Efficient learning by implicit exploration in bandit problems with side observations
arXiv:2604.24555v1 [cs.LG] 27 Apr 2026
Tomáš Kocák Gergely Neu Michal Valko Rémi Munos∗ SequeL team, INRIA Lille – Nord Europe, France {tomas.kocak,gergely.neu,michal.valko,remi.munos}@inria.fr
Abstract We consider online learning problems under a partial observability model capturing situations where the information conveyed to the learner is between full information and bandit feedback. In the simplest variant, we assume that in addition to its own loss, the learner also gets to observe losses of some other actions. The revealed losses depend on the learner’s action and a directed observation system chosen by the environment. For this setting, we propose the first algorithm that enjoys near-optimal regret guarantees without having to know the observation system before selecting its actions. Along similar lines, we also define a new partial information setting that models online combinatorial optimization problems where the feedback received by the learner is between semi-bandit and full feedback. As the predictions of our first algorithm cannot be always computed efficiently in this setting, we propose another algorithm with similar properties and with the benefit of always being computationally efficient, at the price of a slightly more complicated tuning mechanism. Both algorithms rely on a novel exploration strategy called implicit exploration, which is shown to be more efficient both computationally and information-theoretically than previously studied exploration strategies for the problem.
1
Introduction
Consider the problem of sequentially recommending content for a set of users. In each period of this online decision problem, we have to assign content from a news feed to each of our subscribers so as to maximize clickthrough. We assume that this assignment needs to be done well in advance, so that we only observe the actual content after the assignment was made and the user had the opportunity to click. While we can easily formalize the above problem in the classical multi-armed bandit framework [3], notice that we will be throwing out important information if we do so! The additional information in this problem comes from the fact that several news feeds can refer to the same content, giving us the opportunity to infer clickthroughs for a number of assignments that we did not actually make. For example, consider the situation shown on Figure 1a. In this simple example, we want to suggest one out of three news feeds to each user, that is, we want to choose a matching on the graph shown on Figure 1a which covers the users. Assume that news feeds 2 and 3 refer to the same content, so whenever we assign news feed 2 or 3 to any of the users, we learn the value of both of these assignments. The relations between these assignments can be described by a graph structure (shown on Figure 1b), where nodes represent user-news feed assignments, and edges mean that the corresponding assignments reveal the clickthroughs of each other. For a more compact representation, we can group the nodes by the users, and rephrase our task as having to choose one node from each group. Besides its own reward, each selected node reveals the rewards assigned to all their neighbors. ∗
Current affiliation: Google DeepMind
1
user1
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content2 e1,2 e1,3
e2,3
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Figure 1a: Users and news feeds. The thick edges represent one potential matching of users to feeds, grouped news feeds show the same content.
Figure 1b: Users and news feeds. Connected feeds mutually reveal each others clickthroughs.
The problem described above fits into the framework of online combinatorial optimization where in each round, a learner selects one of a very large number of available actions so as to minimize the losses associated with its sequence of decisions. Various instances of this problem have been widely studied in recent years under different feedback assumptions [7, 2, 8], notably including the so-called full-information [13] and semi-bandit [2, 16] settings. Using the example in Figure 1a, assuming full information means that clickthroughs are observable for all assignments, whereas assuming semibandit feedback, clickthroughs are only observable on the actually realized assignments. While it is unrealistic to assume full feedback in this setting, assuming semi-bandit feedback is far too restrictive in our example. Similar situations arise in other practical problems such as packet routing in computer networks where we may have additional information on the delays in the network besides the delays of our own packets. In this paper, we generalize the partial observability model first proposed by Mannor and Shamir [15] and later revisited by Alon et al. [1] to accommodate the feedback settings situated between the full-information and the semi-bandit schemes. Formally, we consider a sequential decision making problem where in each time step t the (potentially adversarial) environment assigns a loss value to each out of d components, and generates an observation system whose role will be clarified soon. Obliviously of the environment’s choices, the learner chooses an action Vt from a fixed action set S ⊂ {0, 1}d represented by a binary vector with at most m nonzero components, and incurs the sum of losses associated with the nonzero components of Vt . At the end of the round, the learner observes the individual losses along the chosen components and some additional feedback based on its action and the observation system. We represent this observation system by a directed observability graph with d nodes, with an edge connecting i → j if and only if the loss associated with j is revealed to the learner whenever Vt,i = 1. The goal of the learner is to minimize its total loss obtained over T repetitions of the above procedure. The two most well-studied variants of this general framework are the multi-armed bandit problem [3] where each action consists of a single component and the observability graph is a graph without edges, and the problem of prediction with expert advice [17, 14, 5] where each action consists of exactly one component and the observability graph is complete. In the true combinatorial setting where m > 1, the empty and complete graphs correspond to the semi-bandit and full-information settings respectively. Our model directly extends the model of Alon et al. [1], whose setup coincides with m = 1 in our framework. Alon et al. themselves were motivated by the work of Mannor and Shamir [15], who considered undirected observability systems where actions mutually uncover each other’s losses. Mannor √ and Shamir proposed an algorithm based on linear programming that achieves a regret of Õ( cT ), where c is the number into which the graph can be split. While this bound √ of cliques √ recovers the usual bounds of Õ( T ) and Õ( dT ) in the experts and bandit settings respectively, it fails to exploit the opportunities offered by observation systems with large number of connections but only small in such situations we would reasonably expect a regret guarantee √ √ cliques. While closer to Õ( T ) than Õ( dT ), the bound of Mannor and Shamir [15] only delivers the latter with c = Ω(d). This issue was addressed by Alon et al. [1] who proposed an algorithm called E XP 3-SET √ that guarantees a regret of O( αT log d), where α is an upper bound on the independence numbers of the observability graphs assigned by the environment. In particular, E XP 3-SET is much more efficient than the algorithm of Mannor and Shamir as it only requires running the E XP 3 algorithm of Auer et al. [3] on the decision set, which runs in time linear in d. Alon et al. [1] also extend the model of Mannor and Shamir in allowing the observability graph to be directed. For this setting, they offer another algorithm called E XP 3-DOM with similar guarantees, although with the serious drawback 2
that it requires access to the observation system before choosing its actions. This assumption poses severe limitations to the practical applicability of E XP 3-DOM, which also needs to solve a sequence of set cover problems as a subroutine. In the present paper, we offer two computationally and information-theoretically efficient algorithms for bandit problems with directed observation systems. Both of our algorithms circumvent the costly exploration phase required by E XP 3-DOM by a trick that we will refer to IX as in Implicit eXploration. Accordingly, we name our algorithms E XP 3-IX and FPL-IX, which are variants of the well-known E XP 3 [3] and FPL [12] algorithms enhanced with implicit exploration. Our first algorithm E XP 3-IX is specifically designed1 to work in the setting of Alon et al. [1] with m = 1 and does not need to solve any set cover problems or have any sort of prior knowledge concerning the observation systems chosen by the adversary.2 FPL-IX, on the other hand, does need either to solve set cover problems or have a prior upper bound on the independence numbers of the observability graphs, but can be computed efficiently for a wide range of true combinatorial problems with m > 1. We note that our algorithms do not even need to know the number of rounds T and our regret bounds scale with the average independence number ᾱ of the graphs played by the adversary rather than the largest of these numbers. They both employ adaptive learning rates and unlike E XP 3-DOM, they do not need to use a doubling trick to be anytime or to aggregate outputs of multiple algorithms to √ 3/2 optimally set their learning rates. Both algorithms achieve regret guarantees of Õ(m ᾱT ) in the √ combinatorial setting, which becomes Õ( ᾱT ) in the simple setting. Before diving into the main content, we give an important graph-theoretic statement that we will rely on when analyzing both of our algorithms. The lemma is a generalized version of Lemma 13 of Alon et al. [1] and its proof is given in Appendix A. Lemma 1. Let G be a directed graph with vertex set V = {1, . . . , d}. Let Ni− be the inneighborhood of node i, i.e., the set of nodes j such that (j → i) ∈ G. Let α be the independence Pd number of G and p1 ,. . . ,pd are numbers from [0, 1] such that i=1 pi ≤ m. Then d X pi m⌈d2 /c⌉ + d + 2m, ≤ 2mα log 1 + 1 1 α p +m Pi + c i=1 m i P where Pi = j∈N − pj and c is a positive constant. i
2
Multi-armed bandit problems with side information
In this section, we start by the simplest setting fitting into our framework, namely the multi-armed bandit problem with side observations. We provide intuition about the implicit exploration procedure behind our algorithms and describe E XP 3-IX, the most natural algorithm based on the IX trick. The problem we consider is defined as follows. In each round t = 1, 2, . . . , T , the environment assigns a loss vector ℓt ∈ [0, 1]d for d actions and also selects an observation system described by the directed graph Gt . Then, based on its previous observations (and likely some external source of randomness) the learner selects action It and subsequently incurs and observes loss ℓt,It . Furthermore, the learner also observes the losses ℓt,j for all j such that (It → j) ∈ Gt , denoted by the indicator Ot,i . Let Ft−1 = σ(It−1 , . . . , I1 ) capture the interaction history up to time t. As usual in online settings [6], the performance is measured in terms of (total expected) regret, which is the difference between a total loss received and the total loss of the best single action chosen in hindsight, " T # X RT = max E (ℓt,It − ℓt,i ) , i∈[d]
t=1
where the expectation integrates over the random choices made by the learning algorithm. Alon et al. [1] adapted the well-known E XP 3 algorithm of Auer et al. [3] for this precise problem. Their algorithm, E XP 3-DOM, works by maintaining a weight wt,i for each individual arm i ∈ [d] in each round, and selecting It according to the distribution wt,i P [It = i |Ft−1 ] = (1 − γ)pt,i + γµt,i = (1 − γ) Pd + γµt,i , j=1 wt,j 1
E XP 3-IX can also be efficiently implemented for some specific combinatorial decision sets even with m > 1, see, e.g., Cesa-Bianchi and Lugosi [7] for some examples. 2 However, it is still necessary to have access to the observability graph to construct low bias estimates of losses, but only after the action is selected.
3
where γ ∈ (0, 1) is parameter of the algorithm and µt is an exploration distribution whose role we will shortly clarify. After each round, E XP 3-DOM defines the loss estimates ℓt,i ℓ̂t,i = 1{(It →i)∈Gt } where ot,i = E [Ot,i |Ft−1 ] = P [(It → i) ∈ Gt |Ft−1 ] ot,i for each i ∈ [d]. These loss estimates are then used to update the weights for all i as wt+1,i = wt,i e−γ ℓ̂t,i . It is easy to see that these loss estimates ℓ̂t,i are unbiased estimates of the true losses whenever pt,i > 0 holds for all i. This requirement along with another important technical issue justify the presence of the exploration distribution µt . The key idea behind E XP 3-DOM is to compute a dominating set Dt ⊆ [d] of the observability graph Gt in each round, and define µt as the uniform distribution over Dt . This choice ensures that ot,i ≥ pt,i + γ/|Dt |, a crucial requirement for the analysis of [1]. In what follows, we propose an exploration scheme that does not need any fancy computations but, more importantly, works without any prior knowledge of the observability graphs. 2.1
Efficient learning by implicit exploration
In this section, we propose the simplest exploration scheme imaginable, which consists of merely pretending to explore. Precisely, we simply sample our action It from the distribution defined as wt,i P [It = i |Ft−1 ] = pt,i = Pd , (1) j=1 wt,j without explicitly mixing with any exploration distribution. Our key trick is to define the loss estimates for all arms i as ℓt,i ℓ̂t,i = 1{(It →i)∈Gt } , ot,i + γt where γt > 0 is a parameter of our algorithm. It is easy to check that ℓ̂t,i is a biased estimate of ℓt,i . The nature of this bias,hhowever, is i very special. First, observe that ℓ̂t,i is an optimistic estimate of
ℓt,i in the sense that E ℓ̂t,i |Ft−1 ≤ ℓt,i . That is, our bias always ensures that, on expectation, we underestimate the loss of any fixed arm i. Even more importantly, our loss estimates also satisfy " d # d d X X X ot,i −1 E pt,i ℓ̂t,i Ft−1 = pt,i ℓt,i + pt,i ℓt,i ot,i + γt i=1 i=1 i=1 (2) d d X X pt,i ℓt,i , = pt,i ℓt,i − γt o + γt i=1 i=1 t,i
that is, the bias of the estimated losses suffered by our algorithm is directly controlled by γt . As we will see in the analysis, it is sufficient to control the bias of our own estimated performance as long as we can guarantee that the loss estimates associated with any fixed arm are optimistic—which is precisely what we have. Note that this slight modification ensures that the denominator of ℓ̂t,i is lower bounded by pt,i + γt , which is a very similar property as the one achieved by the exploration scheme used by E XP 3-DOM. We call the above loss estimation method implicit exploration or IX, as it gives rise to the same effect as explicit exploration without actually having to implement any exploration policy. In fact, explicit and implicit explorations can both be regarded as two different approaches for bias-variance tradeoff: while explicit exploration biases the sampling distribution of It to reduce the variance of the loss estimates, implicit exploration achieves the same result by biasing the loss estimates themselves. From this point on, we take a somewhat more predictable course and define our algorithm E XP 3-IX as a variant of E XP 3 using the IX loss estimates. One of the twists is that E XP 3-IX is actually based on the adaptive learning-rate variant of E XP 3 proposed by Auer et al. [4], which avoids the necessity of prior knowledge of the observability graphs in order to set a proper learning rate. This algorithm b t−1,i = Pt−1 ℓ̂s,i and for all i ∈ [d] computing the weights as is defined by setting L s=1 wt,i = (1/d)e−ηt Lt−1,i . b
These weights are then used to construct the sampling distribution of It as defined in (1). The resulting E XP 3-IX algorithm is shown as Algorithm 1. 4
2.2
Performance guarantees for E XP 3-IX
Our analysis follows the footsteps of Auer et al. [3] and Györfi and Ottucsák [9], who provide an improved analysis of the adaptive learningrate rule proposed by Auer et al. [4]. However, a technical subtlety will force us to proceed a little differently than these standard proofs: for achieving the tightest possible bounds and the most efficient algorithm, we need to tune our learning rates according to some random quantities that depend on the performance of E XP 3IX. In fact, the key quantities in our analysis are the terms Qt =
Algorithm 1 E XP 3-IX 1: Input: Set of actions S = [d], 2: parameters γt ∈ (0, 1), ηt > 0 for t ∈ [T ]. 3: for t = 1 to T do b t−1,i ) for i ∈ [d] 4: wt,i ← (1/d) exp (−ηt L 5: An adversary privately chooses losses ℓt,i 6: 7: 8: 9: 10: 11:
for i ∈ [d] and generates a graph Gt Pd Wt ← i=1 wt,i pt,i ← wt,i /Wt Choose It ∼ pt = (pt,1 , . . . , pt,d ) Observe graph Gt ObservePpairs {i, ℓt,i } for (It → i) ∈ Gt ot,i ← (j→i)∈Gt pt,j for i ∈ [d] ℓ
d X
12: ℓ̂t,i ← ot,it,i +γt 1{(It →i)∈Gt } for i ∈ [d] 13: end for
pt,i , o + γt i=1 t,i
which depend on the interaction history Ft−1 for all t. Our theorem below gives the performance guarantee for E XP 3-IX using a parameter setting adaptive to the values of Qt . A full proof of the theorem is given in the supplementary material. q Pt−1 Theorem 1. Setting ηt = γt = (log d)/(d + s=1 Qs ) , the regret of E XP 3-IX satisfies "r # PT d + t=1 Qt log d . RT ≤ 4E (3) Proof sketch. Following the proof of Lemma 1 in Györfi and Ottucsák [9], we can prove that d d 2 log W X log Wt+1 ηt X t pt,i ℓ̂t,i + pt,i ℓ̂t,i ≤ − . 2 i=1 ηt ηt+1 i=1
(4)
Taking conditional expectations, using Equation (2) and summing up both sides, we get T T d T X X X X log Wt ηt log Wt+1 + γt Qt + E Ft−1 . pt,i ℓt,i ≤ − 2 ηt ηt+1 t=1 t=1 t=1 i=1 Using Lemma 3.5 of Auer et al. [4] and plugging in ηt and γt , this becomes r d T X T X X PT log Wt log Wt+1 pt,i ℓt,i ≤ 3 d + t=1 Qt log d + E − Ft−1 . ηt ηt+1 t=1 i=1 t=1 Taking expectations on both sides, the second term on the right hand side telescopes into h i log W1 log WT +1 log wT +1,j log d E − ≤E − =E + E L̂T,j η1 ηT +1 ηT +1 ηT +1 for any j ∈ [d], giving the desired result as T X d X t=1 i=1
pt,i ℓt,i ≤
T X
ℓt,j + 4E
"r
d+
PT
t=1 Qt
# log d ,
t=1
where we used the definition of ηT and the optimistic property of the loss estimates. Setting m = 1 and c = γt in Lemma 1, gives the following deterministic upper bound on each Qt . Lemma 2. For all t ∈ [T ], d X pt,i ⌈d2 /γt ⌉ + d Qt = ≤ 2αt log 1 + + 2. o + γt αt i=1 t,i 5
Combining Lemma 2 with Theorem 1 we prove our main result concerning the regret of E XP 3-IX. Corollary 1. The regret of E XP 3-IX satisfies r PT d + 2 t=1 (Ht αt + 1) log d, RT ≤ 4 where Ht = log 1 +
3
⌈d2
p
td/ log d⌉ + d αt
! = O(log(dT )).
Combinatorial semi-bandit problems with side observations
We now turn our attention to the setting of online combinatorial optimization (see [13, 7, 2]). In this variant of the online learning problem, the learner has access to a possibly huge action set d S ⊆ {0, 1} where each action is represented by a binary vector v of dimensionality d. In what follows, we assume that ∥v∥1 ≤ m holds for all v ∈ S and some 1 ≤ m ≪ d, with the case m = 1 corresponding to the multi-armed bandit setting considered in the previous section. In each round t = 1, 2, . . . , T of the decision process, the learner picks an action Vt ∈ S and incurs a loss of VtT ℓt . At the end of the round, the learner receives some feedback based on its decision Vt and the loss vector ℓt . The regret of the learner is defined as " T # X T RT = max E (Vt − v) ℓt . v∈S
t=1
Previous work has considered the following feedback schemes in the combinatorial setting: • The full information scheme where the learner√gets to observe ℓt regardless of the chosen action. The minimax optimal regret of order m T log d here is achieved by C OMPONENT H EDGE algorithm of [13], while the√Follow-the-Perturbed-Leader (FPL) [12, 10] was shown to enjoy a regret of order m3/2 T log d by [16]. • The semi-bandit scheme where the learner gets to observe the components ℓt,i of the loss vector where Vt,i = 1, that is, the losses along the components chosen by √ the learner at mdT log d) time t. As shown by [2], C OMPONENT H EDGE achieves a near-optimal O( √ regret guarantee, while [16] show that FPL enjoys a bound of O(m dT log d). • The bandit scheme where the learner only observes its own loss VtT ℓt . There are currently no known efficient algorithms that get close to the minimax regret in this setting—the reader is referred to Audibert et al. [2] for an overview of recent results. In this section, we define a new feedback scheme situated between the semi-bandit and the fullinformation schemes. In particular, we assume that the learner gets to observe the losses of some other components not included in its own decision vector Vt . Similarly to the model of Alon et al. [1], the relation between the chosen action and the side observations are given by a directed observability graph Gt (see example in Figure 1). We refer to this feedback scheme as semi-bandit with side observations. While our theoretical results stated in the previous section continue to hold in this setting, combinatorial E XP 3-IX could rarely be implemented efficiently—we refer to [7, 13] for some positive examples. As one of the main concerns in this paper is computational efficiency, we take a different approach: we propose a variant of FPL that efficiently implements the idea of implicit exploration in combinatorial semi-bandit problems with side observations. 3.1
Implicit exploration by geometric resampling
b t−1 = Pt−1 ℓ̂s of the total losses In each round t, FPL bases its decision on some estimate L s=1 Pt−1 Lt−1 = s=1 ℓs as follows: b t−1 − Zt . Vt = arg min v T ηt L (5) v∈S
Here, ηt > 0 is a parameter of the algorithm and Zt is a perturbation vector with components drawn independently from an exponential distribution with unit expectation. The power of FPL lies in that it only requires an oracle that solves the (offline) optimization problem minv∈S v T ℓ and thus 6
can be used to turn any efficient offline solver into an online optimization algorithm with strong guarantees. To define our algorithm precisely, we need some further notation. We redefine Ft−1 to be σ(Vt−1 , . . . , V1 ), Ot,i to be the indicator of the observed component and let qt,i = E [Vt,i |Ft−1 ]
and
ot,i = E [Ot,i |Ft−1 ] .
The most crucial point of our algorithm is the construction of our loss estimates. To implement the idea of implicit exploration by optimistic biasing, we apply a modified version of the geometric resampling method of Neu and Bartók [16] constructed as follows: Let Ot′ (1), Ot′ (2), . . . be independent copies3 of Ot and let Ut,i be geometrically distributed random variables for all i = [d] with parameter γt . We let ′ Kt,i = min k : Ot,i (k) = 1 ∪ {Ut,i } (6) and define our loss-estimate vector ℓ̂t ∈ Rd with its i-th element as ℓ̂t,i = Kt,i Ot,i ℓt,i .
(7)
By definition, we have E [Kt,i |Ft−1 ] = 1/(ot,i + (1 − ot,i )γt ), implying that our loss estimates are optimistic in the sense that they lower bound the losses in expectation: h i ot,i E ℓ̂t,i Ft−1 = ℓt,i ≤ ℓt,i . ot,i + (1 − ot,i )γt Here we used the fact that Ot,i is independent of Kt,i and has expectation ot,i given Ft−1 . We call this algorithm Follow-the-Perturbed-Leader with Implicit eXploration (FPL-IX, Algorithm 2). Note that the geometric resampling procedure can be terminated as soon as Kt,i becomes welldefined for all i with Ot,i = 1. As noted by Neu and Bartók [16], this requires generating at most d copies of Ot on expectation. As each of these copies requires one access to the linear optimization oracle over S, we conclude that the expected running time of FPL-IX is at most d times that of the expected running time of the oracle. A high-probability guarantee of the running time can be obtained by observing that Ut,i ≤ log 1δ /γt holds with probability at least 1 − δ and thus we can stop sampling after at most d log dδ /γt steps with probability at least 1 − δ. 3.2
Performance guarantees for FPL-IX
The analysis presented in this section com- Algorithm 2 FPL-IX bines some techniques used by Kalai and Vem1: Input: Set of actions S, pala [12], Hutter and Poland [11], and Neu 2: parameters γt ∈ (0, 1), ηt > 0 for t ∈ [T ]. and Bartók [16] for analyzing FPL-style learn- 3: for t = 1 to T do ers. Our proofs also heavily rely on some spe4: An adversary privately chooses losses ℓt,i cific properties of the IX loss estimate defined for all i ∈ [d] and generates a graph Gt in Equation 7. The most important difference 5: Draw Zt,i ∼ Exp(1) for all i ∈ [d] from the analysis presented in Section 2.2 is T b t−1 − Zt 6: Vt ← arg minv∈S v ηt L that now we are not able to use random learnT 7: Receive loss Vt ℓt ing rates as we cannot compute the values cor8: Observe graph Gt responding to Qt efficiently. In fact, these val9: Observe pairs {i, ℓt,i } for all i, such that ues are observable in the information-theoretic (j → i) ∈ Gt and v(It )j = 1 sense, so we could prove bounds similar to Theorem 1 had we had access to infinite compu- 10: Compute Kt,i for all i ∈ [d] using Eq. (6) tational resources. As our focus in this paper 11: ℓ̂t,i ← Kt,i Ot,i ℓt,i is on computationally efficient algorithms, we 12: end for choose to pursue a different path. In particular, our learning rates will be tuned according to efficiently computable approximations α et of the respective independence numbers αt that satisfy αt /C ≤ α et ≤ αt ≤ d for some C ≥ 1. For the sake of simplicity, we analyze the algorithm in the oblivious adversary model. The following theorem states the performance guarantee for FPL-IX in terms of the learning rates and random variables of the form d X qt,i e t (c) = Q . o +c i=1 t,i 3 Such independent copies can be simply generated by sampling independent copies of Vt using the FPL rule (5) and then computing Ot′ (k) using the observability Gt . Notice that this procedure requires no interaction between the learner and the environment, although each sample requires an oracle access.
7
Theorem 2. Assume γt ≤ 1/2 for all t and η1 ≥ η2 ≥ · · · ≥ ηT . The regret of FPL-IX satisfies X T T h i X m (log d + 1) γt et e t (γt ) . RT ≤ + 4m + ηt E Q γt E Q ηT 1 − γt t=1 t=1 Proof sketch. As usual for analyzing FPL methods [12, 11, 16], we first define a hypothetical learner e ∼ Z1 and has access to ℓ̂t on top of L b t−1 that uses a time-independent perturbation vector Z bt − Z e . Vet = arg min v T ηt L v∈S
Clearly, this learner is infeasible as it uses observations from the future. Also, observe that this learner does not actually interact with the environment and depends on the predictions made by the actual learner only through the loss estimates. By standard arguments, we can prove " T # T X m (log d + 1) e . E Vt − v ℓ̂t ≤ ηT t=1 Using the techniques of Neu and Bartók [16], we can relate the performance of Vt to that of Vet , which we can further upper bound after a long and tedious calculation as h i 2 γ T T e e e Ft−1 . E (Vt − Vt ) ℓ̂t Ft−1 ≤ ηt E Vt−1 ℓ̂t Ft−1 ≤ 4mηt E Qt 1−γ h i The result follows by observing that E v T ℓ̂t Ft−1 ≤ v T ℓt for any fixed v ∈ S by the optimistic property of the IX estimate and also from the fact that by the definition of the estimates we infer that h i i h T e t (γt ) . E Vet−1 ℓ̂t Ft−1 ≥ E [ VtT ℓt | Ft−1 ] − γt E Q The next lemma shows a suitable upper bound P for the last two terms in the bound of Theorem 2. It follows from observing that ot,i ≥ (1/m) j∈{N − ∪{i}} qt,j and applying Lemma 1. t,i
Lemma 3. For all t ∈ [T ] and any c ∈ (0, 1), e t (c) = Q
d X
m⌈d2 /c⌉ + d qt,i ≤ 2mαt log 1 + + 2m. o +c αt i=1 t,i
We are now ready to state the main result of this section, which is obtained by combining Theorem 2, Lemma 3, and Lemma 3.5 of Auer et al. [4] applied to the following upper bound q q T T X X PT PT αt αt q q ≤ ≤ 2 C α ≤ 2 d + C t=1 αt . t=1 t Pt Pt−1 t=1 t=1 d + s=1 α es s=1 αs /C Corollary 2. Assume that for ≤ α et ≤ αt ≤ d for some C > 1, and assume r all t ∈ [T ], αt /C Pt−1 es , the regret of FPL-IX satisfies md > 4. Setting ηt = γt = (log d + 1) / m d + s=1 α 3/2
RT ≤ Hm
r
d+C
PT
t=1 αt
(log d + 1),
where H is O(log(mdT )).
Conclusion We presented an efficient algorithm for learning with side observations based on implicit exploration. This technique gave rise to multitude of improvements. Remarkably, our algorithms no longer need to know the observation system before choosing the action unlike the method of [1]. Moreover, we extended the partial observability model of [15, 1] to accommodate problems with large and structured action sets and also gave an efficient algorithm for this setting. Acknowledgements The research presented in this paper was supported by French Ministry of Higher Education and Research, by European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no 270327 (CompLACS), and by FUI project Hermès. 8
References [1] Alon, N., Cesa-Bianchi, N., Gentile, C., and Mansour, Y. (2013). From Bandits to Experts: A Tale of Domination and Independence. In Neural Information Processing Systems (NeurIPS). [2] Audibert, J. Y., Bubeck, S., and Lugosi, G. (2014). Regret in Online Combinatorial Optimization. Mathematics of Operations Research, 39:31–45. [3] Auer, P., Cesa-Bianchi, N., Freund, Y., and Schapire, R. E. (2002a). The nonstochastic multiarmed bandit problem. SIAM J. Comput., 32(1):48–77. [4] Auer, P., Cesa-Bianchi, N., and Gentile, C. (2002b). Adaptive and self-confident on-line learning algorithms. Journal of Computer and System Sciences, 64:48–75. [5] Cesa-Bianchi, N., Freund, Y., Haussler, D., Helmbold, D., Schapire, R., and Warmuth, M. (1997). How to use expert advice. Journal of the ACM, 44:427–485. [6] Cesa-Bianchi, N. and Lugosi, G. (2006). Prediction, Learning, and Games. Cambridge University Press, New York, NY, USA. [7] Cesa-Bianchi, N. and Lugosi, G. (2012). Combinatorial bandits. Journal of Computer and System Sciences, 78:1404–1422. [8] Chen, W., Wang, Y., and Yuan, Y. (2013). Combinatorial Multi-Armed Bandit: General Framework and Applications. In International Conference on Machine Learning (ICML), pages 151– 159. [9] Györfi, L. and Ottucsák, G. (2007). Sequential prediction of unbounded stationary time series. IEEE Transactions on Information Theory, 53(5):1866–1872. [10] Hannan, J. (1957). Approximation to Bayes Risk in Repeated Play. Contributions to the Theory of Games, 3:97–139. [11] Hutter, M. and Poland, J. (2004). Prediction with Expert Advice by Following the Perturbed Leader for General Weights. In Algorithmic Learning Theory (ALT), pages 279–293. [12] Kalai, A. and Vempala, S. (2005). Efficient algorithms for online decision problems. Journal of Computer and System Sciences, 71:291–307. [13] Koolen, W. M., Warmuth, M. K., and Kivinen, J. (2010). Hedging structured concepts. In Conference on Learning Theory (COLT), pages 93–105. [14] Littlestone, N. and Warmuth, M. (1994). The weighted majority algorithm. Information and Computation, 108:212–261. [15] Mannor, S. and Shamir, O. (2011). From Bandits to Experts: On the Value of SideObservations. In Neural Information Processing Systems (NeurIPS). [16] Neu, G. and Bartók, G. (2013). An Efficient Algorithm for Learning with Semi-bandit Feedback. In Algorithmic Learning Theory (ALT), volume 8139 of Lecture Notes in Computer Science, pages 234–248. [17] Vovk, V. (1990). Aggregating strategies. In Computational Learning Theory (COLT), pages 371–386.
A
Proof of Lemma 1
The proof relies on the following two statements borrowed from Alon et al. [1]. Lemma 4. (cf. Lemma 10 of [1]) Let G be a directed graph, with V = {1, . . . , d}. Let d− i be the indegree of the node i and α = α(G) be the independence number of G. Then d X
1 d . ≤ 2α log 1 + α 1 + d− i i=1
Lemma 5. (cf. Lemma 12 of [1]) If a, b ≥ 0 and a + b ≥ B > A > 0, then a a A ≤ + a+b−A a+b B−A 9
Proof. a a aA A A − = ≤ ≤ a+b−A a+b (a + b)(a + b − A) a+b−A B−A
We are now ready to prove Lemma 1. Our proof is obtained as a generalization of the proof of Lemma 13 by Alon et al. [1]. Let M = ⌈d2 /c⌉ and d− i be the indegree of node i. We begin by constructing a discretization of the values pi for all i such that the discretized version of pi satisfies p̂i = k/M for some integer k and p̂i − 1/M < pi ≤ p̂i . By straightforward algebraic manipulations and the fact that x/(x + a) is increasing in x for nonnegative x and a, we obtain the bound d X
pi
1 1 p +m Pi + c i=1 m i
≤m
≤m
d X
p̂ + i=1 i
P
j∈Ni−
d X
P
j∈Ni− pj + mc
p̂i p̂j + mc − d− i /M
p̂i p̂i +
P
j∈Ni− p̂j + mc
d X i=1
pi
p + i=1 i
d X
i=1
≤m
=m
M p̂i +
M p̂ P i
j∈Ni− M p̂j
+m
d X
d− i /M mc − d− i /M i=1
+ 2m,
P where the second to last inequality holds by Lemma 5 with a = p̂i , b = j∈N − p̂j , A = d− i /M , i and B = mc. It remains to find a suitable upper bound for the first sum on the right hand side. To this end, we construct a graph G′ from our original graph G, where that we replace each node i of G by a clique Ci with M pi nodes. In this expanded graph, we connect all vertices in clique Ci with all vertices in Cj if and only if there is an edge from i to j in original graph G. Note that our new graph G′ has the same independence number α as the P original graph G. Also observe that the indegree dˆ− k of a node k in clique Ci is equal to M pi − 1 + j∈N − M pj . Therefore, the remaining term can be i rewritten as d X
M p̂i + i=1
M p̂ P i
j∈Ni−
M p̂j
=
d X X
1
ˆ− i=1 k∈Ci 1 + dk
which in turn can be bounded using Lemma 4 by ! Pd M p̂ mM + d i i=1 2α ln 1 + ≤ 2α ln 1 + . α α Using this bound we get d X
mM + d ≤ 2mα ln 1 + + 2m 1 1 α p +m Pi + c i=1 m i pi
as advertised.
B
Full proof of Theorem 1
Proof (Theorem 1). We start by introducing some notation. Let b t−1,i = L
t−1 X
d
ℓ̂s,i
Wt′ =
and
s=1
10
1 X −ηt−1 Lbt−1,i e . d i=1
′ Following the proof of Lemma 1 of Györfi and Ottucsák [9], we track the evolution of log Wt+1 /Wt to control the regret. We have b t,i d 1 −ηt L d X X W′ 1 1 1 wt,i e−ηt ℓ̂t,i de log t+1 = log = log ηt Wt ηt Wt ηt Wt i=1 i=1 d d X X 1 1 1 log log = pt,i e−ηt ℓ̂t,i ≤ pt,i 1 − ηt ℓ̂t,i + (ηt ℓ̂t,i )2 ηt ηt 2 i=1 i=1 ! d d X 1 η2 X log 1 − ηt pt,i (ℓ̂t,i )2 , = pt,i ℓ̂t,i + t ηt 2 i=1 i=1
where we used the inequality exp(−x) ≤ 1 − x + x2 /2 that holds for x ≥ 0. Using the inequality log(1 − x) ≤ −x that holds for all x, we get X d d ′ X log Wt+1 log Wt ηt pt,i ℓ̂t,i ≤ − + pt,i (ℓ̂t,i )2 η η 2 t t i=1 i=1 X d ′ log Wt+1 log Wt+1 log Wt log Wt+1 ηt pt,i (ℓ̂t,i )2 . = − − + + ηt ηt+1 ηt+1 ηt 2 i=1 The second term in brackets on the right hand side can be bounded as Wt+1 =
d X 1 i=1
d
e−ηt+1 Lt,i = b
d X 1 i=1
d
e−ηt Lt,i b
ηt+1 η t
≤
d X 1 i=1
d
! ηt+1 ηt e−ηt Lt,i b
ηt+1
′ = (Wt+1 ) ηt ,
ηt+1
where we applied Jensen’s inequality to the concave function x ηt for x ∈ R. The function is concave since ηt+1 ≤ ηt by definition. Taking logarithms in the above inequality, we get ′ log Wt+1 log Wt+1 − ≤ 0. ηt+1 ηt Using this inequality, we prove Equation (4) as d d 2 log W X ηt X log Wt+1 t pt,i ℓ̂t,i + pt,i ℓ̂t,i ≤ − . 2 i=1 ηt ηt+1 i=1 Taking conditional expectations and summing up both sides over the time, we get " T d # " T # T d 2 XX X ηt X X log Wt log Wt+1 E Ft−1 . pt,i ℓ̂t,i Ft−1 ≤ E pt,i ℓ̂t,i Ft−1 + − E 2 i=1 ηt ηt+1 t=1 i=1 t=1 t=1 The first term in the above inequality is controlled as " d # d d X X X E pt,i ℓ̂t,i Ft−1 = pt,i ℓt,i + pt,i ℓt,i
ot,i −1 o t,i + γt i=1 i=1 d d X X γt = pt,i ℓt,i − pt,i ℓt,i ot,i + γt i=1 i=1
i=1
≥
d X
pt,i ℓt,i − γt Qt ,
i=1
while the first one on the right hand side as # " d d X X 2 E pt,i (ℓ̂t,i ) Ft−1 = pt,i i=1
i=1
≤
d X i=1
pt,i
d X ℓ2t,i ℓ2t,i o ≤ p ot,i t,i t,i (ot,i + γt )2 (ot,i + γt )ot,i i=1 d X 1 pt,i ot,i = = Qt . (ot,i + γt )ot,i o + γt i=1 t,i
11
Combining these bounds yields T X d X
pt,i ℓt,i ≤
t=1 i=1
T X log Wt+1 log Wt Ft−1 . + γt Qt + − E 2 ηt ηt+1 t=1
T X ηt t=1
q Pt−1 To proceed, we substitute the parameter choice ηt = γt = (log d)/(d + s=1 Qs ) and use a standard algebraic lemma [4, Lemma 3.5] to get r T X T d X X PT log Wt+1 log Wt Ft−1 . d + t=1 Qt log d + − E pt,i ℓt,i ≤ 3 ηt ηt+1 t=1 i=1 t=1 Taking expectation on both sides, the second term on the right hand side telescopes into " T # X log Wt log Wt+1 log W1 log WT +1 log wT +1,j E − =E − ≤E − ηt ηt+1 η1 ηT +1 ηT +1 t=1 h i 1 −ηT +1 L̂T ,j log d −1 e log =E + E L̂T,j , =E ηT +1 d ηT +1 for any j ∈ [d], where we used that WT +1 ≥ wT +1,j and W1 = 1 since w1,i = 1/d by definition for all i ∈ [d]. Substituting ηT +1 , we get " T d # "r # # "r h i XX PT PT log d d + t=1 Qt + E log d d + t=1 Qt + E L̂T,j , E pt,i ℓt,i ≤ 3E t=1 i=1
which together with the fact that our estimates L̂T,j are optimistic yields the theorem.
C
Full proof of Theorem 2
We begin with a statement that concerns the performance of the imaginary learner that predicts Vet in round t. Lemma 6. Assume η1 ≥ η2 ≥ · · · ≥ ηT . For any sequence of loss estimates, the expected regret of the hypothetical learner against any fixed action v ∈ S satisfies " T # T X m (log d + 1) e Vt − v ℓ̂t ≤ E . ηT t=1 Proof. For simplicity, define βt = 1/ηt for t ≥ 1 and β0 = 0. We start by applying the classical follow-the-leader/be-the-leader lemma (see, e.g., [6, Lemma 3.1]) to the loss sequence defined as e 1 , ℓ̂2 − Z(β e 2 − β1 ), . . . , ℓ̂T − Z(β e T − βT −1 )) to obtain (ℓ̂1 − Zβ T X
e (βt − βt−1 ) ≤ Ve T L b T − Zβ e T ≤ vT L b T − Zβ e T . VetT ℓ̂t − Z T
t=1
e ≤ 0, we get After reordering and observing that −v T Z T X t=1
Vet − v
T
ℓ̂t ≤
T X
e (βt − βt−1 )VetT Z
t=1
e ≤ Vet 1 Z ∞
T X e βT . (βt − βt−1 ) = Vet 1 Z ∞ t=1
The h result ifollows from using our uniform upper bound on ∥v∥1 for all v and the well-known bound e E Z ≤ log d + 1. ∞ 12
The following result can be extracted from the proof of Theorem 1 of Neu and Bartók [16]. Lemma 7. For any sequence of nonnegative loss estimates,
i h 2 T E (Vet−1 − Vet )T ℓ̂t Ft ≤ ηt E Vet−1 ℓ̂t Ft .
Using these two lemmas, we can prove the following lemma that upper bounds the total expected regret of FPL-IX in terms of the sum of the variables
e t (c) = Q
d X
qt,i . o +c t,i i=1
Lemma 8. Assume that γt ≤ 1/2 for all t. Then,
T X
E [ VtT ℓt | Ft−1 ] ≤
t=1
X T T T h i h i X X γt et e t (γt ) . E VetT ℓ̂t Ft−1 + 4m ηt E Q γt E Q + 1 − γt t=1 t=1 t=1
Proof. First, note that Lemma 7 implies
i 2 T T e e e E (Vt−1 − Vt ) ℓ̂t Ft−1 ≤ ηt E Vt−1 ℓ̂t Ft−1 h
by the tower rule of expectation. We start by observing that
h
i
T E Vet−1 ℓ̂t Ft−1 = E
" d X i=1
# qt,i ℓ̂t,i Ft−1
" d X
ℓt,i Ot,i Ft−1 =E qt,i o + (1 − ot,i )γt t,i i=1
" d X
#
d X ℓt,i 1 − ot,i (Ot,i + (1 − ot,i )γt ) − γt qt,i ≥E qt,i Ft−1 o + (1 − o )γ o + (1 − ot,i )γt t,i t,i t t,i i=1 i=1 " d # d X X qt,i (1 − ot,i ) qt,i ℓt,i − γt E Ft−1 ≥ o + (1 − ot,i )γt i=1 i=1 t,i " d # d X X qt,i e t (γt ). ≥ qt,i ℓt,i − γt E Ft−1 = E [ VtT ℓt | Ft−1 ] − γt Q o + γ t,i t i=1 i=1
13
#
To simplify some notation, let us fix a time t and define V = Vet−1 . We deduce that 2 T E Vet−1 ℓ̂t Ft−1 d X d X = E Vj ℓ̂t,j Vk ℓ̂t,k Ft−1 j=1 k=1
= E
d X d X
(Vj Kt,j Ot,j ℓt,j ) (Vk Kt,k Ot,k ℓt,k ) Ft−1
(def. of ℓ̂t )
j=1 k=1
≤ E
d X d 2 2 X Kt,j + Kt,k j=1 k=1
≤ E
d X d X
2
(Vj Ot,j ℓt,j ) (Vk Ot,k ℓt,k ) Ft−1
2 2 (2Kt,j Kt,k ≤ Kt,j + Kt,k )
2 Kt,j (Vj Ot,j ℓt,j ) (Vk Ot,k ℓt,k ) Ft−1
(symmetry of j and k)
j=1 k=1
d X 1 ≤ 2E Vk ℓt,k Ft−1 (def. of Kt,j and Ot,k ≤ 1) (Vj Ot,j ℓt,j ) 2 (o + (1 − o )γ ) t,j t,j t j=1 k=1 d X Vj ℓt,j ≤ 2mE Ft−1 o + (1 − o )γ t,j t,j t j=1 d X
d qt,j 2m X qt,j = o + (1 − ot,j )γt 1 − γt j=1 ot,j + γt /(1 − γt ) j=1 t,j γt 2m e γt et Qt = ≤ 4mQ , 1 − γt 1 − γt 1 − γt
≤ 2m
d X
where we used our assumption on γt in the h last line. The i first statement follows from combining the above terms with Lemma 7 and using E v T ℓ̂t Ft−1 ≤ v T ℓt by the optimistic property of the loss estimates ℓ̂t .
D
Proof of Lemma 3
We start with proving the lower bound on ot,i ≥
1 m
X
qt,j .
− j∈{Nt,i ∪{i}}
P We prove this by first proving Ot,i ≥ (1/m) j∈{N − ∪{i}} Vt,j as follows: First, assume that t,i Ot,j = 0, in which case the bound trivially holds, since both sides evaluate to zero by definition of Ot,i . Otherwise, we have 1 m
d
X − j∈{Nt,i ∪{i}}
Vt,j ≤
1 X Vt,j ≤ 1 = Ot,i , m j=1
P where we used j∈V Vt,j ≤ m in the last inequality. Taking expectations gives the desired lower bound on ot,i . Then we get d X
d X qt,i q P t,i ≤ . o + c q + t,i t,i j∈N − qt,j + c i=1 i=1 t,i
The proof is completed using Lemma 1. 14