# dynamic_regret_of_strongly_adaptive_methods__99d89d38.pdf

Dynamic Regret of Strongly Adaptive Methods

Lijun Zhang 1 Tianbao Yang 2 Rong Jin 3 Zhi-Hua Zhou 1

To cope with changing environments, recent developments in online learning have introduced the concepts of adaptive regret and dynamic regret independently. In this paper, we illustrate an intrinsic connection between these two concepts by showing that the dynamic regret can be expressed in terms of the adaptive regret and the functional variation. This observation implies that strongly adaptive algorithms can be directly leveraged to minimize the dynamic regret. As a result, we present a series of strongly adaptive algorithms that have small dynamic regrets for convex functions, exponentially concave functions, and strongly convex functions, respectively. To the best of our knowledge, this is the first time that exponential concavity is utilized to upper bound the dynamic regret. Moreover, all of those adaptive algorithms do not need any prior knowledge of the functional variation, which is a significant advantage over previous specialized methods for minimizing dynamic regret.

1. Introduction

Online convex optimization is a powerful paradigm for sequential decision making (Zinkevich, 2003). It can be viewed as a game between a learner and an adversary: In the t-th round, the learner selects a decision wt Ω, simultaneously the adversary chooses a function ft( ) : Ω7 R, and then the learner suffers an instantaneous loss ft(wt). This study focuses on the full-information setting, where the learner can query the value and gradient of ft (Cesa-Bianchi & Lugosi, 2006). The goal of the learner is to minimize the cumulative loss over T periods . The standard performance measure is regret, which is the difference between the loss

1National Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, China 2Department of Computer Science, The University of Iowa, Iowa City, USA 3Alibaba Group, Seattle, USA. Correspondence to: Lijun Zhang <zhanglj@lamda.nju.edu.cn>.

Proceedings of the 35 th International Conference on Machine Learning, Stockholm, Sweden, PMLR 80, 2018. Copyright 2018 by the author(s).

incurred by the learner and that of the best fixed decision in hindsight, i.e.,

Regret(T) =

t=1 ft(wt) min w Ω

The above regret is typically referred to as static regret in the sense that the comparator is time-invariant. The rationale behind this evaluation metric is that one of the decision in Ωis reasonably good over the T rounds. However, when the underlying distribution of loss functions changes, the static regret may be too optimistic and fails to capture the hardness of the problem.

To address this limitation, new forms of performance measure, including adaptive regret (Hazan & Seshadhri, 2007; 2009) and dynamic regret (Zinkevich, 2003; Hall & Willett, 2013), were proposed and received significant interest recently. Following the terminology of Daniely et al. (2015), we define the strongly adaptive regret as the maximum static regret over intervals of length τ, i.e.,

SA-Regret(T, τ)

= max [s,s+τ 1] [T ]

t=s ft(wt) min w Ω

Minimizing the adaptive regret enforces the learner to have a small static regret over any interval of length τ. Since the best decision for different intervals could be different, the learner is essentially competing with a changing comparator.

A parallel line of research introduces the concept of dynamic regret, where the cumulative loss of the learner is compared against a comparator sequence u1, . . . , u T Ω, i.e.,

D-Regret(u1, . . . , u T ) =

t=1 ft(ut). (2)

It is well-known that in the worst case, a sublinear dynamic regret is impossible unless we impose some regularities on the comparator sequence or the function sequence (Jadbabaie et al., 2015). A representative example is the functional variation defined below

t=2 max w Ω|ft(w) ft 1(w)|. (3)

Dynamic Regret of Strongly Adaptive Methods

Besbes et al. (2015) have proved that as long as VT is sublinear in T, there exists an algorithm that achieves a sublinear dynamic regret. Furthermore, a general restarting procedure is developed, and it enjoys O(T 2/3V 1/3 T ) and O(log T TVT ) rates for convex functions and strongly convex functions, respectively. However, the restarting procedure can only be applied when an upper bound of VT is known beforehand, thus limiting its application in practice.

While both the adaptive and dynamic regrets aim at coping with changing environments, little is known about their relationship. This paper makes a step towards understanding their connections. Specifically, we show that the strongly adaptive regret in (1), together with the functional variation, can be used to upper bound the dynamic regret in (2). Thus, an algorithm with a small strongly adaptive regret is automatically equipped with a tight dynamic regret. As a result, we obtain a series of algorithms for minimizing the dynamic regret that do not need any prior knowledge of the functional variation. The main contributions of this work are summarized below.

We provide a general theorem that upper bounds the dynamic regret in terms of the strongly adaptive regret and the functional variation. For convex functions, we show that the strongly adaptive algorithm of Jun et al. (2017) has a dynamic regret of O(T 2/3V 1/3 T log1/3 T), which matches the minimax rate (Besbes et al., 2015), up to a polylogarithmic factor. For exponentially concave functions, we propose a strongly adaptive algorithm that allows us to control the tradeoff between the adaptive regret and the computational cost explicitly. Then, we demonstrate that its dynamic regret is O(d TVT log T), where d is the dimensionality. To the best of our knowledge, this is the first time that exponential concavity is utilized in the analysis of dynamic regret. For strongly convex functions, our proposed algorithm can also be applied and yields a dynamic regret of O( TVT log T), which is also minimax optimal up to a polylogarithmic factor.

2. Related Work

We give a brief introduction to previous work on static, adaptive, and dynamic regrets in the context of online convex optimization.

2.1. Static Regret

The majority of studies in online learning are focused on static regret (Shalev-Shwartz & Singer, 2007; Langford et al., 2009; Shalev-Shwartz, 2011; Zhang et al., 2013). For general convex functions, the classical online gradient

descent achieves O(

T) and O(log T) regret bounds for convex and strongly convex functions, respectively (Zinkevich, 2003; Hazan et al., 2007; Shalev-Shwartz et al., 2007). Both the O(

T) and O(log T) rates are known to be minimax optimal (Abernethy et al., 2009). When functions are exponentially concave, a different algorithm, named online Newton step, is developed and enjoys an O(d log T) regret bound, where d is the dimensionality (Hazan et al., 2007).

2.2. Adaptive Regret

The concept of adaptive regret is introduced by Hazan & Seshadhri (2007), and later strengthened by Daniely et al. (2015). Specifically, Hazan & Seshadhri (2007) introduce the weakly adaptive regret

WA-Regret(T)

= max [s,q] [T ]

t=s ft(wt) min w Ω

To minimize the adaptive regret, Hazan & Seshadhri (2007) have developed two meta-algorithms: an efficient algorithm with O(log T) computational complexity per iteration and an inefficient one with O(T) computational complexity per iteration. These meta-algorithms use an existing online method (that was possibly designed to have small static regret) as a subroutine.1 For convex functions, the efficient and inefficient meta-algorithms have O( p

T log3 T) and O( p

T log T) regret bounds, respectively. For exponentially concave functions, those rates are improved to O(d log2 T) and O(d log T), respectively. We can see that the price paid for the adaptivity is very small: The rates of weakly adaptive regret differ from those of static regret only by logarithmic factors.

A major limitation of weakly adaptive regret is that it does not respect short intervals well. Taking convex functions as an example, the O( p

T log3 T) and O( p

T log T) bounds are meaningless for intervals of length O(

T). To overcome this limitation, Daniely et al. (2015) proposed the strongly adaptive regret SA-Regret(T, τ) which takes the length of the interval τ as a parameter, as indicated in (1). From the definitions, we have SA-Regret(T, τ) WA-Regret(T), but it does not mean the notation of weakly adaptive regret is stronger, because an upper bound for WA-Regret(T) could be very loose for SA-Regret(T, τ) when τ is small.

If the strongly adaptive regret is small for all τ < T, we can guarantee the learner has a small regret over any interval of

1For brevity, we ignored the factor of subroutine in the statements of computational complexities. The O( ) computational complexity should be interpreted as O( ) s space complexity and O( ) t time complexity, where s and t are space and time complexities of the subroutine per iteration, respectively.

Dynamic Regret of Strongly Adaptive Methods

any length. In particular, Daniely et al. (2015) introduced the following definition.

Definition 1 Let R(τ) be the minimax static regret bound of the learning problem over τ periods. An algorithm is strongly adaptive, if

SA-Regret(T, τ) = O(poly(log T) R(τ)), τ.

It is easy to verify that the meta-algorithms of Hazan & Seshadhri (2007) are strongly adaptive for exponentially concave functions,2 but not for convex functions. Thus, Daniely et al. (2015) developed a new meta-algorithm that satisfies SA-Regret(T, τ) = O( τ log T) for convex functions, and thus is strongly adaptive. The algorithm is also efficient and the computational complexity per iteration is O(log T). Later, the strongly adaptive regret of convex functions was improved to O( τ log T) by Jun et al. (2017), and the computational complexity remains O(log T) per iteration. All the previously mentioned algorithms for minimizing adaptive regret need to query the gradient of the loss function at least O(log t) times in the t-th iteration. In a recent study, Wang et al. (2018) demonstrate that the number of gradient evaluations per iteration can be reduced to 1 by introducing the surrogate loss.

2.3. Dynamic Regret

In a seminal work, Zinkevich (2003) proposed to use the path-length defined as

P(u1, . . . , u T ) =

t=2 ut ut 1 2

to upper bound the dynamic regret, where u1, . . . , u T Ω is a comparator sequence. Specifically, Zinkevich (2003) proved that for any sequence of convex functions, the dynamic regret of online gradient descent can be upper bounded by O(

TP(u1, . . . , u T )). Another regularity of the comparator sequence, which is similar to the path-length, is defined as

P (u1, . . . , u T ) =

t=2 ut Φt(ut 1) 2

where Φt( ) is a dynamic model that predicts a reference point for the t-th round. Hall & Willett (2013) developed a novel algorithm named dynamic mirror descent and proved that its dynamic regret is on the order of

TP (u1, . . . , u T ). The advantage of P (u1, . . . , u T ) is that when the comparator sequence follows the dynamical

2That is because (i) SA-Regret(T, τ) WA-Regret(T), and (ii) there is a poly(log T) factor in the definition of strong adaptivity.

model closely, it can be much smaller than the path-length P(u1, . . . , u T ).

Let w t argminw Ωft(w) be a minimizer of ft( ). For any sequence of u1, . . . , u T Ω, we have

D-Regret(u1, . . . , u T ) =

D-Regret(w 1, . . . , w T ) =

t=1 min w Ωft(w).

Thus, D-Regret(w 1, . . . , w T ) can be treated as the worst case of the dynamic regret, and there are many works that were devoted to minimizing D-Regret(w 1, . . . , w T ) (Jadbabaie et al., 2015; Mokhtari et al., 2016; Yang et al., 2016; Zhang et al., 2017).

When a prior knowledge of P(w 1, . . . , w T ) is available, D-Regret(w 1, . . . , w T ) can be upper bounded by O( p

TP(w 1, . . . , w T )) (Yang et al., 2016). If all the functions are strongly convex and smooth, the upper bound can be improved to O(P(w 1, . . . , w T )) (Mokhtari et al., 2016). The O(P(w 1, . . . , w T )) rate is also achievable when all the functions are convex and smooth, and all the minimizers w t s lie in the interior of Ω(Yang et al., 2016). In a recent study, Zhang et al. (2017) introduced a new regularity squared path-length

S(w 1, . . . , w T ) =

t=2 w t w t 1 2 2

which could be much smaller than the path-length P(w 1, . . . , w T ) when the difference between successive minimizers is small. Zhang et al. (2017) developed a novel algorithm named online multiple gradient descent, and proved that D-Regret(w 1, . . . , w T ) is on the order of min(P(w 1, . . . , w T ), S(w 1, . . . , w T )) for (semi-) strongly convex and smooth functions.

Discussions Although closely related, adaptive regret and dynamic regret are studied independently and there are few discussions of their relationships. In the literature, dynamic regret is also referred to as tracking regret or shifting regret (Littlestone & Warmuth, 1994; Herbster & Warmuth, 1998; 2001). In the setting of prediction with expert advice , Adamskiy et al. (2012) have shown that the tracking regret can be derived from the adaptive regret. In the setting of online linear optimization in the simplex , Cesa-bianchi et al. (2012) introduced a generalized notion of shifting regret which unifies adaptive regret and shifting regret. Different from previous work, this paper considers the setting of online convex optimization, and illustrates that the dynamic regret can be upper bounded by the adaptive regret and the functional variation.

Dynamic Regret of Strongly Adaptive Methods

3. A Unified Adaptive Algorithm

In this section, we introduce a unified approach for minimizing the adaptive regret of exponentially concave functions, as well as strongly convex functions.

3.1. Motivation

We first provide the definition of exponentially concave (abbr. exp-concave) functions (Cesa-Bianchi & Lugosi, 2006).

Definition 2 A function f( ) : Ω7 R is α-exp-concave if exp( αf( )) is concave over Ω.

For exp-concave functions, Hazan & Seshadhri (2007) have developed two meta-algorithms that take the online Newton step as its subroutine, and proved the following properties.

The inefficient one has O(T) computational complexity per iteration, and its adaptive regret is O(d log T). The efficient one has O(log T) computational complexity per iteration, and its adaptive regret is O(d log2 T).

As can be seen, there is a tradeoff between the computational complexity and the adaptive regret: A lighter computation incurs a looser bound and a tighter bound requires a higher computation. Our goal is to develop a unified approach, that allows us to trade effectiveness for efficiency explicitly.

3.2. Improved Following the Leading History (IFLH)

Let E be an online learning algorithm that is designed to minimize the static regret of exp-concave functions or strongly convex functions, e.g., online Newton step (Hazan et al., 2007) or online gradient descent (Zinkevich, 2003). Similar to the approach of following the leading history (FLH) (Hazan & Seshadhri, 2007), at any time t, we will instantiate an expert by applying the online learning algorithm E to the sequence of loss functions ft, ft+1, . . ., and utilize the strategy of learning from expert advice to combine solutions of different experts (Herbster & Warmuth, 1998). Our method is named as improved following the leading history (IFLH), and is summarized in Algorithm 1.

Let Et be the expert that starts to work at time t. To control the computational complexity, we will associate an ending time et for each Et. The expert Et is alive during the period [t, et 1]. In each round t, we maintain a working set of experts St, which contains all the alive experts, and assign a probability pj t for each Ej St. In Steps 6 and 7, we remove all the experts whose ending times are no larger than t. Since the number of alive experts has changed, we need to update the probability assigned to them, which is performed in Steps 12 to 14. In Steps 15 and 16, we add a new expert Et to St, calculate its ending time according to Definition 3 introduced below, and set pt t = 1

t . It is easy

Algorithm 1 Improved Following the Leading History (IFLH)

1: Input: An integer K 2: Initialize S0 = . 3: for t = 1, . . . , T do 4: Set Zt = 0 {Remove some existing experts} 5: for Ej St 1 do 6: if ej t then 7: Update St 1 St 1 \ {Ej} 8: else 9: Set Zt = Zt + bpj t 10: end if 11: end for {Normalize the probability} 12: for Ej St 1 do

13: Set pj t = bpj t Zt 1 1

14: end for {Add a new expert Et} 15: Set St = St 1 {Et} 16: Compute the ending time et = EK(t) according to Definition 3 and set pt t = 1

t {Compute the final predicted model} 17: Submit the solution

Ej St pj twj t

and suffer loss ft(wt) {Update weights and expert} 18: Set Zt+1 = 0 19: for Ej St do 20: Compute pj t+1 = pj t exp( αft(wj t)) and Zt+1 = Zt+1 + pj t+1 21: Pass the function ft( ) to Ej

22: end for 23: for Ej St do

24: Set bpj t+1 = pj t+1 Zt+1 25: end for 26: end for

to verify P

Ej St pj t = 1. Let wj t be the output of Ej at the t-th round, where t j. In Step 17, we submit the weighted average of wj t with coefficient pj t as the output wt, and suffer the loss ft(wt). From Steps 18 to 25, we use the exponential weighting scheme to update the weight for each expert Ej based on its loss ft(wj t). In Step 21, we pass the loss function to all the alive experts such that they can update their predictions for the next round.

The difference between our IFLH and the original FLH is how to decide the ending time et of expert Et. In this paper, we propose the following base-K ending time.

Dynamic Regret of Strongly Adaptive Methods

Definition 3 (Base-K Ending Time) Let K be an integer, and the representation of t in the base-K number system as

where 0 βτ < K, for all τ 0. Let k be the smallest integer such that βk > 0, i.e., k = min{τ : βτ > 0}. Then, the base-K ending time of t is defined as

τ k+1 βτKτ + Kk+1.

In other words, the ending time is the number represented by the new sequence obtained by setting the first nonzero element in the sequence β0, β1, . . . to be 0 and adding 1 to the element after it.

Let s take the decimal system as an example (i.e., K = 10). Then,

E10(1) = E10(2) = = E10(9) = 10,

E10(11) = E10(12) = = E10(19) = 20,

E10(10) = E10(20) = = E10(90) = 100.

3.3. Theoretical Guarantees

When the base-K ending time is used in Algorithm 1, we have the following properties.

Lemma 1 Suppose we use the base-K ending time in Algorithm 1.

1. For any t 1, we have

|St| ( log K t + 1) (K 1) = O K log t

2. For any interval I = [r, s] [T], we can always find m segments Ij = [tj, etj 1], j [m] with m log K(s r + 1) + 1, such that t1 = r, etj = tj+1, j [m 1], and etm > s.

The first part of Lemma 1 implies that the size of St is O(K log t/ log K). An example of St in the decimal system is given below.

481, 482, . . . , 486,

410, 420, . . . , 480,

100, 200, . . . , 400

The second part of Lemma 1 implies that for any interval I = [r, s], we can find O(log s/ log K) experts such that their survival periods cover I. Again, we present an example in the decimal system: The interval [111, 832] can be covered by

[111, 119], [120, 199], and [200, 999]

which are the survival periods of experts E111, E120, and E200, respectively. Recall that E10(111) = 120, E10(120) = 200, and E10(200) = 1000.

We note that a similar strategy for deciding the ending time was proposed by Gy orgy et al. (2012) in the study of prediction with expert advice . The main difference is that their strategy is built upon base-2 number system and introduces an additional parameter g to compromise between the computational complexity and the regret, in contrast our method relies on base-K number system and uses K to control the tradeoff. Lemma 2 of Gy orgy et al. (2012) indicates an O(g log t) bound on the number of alive experts, which is worse than our O(K log t/ log K) bound by a logarithmic factor.

To present adaptive regret bounds, we introduce the following common assumption.

Assumption 1 Both the gradient and the domain are bounded.

The gradients of all the online functions are bounded by G, i.e., maxw Ω ft(w) G for all ft. The diameter of the domain Ωis bounded by B, i.e., maxw,w Ω w w B.

Based on Lemma 1, we have the following theorem regarding the adaptive regret of exp-concave functions.

Theorem 1 Suppose Assumption 1 holds, Ω Rd, and all the functions are α-exp-concave. If online Newton step is used as the subroutine in Algorithm 1, we have

t=r ft(wt) min w Ω

(5d + 1)m + 2

α + 5dm GB log T

where [r, s] [T] and m log K(s r + 1) + 1. Thus,

SA-Regret(T, τ)

(5d + 1) m + 2

α + 5d m GB log T = O d log2 T

where m = log K τ + 1.

From Lemma 1 and Theorem 1, we observe that the adaptive regret is a decreasing function of K, while the computational cost is an increasing function of K. Thus, we can control the tradeoff by tuning the value of K. Specifically, Lemma 1 indicates the proposed algorithm has

( log K T + 1) (K 1) = O K log T

computational complexity per iteration. On the other hand, Theorem 1 implies that for α-exp-concave functions that

Dynamic Regret of Strongly Adaptive Methods

Table 1. Efficiency and Effectiveness Tradeoff

K Complexity Adaptive Regret

2 O(log T) O(d log2 T) T 1/γ O(γT 1/γ) O(γd log T) T O(T) O(d log T)

satisfy Assumption 1, the strongly adaptive regret of Algorithm 1 is

(5d + 1) m + 2

α + 5d m GB log T = O d log2 T

where d is the dimensionality and m = log K(τ) + 1.

We list several choices of K and the resulting theoretical guarantees in Table 1, and have the following observations.

When K = 2, we recover the guarantee of the efficient algorithm of Hazan & Seshadhri (2007), and when K = T, we obtain the inefficient one. By setting K = T 1/γ where γ > 1 is a small constant, such as 10, the strongly adaptive regret can be viewed as O(d log T), and at the same time, the computational complexity is also very low for a large range of T.

Next, we consider strongly convex functions.

Definition 4 A function f( ) : Ω7 R is λ-strongly convex if

f(y) f(x)+ f(x), y x + λ

2 y x 2 2, x, y Ω.

It is easy to verify that strongly convex functions with bounded gradients are also exp-concave (Hazan et al., 2007).

Lemma 2 Suppose f( ) : Ω7 R is λ-strongly convex and f(w) G for all w Ω. Then, f( ) is λ G2 -expconcave.

According to the above lemma, we still use Algorithm 1 as the meta-algorithm, but choose online gradient descent as the subroutine. In this way, the adaptive regret does not depend on the dimensionality d.

Theorem 2 Suppose Assumption 1 holds, and all the functions are λ-strongly convex. If online gradient descent is used as the subroutine in Algorithm 1, we have

t=r ft(wt) min w Ω

t=r ft(w) G2

2λ m + (3m + 4) log T

where [r, s] [T] and m log K(s r + 1) + 1. Thus

SA-Regret(T, τ)

2λ m + (3 m + 4) log T = O log2 T

where m = log K τ + 1.

4. From Adaptive to Dynamic

In this section, we first introduce a general theorem that bounds the dynamic regret by the adaptive regret, and then derive specific regret bounds for convex functions, exponentially concave functions, and strongly convex functions.

4.1. Adaptive-to-Dynamic Conversion

Let I1 = [s1, q1], I2 = [s2, q2], . . . , Ik = [sk, qk] be a partition of [1, T]. That is, they are successive intervals such that

s1 = 1, qi + 1 = si+1, i [k 1], and qk = T. (4)

Define the local functional variation of the i-th interval as

t=si+1 max w Ω|ft(w) ft 1(w)|

and it is obvious that Pk i=1 VT (i) VT .3 Then, we have the following theorem for bounding the dynamic regret in terms of the strongly adaptive regret and the functional variation.

Theorem 3 Let w t argminw Ωft(w). For all integer k [T], we have

D-Regret(w 1, . . . , w T )

min I1,...,Ik

SA-Regret(T, |Ii|) + 2|Ii| VT (i)

where the minimization is taken over any sequence of intervals that satisfy (4).

The above theorem is analogous to Proposition 2 of Besbes et al. (2015), which provides an upper bound for a special choice of the interval sequence. The main difference is that there is a minimization operation in our bound, which allows us to get rid of the issue of parameter selection. For a specific type of problems, we can plug in the corresponding

3Note that in certain cases, the sum of local functional variation Pk i=1 VT (i) can be much smaller than the total functional variation VT . For example, when the sequence of functions only changes k times, we can construct the intervals based on the changing rounds such that Pk i=1 VT (i) = 0.

Dynamic Regret of Strongly Adaptive Methods

upper bound of strongly adaptive regret, and then choose any sequence of intervals to obtain a concrete upper bound. In particular, the choice of the intervals may depend on the (possibly unknown) functional variation.

4.2. Convex Functions

For convex functions, we choose the meta-algorithm of Jun et al. (2017) and take the online gradient descent as its subroutine. The following theorem regarding the adaptive regret can be obtained from that paper.

Theorem 4 Under Assumption 1, the meta-algorithm of Jun et al. (2017) is strongly adaptive with

SA-Regret(T, τ)

7 log T + 5 τ = O( p

From Theorems 3 and 4, we derive the following bound for the dynamic regret.

Corollary 5 Under Assumption 1, the meta-algorithm of Jun et al. (2017) satisfies

D-Regret(w 1, . . . , w T )

7 log T + 5)

5)T 2/3V 1/3 T log1/6 T + 24T 2/3V 1/3 T log1/3 T

T log T, T 2/3V 1/3 T log1/3 T o

where c = 12BG/(

According to Theorem 2 of Besbes et al. (2015), we know that the minimax dynamic regret of convex functions is O(T 2/3V 1/3 T ). Thus, our upper bound is minimax optimal up to a polylogarithmic factor. Although the restarted online gradient descent of Besbes et al. (2015) achieves a dynamic regret of O(T 2/3V 1/3 T ), it requires to know an upper bound of the functional variation VT . In contrast, the metaalgorithm of Jun et al. (2017) does not need any prior knowledge of VT . We note that the meta-algorithm of Daniely et al. (2015) can also be used here, and its dynamic regret is on the order of max n

T log T, T 2/3V 1/3 T log2/3 T o .

4.3. Exponentially Concave Functions

We proceed to consider exp-concave functions, defined in Definition 2. Exponential concavity is stronger than convexity but weaker than strong convexity. It can be used to model many popular losses used in machine learning, such as the square loss in regression, logistic loss in classification and negative logarithm loss in portfolio management (Koren, 2013).

For exp-concave functions, we choose Algorithm 1 in this paper, and take the online Newton step as its subroutine. Based on Theorems 1 and 3, we derive the dynamic regret of the proposed algorithm.

Corollary 6 Let K = T 1/γ , where γ > 1 is a small constant. Suppose Assumption 1 holds, Ω Rd, and all the functions are α-exp-concave. Algorithm 1, with online Newton step as its subroutine, is strongly adaptive with

SA-Regret(T, τ)

(5d + 1)(γ + 1) + 2

α + 5d(γ + 1)GB log T

=O (γd log T) = O (d log T)

and its dynamic regret satisfies

D-Regret(w 1, . . . , w T )

(5d + 1)(γ + 1) + 2

α + 5d(γ + 1)GB + 2

max n log T, p

TVT log T o

=O d max n log T, p

TVT log T o .

To the best of our knowledge, this is the first dynamic regret that exploits exponential concavity. Furthermore, according to the minimax dynamic regret of strongly convex functions (Besbes et al., 2015), our upper bound is minimax optimal, up to a polylogarithmic factor.

4.4. Strongly Convex Functions

Finally, we study strongly convex functions. According to Lemma 2, we know that strongly convex functions with bounded gradients are also exp-concave. Thus, Corollary 6 can be directly applied to strongly convex functions, and yields a dynamic regret of O(d TVT log T). However, the upper bound depends on the dimensionality d. To address this limitation, we use online gradient descent as the subroutine in Algorithm 1.

From Theorems 2 and 3, we have the following theorem, in which both the adaptive and dynamic regrets are independent from d.

Corollary 7 Let K = T 1/γ , where γ > 1 is a small constant. Suppose Assumption 1 holds, and all the functions are λ-strongly convex. Algorithm 1, with online gradient descent as its subroutine, is strongly adaptive with

SA-Regret(T, τ)

2λ γ + 1 + (3γ + 7) log T

=O (γ log T) = O (log T)

Dynamic Regret of Strongly Adaptive Methods

and its dynamic regret satisfies

D-Regret(w 1, . . . , w T )

λ + 2 log T

TVT log T + 5γG2

=O max n log T, p

TVT log T o .

According to Theorem 4 of Besbes et al. (2015), the minimax dynamic regret of strongly convex functions is O( TVT ), which implies our upper bound is almost minimax optimal. By comparison, the restarted online gradient descent of Besbes et al. (2015) has a dynamic regret of O(log T TVT ), but it requires to know an upper bound of VT .

5. Analysis

We here present the proof of Theorem 3. The omitted proofs are provided in the supplementary.

5.1. Proof of Theorem 3

First, we upper bound the dynamic regret in the following way

D-Regret(w 1, . . . , w T )

t=si ft(wt)

t=si min w Ωft(w)

t=si ft(wt) min w Ω

| {z } :=ai

t=si min w Ωft(w)

| {z } :=bi

From the definition of strongly adaptive regret, we can upper bound ai by

t=si ft(wt) min w Ω

t=si ft(w) SA-Regret(T, |Ii|).

To upper bound bi, we follow the analysis of Proposition 2

of Besbes et al. (2015):

t=si min w Ωft(w)

t=si ft(w t )

t=si ft(w si)

t=si ft(w t )

|Ii| max t [si,qi]

ft(w si) ft(w t ) .

Furthermore, for any t [si, qi], we have

ft(w si) ft(w t )

=ft(w si) fsi(w si) + fsi(w si) ft(w t )

ft(w si) fsi(w si) + fsi(w t ) ft(w t )

Combining (6) with (7), we have

t=si min w Ωft(w) 2|Ii| VT (i).

Substituting the upper bounds of ai and bi into (5), we arrive at

D-Regret(w 1, . . . , w T )

i=1 (SA-Regret(T, |Ii|) + 2|Ii| VT (i)) .

Since the above inequality holds for any partition of [1, T], we can take minimization to get a tight bound.

6. Conclusions and Future Work

In this paper, we demonstrate that the dynamic regret can be upper bounded by the adaptive regret and the functional variation, which implies strongly adaptive algorithms are automatically equipped with tight dynamic regret bounds. As a result, we are able to derive dynamic regret bounds for convex functions, exp-concave functions, and strongly convex functions. Moreover, we provide a unified approach for minimizing the adaptive regret of exp-concave functions, as well as strongly convex functions.

The adaptive-to-dynamic conversion leads to a series of dynamic regret bounds in terms of the functional variation. As we mentioned before, dynamic regret can also be upper bounded by other regularities such as the path-length. It is interesting to investigate whether those kinds of upper bounds can also be established for strongly adaptive algorithms.

Dynamic Regret of Strongly Adaptive Methods

Acknowledgements

This work was partially supported by the National Key R&D Program of China (2018YFB1004300), NSFC (61603177, 61333014), Jiangsu SF (BK20160658), YESS (2017QNRC001), NSF (IIS-1545995), and the Collaborative Innovation Center of Novel Software Technology and Industrialization.

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