Yadkori et al., 2011). For the RBF kernel, we get  $R_n^{dual} \leq \mathcal{O}\left(\rho \sqrt{n} (\log(n)^{2d+2} + \log\frac{1}{\delta})\right)$ .
+
+Applied to the confounded setting with either the one- or two-point reduction, we get the following result. Note that depending on whether the learner requires one or two evaluations per round, the timescale differs by a factor of two.
+
+Corollary 1. In the confounded setting (1) with  $\sigma^2$ -sub-Gaussian observation noise and dueling feedback obtained via the one-point reduction (4), the regret of Algorithm 1 satisfies with probability at least  $1 - \delta$
+
+$$
+R _ {n} \leq \mathcal {O} \left(\left(C _ {\max } + \sigma\right) \sqrt {n \beta_ {n , \delta} \left(\gamma_ {n} + \log \frac {1}{\delta}\right)}\right),
+$$
+
+assuming that  $\max_{t\in [n]}b_t\leq C_{\mathrm{max}}$  and the adversary is allowed to choose  $b_{t}$  depending on all previous actions and observations,  $\{x_s,y_s\}_{s = 1}^{t - 1}$
+
+With the two-point reduction (3), Algorithm 1 satisfies with probability at least  $1 - \delta$ ,
+
+$$
+R _ {n} \leq \mathcal {O} \left(\left(D _ {\max } + \sigma\right) \sqrt {n \beta_ {n , \delta} \left(\gamma_ {n} + \log \frac {1}{\delta}\right)}\right),
+$$
+
+assuming that  $\max_{t\in [n]}|b_{2t} - b_{2t + 1}|\leq D_{\mathrm{max}}$  and the adversary is allowed to choose  $b_{t}$  depending on all but the last two actions and observations,  $\{x_s,y_s\}_{s = 1}^{t - 2}$ .
+
+Note that the algorithm requires knowledge of the bound  $C_{\mathrm{max}}$  or  $D_{\mathrm{max}}$  respectively, which is needed to compute  $\beta_{t,\delta}$ . This is in line with the previous work in the linear setting (Krishnamurthy et al., 2018; Kim & Paik, 2019). Removing or weakening this assumption is an interesting direction for future work. Note that in the linear case with the one-point reduction method, our regret bound is  $R_n \leq \tilde{\mathcal{O}}(C_{\mathrm{max}} d\sqrt{n})$ , which matches the result by Krishnamurthy et al. (2018). The dependence on  $d$  and  $n$  cannot be improved even in the un-confounded linear bandit setting for general  $\mathcal{X}$  (Lattimore & Szepesvári, 2020, Theorem 24.1). The results for the two-point reduction require a stronger assumption on the sequence  $(b_t)_{t=1}^{2n}$ , but the regret only depends on the differences  $|b_{2t} - b_{2t+1}|$ . Hence, the result assures sub-linear regret even for settings where the confounding terms are unbounded, for example when the objective function is subject to drift.
+
+Proof of Theorem 1. First, by Lemma 1 with probability at last  $1 - \delta$ ,  $\Delta(x_{t}) \leq 2\hat{\Delta}_{t}(x_{t})$ . We extend the definition of the gap estimate to two points,  $\hat{\Delta}_{t}(x^{1}, x^{2}) = \hat{\Delta}(x^{1}) + \hat{\Delta}(x^{2})$ . For a sampling distribution  $\mu \in \mathcal{P}(\mathcal{X} \times \mathcal{X})$ , denote the expected gap  $\hat{\Delta}_{t}(\mu) \triangleq \mathbb{E}_{x^{1}, x^{2} \sim \mu}[\hat{\Delta}_{t}(x^{1}, x^{2})]$  and the expected information gain  $I_{t}(\mu) = \mathbb{E}_{x^{1}, x^{2} \sim \mu}[I_{t}(x^{1}, x^{2})]$ . The information ratio is defined as follows:
+
+$$
+\Psi_ {t} (\mu) \triangleq \frac {\hat {\Delta} _ {t} (\mu) ^ {2}}{I _ {t} (\mu)}. \tag {12}
+$$
+
+Let  $(\mu_t)_{t=1}^n$  be the sequence of sampling distributions  $\mu_t \in \mathcal{P}(\mathcal{X} \times \mathcal{X})$  defined by Algorithm 1,
+
+$$
+\mu_ {t} = (1 - p _ {t}) e _ {(\hat {x} _ {t} ^ {*}, \hat {x} _ {t} ^ {*})} + p _ {t} e _ {(\hat {x} _ {t} ^ {*}, z _ {t})},
+$$
+
+where  $e_x$  denotes a Dirac on  $x \in \mathcal{X}$ . By (Lemma 1, Kirschner et al., 2020b), with probability  $1 - \delta$ ,
+
+$$
+R _ {n} ^ {\text {d u e l}} \leq \sqrt {\sum_ {t = 1} ^ {n} \Psi_ {t} (\mu_ {t}) \left(\gamma_ {n} + \mathcal {O} \left(\log \frac {1}{\delta}\right)\right)} + \mathcal {O} \left(\log \frac {n}{\delta}\right).
+$$
+
+The claim in the theorem follows if we show that the information ratio is bounded such that  $\Psi_t(\mu_t) \leq \mathcal{O}(\beta_{n,\delta})$ . To this end, note that
+
+$$
+\begin{array}{l} \Psi_ {t} (\mu_ {t}) = \frac {\left((1 - p _ {t}) 2 \delta_ {t} + p _ {t} (\hat {\Delta} _ {t} (z _ {t}) + \delta_ {t})\right) ^ {2}}{p I _ {t} (\hat {x} _ {t} ^ {*} , z _ {t})} \\ \leq \min  _ {x \in \mathcal {X}, p \in [ 0, 1 ]} \frac {4 \left((1 - p) \delta_ {t} + p \hat {\Delta} _ {t} (x)\right) ^ {2}}{p I _ {t} \left(\hat {x} _ {t} ^ {*} , x\right)} \\ \leq \min  _ {x \in \mathcal {X}} \frac {4 \hat {\Delta} _ {t} (x) ^ {2}}{I _ {t} \left(\hat {x} _ {t} ^ {*} , x\right)}, \tag {13} \\ \end{array}
+$$
+
+where the first inequality follows from  $\delta_t \leq \hat{\Delta}_t(x)$  and the definition of  $z_t$  and  $p_t$ , and the second inequality sets  $p = 1$ .
+
+On the other hand, using that the kernel is bounded,  $k(x,x)\leq 1$ , one easily checks that  $\psi_t(x^1,x^2)\leq 4$ . With  $a\leq 3\log (1 + a)$  for all  $a\in [0,4]$  we find for  $x^{1},x^{2}\in \mathcal{X}$
+
+$$
+\psi_ {t} \left(x ^ {1}, x ^ {2}\right) \leq 3 \log \left(1 + \psi_ {t} \left(x ^ {1}, x ^ {2}\right)\right) = 3 I _ {t} \left(x ^ {1}, x ^ {2}\right). \tag {14}
+$$
+
+Next, define  $\tilde{z}_t = \arg \max_{x\in \mathcal{X}}\hat{f}_t(x) + \beta_{t,\delta}^{1 / 2}\psi_t(\hat{x}_t,x)^{1 / 2}$  and observe that  $\hat{\Delta}_{t}(\tilde{z}_{t}) = \beta_{t,\delta}^{1 / 2}\psi_{t}(\hat{x}_{t},\tilde{z}_{t})^{1 / 2}$ . The claim follows with (13) from noting that
+
+$$
+\Psi_ {t} (\mu_ {t}) \leq \frac {4 \hat {\Delta} _ {t} \left(\tilde {z} _ {t}\right) ^ {2}}{I _ {t} \left(\tilde {z} _ {t}\right)} \leq 1 2 \frac {\beta_ {t , \delta} \psi_ {t} \left(\hat {x} _ {t} , \tilde {z} _ {t}\right)}{\psi_ {t} \left(\hat {x} _ {t} , \tilde {z} _ {t}\right)} = 1 2 \beta_ {t, \delta}. \tag {15}
+$$
+
+The result follows from the fact that  $\beta_{t,\delta}$  is monotonically increasing.
+
+Algorithm 1 also satisfies a gap-dependent bound for finite action sets  $\mathcal{X}$ . Let  $\Delta_{\mathrm{min}} = \min_{x \neq x^*} \Delta(x)$  be the smallest non-zero gap and assume that  $x^*$  is unique. The theorem applies to the confounded setting via the reduction method similar to Corollary 1.
+
+Theorem 2. Assuming that  $x^{*}$  is unique, the regret of Algorithm 1 satisfies with probability at least  $1 - \delta$ ,
+
+$$
+R _ {n} ^ {\text {d u e l}} \leq \mathcal {O} \left(\Delta_ {\min } ^ {- 1} \beta_ {n, \delta} \left(\gamma_ {n} + \log \frac {n}{\delta}\right)\right).
+$$
+
+For linear bandits, the regret bound reads  $R_{n}^{dual} \leq \mathcal{O}(\Delta_{\min}^{-1} d^{2} \log(n)^{2})$  and for the RKHS setting with RBF kernel, the bound is  $R_{n}^{dual} \leq \mathcal{O}(\Delta_{\min}^{-1} \log(n)^{2d+2})$ .
+
+Proof of Theorem 2. Our proof uses the strategy introduced by Kirschner et al. (2020c) that relies on finding an instance-dependent bound on the information ratio. Let  $\Psi_t(\mu_t)$  be the information ratio defined in (12). We apply (Kirschner et al., 2020b, Lemma 1 with  $\eta = 0$ ) to the scaled information gain  $\tilde{I}_t = \beta_{t,\delta}I_t$ , to find with probability  $1 - \delta$
+
+$$
+\sum_ {t = 1} ^ {n} \hat {\Delta} _ {t} \left(x _ {t} ^ {1}, x _ {t} ^ {2}\right) \leq \sqrt {\sum_ {t = 1} ^ {n} \frac {\Psi_ {t} \left(\mu_ {t}\right)}{\beta_ {t , \delta}} \left(\beta_ {n , \delta} \left(\gamma_ {n} + \mathcal {O} \left(\log \frac {n}{\delta}\right)\right)\right)}
+$$
+
+We condition now on the event that the previous equation and the confidence estimate in (7) hold simultaneously. As before, let  $\tilde{z}_t = \arg \max_{x\in \mathcal{X}}\hat{f}_t(x) + \beta_{t,\delta}^{1 / 2}\psi_t(\hat{x}_t,x)^{1 / 2}$ . We may assume that  $\tilde{z}_t\neq \hat{x}_t$ , since otherwise  $\delta_t = 0$  and therefore  $\Psi_t(\mu_t) = 0$ . On the other hand, this implies that  $2\hat{\Delta}_{t}(\tilde{z}_{t})\geq \Delta_{\mathrm{min}}$  by (1) and using that  $x^{*}$  is unique.
+
+Next, we reuse the inequality leading to (13) to find
+
+$$
+\Psi_ {t} (\mu_ {t}) \leq \min _ {p \in [ 0, 1 ]} \frac {4 \left((1 - p) \delta_ {t} + p \hat {\Delta} _ {t} (\tilde {z} _ {t})\right) ^ {2}}{p I _ {t} (\hat {x} _ {t} , \tilde {z} _ {t})}.
+$$
+
+First, consider the case where  $2\delta \leq \hat{\Delta}_t(\tilde{z}_t)$ . Computing the minimizer of the previous display and using (14), we find
+
+$$
+\Psi_ {t} (\mu_ {t}) \leq \frac {1 6 \delta_ {t} \hat {\Delta} _ {t} (\tilde {z} _ {t})}{I _ {t} (\hat {x} _ {t} , \tilde {z} _ {t})} \leq \frac {4 8 \delta_ {t} \beta_ {t , \delta}}{\hat {\Delta} _ {t} (\tilde {z} _ {t})} \leq \frac {9 6 \delta_ {t} \beta_ {t , \delta}}{\Delta_ {\mathrm {m i n}}}.
+$$
+
+For the other case where  $2\delta_{t} > \hat{\Delta}_{t}(\tilde{z}_{t})$ , using (15) directly gives
+
+$$
+\Psi_ {t} (\mu_ {t}) \leq 1 2 \beta_ {t, \delta} \leq \frac {2 4 \delta_ {t} \beta_ {t , \delta}}{\hat {\Delta} _ {t} (\tilde {z} _ {t})} \leq \frac {4 8 \delta_ {t} \beta_ {t , \delta}}{\Delta_ {\min }}.
+$$
+
+Finally, note that  $\delta_t \leq \frac{1}{2} \hat{\Delta}_t(x_t^1, x_t^2)$ . Using the bound on the information ratio, and solving for the regret, we find
+
+$$
+\sum_ {t = 1} ^ {n} \hat {\Delta} _ {t} \left(x _ {t} ^ {1}, x _ {t} ^ {2}\right) \leq \mathcal {O} \left(\Delta_ {\min } ^ {- 1} \beta_ {n, \delta} \left(\gamma_ {n} + \log \frac {n}{\delta}\right)\right).
+$$
+
+The claim follows with Lemma 1 by noting that,
+
+$$
+R _ {n} ^ {\text {d u e l}} \leq \sum_ {t = 1} ^ {n} 2 \hat {\Delta} _ {t} \left(x _ {t} ^ {1}, x _ {t} ^ {2}\right).
+$$
+
+![](images/c3821c6b124f2558947e37477a94985a1801ba6c237d8b343d968db48e71d9d5.jpg)
+
+# 4.2. A Connection to Doubly-Robust Estimation
+
+In the finite-dimensional, linear case, the objective function is  $f(x) = \langle x,\theta \rangle$  for a fixed parameter  $\theta \in \mathbb{R}^d$ . To obtain an estimate  $\hat{\theta}_t$  of the unknown parameter  $\theta$  directly from confounded data  $\{(x_s,y_s = \langle x_s,\theta \rangle +b_s + \epsilon_s)\}_{s = 1}^{t - 1}$ , Krishnamurthy et al. (2018) use a randomized policy  $x_{t}\sim \mu_{t}$  and a doubly-robust estimation approach. For centered feature vectors  $\bar{x}_t = \mathbb{E}_{x\sim \mu_t}[x]$  and regularizer  $\lambda >0$ , they define
+
+$$
+\Gamma_ {t} \triangleq \sum_ {s = 1} ^ {t - 1} \left(x _ {t} - \bar {x} _ {t}\right) \left(x _ {t} - \bar {x} _ {t}\right) ^ {\top} + \lambda \mathbf {1} _ {d},
+$$
+
+$$
+\hat {\theta} _ {t} \triangleq \Gamma_ {t} ^ {- 1} \sum_ {s = 1} ^ {t - 1} \left(x _ {t} - \bar {x} _ {t}\right) y _ {t}. \tag {16}
+$$
+
+They further derive the following high-probability bound for doubly-robust estimator:
+
+$$
+\left\| \hat {\theta} _ {t} - \theta \right\| _ {\Gamma_ {t}} ^ {2} \leq \mathcal {O} (d \log (n) + \log (n / \delta) + \lambda). \tag {17}
+$$
+
+Interestingly, when  $\mu_t = \mathrm{Uniform}(\{x^1, x^2\})$  is chosen to randomize between two actions  $x^1, x^2 \in \mathcal{X}$ , then this estimator coincides with the least-squares estimator that we obtain for the dueling bandit feedback. This follows immediately from noting that  $2(x_t - \bar{x}_t) = x^1 - x^2$ . Further, the concentration bounds are on the same quantity, but the reduction avoids the detour to prove the (more general) concentration bound (17) and leads tighter bounds. We remark that in general, the BOSE algorithm by Krishnamurthy et al. (2018) requires to compute a sampling distributions  $\mu_t$  supported on  $d + 1$  points. We also note that Kim & Paik (2019) propose another variant of the doubly-robust estimator (16). This estimator coincides with our estimation scheme in the same way.
+
+![](images/f5beca59ea8c55dd7c7661bd0e169366a74c00008785f60c6ca56f18e30ad50d.jpg)  
+Figure 1. Benchmarks with randomly sampled action sets of size 20 and  $d = 4$ . The confidence region shows  $2 \times$  standard error over 50 repetition. Without confounding, UCB performs best, and is closely followed by IDS and Semi-TS. With additive bias the performance of UCB degrades significantly, whereas the robust methods maintain sublinear regret. Note that in the drift experiment (bottom left), the bias is unbounded, which leads to linear regret for all methods except IDS-two. With bounded bias (right column), IDS-one and Semi-TS show similar performance, whereas the performance of IDS-two is varying. The BOSE algorithm shows sublinear behavior but is much more conservative.
+
+# 5. Experiments
+
+We evaluate the proposed method with the one-point reduction (IDS-one) and the two-point reduction (IDS-two) in two numerical experiments with confounded observations. To allow a fair comparison with the two-sample scheme, we account for the regret of both evaluations and scale the x-axis appropriately.
+
+# 5.1. Baselines
+
+UCB For the linear setting, LinUCB (Auer, 2003) is implemented as in (Abbasi-Yadtori et al., 2011, Figure 1) using a regularizer  $\lambda = 1$  and confidence coefficient  $\beta_{t,\delta}^{1/2} = \sqrt{\log\det V_t + 2\log\frac{1}{\delta}} + 1$ . In the kernelized setting, we use GPUCB (Srinivas et al., 2010) with an empirically tuned confidence coefficient  $\beta_{n,\delta} = 1$ . As shown by Bogunovic et al. (2020a), increasing the confidence coefficient to a larger value as required in the stochastic setting can lead to better robustness, although we did not see an improvement of performance in our experiments.
+
+BOSE The BOSE algorithm (Krishnamurthy et al., 2018, Algorithm 1) uses the doubly-robust least-squares estimator
+
+![](images/7e104aa1c6f33e871dd043f67029a4766f6ab697a6232aee48bc487b0a489268.jpg)  
+Figure 2. Performance of kernelized algorithms on the camelback benchmark. The confidence region shows  $2 \times$  standard error over 50 repetitions. With bounded confounding that simulates a recalibration process (left), GP-UCB is surprisingly competitive with IDS-two, followed by IDS-one. When the feedback is confounded by an unbounded, periodic drift (right), we observe linear regret of GP-UCB and IDS-one (because the bias is unbounded), whereas IDS-two maintains good performance despite the confounding.
+
+(16). We set the required concentration coefficient  $\beta_{t,\delta}^{\mathrm{DR}}$  to
+
+$$
+\beta_ {t, \delta} ^ {\mathrm {D R}} = \sqrt {d \log (1 + t / d) + 2 \log \left(\frac {t}{\delta}\right)} + 1,
+$$
+
+where we drop (conservative) constants required for the theoretical results in favor of better empirical performance. BOSE requires to solve a convex-quadratic feasibility problem on the space of sampling distributions over the remaining plausible actions, and no specific computation method was recommended by the authors. We compute the sampling distribution by solving the saddle point problem stated in (Krishnamurthy et al., 2018, Appendix D) using exponentiated gradient descent.
+
+SemiTS The semi-parametric Thompson sampling is implemented as in (Kim & Paik, 2019, Algorithm 1), with a less conservative over-sampling parameter  $v = \sqrt{2\log(t / \delta)}$ . Our choice improves performance over the theoretical value. We also remark that SemiTS requires to compute the probability of each action being optimal under a Gaussian perturbation of the mean parameter. We do so by computing the empirical sampling probabilities from 1000 random samples per round, the alternative being to compute Gaussian integrals over  $d$ -dimensional polytopes  $\mathcal{C}_x = \{\nu \in \mathbb{R}^d : \langle x, \nu \rangle \geq \max_{x' \in \mathcal{X}} \langle x', \nu \rangle\}$ . While SemiTS is significantly faster than BOSE in our implementation, computing the posterior probabilities accurately for larger action sets remains challenging.
+
+In all experiments we set confidence level  $\delta = 0.05$ .
+
+# 5.2. Environments
+
+Linear Reward In the first experiment, we use a linear reward function  $f(x) = \langle x, \theta \rangle$ . For each repetition we sample  $k = 20$  actions uniformly on the  $d = 4$  dimensional
+
+unit sphere. We add Gaussian observation noise with variance  $\sigma^2 = 1$ , that is  $\epsilon_t \sim \mathcal{N}(0,1)$  in (1). In this setting we compare to BOSE, SemiTS and LinUCB (Auer, 2003; Abbasi-Yadkori et al., 2011), where the latter does not directly deal with the confounding. We consider four different types of confounding: a) no bias; b) the adversary repeats the last observation with a minus sign,  $b_{t} = -y_{t - 1}$ , which makes it much harder to identify the best action (negative repeat); c) a continues drift,  $b_{t} = -0.1t$ , i.e. unbounded confounding; and d) same as the previous, but with compensated drift,  $b_{t} = -0.1t + y_{t - 1}$ , thereby making the bias terms bounded but dependent on the previous observation.
+
+The result is shown in Figure 1. As expected, in the unconfounded setting UCB works best, followed by both IDS variants and SemiTS with reasonable performance. With confounding, the regret of UCB is increased by a lot, whereas BOSE shows sublinear behaviour but is relatively inefficient. In the example with unbounded bias (drift) only IDS-two performs well, as in fact the theoretical assumptions for all other methods are invalidated. With bounded bias (i.e. negative repeat and compensated drift), SemiTS and IDS-one are competitive, while IDS-two clearly outperforms the baselines in the negative repeat experiment.
+
+Camelback Our second experiment is in the non-linear, kernelized setting with observation noise variance  $\sigma^2 = 0.1$ . As benchmark we choose the camelback function on the domain  $[-2,2] \times [-1,1]$ ,
+
+$$
+\begin{array}{l} f (x _ {1}, x _ {2}) = \\ - \min  \left(x _ {1} ^ {2} \left(4 - 2. 1 x _ {1} ^ {2} + \frac {x _ {1} ^ {4}}{3 .}\right) + x _ {1} x _ {2} + x _ {2} ^ {2} \left(4 x _ {2} ^ {2} - 4\right), 2. 5\right) \\ \end{array}
+$$
+
+We discretize the input space using 30 points per dimension. The only direct competitor that we are aware of is the method of Bogunovic et al. (2020a). This method is, however, equivalent to GP-UCB (Srinivas et al., 2010) with an up-scaled confidence coefficient. This suggests that the UCB approach is inherently robust up to a certain degree of corruption, which is also visible in our experiment. For both algorithms, we use an RBF kernel with lengthscale 0.2 and regularizer  $\lambda = 1$ , and set  $\beta_{n,\delta} = 1$  in favor of better empirical performance. We use two types of confounding that we expect is relevant in applications:  $a$ ) a calibration process, which monitors a moving average over the last 10 observations and adjusts the output range to  $[-0.1, 0.1]$  whenever the average is no longer in this range; and  $b$ ) periodic drift of the objective,  $b_{t} = \sin(0.2t) - 0.1t$ . Results are shown in Figure 2. In the first variant GPUCB works surprisingly well despite the confounding and is on-par with IDS-two. With unbounded drift, both GPUCB and IDS-one obtain linear regret, whereas the performance of IDS-two in unaffected.
+
+# 6. Conclusion
+
+We introduced randomized evaluation schemes based on pair-wise comparisons that makeueling bandit algorithms applicable to robust optimization with additive confounding. Moreover, we derived a kernelizedueling bandit algorithm based on recent ideas by Kirschner et al. (2020b). The resulting algorithm satisfies worst-case and gap-dependent regret bounds on the cumulative regret and could be of broader interest in theueling bandit setting. Our numerical experiments validate the theoretical findings.
+
+# Acknowledgements
+
+This research has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme grant agreement No 815943.
+
+# References
+
+Abbasi-Yadkori, Y. Online Learning for Linearly Parametrized Control Problems. PhD thesis, 2012.  
+Abbasi-Yadkori, Y., Pál, D., and Szepesvári, C. Improved algorithms for linear stochastic bandits. In Advances in Neural Information Processing Systems, pp. 2312-2320, 2011.  
+Abe, N. and Long, P. M. Associative reinforcement learning using linear probabilistic concepts. In Proceedings of the Sixteenth International Conference on Machine Learning, ICML '99, pp. 3-11, San Francisco, CA, USA, 1999. Morgan Kaufmann Publishers Inc. ISBN 1-55860-612-2.  
+Agarwal, A., Agarwal, S., and Patil, P. Stochastic dueling bandits with adversarial corruption. In Algorithmic Learning Theory, pp. 217-248. PMLR, 2021.  
+Auer, P. Using confidence bounds for exploitation-exploration trade-offs. J. Mach. Learn. Res., 3:397-422, March 2003. ISSN 1532-4435.  
+Bengs, V., Busa-Fekete, R., El Mesaoudi-Paul, A., and Hüllermeier, E. Preference-based online learning with dueling bandits: A survey. Journal of Machine Learning Research, 22(7):1-108, 2021.  
+Bogunovic, I., Scarlett, J., Jegelka, S., and Cevher, V. Adversarily robust optimization with gaussian processes. arXiv preprint arXiv:1810.10775, 2018.  
+Bogunovic, I., Krause, A., and Scarlett, J. Corruption-tolerant gaussian process bandit optimization. In International Conference on Artificial Intelligence and Statistics, pp. 1071-1081. PMLR, 2020a.
+
+Bogunovic, I., Losalka, A., Krause, A., and Scarlett, J. Stochastic linear bandits robust to adversarial attacks. arXiv preprint arXiv:2007.03285, 2020b.  
+Bull, A. D. Convergence rates of efficient global optimization algorithms. Journal of Machine Learning Research, 12(10), 2011.  
+Chowdhury, S. R. and Gopalan, A. On kernelized multiarmed bandits. In International Conference on Machine Learning, 2017.  
+Dani, V., Hayes, T. P., and Kakade, S. M. Stochastic Linear Optimization under Bandit Feedback. In  $COLT$ , pp. 355-366. Omnipress, 2008.  
+Duris, J., Kennedy, D., Hanuka, A., Shtalenkova, J., Edelen, A., Baxevanis, P., Egger, A., Cope, T., McIntire, M., Ermon, S., et al. Bayesian optimization of a free-electron laser. Physical review letters, 124(12):124801, 2020.  
+Frazier, P. I. A tutorial on bayesian optimization. arXiv preprint arXiv:1807.02811, 2018.  
+Gupta, A., Koren, T., and Talwar, K. Better algorithms for stochastic bandits with adversarial corruptions. In Conference on Learning Theory, pp. 1562-1578. PMLR, 2019.  
+Juranic, P., Rehanek, J., Arrell, C. A., Pradervand, C., Ischebeck, R., Erny, C., Heimgartner, P., Gorgisyan, I., Thominet, V., Tiedtke, K., et al. Swissfel aramis beamline photon diagnostics. Journal of synchrotron radiation, 25(4):1238-1248, 2018.  
+Kandasamy, K., Schneider, J., and Poczos, B. High dimensional bayesian optimisation and bandits via additive models. In International conference on machine learning, pp. 295-304. PMLR, 2015.  
+Kim, G.-S. and Paik, M. C. Contextual multi-armed bandit algorithm for semiparametric reward model. In International Conference on Machine Learning, pp. 3389-3397, 2019.  
+Kirschner, J., Mutny, M., Hiller, N., Ischebeck, R., and Krause, A. Adaptive and safe bayesian optimization in high dimensions via one-dimensional subspaces. In International Conference on Machine Learning, pp. 3429-3438, 2019.  
+Kirschner, J., Bogunovic, I., Jegelka, S., and Krause, A. Distributionally robust bayesian optimization. In International Conference on Artificial Intelligence and Statistics, pp. 2174-2184. PMLR, 2020a.  
+Kirschner, J., Lattimore, T., and Krause, A. Information directed sampling for linear partial monitoring. volume
+
+125 of Proceedings of Machine Learning Research, pp. 2328-2369. PMLR, 09-12 Jul 2020b.  
+Kirschner, J., Lattimore, T., Vernade, C., and Szepesvári, C. Asymptotically optimal information-directed sampling. arXiv preprint arXiv:2011.05944, 2020c.  
+Krishnamurthy, A., Wu, Z. S., and Syrgkanis, V. Semiparametric contextual bandits. In International Conference on Machine Learning, pp. 2776-2785, 2018.  
+Lattimore, T. and Szepesváři, C. Bandit algorithms. Cambridge University Press, 2020.  
+Li, Y., Lou, E. Y., and Shan, L. Stochastic linear optimization with adversarial corruption. arXiv preprint arXiv:1909.02109, 2019.  
+Lin, T., Abrahao, B., Kleinberg, R., Lui, J., and Chen, W. Combinatorial partial monitoring game with linear feedback and its applications. In International Conference on Machine Learning, pp. 901-909, 2014.  
+Liu, G. and Lai, L. Action-manipulation attacks on stochastic bandits. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3112-3116. IEEE, 2020.  
+Lykouris, T., Mirrokni, V., and Paes Leme, R. Stochastic bandits robust to adversarial corruptions. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pp. 114-122, 2018.  
+Mockus, J. The bayesian approach to global optimization. System Modeling and Optimization, pp. 473-481, 1982.  
+Russo, D. and Van Roy, B. Learning to optimize via information-directed sampling. In Advances in Neural Information Processing Systems, pp. 1583-1591, 2014.  
+Saha, A. and Gopalan, A. Regret minimization in stochastic contextual dueling bandits. arXiv preprint arXiv:2002.08583, 2020.  
+Shahriari, B., Swersky, K., Wang, Z., Adams, R. P., and de Freitas, N. Taking the human out of the loop: A review of bayesian optimization. Proceedings of the IEEE, 104 (1):148-175, 2016.  
+Sorokin, A. A., Bican, Y., Bonfigt, S., Brachmanski, M., Braune, M., Jastrow, U. F., Gottwald, A., Kaser, H., Richter, M., and Tiedtke, K. An x-ray gas monitor for free-electron lasers. Journal of synchrotron radiation, 26 (4):1092-1100, 2019.  
+Srinivas, N., Krause, A., Kakade, S. M., and Seeger, M. Gaussian process optimization in the bandit setting: No regret and experimental design. International Conference on Machine Learning, 2010.
+
+Sui, Y., Yue, Y., and Burdick, J. W. Correlational dueling bandits with application to clinical treatment in large decision spaces. arXiv preprint arXiv:1707.02375, 2017a.  
+Sui, Y., Zhuang, V., Burdick, J. W., and Yue, Y. Multidueling bandits with dependent arms. arXiv preprint arXiv:1705.00253, 2017b.  
+Sui, Y., Zoghi, M., Hofmann, K., and Yue, Y. Advancements in dueling bandits. In IJCAI, pp. 5502-5510, 2018.  
+Tewari, A. and Murphy, S. A. From ads to interventions: Contextual bandits in mobile health. In Mobile Health, pp. 495-517. Springer, 2017.  
+Xu, Y., Joshi, A., Singh, A., and Dubrawski, A. Zeroth order non-convex optimization with dueling-choice bandits. In Conference on Uncertainty in Artificial Intelligence, pp. 899–908. PMLR, 2020.  
+Yue, Y. and Joachims, T. Interactively optimizing information retrieval systems as a dueling bandits problem. In Proceedings of the 26th Annual International Conference on Machine Learning, pp. 1201-1208, 2009.  
+Yue, Y., Broder, J., Kleinberg, R., and Joachims, T. The k-armed dueling bandits problem. Journal of Computer and System Sciences, 78(5):1538-1556, 2012.

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+# Bias-Variance Reduced Local SGD for Less Heterogeneous Federated Learning
+
+Tomoya Murata  $^{1,2}$  Taiji Suzuki  $^{2,3}$
+
+# Abstract
+
+Recently, local SGD has got much attention and been extensively studied in the distributed learning community to overcome the communication bottleneck problem. However, the superiority of local SGD to minibatch SGD only holds in quite limited situations. In this paper, we study a new local algorithm called Bias-Variance Reduced Local SGD (BVR-L-SGD) for nonconvex distributed optimization. Algorithmically, our proposed bias and variance reduced local gradient estimator fully utilizes small second-order heterogeneity of local objectives and suggests randomly picking up one of the local models instead of taking the average of them when workers are synchronized. Theoretically, under small heterogeneity of local objectives, we show that BVR-L-SGD achieves better communication complexity than both the previous non-local and local methods under mild conditions, and particularly BVR-L-SGD is the first method that breaks the barrier of communication complexity  $\Theta(1/\varepsilon)$  for general nonconvex smooth objectives when the heterogeneity is small and the local computation budget is large. Numerical results are given to verify the theoretical findings and give empirical evidence of the superiority of our method.
+
+# 1. Introduction
+
+Nowadays, optimization problems arising in machine learning are often large and require huge computational time. Distributed learning is one of the attractive approaches to reduce the computational time by utilizing parallel computing. In classical distributed learning, each worker has the whole dataset used in optimization or a random sub
+
+$^{1}$ NTT DATA Mathematical Systems Inc., Tokyo, Japan  $^{2}$ Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan  $^{3}$ Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan. Correspondence to: Tomoya Murata <murata@msi.co.jp>, Taiji Suzuki <taiji@mist.i.u-tokyo.ac.jp>.
+
+Proceedings of the  $38^{th}$  International Conference on Machine Learning, PMLR 139, 2021. Copyright 2021 by the author(s).
+
+set of the whole dataset which is not explicitly exchanged. In recent federated learning, introduced by Konečný et al. (2015); Shokri & Shmatikov (2015); McMahan et al. (2017), we build a global model across multiple devices or servers without explicitly exchanging their own local datasets, and local datasets can be heterogeneous, i.e., each local dataset may be generated from a different distribution. There are various federated learning scenarios (e.g., personalization, preservation of the privacy of local information, robustness to attacks and failures, guarantees of fairness) and refer to the extensive survey (Kairouz et al., 2019) for these topics.
+
+One of the most naive and widely used approaches to distributed learning is minibatch Stochastic Gradient Descent (SGD) (Dekel et al., 2012), which is also called as Federated Averaging (FedAvg) (McMahan et al., 2017). Each worker computes minibatch stochastic gradient of the own local objective and then their average is used to update the global model. Also, more computationally efficient methods including minibatch Stochastic Variance Reduced Gradient (SVRG) (Johnson & Zhang, 2013; Allen-Zhu & Hazan, 2016; Reddi et al., 2016a) and its variant (Lei et al., 2017), minibatch StochAstic Recursive grAdient algorithmHm (SARAH) (Nguyen et al., 2017; 2019) and its variants (Fang et al., 2018; Zhou et al., 2018) are applicable to the problem. Particularly, SARAH achieves the optimal total computational complexity in nonconvex optimization.
+
+Unfortunately, minibatch methods suffer from their communication cost because of the necessity to communicate local gradients for every single global update. One of the possible solutions is using a large batch to compute local gradients (Goyal et al., 2017), but the communication complexity, that is the necessary number of communication rounds to optimize, is theoretically never smaller than the one of deterministic GD and thus communication cost is still problematic.
+
+To overcome the communication bottleneck problem, local methods have got much attention due to its empirical effectiveness (Konečný et al., 2015; Lin et al., 2018). In local SGD (also called Parallel Restart SGD), each worker independently updates the local model based on his own local dataset, and periodically communicates and averages the local models. Many papers (Stich, 2018; Yu et al., 2019; Haddadpour & Mahdavi, 2019; Haddadpour et al., 2019;
+
+Koloskova et al., 2020; Khaled et al., 2020) have stated the superiority of local SGD to minibatch SGD, but these results are based on unfair comparisons and hence not satisfactory. Concretely, they have compared the two algorithms with the same local minibatch size  $b$ , which means that local SGD with  $K$  local updates requires  $K$  times larger number of local computations per communication round than minibatch SGD. $^{1}$  If we fix the number of single stochastic gradient computations per communication round to  $\mathcal{B} := Kb$  for each worker, their results indicate that the communication complexity of local SGD with  $K$  local updates and  $b$  local minibatch size are never better than the one of minibatch SGD with  $\mathcal{B}$  local minibatch size for any  $\mathcal{B}$ . This point is quite important, but many papers have overlooked it.
+
+Recently, Woodworth et al. (2020b;a) have shown that, for the first time, theoretical superiority of local SGD to minibatch SGD under fair comparison for convex optimization when the heterogeneity of local objectives is small. On the other hand, their derived lower bound of local SGD also suggests the limitation of local SGD. Specifically, they have shown that if the first-order heterogeneity of local objectives $^{2}$  is greater than  $\sqrt{\varepsilon}$ , where  $\varepsilon$  is the desired optimization accuracy, the communication complexity of local SGD is even worse than the one of minibatch SGD. In other words, the quite small heterogeneity of local objectives is essential for the superiority of local SGD to minibatch SGD, which is a clear limitation of local SGD. SCAFFOLD (Karimireddy et al., 2020b) is a new local algorithm based on the idea of reducing their called client-drift, which uses a similar formulation to the variance reduction technique. However, the communication complexity is the same as minibatch SGD for general nonconvex objectives and it requires small heterogeneity and quadraticity of local objectives to surpass minibatch SGD, which is also quite limited. Inexact DANE (Reddi et al., 2016b) is another variant of local methods that uses a general local subsolver that returns an approximate minimizer of the regularized local objective. Again, the superiority to non-local methods has been only shown for quadratic convex objectives.
+
+In summary, both in classical distributed learning and recent federated learning, naive minibatch (i.e., non-local) methods often suffer from their communication cost. Several local methods surpass non-local ones in terms of communication complexity. However, the necessary conditions for the superiority of the previous local algorithms to non-local ones are quite limited (i.e., extremely small heterogeneity
+
+or quadraticity of local objectives). A natural question is that: is there a local algorithm which surpasses non-local (and existing local) ones in terms of communication complexity with a fixed local computation budget under more relaxed conditions?
+
+# Main Contributions
+
+We propose a new local algorithm called Bias-Variance Reduced Local SGD (BVR-L-SGD) for nonconvex distributed learning. The main features of our method are as below.
+
+Algorithmic Features. The algorithm is based on our bias and variance reduced gradient estimator that simultaneously reduces the bias caused by local updates and the variance caused by stochasticization based on the idea of SARAH like variance reduction technique. Importantly, to fully utilize the second-order heterogeneity of local objectives, a randomly picked local model is used as a synchronized global model instead of taking the average of them, which is typical in the previous local methods.
+
+Theoretical Features. We analyse BVR-L-SGD for general nonconvex smooth objectives under second-order heterogeneity assumption, which interpolates the heterogeneity of local objectives between the identical case and the extremely non-III case, and plays a critical role in our nonconvex analysis. The comparison of the communication complexities of our method with the most relevant existing results is given in Table 1. The communication complexity of BVR-L-SGD has a better dependence on  $\varepsilon$  than minibatch SGD, local SGD and SCAFFOLD. When  $\mathcal{BP} \gg \sqrt{n \wedge 1 / \varepsilon}$  and the second-order heterogeneity  $\zeta$  of local objectives is small relative to the smoothness  $L$ , BVR-L-SGD strictly surpasses minibatch SARAH. Furthermore, BVR-L-SGD is the first method that breaks the barrier of communication complexity  $1 / \varepsilon$  when local computation budget  $\mathcal{B} \to \infty$ , for general smooth nonconvex objectives with small heterogeneity  $\zeta$ . Importantly, even when the heterogeneity is high, the communication complexity of our method is never worse than the ones of the existing methods since the second-order heterogeneity  $\zeta$  is bounded by two times the smoothness  $L$  of local objectives<sup>4</sup>.
+
+As a result, BVR-L-SGD is a novel and promising communication efficient method for nonconvex optimization both in classical distributed learning (i.e., local data distributions are nearly identical) and recent federated learning (i.e., local data distributions can be highly heterogeneous).
+
+<table><tr><td>Algorithm</td><td>Communication Complexity</td><td>Extra Assumptions</td></tr><tr><td>Minibatch SGD</td><td>1/ε + 1/BPε2</td><td>None</td></tr><tr><td>Minibatch SARAH (Nguyen et al., 2019)</td><td>1/ε + √n ∧ 1/BPε</td><td>None</td></tr><tr><td>Local SGD (Yu et al., 2019)</td><td>1/Bε + 1/BPε2 + 1/ε3</td><td>G gradient boundedness</td></tr><tr><td>Local SGD (Khaled et al., 2020)3</td><td>1+σdif/dif/BPε2 + σdif2dBp/ε</td><td>convexity
+B ≤ 1+σdif/dif or σdif2≥ε</td></tr><tr><td>Local SGD (Woodworth et al., 2020a)</td><td>1/Bε + 1/BPε2 + 1/√Bε3/2 + ζ1/ε3</td><td>convexity, 
+1st-order ζ1 heterogeneity</td></tr><tr><td>Local SGD (Woodworth et al., 2020a) 
+(Lower bound)</td><td>1/Bε3/2 + 1/BPε2 + (1/ε ∧ ζ1/ε3)</td><td>convexity, 
+1st-order ζ1 heterogeneity</td></tr><tr><td>SCAFFOLD (Karimireddy et al., 2020b)</td><td>1/ε + 1/BPε2</td><td>None</td></tr><tr><td>SCAFFOLD (Karimireddy et al., 2020b)</td><td>1/Bε + 1/BPε2 + ζ/ε</td><td>quadraticity, 
+2nd-order ζ heterogeneity</td></tr><tr><td>BVR-L-SGD (this paper)</td><td>1/√Bε + √n ∧ 1/BPε + ζ/ε</td><td>2nd-order ζ heterogeneity</td></tr></table>
+
+Table 1. Comparison of the order of the necessary number of communication rounds to satisfy  $\mathbb{E}\| f(x_{\mathrm{out}})\| ^2\leq \varepsilon$  (or  $f(x_{\mathrm{out}}) - f(x_{*})\leq \varepsilon$  for convex  $f$ ). "Extra Assumptions" indicates the necessary assumptions to derive the results other than Assumptions 2, 3 and 4 in Section 2.  $\varepsilon$  is the desired optimization accuracy.  $\mathcal{B}$  is the local computation budget that is the allowed number of single stochastic gradient computations per communication round for each worker.  $P$  is the number of workers.  $n$  is the total number of samples and is possibly  $\infty$  in online (i.e., expected risk minimization) settings. The smoothness of local objectives  $L$ , the variance of a single stochastic gradient  $\sigma^2$  and gradient boundedness  $G$  are regarded as  $\Theta (1)$  for ease of presentation. Note that in this notation, second-order heterogeneity  $\zeta$  always satisfies  $\zeta \leq \Theta (L) = \Theta (1)$ .  $\sigma_{\mathrm{dif}}^2$  is the squared local gradient norm at an optimum (for the precise definition, see (Khaled et al., 2020)).
+
+Other Related Work. Several recent papers have also studied local algorithms combined with variance reduction technique (Sharma et al., 2019; Das et al., 2020; Karimireddy et al., 2020a). Sharma et al. (2019) have considered Parallel Restart SPIDER (PR-SPIDER), that is a local variant of SPIDER (Fang et al., 2018) and shown that the proposed algorithm achieves the optimal total computational complexity and the communication complexity of  $1 / \varepsilon$  for noncovnex smooth objectives. However, these rates essentially match the ones of non-local SARAH and no advantage of localization has been shown. Also, Das et al. (2020) have considered a SPIDER like local algorithm called FedGLOMO but the derived communication complexity is only  $1 / \varepsilon^{3 / 2}$  in general and the rate is even worse than minibatch SARAH. Karimireddy et al. (2020a) have proposed MIME, which is essentially a combination of local SGD and SVRG-like variance reduction technique. They have shown that MIME achieves the communication complexity of  $1 / (\mathcal{B}\varepsilon) + 1 / (P^{3 / 4}\varepsilon^{3 / 2}) + \zeta /\varepsilon$  for  $\zeta$  second-order heterogeneous nonconvex smooth objectives. Importantly, the second term of the rate of BVR-L-SGD has better dependencies on  $P$  and  $\mathcal{B}$  than the one of MIME. Particularly, BVR-L-SGD achieves  $\zeta /\varepsilon$  when  $\mathcal{B}\to \infty$  but MIME does not possess this property.
+
+# 2. Problem Definition and Assumptions
+
+In this section, we first introduce the notations used in this paper. Then, the problem setting considered in this paper is
+
+illustrated and theoretical assumptions are given.
+
+Notation.  $\| \cdot \|$  denotes the Euclidean  $L_{2}$  norm  $\| \cdot \| _2\colon \| x\| =$ $\sqrt{\sum_{i}x_{i}^{2}}$  for vector  $x$ . For a matrix  $X$ ,  $\| X\|$  denotes the induced norm by the Euclidean  $L_{2}$  norm. For a natural number  $m$ ,  $[m]$  denotes the set  $\{1,2,\dots,m\}$ . For a set  $A$ ,  $\# A$  means the number of elements, which is possibly  $\infty$ . For any number  $a,b$ ,  $a\vee b$  and  $a\wedge b$  denote  $\max \{a,b\}$  and  $\min \{a,b\}$  respectively. We denote the uniform distribution over  $A$  by  $\operatorname {Unif}(A)$ .
+
+# 2.1. Problem Setting
+
+We want to minimize nonconvex smooth objective
+
+$$
+f (x) := \frac {1}{P} \sum_ {p = 1} ^ {P} f _ {p} (x), \text {w h e r e} f _ {p} (x) := \mathbb {E} _ {z \sim D _ {p}} [ \ell (x, z) ]
+$$
+
+for  $x \in \mathbb{R}^d$ , where  $D_p$  is the data distribution associated with worker  $p$ . Although we consider both offline (i.e.,  $\# \mathrm{supp}(D_p) < \infty$  for every  $p \in [P]$ ) and online (i.e.,  $\# \mathrm{supp}(D_p) = \infty$  for some  $p \in [P]$ ) settings, it is assumed for offline settings that each local dataset has an equal number of samples, i.e.,  $\# \mathrm{supp}(D_p) = \# \mathrm{supp}(D_{p'})$  for every  $p, p' \in [P]$  just for simplicity. We assume that each worker  $p$  can only access the own data distribution  $D_p$  without communication. Aggregation (e.g., summation) of all the worker's  $d$ -dimensional parameters or broadcast of a  $d$ -dimensional parameter from one worker to the other workers can be realized by single communication. In typical
+
+situations, single communication is more time-consuming than single stochastic gradient computation. Let  $\mathcal{C}$  denotes the single communication cost and  $\mathcal{G}$  does the single stochastic gradient computation. Using these notations, we assume  $\mathcal{C} \geq \mathcal{G}$ . Since we expect that a larger number of available stochastic gradients in a communication round leads to faster optimization, we can increase the number of stochastic gradient computations unless the total gradient computational time exceeds  $\mathcal{C}$ . This motivates the concept of local computation budget  $\mathcal{B} (\leq \mathcal{C} / \mathcal{G})$ : given a communication and computational environment, it is assumed that each worker can only compute at most  $\mathcal{B}$  single stochastic gradients per communication round on average. Then, we compare the communication complexity, that is the total number of communication rounds of a distributed optimization algorithm to achieve the desired optimization accuracy. From the definition, given a communication and computational environment, the communication complexity with a fixed local computation budget  $\mathcal{B} := \mathcal{C} / \mathcal{G}$  captures the best achievable total execution time of an algorithm. Generally, for a larger budget, we expect smaller communication complexity.
+
+# 2.2. Theoretical Assumptions
+
+In this paper, we always assume the following four assumptions. Assumptions 2, 3 and 4 are fairly standard in first-order nonconvex optimization.
+
+Assumption 1 (Heterogeneity).  $\{f_p\}_{p = 1}^P$  is second-order  $\zeta$ -heterogeneous, i.e., for any  $p,p^{\prime}\in [P]$
+
+$$
+\left| \left| \nabla^ {2} f _ {p} (x) - \nabla^ {2} f _ {p ^ {\prime}} (x) \right| \right| \leq \zeta , \forall x \in \mathbb {R} ^ {d}.
+$$
+
+Assumption 1 characterizes the heterogeneity of local objectives  $\{f_p\}_{p = 1}^P$  and has a critical role in our analysis. We expect that relatively small heterogeneity to the smoothness reduces the necessary number of communication to optimize global objective  $f = (1 / P)\sum_{p = 1}^{P}f_{p}$ . If the local objectives are identical, i.e.,  $D_{p} = D_{p^{\prime}}$  for every  $p,p^{\prime}\in [P]$ ,  $\zeta$  becomes zero. When each  $D_{p}$  is the empirical distribution of  $n / P$  IID samples from common data distribution  $D$ , we have  $\| \nabla^2 f_p(x) - \nabla^2 f_{p'}(x)\| \leq \widetilde{\Theta} (\sqrt{P / n} L)$  with high probability by matrix Hoeffding's inequality under Assumption 2 for fixed  $x^5$ . Hence, in classical distributed learning regimes, Assumption 1 naturally holds. An important remark is that Assumption 2 implies  $\zeta \leq 2L$ , i.e., the heterogeneity is bounded by the smoothness. This means that Assumption 1 gives an interpolation between the identical data setting  $\zeta = 0$  and the extremely non-IID setting  $\zeta = 2L$ . Even in federated learning regimes  $\zeta \gg \sqrt{P / n} L$ , we can expect  $\zeta \ll 2L$  for some problems.
+
+# Algorithm 1 Local  $\mathrm{GD}(\tilde{x}_0,\eta ,B,b,K,T)$
+
+1: for  $t = 1$  to  $T$  do
+
+2: for  $p = 1$  to  $P$  in parallel do
+
+3: Set  $x_0^{(p)} = \widetilde{x}_{t - 1}$
+
+4: for  $k = 1$  to  $K$  do
+
+5: Update  $x_{k}^{(p)} = x_{k - 1}^{(p)} - \eta \nabla f_{p}(x_{k - 1}^{(p)})$
+
+6: end for
+
+7: end for
+
+8: Communicate  $\{x_{t}^{(p)}\}_{p = 1}^{P}$ .
+
+9:  $\widetilde{x}_t = \frac{1}{P}\sum_{p = 1}^P x_{\hat{k}}^{(p)}$  ( $\hat{k} \sim \mathrm{Unif}[K]$ ).
+
+10: end for
+
+11: Return:  $\tilde{x}_{\hat{t}}$  ( $\hat{t} \sim \mathrm{Unif}[T]$ ).
+
+Assumption 2 (Smoothness). For any  $p \in [P]$  and  $z \in \operatorname{supp}(D_p)$ ,  $\ell(\cdot, z)$  is  $L$ -smooth, i.e.,
+
+$$
+\| \nabla \ell (x, z) - \nabla \ell (y, z) \| \leq L \| x - y \|, \forall x, y \in \mathbb {R} ^ {d}.
+$$
+
+We assume  $L$ -smoothness of loss  $\ell$  rather than risk  $f$ . This assumption is a bit strong, but is typically necessary in the analysis of variance reduced gradient estimators.
+
+Assumption 3 (Existence of global optimum).  $f$  has a global minimizer  $x_{*}\in \mathbb{R}^{d}$
+
+Assumption 4 (Bounded gradient variance). For every  $p \in [P]$ ,
+
+$$
+\mathbb {E} _ {z \sim D _ {p}} \| \nabla \ell (x, z) - \nabla f _ {p} (x) \| ^ {2} \leq \sigma^ {2}.
+$$
+
+Assumption 4 says that the variance of stochastic gradient is bounded for every local objective.
+
+# 3. Approach and Proposed Algorithms
+
+In this section, we introduce our approach and provide details of the proposed algorithms.
+
+# 3.1. Core Concepts and Approach
+
+Here, we describe four main building blocks of our algorithm, that are localization, bias reduction, stochasticization and variance reduction. Although our algorithm relies on SARAH like variance reduction technique, in this subsection we will describe our approach using SVRG like variance reduction rather than SARAH like one to simply convey the core ideas.
+
+Localization. One of the promising methods for reducing communication cost is local methods. In local methods, each worker independently optimizes the local objective and periodically communicate the current solution. For example, the algorithm of local GD, which is a deterministic variant of local SGD, is given in Algorithm 1. In some sense, the local gradient  $\nabla f_{p}(x)$  can be regarded as a biased
+
+estimator of the global gradient  $\nabla f(x)$ . One of the limitations of local GD is the existence of the potential bias of the local gradient  $\nabla f_{p}(x)$  to approximate the global one  $\nabla f(x)$  for heterogeneous local objectives  $\{f_p\}_{p = 1}^P$ . The bias  $\| \nabla f_p(x) - \nabla f(x)\|$  critically affects the convergence speed and can be bounded as  $\| \nabla f_p(x) - \nabla f(x)\| \leq \zeta_1$  under the first order  $\zeta_{1}$ -heterogeneity condition. This implies that the bias heavily depends on the heterogeneity parameter  $\zeta_{1}$  and does not converge to zero as  $x\to x_{*}$ . Hence, the existing analysis of local methods requires extremely small  $\zeta_{1}$  that typically depends on the optimization accuracy  $\varepsilon$  to surpass non-local methods including GD and minibatch SGD in terms of communication complexity, which is quite limited in many situations.
+
+Bias Reduction. To reduce the bias of local gradient, we introduce bias reduction technique. Concretely, we construct the local estimator  $\nabla f_{p}(x) - \nabla f_{p}(x_{0}) + \nabla f(x_{0})$  to approximate  $\nabla f(x)$ . Here,  $x_0$  is the previously communicated solution. This construction evokes the famous variance reduction technique. Analogically to the analysis of variance reduced gradient estimators, under the second order  $\zeta$ -heterogeneity, the bias can be bounded as  $\| \nabla f_p(x) - \nabla f_p(x_0) + \nabla f(x_0) - \nabla f(x)\| \leq \zeta \| x - x_0\|$ . This means that the bias converges to zero as  $x$  and  $x_0\to x_*$ . Hence, the bias of the introduced estimator is reduced by utilizing the periodically computed global gradient  $\nabla f(x)$ . This enables us to show faster convergence than vanilla non-local and local GD even for not too small  $\zeta$ .
+
+Stochastization. Generally, deterministic methods require huge computational cost for single update in large scale optimization. The classical idea to handle this problem is stochasticization. For example, non-distributed SGD naively uses  $\nabla \ell (x,z)$  with single sample  $z\sim D_p$  to approximate  $\nabla f(x) = \mathbb{E}_{z\sim D}[\nabla \ell (x,z)]$ . Although stochasticization reduces the computational cost per update, the variance due to it generally slows down the convergence speed. Similar to standard SGD, we can naively stochastize our bias reduced estimator as  $\ell (x,z) - \ell (x_0,z) + (1 / P)\sum_{p' = 1}^{P}\ell (x_0,z_{p'})$  where  $z\sim D_p$  and  $z_{p'}\sim D_{p'}$  for  $p'\sim [P]$ . Here,  $\{z_{p'}\}_{p' = 1}^P$  is sampled only at communication time. As pointed out before, the variance  $\mathbb{E}_{z,z'\sim D_p}\| \ell (x,z) - \ell (x_0,z) + (1 / P)\sum_{p' = 1}^{P}\ell (x_0,z_{p'}) - (\nabla f_p(x) - \nabla f_p(x_0) + \nabla f(x_0))\|^2$  caused by stochasticization may leads to slow convergence.
+
+Variance Reduction. To reduce the variance of the gradient estimator due to stochasticization, we introduce variance reduction technique. Variance reduction is also classical technique and has been extensively analysed both in convex and nonconvex optimization. The essence of variance reduction is the utilization of periodically computed full gradient  $\nabla f(x)$ . In non-distributed cases, a variance reduced estimator is defined as  $\nabla \ell (x,z) - \nabla \ell (x_0,z) + \nabla f(x_0)$  with  $z\sim D$ . This estimator is unbiased and the vari-
+
+Algorithm 2 BVR-L-SGD  $(\widetilde{x}_0,\eta ,b,\widetilde{b},K,T,S)$  
+1: for  $s = 1$  to  $S$  do  
+2: for  $p = 1$  to  $P$  in parallel do  
+3: if  $\widetilde{b} \geq \frac{1}{P}\sum_{p=1}^{P}\# \mathrm{supp}(D_p)$  then  
+4:  $\widetilde{\nabla}^{(p)} = \nabla f_p(\widetilde{x}_{s-1})$ .  
+5: else  
+6:  $\widetilde{\nabla}^{(p)} = \frac{1}{b}\sum_{l=1}^{\widetilde{b}}\nabla \ell(\widetilde{x}_{s-1},z_l)(z_l \stackrel{i.i.d.}{\sim} D_p)$ .  
+7: end if  
+8: end for  
+9: Communicate  $\{\widetilde{\nabla}^{(p)}\}_{p=1}^{P}$ , set  $\widetilde{v}_0 = \frac{1}{P}\sum_{p=1}^{P}\widetilde{\nabla}^{(p)}$ .  
+10: Set  $x_0 = x_{-1} = \widetilde{x}_{s-1}$ .  
+11: for  $t = 1$  to  $T$  do  
+12: for  $p = 1$  to  $P$  in parallel do  
+13:  $g_t^{(p)}(x_{t-1}) = \frac{1}{Kb}\sum_{l=1}^{Kb}\nabla \ell(x_{t-1},z_l)$ ,  
+14:  $g_t^{(p)}(x_{t-2}) = \frac{1}{Kb}\sum_{l=1}^{Kb}\nabla \ell(x_{t-2},z_l)$   
+15: for  $z_l \stackrel{i.i.d.}{\sim} D_p$ .  
+16:  $\widetilde{v}_t^{(p)} = g_t^{(p)}(x_{t-1}) - g_t^{(p)}(x_{t-2}) + \widetilde{v}_{t-1}^{(p)}$ .  
+17: end for  
+18: Communicate  $\{\widetilde{v}_t^{(p)}\}_{p=1}^P$ , set  $\widetilde{v}_t = \frac{1}{P}\sum_{p=1}^{P}\widetilde{v}_t^{(p)}$ .  
+19: for  $p = 1$  to  $P$  in parallel do  
+20:  $x_t^{(p)}, x_t^{(p),\mathrm{out}} = \text{Local-Routine}(p,x_{t-1},\eta,\widetilde{v}_t,b,K)$ .  
+21: end for  
+22: Communicate  $\{x_t^{(p)}\}_{p=1}^P$  and  $\{x_t^{(p),\mathrm{out}}\}_{p=1}^P$ .  
+23: Set  $x_t = x_t^{(\hat{p})}$  and  $x_t^{\mathrm{out}} = x_t^{(\hat{p}),\mathrm{out}}$  ( $\hat{p} \sim \mathrm{Unif}[P]$ ).  
+24: end for  
+25: Set  $\widetilde{x}_s = x_T$  and  $\widetilde{x}_s^{\mathrm{out}} = x_t^{\mathrm{out}}$  ( $\hat{t} \sim \mathrm{Unif}[T]$ ).  
+26: Set  $\widetilde{x}_s = x_T$  and  $\widetilde{x}_s^{\mathrm{out}} = x_t^{\mathrm{out}}$  ( $\hat{t} \sim \mathrm{Unif}[S]$ ).  
+27: end for  
+28: Return:  $\widetilde{x}^{\mathrm{out}} = \widetilde{x}_s^{\mathrm{out}}$  ( $\hat{s} \sim \mathrm{Unif}[S]$ ).
+
+once  $\mathbb{E}_{z\sim D}\| \nabla \ell (x,z) - \nabla \ell (x_0,z) + \nabla f(x_0) - \nabla f(x)\| ^2$  can be bounded by  $L^2\| x - x_0\|^2$ , where  $L$  is the smoothness parameter of  $\ell$ . If  $x$  and  $x_0\to x_*$ , the variance converges to zero. In this mean, the estimator reduces the variance caused by stochasticization and also maintains computational efficiency by using periodically computed global full gradients. Analogous to this formulation, each worker  $p$  computes a variance reduced local gradient estimator  $\nabla \ell (x,z) - \nabla \ell (x_0,z) + \nabla f(x_0)$  with  $z\sim D_p$ .
+
+# Concrete Algorithm
+
+In this paragraph, we illustrate the concrete procedure of our proposed algorithm based on the concepts described in the previous paragraph.
+
+The proposed algorithm for nonconvex objectives is provided in Algorithm 2. In line 2-9, worker  $p$  computes the full gradient of local objective  $f_{p}$  (or a large batch stochastic gradient of  $f_{p}$  if the learning problem is on-line, which means that  $\# \mathrm{supp}(D_p) = \infty$  for some  $p$ ). Then, each
+
+Algorithm 3 Local-Routine  $(p, x_0, \eta, v_0, b, K)$
+
+1: Set  $x_0^{(p)} = x_{-1}^{(p)} = x_0$ .  
+2: for  $k = 1$  to  $K$  do  
+3:  $g_{k}^{(p)}(x_{k - 1}^{(p)}) = \frac{1}{b}\sum_{l = 1}^{b}\ell (x_{k - 1}^{(p)},z_{l}),g_{k}^{(p)}(x_{k - 2}^{(p)}) =$
+
+$$
+\frac {1}{b} \sum_ {l = 1} ^ {b} \ell (x _ {k - 2} ^ {(p)}, z _ {l}) \left(z _ {l} \begin{array}{c c} i. i. d. \\ \sim \end{array} D _ {p}\right).
+$$
+
+4:  $v_{k}^{(p)} = g_{k}^{(p)}(x_{k - 1}^{(p)}) - g_{k}^{(p)}(x_{k - 2}^{(p)}) + v_{k - 1}^{(p)}.$  
+5: Update  $x_{k}^{(p)} = x_{k - 1}^{(p)} - \eta v_{k}^{(p)}$  
+6: end for  
+7: Return:  $x_{K}^{(p)}$ ,  $x_{\hat{k}}^{(p)}$  ( $\hat{k} \sim \mathrm{Unif}[K]$ ).
+
+worker broadcasts it and the gradient of global objective  $f$  is executed by averaging the communicated local gradients (line 9). Then, for each iteration  $t$ , each worker computes variance reduced local gradient  $\widetilde{v}_t^{(p)}$  that approximates the full local gradient using Kb IID samples (line 13-16), that is an important process for computational efficiency. Then,  $\{\widetilde{v}_t^{(p)}\}_{p=1}^P$  is communicated and  $\widetilde{v}_t$  is obtained by averaging them. Using previous solution  $x_{t-1}$  and  $\widetilde{v}_t$  as inputs, each worker runs Local-Routine (Algorithm 3) (line 21). The next solution  $x_t$  at iteration  $t$  is set to the randomly chosen solutions from Local-Routine's outputs  $\{x_t\}_{p=1}^P$  rather than averaging them. When we terminate the loop from line 11 to 22, the next solution at stage  $s$  is set to the randomly chosen solutions from  $\{x_t\}_{p=1,t=1}^P$  (line 24) rather than averaging them again. Although the model averaging process has a critical role in all the previous local algorithms, the random picking process is essential in our analysis to fully utilize the second-order heterogeneity and is one of the algorithmic novelties of our method.
+
+The local computation algorithm Local-Routine is illustrated in Algorithm 3. In lines 3-5, we again use variance reduction with  $v_{0}$  as a snapshot gradient. Here, we adopt the SARAH like variance reduction rather than the SVRG like one because SARAH achieves the optimal computational complexity for non-distributed nonconvex optimization.
+
+Remark (Communication and computational complexity). The communication complexity is  $\Theta(ST)$  and the averaged number of single gradient computations per communication round for each worker is  $\Theta((Kb + \widetilde{b}/T))$ .
+
+Remark (Generalization of SARAH). When  $K = 1$ , BVR-L-SGD exactly matches to minibatch SARAH. In this sense, BVR-L-SGD is a generalization of minibatch SARAH.
+
+Remark (Practical Implementation). Practically, in line 19-24 of Algorithm 2, we randomly choose worker  $\hat{p}$  at first and execute Local-Routine only for worker  $\hat{p}$ . Note that this procedure gives an equivalent algorithm to the original one but reduces the computational and communication cost. More specific procedures of the practical implementation are found in the supplementary material (Section C).
+
+# 4. Convergence Analysis
+
+In this section, we provide theoretical convergence analysis of our proposed algorithm. For the proofs, see the supplementary material (Section A and B).
+
+# 4.1. Analysis of Local-Routine
+
+Here, we analyse Local-Routine (Algorithm 3).
+
+Lemma 4.1 (Descent Lemma). Suppose that Assumption 2 holds. There exists  $\eta_1 = \Theta(1/L)$  such that for any  $\eta \leq \eta_1$ , Local-Routine  $(p, x_0, \eta, v_0, b, K)$  satisfies for  $k \in [K]$ ,
+
+$$
+\begin{array}{l} \mathbb {E} \| \nabla f (x _ {k - 1} ^ {(p)}) \| ^ {2} \leq \Theta \left(\frac {1}{\eta}\right) (\mathbb {E} f (x _ {k - 1} ^ {(p)}) - \mathbb {E} f (x _ {k} ^ {(p)})) \\ + \frac {5}{2} \mathbb {E} \| v _ {k} ^ {(p)} - \nabla f (x _ {k - 1} ^ {(p)}) \| ^ {2}. \\ \end{array}
+$$
+
+The deviation of  $v_{k}^{(p)}$  from  $\nabla f(x_{k - 1}^{(p)})$  can be bounded by the following lemma.
+
+Lemma 4.2. Suppose that Assumptions 1 and 2 hold. Then, there exists  $\eta_{2} = \Theta (1 / (K\zeta)\wedge \sqrt{b} /(\sqrt{K} L))$  such that for  $\eta \leq \eta_{2}$ , Local-Routine  $(p,x_0,\eta ,v_0,b,K)$  satisfies
+
+$$
+\begin{array}{l} \frac {1}{K} \sum_ {k = 1} ^ {K} \mathbb {E} \| v _ {k} ^ {(p)} - \nabla f (x _ {k - 1} ^ {(p)}) \| ^ {2} \\ \leq \Theta \left(C _ {\eta}\right) \sum_ {k = 1} ^ {K} \mathbb {E} \left\| \nabla f (x _ {k - 1} ^ {(p)}) \right\| ^ {2} + \Theta (1) \| v _ {0} - \nabla f (x _ {0}) \| ^ {2}, \\ \end{array}
+$$
+
+where  $C_{\eta}\coloneqq \eta^{2}L^{2} / b + \eta^{2}\zeta^{2}K$
+
+Combining Lemma 4.1 and 4.2 results in the following proposition.
+
+Proposition 4.3. Suppose that Assumptions 1 and 2 hold. There exists  $\eta_3 = \Theta(1 / L \wedge 1 / (K\zeta) \wedge \sqrt{b} / (\sqrt{K}L))$  such that for  $\eta \leq \eta_3$ , Local-Routine  $(p, x_0, \eta, v_0, b, K)$  satisfies
+
+$$
+\begin{array}{l} \mathbb {E} \| \nabla f (x _ {\hat {k}} ^ {(p)}) \| ^ {2} \leq \Theta \left(\frac {1}{\eta K}\right) (\mathbb {E} f (x _ {0}) - \mathbb {E} f (x _ {K} ^ {(p)})) \\ + \Theta (1) \| v _ {0} - \nabla f (x _ {0}) \| ^ {2}. \\ \end{array}
+$$
+
+# 4.2. Analysis of BVR-L-SGD
+
+Here, we analyse BVR-L-SGD (Algorithm 2). The following lemma bounds the variance of  $\widetilde{v}_t$ , which arises in Proposition 4.3.
+
+Lemma 4.4. Suppose that Assumptions 1, 2 and 4 hold. Then, there exists \(\eta_4 = \Theta(1/(K\zeta) \wedge \sqrt{b}/(\sqrt{K}L) \wedge \sqrt{Pb}/(\sqrt{KT}L))\) such that for \(\eta \leq \eta_4\), \(BVR-L-SGD(\widetilde{x}_0,
+
+$\eta, b, \widetilde{b}, K, T, S)$  satisfies
+
+$$
+\begin{array}{l} \frac {1}{T} \sum_ {t = 1} ^ {T} \mathbb {E} \| \widetilde {v} _ {t} - \nabla f (x _ {t - 1}) \| ^ {2} \\ \leq \Theta \left(C _ {\eta} ^ {\prime}\right) \frac {1}{T} \sum_ {t ^ {\prime} = 1} ^ {T} \frac {1}{P} \sum_ {p = 1} ^ {P} \mathbb {E} [ G _ {t ^ {\prime}} ^ {(p)} ] + \mathbb {1} _ {\widetilde {b} <   \frac {n}{P}} \frac {\sigma^ {2}}{P \widetilde {b}}. \\ \end{array}
+$$
+
+where  $C_{\eta}^{\prime} \coloneqq \eta^{2}L^{2}KT / (Pb) + \eta^{2}L^{2}K / b + \eta^{2}\zeta^{2}K^{2}$ ,  $n \coloneqq \sum_{p = 1}^{P}\# \mathrm{supp}(D_p)$  and  $G_{t'}^{(p)} \coloneqq \| \nabla f(x_{t' - 1}^{(p),\mathrm{out}})\| ^2$ .
+
+Using the result of Local-Routine (Proposition 4.3) and Lemma 4.4, we obtain the following theorem.
+
+Theorem 4.5. Suppose that Assumptions 1, 2, 3 and 4 hold. Then, there exists  $\eta = \Theta(1 / L \wedge 1 / (K\zeta) \wedge \sqrt{b} / (\sqrt{K}L) \wedge \sqrt{Pb} / (\sqrt{KT}L))$  such that BVR-L-SGD( $\widetilde{x}_0$ ,  $\eta$ ,  $b$ ,  $\widetilde{b}$ ,  $K$ ,  $T$ ,  $S$ ) satisfies
+
+$$
+\begin{array}{l} \mathbb {E} \| \nabla f (\widetilde {x} ^ {\text {o u t}}) \| ^ {2} \leq \Theta \left(\frac {1}{\eta K T S}\right) \left(\mathbb {E} f (\widetilde {x} _ {0}) - f (x _ {*})\right) \\ + \Theta (1) \mathbb {1} _ {\widetilde {b} <   \frac {n}{P}} \frac {\sigma^ {2}}{P \widetilde {b}}, \\ \end{array}
+$$
+
+where  $n\coloneqq \sum_{p = 1}^{P}\# \mathrm{supp}(D_p)$
+
+Theorem 4.5 immediately implies the following corollary which characterises the communication complexity of BVR-L-SGD.
+
+Corollary 4.6. Suppose that Assumptions 1, 2, 3 and 4 hold. We denote  $n = \sum_{p=1}^{P} \# \operatorname{supp}(D_p)$ . Let  $\widetilde{b} = \Theta((n/P) \wedge (\sigma^2/(P\varepsilon)))$ . Then, there exists  $\eta = \Theta(1/L \wedge 1/(K\zeta) \wedge \sqrt{b}/(\sqrt{KL}) \wedge \sqrt{Pb}/(\sqrt{KT}L))$  such that BVR-L-SGD( $\widetilde{x}_0, \eta, b, \widetilde{b}, K, T, S$ ) with  $S = \Theta(1+1/(\eta KT\varepsilon))$  satisfies  $\mathbb{E}\|\nabla f(\widetilde{x}^{\mathrm{out}})\|^2 \leq \Theta(\varepsilon)$  with communication complexity
+
+$$
+S T = \Theta \left(\frac {L}{K \varepsilon} + \frac {\zeta}{\varepsilon} + \frac {L}{\sqrt {K b} \varepsilon} + \sqrt {\frac {T}{K b P}} \frac {L}{\varepsilon} + T\right).
+$$
+
+Remark (Communication efficiency). Given local computation budget  $\mathcal{B}$ , we set  $T = \Theta(1 + \widetilde{b} / \mathcal{B})$  and  $Kb = \Theta(\mathcal{B})$  with  $b \leq \Theta(\sqrt{\mathcal{B}})$ , where  $\widetilde{b}$  was defined in Corollary 4.6. Then, we have the averaged number of local computations per communication round  $Kb + \widetilde{b} / T = \Theta(\mathcal{B})$  and the total communication complexity with budget  $\mathcal{B}$  becomes  $\Theta((L / (\sqrt{\mathcal{B}}\varepsilon) + \sqrt{n \wedge (\sigma^2 / \varepsilon)L / (\mathcal{BP}\varepsilon) + (n \wedge (\sigma^2 / \varepsilon)) / (\mathcal{BP}) + \zeta / \varepsilon)}$ .
+
+# 5. Numerical Results
+
+In this section, we provide several experimental results to verify our theoretical findings.
+
+We conducted a ten-class classification on CIFAR10 $^{6}$  dataset. Several heterogeneity patterns of local datasets
+
+![](images/cdd42d6ae5392bc6daeac54cd60395a64d8fc2358277bef0ff026ebf16968de5.jpg)  
+Figure 1. An illustration of our local datasets generation process.
+
+were artificially created. For each heterogeneity, we compared the empirical performances of our method and several existing methods.
+
+Data Preparation. We first equalized the number of data per class by randomly removing the excess data for both the train and test datasets for only simplicity. Then, for fixed  $q \in \{0.1, 0.35, 0.6, 0.85\}$ ,  $q \times 100\%$  of the data of class  $p$  was assigned to worker  $p$  for  $p \in [P]$ . Here, we set the number of workers to the number of classes. Then, for each class  $p$ , we equally divided the remained  $(1 - q) \times 100\%$  data of class  $p$  into  $P - 1$  sets and distributed them to correspondence worker  $p' \neq p$ . As a result, we obtained several patterns of class imbalanced local datasets with various heterogeneity (we expect smaller heterogeneity for smaller  $q$  and particularly  $\zeta \approx 0$  when  $q = 0.1$  since  $P = 10$ ). An illustration of this process for  $P = 3$  is given in Figure 1. From this process, we fixed the number of workers  $P$  to ten. Finally, we normalized each channel of the data to be mean and standard deviation 0.5.
+
+Models. We conducted our experiments using an one-hidden layer fully connected neural network with 100 hidden units and softmax activation. For loss function, we used the standard cross-entropy loss. We initialized parameters by uniformly sampling the parameters from  $\left[-\sqrt{6 / (n_{\mathrm{in}} + n_{\mathrm{out}})},\sqrt{6 / (n_{\mathrm{in}} + n_{\mathrm{out}})}\right]$  (Glorot & Bengio, 2010), where  $n_{\mathrm{in}}$  and  $n_{\mathrm{out}}$  are the number of units in the input and output layers respectively. Furthermore, we add  $L_{2}$ -regularizer to the empirical risk with fixed regularization parameter  $5\times 10^{-3}$ .
+
+Implemented Algorithms. We implemented minibatch SGD, Local SGD, SARAH, SCAFFOLD and our BVR-L-SGD. For each local computation budget  $\mathcal{B} \in \{256, 512, 1024\}$ , we set  $K = \mathcal{B} / 16$  and  $b = 16$  for local methods (Local SGD, SCAFFOLD and BVR-L-SGD), and  $b = \mathcal{B}$  for non-local ones (minibatch SGD and SARAH). Note that each algorithm requires the same order of stochastic gradient computations per communication. For each algorithm, we tuned learning rate  $\eta$  from  $\{0.005, 0.01, 0.05, 0.1, 0.5, 1.0\}$ . The details of the tuning procedure are found in the supplementary material (Section D).
+
+Evaluation. We compared the implemented algorithms using four criteria of train loss; train accuracy; test loss and test
+
+![](images/cec26e871a58866bdb51d4bca85fca81258e4920e5d95bd8cacac1c5b4494110.jpg)  
+(a) Best Train Loss
+
+![](images/2515617b9b245d4403309cb7e4a4dac68a0fd16ba0edb8cc5182a2b21fc0d243.jpg)  
+(b) Best Test Accuracy  
+Figure 2. Comparison of the best train loss and test accuracy against heterogeneity parameter  $q$ .
+
+![](images/7f5959461f7a1cbe1a3238ae9d78d237c13e555b0ac0dc4ab5f8a2355f2b22db.jpg)  
+(a) Best Train Loss  
+Figure 3. Comparison of the best train loss and test accuracy against local computation budget  $\mathcal{B}$ .
+
+![](images/01fabbc7830102fa51bb3320d752a5d68a6ed6554b15d9f2a5dedde4148dd4d2.jpg)  
+(b) Best Test Accuracy
+
+accuracy against (i) heterogeneity  $q$ ; (ii) local computation budgets  $\mathcal{B}$  and (iii) the number of communication rounds. The total number of communication rounds was fixed to 3,000 for each algorithm. We independently repeated the experiments 5 times and report the mean and standard deviation of the above criteria. Due to the space limitation, we will only report train loss and test accuracy in the main paper. The full results are found in the supplementary material (Section D).
+
+Results 1: Effect of the heterogeneity. Here, we investigate the effect of the heterogeneity on the convergence speed of the algorithms. To clarify the pure effect of the heterogeneity, we fixed the local computation budget  $\mathcal{B}$  to 1,024, which was the largest one in our experiments. Figure 2 shows the comparison of the best-achieved train loss and test accuracy in 3,000 communication rounds against heterogeneity parameter  $q$ . From this, we can see that the convergence speed of the local methods deteriorated as heterogeneity parameter  $q$  increased. Particularly, the degree of the performance degradation of L-SGD and SCAFFOLD was serious. In contrast, this phenomenon was not observed for non-local methods, because the convergence rates of non-local methods do not depend on heterogeneity  $\zeta$  as in Table 1. Importantly, even for the largest  $q$ , BVR-L-SGD significantly outperformed the other methods.
+
+Results 2: Effect of the local computation budget size. Now, we examine the effect of the size of the local computation budget  $\mathcal{B}$  to the convergence speed. For this purpose, we fixed heterogeneity parameter  $q$  to the smallest one. Figure 3 shows the comparison of the best-achieved train loss and test accuracy against local computation budget  $\mathcal{B}$ . We can see that the local methods improved their performances as local computation budget  $\mathcal{B}$  increased, but non-local methods did not. This is because local methods can potentially achieve a smaller communication complexity than  $1 / \varepsilon$  by increasing  $\mathcal{B}$  for small  $\zeta$ , but non-local methods can not break the barrier of  $1 / \varepsilon$  for any  $\mathcal{B}$  as in Table 1.
+
+Results 3: Effect of the number of communication rounds. Finally, to see the trends of train loss and test accuracy during optimization, we give the comparison of
+
+the train loss and test accuracy against the number of communication rounds (Figure 4). For the space limitation, we only report the case  $\mathcal{B} = 1,024$ . From these results, it can be seen that our proposed BVR-L-SGD consistently outperformed the other methods from beginning to end.
+
+In summary, for small heterogeneity, local methods significantly surpassed non-local methods. For relatively large heterogeneity, the performances of the existing local methods were seriously degraded. In contrast, the degree of deterioration of BVR-L-SGD was small and BVR-L-SGD consistently out-performed both the existing non-local and local methods. These observations strongly verify the theoretical findings and showed the empirical superiority of our method.
+
+# 6. Conclusion and Future Work
+
+In this paper, we studied our proposed BVR-L-SGD for nonconvex distributed learning, which is based on the bias-variance reduced gradient estimator to fully utilize the small second-order heterogeneity of local objectives and suggests randomly picking up one of the local models instead of taking the average of them when workers are synchronized. Our theory implies the superiority of BVR-L-SGD to previous non-local and local methods in terms of communication complexity. The numerical results strongly encouraged our theoretical results and suggested the empirical superiority of the proposed method.
+
+One promising and challenging future work is to extend our algorithm and theory to the problem of finding second-order stationary points. Although there are many papers for finding second-order stationary points for general nonconvex problems (Ge et al., 2015; Allen-Zhu, 2017; Jin et al., 2017; Li, 2019), it might be inherently difficult for local algorithms to efficiently find a local minima due to the nature of local updates. An open question is that: can we construct a local algorithm that guarantees to find second-order stationary points and is more communication efficient than non-local methods for local objectives with small heterogeneity?
+
+![](images/2a24a33046c4d217249c44cb99ea8bba6c4af679d8c0467c751f4e9f3341d5fb.jpg)  
+(a) Train Loss
+
+![](images/49ff931405677cdcfbeabb25758fa4aa86f9047f033b2488c5f2447621980664.jpg)  
+(b) Test Accuracy
+
+![](images/9cd7f4fbbbcab47bc73674ea06cdbb199cacd5c783b7da758b42f9e9f1cc5777.jpg)  
+(c) Train Loss
+
+![](images/893b2d974c2868ea9510f1e0202cbfb3faa8f0847f055417f66e3c67557d1138.jpg)  
+(d) Test Accuracy
+
+![](images/4f4ea408e9dfc8b339d556a8628cb56c8a3ab646be65ca54b1b71e1a1e0b8210.jpg)  
+(e) Train Loss
+
+![](images/a8f3770d53f89fe38298a02b33807d4a45450ddcffa2ff34c8d52b98e37cd548.jpg)  
+(f) Test Accuracy
+
+![](images/14bffc73d456e135460f2b9e9c083e52561ea6f33c81f7c9ed2ef9e25dba2835.jpg)  
+(g) Train Loss
+
+![](images/f9ab137e9ebdb5ac056ca9ed2c00a4ae16f59680ca5e1ecd42fc263f4bb0b50a.jpg)  
+(h) Test Accuracy  
+Figure 4. Comparison of the train loss and test accuracy against the number of communication rounds.
+
+# Acknowledgement
+
+TS was partially supported by JSPS KAKENHI (18K19793, 18H03201, and 20H00576), Japan DigitalDesign, and JST CREST.
+
+# References
+
+Allen-Zhu, Z. Natasha 2: Faster non-convex optimization than sgd. arXiv preprint arXiv:1708.08694, 2017.  
+Allen-Zhu, Z. and Hazan, E. Variance reduction for faster non-convex optimization. In International conference on machine learning, pp. 699–707. PMLR, 2016.  
+Das, R., Acharya, A., Hashemi, A., Sanghavi, S., Dhillon,
+
+I. S., and Topcu, U. Faster non-convex federated learning via global and local momentum. arXiv preprint arXiv:2012.04061, 2020.  
+Dekel, O., Gilad-Bachrach, R., Shamir, O., and Xiao, L. Optimal distributed online prediction using mini-batches. Journal of Machine Learning Research, 13(1), 2012.  
+Fang, C., Li, C. J., Lin, Z., and Zhang, T. Spider: Near-optimal non-convex optimization via stochastic path integrated differential estimator. arXiv preprint arXiv:1807.01695, 2018.  
+Ge, R., Huang, F., Jin, C., and Yuan, Y. Escaping from saddle points—online stochastic gradient for tensor decomposition. In Conference on learning theory, pp. 797-842. PMLR, 2015.  
+Glorot, X. and Bengio, Y. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pp. 249-256. JMLR Workshop and Conference Proceedings, 2010.  
+Goyal, P., Dollar, P., Girshick, R., Noordhuis, P., Wesolowski, L., Kyrola, A., Tulloch, A., Jia, Y., and He, K. Accurate, large minibatch sgd: Training imagenet in 1 hour. arXiv preprint arXiv:1706.02677, 2017.  
+Haddadpour, F. and Mahdavi, M. On the convergence of local descent methods in federated learning. arXiv preprint arXiv:1910.14425, 2019.  
+Haddadpour, F., Kamani, M. M., Mahdavi, M., and Cadambe, V. R. Local sgd with periodic averaging: Tighter analysis and adaptive synchronization. arXiv preprint arXiv:1910.13598, 2019.  
+Jin, C., Ge, R., Netrapalli, P., Kakade, S. M., and Jordan, M. I. How to escape saddle points efficiently. In International Conference on Machine Learning, pp. 1724-1732. PMLR, 2017.  
+Johnson, R. and Zhang, T. Accelerating stochastic gradient descent using predictive variance reduction. Advances in neural information processing systems, 26:315-323, 2013.  
+Kairouz, P., McMahan, H. B., Avent, B., Bellet, A., Dennis, M., Bhagoji, A. N., Bonawitz, K., Charles, Z., Cormode, G., Cummings, R., et al. Advances and open problems in federated learning. arXiv preprint arXiv:1912.04977, 2019.  
+Karimireddy, S. P., Jaggi, M., Kale, S., Mohri, M., Reddi, S. J., Stich, S. U., and Suresh, A. T. Mime: Mimicking centralized stochastic algorithms in federated learning. arXiv preprint arXiv:2008.03606, 2020a.
+
+Karimireddy, S. P., Kale, S., Mohri, M., Reddi, S., Stich, S., and Suresh, A. T. Scaffold: Stochastic controlled averaging for federated learning. In International Conference on Machine Learning, pp. 5132-5143. PMLR, 2020b.  
+Khaled, A., Mishchenko, K., and Richtárik, P. Tighter theory for local sgd on identical and heterogeneous data. In International Conference on Artificial Intelligence and Statistics, pp. 4519-4529. PMLR, 2020.  
+Koloskova, A., Loizou, N., Boreiri, S., Jaggi, M., and Stich, S. A unified theory of decentralized sgd with changing topology and local updates. In International Conference on Machine Learning, pp. 5381-5393. PMLR, 2020.  
+Konečny, J., McMahan, B., and Ramage, D. Federated optimization: Distributed optimization beyond the datacenter. arXiv preprint arXiv:1511.03575, 2015.  
+Lei, L., Ju, C., Chen, J., and Jordan, M. I. Non-convex finite-sum optimization via scsg methods. arXiv preprint arXiv:1706.09156, 2017.  
+Li, Z. Ssrgd: Simple stochastic recursive gradient descent for escaping saddle points. arXiv preprint arXiv:1904.09265, 2019.  
+Lin, T., Stich, S. U., Patel, K. K., and Jaggi, M. Don't use large mini-batches, use local sgd. arXiv preprint arXiv:1808.07217, 2018.  
+McMahan, B., Moore, E., Ramage, D., Hampson, S., and y Arcas, B. A. Communication-efficient learning of deep networks from decentralized data. In Artificial Intelligence and Statistics, pp. 1273-1282. PMLR, 2017.  
+Nguyen, L. M., Liu, J., Scheinberg, K., and Takáč, M. Sarah: A novel method for machine learning problems using stochastic recursive gradient. In International Conference on Machine Learning, pp. 2613-2621. PMLR, 2017.  
+Nguyen, L. M., van Dijk, M., Phan, D. T., Nguyen, P. H., Weng, T.-W., and Kalagnanam, J. R. Finite-sum smooth optimization with sarah. arXiv preprint arXiv:1901.07648, 2019.  
+Reddi, S. J., Hefny, A., Sra, S., Poczos, B., and Smola, A. Stochastic variance reduction for nonconvex optimization. In International conference on machine learning, pp. 314-323. PMLR, 2016a.  
+Reddi, S. J., Konečny, J., Richtárik, P., Póczós, B., and Smola, A. Aide: Fast and communication efficient distributed optimization. arXiv preprint arXiv:1608.06879, 2016b.
+
+Sharma, P., Kafle, S., Khanduri, P., Bulusu, S., Rajawat, K., and Varshney, P. K. Parallel restarted spider-communication efficient distributed nonconvex optimization with optimal computation complexity. arXiv preprint arXiv:1912.06036, 2019.  
+Shokri, R. and Shmatikov, V. Privacy-preserving deep learning. In Proceedings of the 22nd ACM SIGSAC conference on computer and communications security, pp. 1310-1321, 2015.  
+Stich, S. U. Local sgd converges fast and communicates little. arXiv preprint arXiv:1805.09767, 2018.  
+Woodworth, B., Patel, K. K., and Srebro, N. Minibatch vs local sgd for heterogeneous distributed learning. arXiv preprint arXiv:2006.04735, 2020a.  
+Woodworth, B., Patel, K. K., Stich, S., Dai, Z., Bullins, B., McMahan, B., Shamir, O., and Srebro, N. Is local sgd better than minibatch sgd? In International Conference on Machine Learning, pp. 10334-10343. PMLR, 2020b.  
+Yu, H., Yang, S., and Zhu, S. Parallel restarted sgd with faster convergence and less communication: Demystifying why model averaging works for deep learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 5693-5700, 2019.  
+Zhou, D., Xu, P., and Gu, Q. Stochastic nested variance reduction for nonconvex optimization. arXiv preprint arXiv:1806.07811, 2018.

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