Online Least Squares Estimation with Self-Normalized Processes: An Application to Bandit Problems

The analysis of online least squares estimation is at the heart of many stochastic sequential decision making problems. We employ tools from the self-normalized processes to provide a simple and self-contained proof of a tail bound of a vector-valued martingale. We use the bound to construct a new tighter confidence sets for the least squares estimate. We apply the confidence sets to several online decision problems, such as the multi-armed and the linearly parametrized bandit problems. The confidence sets are potentially applicable to other problems such as sleeping bandits, generalized linear bandits, and other linear control problems. We improve the regret bound of the Upper Confidence Bound (UCB) algorithm of Auer et al. (2002) and show that its regret is with high-probability a problem dependent constant. In the case of linear bandits (Dani et al., 2008), we improve the problem dependent bound in the dimension and number of time steps. Furthermore, as opposed to the previous result, we prove that our bound holds for small sample sizes, and at the same time the worst case bound is improved by a logarithmic factor and the constant is improved.

Authors

• 27 publications
• 7 publications
• 88 publications
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• Improved Confidence Bounds for the Linear Logistic Model and Applications to Linear Bandits

We propose improved fixed-design confidence bounds for the linear logist...
11/23/2020 ∙ by Kwang-Sung Jun, et al. ∙ 17

• Bias no more: high-probability data-dependent regret bounds for adversarial bandits and MDPs

We develop a new approach to obtaining high probability regret bounds fo...
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• A General Framework to Analyze Stochastic Linear Bandit

In this paper we study the well-known stochastic linear bandit problem w...
02/12/2020 ∙ by Nima Hamidi, et al. ∙ 0

• Online Bandit Linear Optimization: A Study

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• The Randomized Elliptical Potential Lemma with an Application to Linear Thompson Sampling

In this note, we introduce a randomized version of the well-known ellipt...
02/16/2021 ∙ by Nima Hamidi, et al. ∙ 0

• Instance-Wise Minimax-Optimal Algorithms for Logistic Bandits

Logistic Bandits have recently attracted substantial attention, by provi...
10/23/2020 ∙ by Marc Abeille, et al. ∙ 0

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1 Introduction

The least squares method forms a cornerstone of statistics and machine learning. It is used as the main component of many stochastic sequential decision problems, such as multi-armed bandit, linear bandits, and other linear control problems. However, the analysis of least squares in these online settings is non-trivial because of the correlations between data points. Fortunately, there is a connection between online least squares estimation and the area of self-normalized processes. Study of self-normalized processes has a long history that goes back to Student and is treated in detail in recent book by

de la Peña et al. (2009). Using these tools we provide a proof of a bound on the deviation for vector-valued martingales. A less general version of the bound can be found already in de la Peña et al. (2004, 2009). Additionally our proof, based on the method of mixtures, is new, simpler and self-contained. The bound improves the previous bound of Rusmevichientong and Tsitsiklis (2010) and it is applicable to virtually any online least squares problem.

The bound that we derive, gives immediately rise to tight confidence sets for the online least squares estimate that can replace the confidence sets in existing algorithms. In particular, the confidence sets can be used in the UCB algorithm for the multi-armed bandit problem, the ConfidenceBall algorithm of Dani et al. (2008) for the linear bandit problem, and LinRel algorithm of Auer (2003)

for the associative reinforcement learning problem. We show that this leads to improved performance of these algorithms. Our hope is that the new confidence sets can be used to improve the performance of other similar linear decision problems.

The multi-armed bandit problem, introduced by Robbins (1952), is a game between the learner and the environment. At each time step, the learner chooses one of actions and receives a reward which is generated independently at random from a fixed distribution associated with the chosen arm. The objective of the learner is to maximize his total reward. The performance of the learner is evaluated by the regret, which is defined as the difference between his total reward and the total reward of the best action. Lai and Robbins (1985) prove a lower bound on the expected regret of any algorithm, where is the number of time steps, and are the reward distributions of the optimal arm and arm respectively, and is the KL-divergence.

Auer et al. (2002)

designed the UCB algorithm and proved a finite-time logarithmic bound on its regret. He used Hoeffding’s inequality to construct confidence intervals and obtained a

bound on the expected regret, where is the difference between the expected rewards of the best and the second best action.color=red, color=red, todo: color=red, Cite high-probability bound for UCB by Bubeck.  We modify UCB so that it uses our new confidence sets and we show a stronger result. Namely, we show that with probability , the regret of the modified algorithm is . Seemingly, this result contradicts the lower bound of Lai and Robbins (1985), however our algorithm depends on which it receives as an input. The expected regret of the modified algorithm with matches the regret of the original algorithm.

In the linear bandit problem, the learner chooses repeatedly actions from a fixed subset of and receives a random reward, expectation of which is a linear function of the action. Dani et al. (2008) proposed the ConfidenceBall algorithm and showed that its regret is at most with probability at most . We modify their algorithm so that it uses our new confidence sets and we show that its regret is at most . Additionally, constants in our bound are smaller, and our bound holds for all , as opposed the previous one which holds only for sufficiently large . Dani et al. (2008) prove also a problem dependent regret bound. Namely, they show that the regret of their algorithm is where is the “gap” as defined in (Dani et al., 2008). For our modified algorithm we prove an improved bound.

1.1 Notation

We use to denote the 2-norm. For a positive definite matrix , the weighted -norm is defined by , where . The inner product is denoted by and the weighted inner-product . We use

to denote the minimum eigenvalue of the positive definite matrix

. We use to denote that is positive definite, while we use to denote that it is positive semidefinite. The same notation is used to denote the Loewner partial order of matrices. We shall use to denote the unit vector, i.e., for all , and .

2 Vector-Valued Martingale Tail Inequalities

Let be a filtration, be an -valued stochastic process adapted to , be a real-valued martingale difference process adapted to . Assume that is conditionally sub-Gaussian in the sense that there exists some such that for any , ,

 E[exp(γηk)|Fk−1]≤exp(γ2R22)a.s. (1)

Consider the martingale

 St=t∑k=1ηkmk−1 (2)

and the matrix-valued processes

 Vt=t∑k=1mk−1m⊤k−1,¯¯¯¯Vt=V+Vt,t≥0, (3)

where is an -measurable, positive definite matrix. In particular, assume that with probability one, the eigenvalues of are larger than and that holds a.s. for any .

The following standard inequality plays a crucial role in the following developments:

Lemma 1.

Consider , as defined above and let be a stopping time with respect to the filtration . Let be arbitrary and consider

 Pλt=exp(t∑k=1[ηk⟨λ,mk−1⟩R−12⟨λ,mk−1⟩2]).

Then is almost surely well-defined and

 E[Pλτ]≤1.
Proof.

The proof is standard (and is given only for the sake of completeness). We claim that is a supermartingale. Let

 Dk=exp(ηk⟨λ,mk−1⟩R−12⟨λ,mk−1⟩2).

Observe that by (1), we have . Clearly, is -adapted, as is . Further,

 E[Pt|Ft−1] =E[D1⋯Dt−1Dt|Ft−1]=D1⋯Dt−1E[Dt|Ft−1]≤Pt−1,

showing that is indeed a supermartingale.

Now, this immediately leads to the desired result when for some deterministic time . This is based on the fact that the mean of any supermartingale can be bounded by the mean of its first element. In the case of , for example, we have .

Now, in order to consider the general case, let .111 is a shorthand notation for . It is well known that is still a supermartingale with . Further, since was nonnegative, so is . Hence, by the convergence theorem for nonnegative supermartingales, almost surely, exists, i.e., is almost surely well-defined. Further, by Fatou’s Lemma. ∎

Before stating our main results, we give some recent results, which can essentially be extracted from the paper by Rusmevichientong and Tsitsiklis (2010).

Theorem 2.

Consider the processes , as defined above and let

 κ=√3+2log((L2+trace(V))/λ0). (4)

Then, for any , , with probability at least ,

 ∥St∥¯¯¯¯V−1t≤2κ2R√logt√dlog(t)+log(1/δ). (5)

We now show how to strengthen the previous result using the method of mixtures, originally used by Robbins and Siegmund (1970) to evaluate boundary crossing probabilities for Brownian motion.

Theorem 3 (Self-normalized bound for vector-valued martingales).

Let , , , , and be as before and let be a stopping time with respect to the filtration . Assume that is deterministic. Then, for any , with probability ,

 (6)
Proof.

Without loss of generality, assume that (by appropriately scaling , this can always be achieved). Let

 Mt(λ) =exp(⟨λ,St⟩−12∥λ∥2Vt).

Notice that by Lemma 1, the mean of is not larger than one.

Let

be a Gaussian random variable which is independent of all the other random variables and whose covariance is

. Define

 Mt=E[Mt(Λ)|F∞].

Clearly, we still have .

Let us calculate : Let denote the density of and for a positive definite matrix let . Then,

 Mt =∫Rdexp(⟨λ,St⟩−12∥λ∥2Vt)f(λ)dλ =∫Rdexp(−12∥∥λ−V−1tSt∥∥2Vt+12∥St∥2V−1t)f(λ)dλ =1c(V)exp(12∥St∥2V−1t)∫Rdexp(−12{∥∥λ−V−1tSt∥∥2Vt+∥λ∥2V})dλ.

Elementary calculation shows that if , ,

 ∥x−a∥2P+∥x∥2Q=∥∥x−(P+Q)−1Pa∥∥2P+Q+∥a∥2P−∥Pa∥2(P+Q)−1.

Therefore,

 ∥∥λ−V−1tSt∥∥2Vt+∥λ∥2V =∥∥λ−(V+Vt)−1St∥∥2V+Vt+∥∥V−1tSt∥∥2Vt−∥St∥2(V+Vt)−1 =∥∥λ−(V+Vt)−1St∥∥2V+Vt+∥St∥2V−1t−∥St∥2(V+Vt)−1,

which gives

 Mt

Now, from , we obtain

 P(∥Sτ∥2(V+Vτ)−1>2log(det(V+Vτ)\nicefrac12det(V)\nicefrac121δ)) =P⎛⎜ ⎜ ⎜ ⎜⎝exp(12∥Sτ∥2(V+Vτ)−1)δ−1(det(V+Vτ)/det(V))12>1⎞⎟ ⎟ ⎟ ⎟⎠ ≤E⎡⎢ ⎢ ⎢ ⎢⎣exp(12∥Sτ∥2(V+Vτ)−1)δ−1(det(V+Vτ)/det(V))12⎤⎥ ⎥ ⎥ ⎥⎦ =E[Mτ]δ≤δ,

thus finishing the proof. ∎

[Uniform Bound] Under the same assumptions as in the previous theorem, for any , with probability ,

 ∀t≥0,∥St∥2¯¯¯¯V−1t≤2R2log(det(¯¯¯¯Vt)\nicefrac12det(V)\nicefrac−12δ). (7)
Proof.

We will use a stopping time construction, which goes back at least to Freedman (1975). Define the bad event

 Bt(δ)={ω∈Ω : ∥St∥2¯V−1t>2R2log(det(¯Vt)1/2det(V)−1/2δ)} (8)

We are interested in bounding the probability that happens. Define , with the convention that . Then, is a stopping time. Further,

 ⋃t≥0Bt(δ)={ω : τ(ω)<∞}.

Thus, by Theorem 3

 P(⋃t≥0Bt(δ)) =P(τ<∞) =P(∥Sτ∥2¯V−1τ>2R2log(det(¯Vτ)1/2det(V)−1/2δ),τ<∞) ≤P(∥Sτ∥2¯V−1τ>2R2log(det(¯Vτ)1/2det(V)−1/2δ)) ≤δ.

Let us now turn our attention to understanding the determinant term on the right-hand side of (6).

Lemma 4.

We have that

 logdet(¯¯¯¯Vt)detV≤t∑k=1∥mk−1∥2¯¯¯¯V−1k−1.

Further, we have that

 t∑k=1(∥mk−1∥2¯¯¯¯V−1k−1∧1) ≤2(logdet(¯¯¯¯Vt)−logdetV)≤2(dlog((trace(V)+tL2)/d)−logdetV).

Finally, if then

 t∑k=1∥mk−1∥2¯¯¯¯V−1k−1≤2logdet(¯¯¯¯Vt)det(V).
Proof.

Elementary algebra gives

 det(¯¯¯¯Vt) =det(¯¯¯¯Vt−1+mt−1m⊤t−1)=det(¯¯¯¯Vt−1)det(I+¯¯¯¯V−\nicefrac12t−1mt−1(¯¯¯¯V−\nicefrac12t−1mt−1)⊤) =det(¯¯¯¯Vt−1)(1+∥mt−1∥2¯¯¯¯V−1t−1)=det(V)t∏k=1(1+∥mk−1∥2¯¯¯¯V−1k−1), (9)

where we used that all the eigenvalues of a matrix of the form are one except one eigenvalue, which is

and which corresponds to the eigenvector

. Using , we can bound by

 logdet(¯¯¯¯Vt)≤logdetV+t∑k=1∥mk−1∥2¯¯¯¯V−1k−1.

Combining , which holds when , and (9), we get

 t∑k=1(∥mk−1∥2¯¯¯¯V−1k−1∧1)≤2t∑k=1log(1+∥mk−1∥2¯¯¯¯V−1k−1)=2(logdet(¯¯¯¯Vt)−logdetV).

The trace of is bounded by , assuming . Hence, and therefore,

 logdet(¯¯¯¯Vt)≤dlog((trace(V)+tL2)/d),

finishing the proof of the second inequality. The sum can itself be upper bounded as a function of provided that is large enough. Notice . Hence, we get that if ,

 logdet(¯¯¯¯Vt)detV≤t∑k=1∥mk−1∥2¯¯¯¯V−1k−1≤2logdet(¯¯¯¯Vt)det(V).

Most of this argument can be extracted from the paper of Dani et al. (2008). However, the idea goes back at least to Lai et al. (1979), Lai and Wei (1982) (a similar argument is used around Theorem 11.7 in the book by Cesa-Bianchi and Lugosi (2006)). Note that Lemmas B.9–B.11 of Rusmevichientong and Tsitsiklis (2010) also give a bound on , with an essentially identical argument. Alternatively, one can use the bounding technique of Auer (2003) (see the proof of Lemma 13 there on pages 412–413) to derive a bound like for a suitable chosen constant .

Remark 5.

By combining Corollary 3 and Lemma 4, we get a simple worst case bound that holds with probability :

 ∀t≥0,∥St∥2¯¯¯¯V−1t≤dR2log(trace(V)+tL2dδ). (10)

Still, the new bound is considerably better than the previous one given by Theorem 2. Note that the factor cannot be removed, as shown by Problem 3, page 203 in the book by de la Peña et al. (2009).

3 Optional Skipping

Consider the case when , , i.e., the case of an optional skipping process. Then, using again , and thus the expression studied becomes

 ∥St∥¯¯¯¯V−1t=|∑tk=1εk−1ηk|√1+Nt.

We also have

 logdet(¯¯¯¯Vt)=t∑k=1log(1+εk−11+Nk)≤t∑k=1εk−11+Nk=Nt+1∑k=11k≤1+∫Nt+11x−1dx=1+log(1+Nt).

Thus, we get, with probability

 ∀s≥0,∣∣ ∣∣s∑k=1εk−1ηk∣∣ ∣∣≤ ⎷(1+Ns)(1+2log((1+Ns)\nicefrac12δ)). (11)

If we apply Doob’s optional skipping and Hoeffding-Azuma, with a union bound (see, e.g., the paper of Bubeck et al. (2008)), we would get, for any , , with probability ,

 ∀s∈{0,…,t},∣∣ ∣∣s∑k=1εk−1ηk∣∣ ∣∣≤√2Nslog(2tδ). (12)

The major difference between these bounds is that (12) depends explicitly on , while (11) does not. This has the positive effect that one need not recompute the bound if does not grow, which helps e.g. in the paper of Bubeck et al. (2008) to improve the computational complexity of the HOO algorithm. Also, the coefficient of the leading term in (11) under the square root is , whereas in (12) it is .

Instead of a union bound, it is possible to use a “peeling device” to replace the conservative factor in the above bound by essentially . This is done e.g. in Garivier and Moulines (2008) in their Theorem 22.222They give their theorem as ratios, which they should not, since their inequality then fails to hold for . However, this is easy to remedy by reformulating their result as we do it here. From their derivations, the following one sided, uniform bound can be extracted (see Remark 24, page 19): For any , , with probability ,

 ∀s∈{0,…,t},s∑k=1εk−1ηk≤√4Ns1.99log(6logtδ). (13)

As noted by Garivier and Moulines (2008), due to the law of iterated logarithm, the scaling of the right-hand side as a function of cannot be improved in the worst-case. However, this leaves open the possibility of deriving a maximal inequality which depends on only through .

4 The Multi-Armed Bandit Problem

Now we turn our attention to the multi-armed bandit problem. Let denote the expected reward of action and , where is the expected reward of the optimal action. We assume that if we choose action in round , we obtain reward . Let denote the number of times that we have played action up to time , and denote the average of the rewards received by action up to time . From (11) with instead of and a union bound over the actions, we have the following confidence intervals that hold with probability at least :

 ∀i∈{1,…,K}, ∀s∈{1,2,…},∣∣¯Xi,s−μi∣∣≤ci,s, (14)

where

 ci,s= ⎷1+Ni,sN2i,s(1+2log(K(1+Ni,s)\nicefrac12δ)).

Modify the UCB Algorithm of Auer et al. (2002) to use the confidence intervals (14) and change the action selection rule accordingly. Hence, at time , we choose the action

 It=argmaxi¯Xi,t+ci,t. (15)

We call this algorithm UCB().

Theorem 6.

With probability at least , the total regret of the UCB() algorithm with the action selection rule (15) is constant and is bounded by

 R(T)≤∑i:Δi>0(3Δi+16Δilog2KΔiδ).

where is the index of the optimal action.

Proof.

Suppose the confidence intervals do not fail. If we play action , the upper estimate of the action is above . Hence,

 ci,s≥Δi2.

Substituting and squaring gives

 N2i,s−1Ni,s+1≤N2i,sNi,s+1≤4Δ2i(1+2logK(1+Ni,s)1/2δ).

By using Lemma 8 of Antos et al. (2010), we get that

 Ni,s≤3+16Δ2ilog2KΔiδ.

Thus, using , we get that with probability at least , the total regret is bounded by

 R(T)≤∑i:Δi>0(3Δi+16Δilog2KΔiδ).

Remark 7.

Lai and Robbins (1985) prove that for any suboptimal arm ,

 E[Ni,t]≥logtD(pj,p∗),

where, and are the reward density of the optimal arm and arm respectively, and is the KD-divergence. This lower bound does not contradict Theorem 6, as Theorem 6 only states a high probability upper bound for the regret. Note that UCB() takes delta as its input. Because with probability , the regret in time can be , on expectation, the algorithm might have a regret of . Now if we select , then we get upper bound on the expected regret.

5 Application to Least Squares Estimation and Linear Bandit Problem

In this section we first apply Theorem 3 to derive confidence intervals for least-squares estimation, where the covariate process is an arbitrary process and then use these confidence intervals to improve the regret bound of Dani et al. (2008) for the linear bandit problem. In particular, our assumption on the data is as follows:

• Assumption A1  Let be a filtration, , , be a sequence of random variables over such that is -measurable, and is -measurable . Assume that there exists such that , i.e., is a martingale difference sequence (, ) and that is sub-Gaussian: There exists such that for any ,

 E[exp(γεi)|Fi−1]≤exp(γ2R2/2).

We shall call the random variables covariates and the random variables the responses. Note that the assumption allows any sequential generation of the covariates.

Let be the -regularized least-squares estimate of with regularization parameter :

 ^θt=(X⊤X+λI)−1X⊤Y,^θ0=0, (16)

where is the matrix whose rows are and . We further let .

We are interested in deriving a confidence bound on the error of predicting the mean response at an arbitrarily chosen random covariate using the least-squares predictor . Using

 ^θt = (X⊤X+λI)−1X⊤(Xθ∗+ε) = (X⊤X+λI)−1X⊤ε+(X⊤X+λI)−1(X⊤X+λI)θ∗−λ(X⊤X+λI)−1θ∗ = (X⊤X+λI)−1X⊤ε+θ∗−λ(X⊤X+λI)−1θ∗,

we get

 x⊤^θt−x⊤θ∗ =x⊤(X⊤X+λI)−1X⊤ε−λx⊤(X⊤X+λI)−1θ∗ =⟨x,X⊤ε⟩V−1t−λ⟨x,θ∗⟩V−1t,

where . Note that is positive definite (thanks to ) and hence so is , so the above inner product is well-defined. Using the Cauchy-Schwartz inequality, we get

 |x⊤^θt−x⊤θ∗| ≤∥x∥V−1t(∥∥X⊤ε∥∥V−1t+λ∥θ∗∥V−1t) ≤∥x∥V−1t(∥∥X⊤ε∥∥V−1t+λ1/2∥θ∗∥),

where we used that . Fix any . By Corollary 3, with probability at least ,

 ∀t≥1,∥∥X⊤ε∥∥V−1t≤R ⎷2log(det(Vt)\nicefrac12det(λI)\nicefrac−12δ).

Therefore, on the event where this inequality holds, one also has

 |x⊤^θt−x⊤θ∗|≤∥x∥V−1t⎛⎜⎝R ⎷2log(det(Vt)\nicefrac12det(λI)\nicefrac−12δ)+λ1/2∥θ∗∥⎞⎟⎠.

Similarly, we can derive a worst-case bound. The result is summarized in the following statement:

Theorem 8.

Let , , satisfy the linear model Assumption 5 with some , and let be the associated filtration. Assume that w.p.1 the covariates satisfy , and . Consider the -regularized least-squares parameter estimate with regularization coefficient (cf. (16)). Let be an arbitary, -valued random variable. Let be the regularized design matrix underlying the covariates. Then, for any , with probability at least ,

 ∀t≥1,|x⊤^θt−x⊤θ∗|≤∥x∥V−1t⎛⎜⎝R ⎷2log(det(Vt)\nicefrac12det(λI)\nicefrac−12δ)+λ1/2S⎞⎟⎠. (17)

Similarly, with probability ,

 ∀t≥1,|x⊤^θt−x⊤θ∗|≤∥x∥V−1t⎛⎜ ⎜⎝R  ⎷dlog⎛⎝1+tLλδ⎞⎠+λ1/2S⎞⎟ ⎟⎠. (18)
Remark 9.

We see that increases the second term (the “bias term”) in the parenthesis of the estimate. In fact, for fixed gives (as it should be). Decreasing , on the other hand increases