Title: How to Boost Any Loss Function
Venue: NeurIPS 2024
Accepted: yes
Keywords: boosting

loss functions

zeroth-order optimisation
Abstract: Boosting is a highly successful ML-born optimization setting in which one is required to computationally efficiently learn arbitrarily good models based on the access to a weak learner oracle, providing classifiers performing at least slightly differently from random guessing. A key difference with gradient-based optimization is that boosting's original model does not requires access to first order information about a loss, yet the decades long history of boosting has quickly evolved it into a first order optimization setting -- sometimes even wrongfully *defining* it as such. Owing to recent progress extending gradient-based optimization to use only a loss' zeroth ($0^{th}$) order information to learn, this begs the question: what loss functions be efficiently optimized with boosting and what is the information really needed for boosting to meet the *original* boosting blueprint's requirements ?

We provide a constructive formal answer essentially showing that *any* loss function can be optimized with boosting and thus boosting can achieve a feat not yet known to be possible in the classical $0^{th}$ order setting, since loss functions are not required to be be convex, nor differentiable or Lipschitz -- and in fact not required to be continuous either. Some tools we use are rooted in quantum calculus, the mathematical field -- not to be confounded with quantum computation -- that studies calculus without passing to the limit, and thus without using first order information.

How to Boost Any Loss Function

Abstract

Boosting is a highly successful ML-born optimization setting in which one is required to computationally efficiently learn arbitrarily good models based on the access to a weak learner oracle, providing classifiers performing at least slightly differently from random guessing. A key difference with gradient-based optimization is that boosting's original model does not requires access to first order information about a loss, yet the decades long history of boosting has quickly evolved it into a first order optimization setting - sometimes even wrongfully defining it as such. Owing to recent progress extending gradient-based optimization to use only a loss' zeroth ( 0 th ) order information to learn, this begs the question: what loss functions can be efficiently optimized with boosting and what is the information really needed for boosting to meet the original boosting blueprint's requirements? We provide a constructive formal answer essentially showing that any loss function can be optimized with boosting and thus boosting can achieve a feat not yet known to be possible in the classical 0 th order setting, since loss functions are not required to be be convex, nor differentiable or Lipschitz - and in fact not required to be continuous either. Some tools we use are rooted in quantum calculus, the mathematical field - not to be confounded with quantum computation - that studies calculus without passing to the limit, and thus without using first order information.

1 Introduction

In ML, zeroth order optimization has been devised as an alternative to techniques that would otherwise require access to ¥ 1 -order information about the loss to minimize, such as gradient descent (stochastic or not, constrained or not, etc., see Section 2). Such approaches replace the access to a so-called oracle providing derivatives for the loss at hand, operations that can be consuming or not available in exact form in the ML world, by the access to a cheaper function value oracle, providing loss values at queried points.

Zeroth order optimization has seen a considerable boost in ML over the past years, over many settings and algorithms, yet, there is one foundational ML setting and related algorithms that, to our knowledge, have not yet been the subject of investigations: boosting [32, 31]. Such a question is very relevant: boosting has quickly evolved as a technique requiring first-order information about the loss optimized [6, Section 10.3], [41, Section 7.2.2] [53]. It is also not uncommon to find boosting reduced to this first-order setting [9]. However, originally, the boosting model did not mandate the access to any first-order information about the loss, rather requiring access to a weak learner providing classifiers at least slightly different from random guessing [31]. In the context of zeroth-order optimization gaining traction in ML, it becomes crucial to understand not just whether differentiability is necessary for boosting, but more generally what are loss functions that can be boosted with a weak learner and in fine where boosting stands with respect to recent formal progress on lifting gradient descent to zeroth-order optimisation.

In this paper , we settle the question: we design a formal boosting algorithm for any loss function whose set of discontinuities has zero Lebesgue measure. With traditional floating point encoding (e.g. float64), any stored loss function would de facto meet this condition; mathematically speaking, we encompass losses that are not necessarily convex, nor differentiable or Lipschitz. This is a key difference with classical zeroth-order optimization results where the algorithms are zeroth-order but their proof of convergence makes various assumptions about the loss at hand, such as convexity, differentiability (once or twice), Lipschitzness, etc. . Our proof technique builds on a simple boosting technique for convex functions that relies on an order-one Taylor expansion to bound the progress between iterations [45]. Using tools from quantum calculus * , we replace this progress using v -derivatives and a quantity related to a generalisation of the Bregman information [7]. The boosting rate involves the classical weak learning assumption's advantage over random guessing and a new parameter bounding the ratio of the expected weights (squared) over a generalized notion of curvature involving v -derivatives. Our algorithm, which learns a linear model, introduces notable generalisations compared to the AdaBoost / gradient boosting lineages, chief among which the computation of acceptable offsets for the v -derivatives used to compute boosting weights, offsets being zero for classical gradient boosting. To preserve readability and save space, all proofs and additional information are postponed to an Appendix.

2 Related work

Over the past years, ML has seen a substantial push to get the cheapest optimisation routines, in general batch [14], online [27], distributed [3], adversarial [20, 18] or bandits settings [2] or more specific settings like projection-free [26, 28, 51] or saddle-point optimisation [25, 38]. We summarize several dozen recent references in Table 1 in terms of assumptions for the analysis about the loss optimized, provided in Appendix, Section A. Zeroth-order optimization reduces the information available to the learner to the "cheapest" one which consists in (loss) function values, usually via a so-called function value oracle . However, as Table 1 shows, the loss itself is always assumed to have some form of "niceness" to study the algorithms' convergence, such as differentiability, Lipschitzness, convexity, etc. . Another quite remarkable phenomenon is that throughout all their diverse settings and frameworks, not a single one of them addresses boosting. Boosting is however a natural candidate for such investigations, for two reasons. First, the most widely used boosting algorithms are first-order information hungry [6, 41, 53]: they require access to derivatives to compute examples' weights and classifiers' leveraging coefficients. Second and perhaps most importantly, unlike other optimization techniques like gradient descent, the original boosting model does not mandate the access to a first-order information oracle to learn, but rather to a weak learning oracle which supplies classifiers performing slightly differently from random guessing [32, 31]. Only few approaches exist to get to "cheaper' algorithms relying on less assumptions about the loss at hand, and to our knowledge do not have boosting-compliant convergence proofs, as for example when alleviating convexity [16, 46] or access to gradients of the loss [54]. Such questions are however important given the early negative results on boosting convex potentials with first-order information [37] and the role of the classifiers in the negative results [39].

Finally, we note that a rich literature has developed in mathematics as well for derivative-free optimisation [34], yet methods would also often rely on assumptions included in the three above ( e.g. [42]). It must be noted however that derivative-free optimisation has been implemented in computers for more than seven decades [24].

3 Definitions and notations

The following shorthands are used: r n s . GLYPH<16> t 1 , 2 , ..., n u for n P N GLYPH<6> , z GLYPH<4> r a, b s . GLYPH<16> r min t za, zb u , max t za, zb us for z P R , a / b P R . In the batch supervised learning setting, one is given a training set of m examples S . GLYPH<16> tp x i , y i q , i P r m su , where x i P X is an observation ( X is called the domain: often, X GLYPH<132> R d ) and y i P Y . GLYPH<16> tGLYPH<1> 1 , 1 u is a label, or class. We study the empirical convergence of boosting, which requires fast convergence on training. Such a setting is standard in zeroth order optimization [42]. Also, investigating generalization would entail specific design choices about the loss at hand and thus would restrict the scope of our result (see e.g. [8]). The objective is to

Bregman secant distortion

Bregman information

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learn a classifier , i.e. a function h : X GLYPH<209> R which belongs to a given set H . The goodness of fit of some h on S is evaluated from a given function F : R GLYPH<209> R called a loss function, whose expectation on training is sought to be minimized:

The set of most popular losses comprises convex functions: the exponential loss ( F EXP p z q . GLYPH<16> exp pGLYPH<1> z q ), the logistic loss ( F LOG p z q . GLYPH<16> log p 1 GLYPH<0> exp pGLYPH<1> z qq ), the square loss ( F SQ p z q . GLYPH<16> p 1 GLYPH<1> z q 2 ), the Hinge loss ( F H p z q . GLYPH<16> max t 0 , 1 GLYPH<1> z u ). These are surrogate losses because they all define upperbounds of the 0/1-loss ( F 0/1 p z q . GLYPH<16> 1 z / 0 , " 1 ' being the indicator variable).

Our ML setting is that of boosting [31]. It consists in having primary access to a weak learner WL that when called, provides so-called weak hypotheses, weak because barely anything is assumed in terms of classification performance relatively to the sample over which they were trained. Our goal is to devise a so-called "boosting" algorithm that can take any loss F as input and training sample S and a target loss value F GLYPH<6> and after some T calls to the weak learner crafts a classifier H T satisfying F p S, H T q / F GLYPH<6> , where T depends on various parameters of the ML problem. Our boosting architecture is a linear model: H T . GLYPH<16> GLYPH<176> t α t h t where each h t is an output from the weak learner and leveraging coefficients α t have to be computed during boosting. Notice that this is substantially more general than the classical boosting formulation where the loss would be fixed or belong to a restricted subset of functions.

4 v -derivatives and Bregman secant distortions

Unless otherwise stated, in this Section, F is a function defined over R .

Definition 4.1. [30] For any z, v P R , we let δ v F p z q . GLYPH<16> p F p z GLYPH<0> v qGLYPH<1> F p z qq{ v denote the v -derivative of F in z .

This expression, which gives the classical derivative when the offset v GLYPH<209> 0 , is called the h -derivative in quantum calculus [30, Chapter 1]. We replaced the notation for the risk of confusion with classifiers. Notice that the v -derivative is just the slope of the secant that passes through points p z, F p z qq and p z GLYPH<0> v, F p z GLYPH<0> v qq (Figure 1). Higher order v -derivatives can be defined with the same offset used several times [30]. Here, we shall need a more general definition that accommodates for variable offsets.

Definition 4.2. Let v 1 , v 2 , ..., v n P R and V . GLYPH<16> t v 1 , v 2 , ..., v n u and z P R . The V -derivative δ V F is:

If v i GLYPH<16> v, @ i P r n s then we write δ p n q v F p z q . GLYPH<16> δ V F p z q .

In the Appendix, Lemma B.1 computes the unravelled expression of δ V F p z q , showing that the order of the elements in V does not matter; n is called the order of the V -derivative.

We can now define a generalization of Bregman divergences called Bregman Secant distortions .

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Proprietary + Confidential

Definition 4.3. For any z, z 1 , v P R , the Bregman Secant distortion S F | v p z 1 } z q with generator F and offset v is:

Even if F is convex, the distortion is not necessarily positive, though it is lowerbounded (Figure 1). There is an intimate relationship between the Bregman Secant distortions and Bregman divergences. We shall use a definition slightly more general than the original one when F is differentiable [11, eq. (1.4)], introduced in information geometry [5, Section 3.4] and recently reintroduced in ML [10].

Definition 4.4. The Bregman divergence with generator F (scalar, convex) between z 1 and z is D F p z 1 } z q . GLYPH<16> F p z 1 q GLYPH<0> F GLYPH<14> p z q GLYPH<1> z 1 z , where F GLYPH<14> p z q . GLYPH<16> sup t tz GLYPH<1> F p t q is the convex conjugate of F .

We state the link between S F | v and D F (proof omitted).

Lemma 4.5. Suppose F strictly convex differentiable. Then lim v GLYPH<209> 0 S F | v p z 1 } z q GLYPH<16> D F p z 1 } F 1 p z qq .

Relaxed forms of Bregman divergences have been introduced in information geometry [43].

Definition 4.6. For any a, b, α P R , denote for short I a,b . GLYPH<16> r min t a, b u , max t a, b us and p uv q α . GLYPH<16> αu GLYPH<0> p 1 GLYPH<1> α q v . The Optimal Bregman Information (OBI) of F defined by triple p a, b, c q P R 3 is:

As represented in Figure 1 (right), the OBI is obtained by drawing the line passing through p a, F p a qq and p b, F p b qq and then, in the interval I a,c , look for the maximal difference between the line and F . We note that Q F is non negative because a P I a,c and for the choice α GLYPH<16> 1 , the RHS in (1) is 0. We also note that when F is convex, the RHS is indeed the maximal Bregman information of two points in [7, Definition 2], where maximality is obtained over the probability measure. The following Lemma follows from the definition of the Bregman secant divergence and the OBI. An inspection of the functions in Figure 1 provides a graphical proof.

Lemma 4.7. For any F ,

and if F is convex,

We shall abbreviate the two possible forms of OBI in the RHS of (2), (3) as:

5 Boosting using only queries on the loss

We make the assumption that predictions of so-called "weak classifiers" are finite and non-zero on training without loss of generality (otherwise a simple tweak ensures it without breaking the weak learning framework, see Appendix, Section B.2). Excluding 0 ensures our algorithm does not make use of derivatives.

Assumption 5.1. @ t ¡ 0 , @ i P r m s , | h t p x i q| P p 0 , GLYPH<0>8q (we thus let M t . GLYPH<16> max i | h t p x i q| ).

For short, we define two edge quantities for i P r m s and t GLYPH<16> 1 , 2 , ... , where α t is a leveraging coefficient for the weak classifiers in an ensemble H T p . q . GLYPH<16> GLYPH<176> t Pr T s α t h t p . q . We observe

Algorithm 1 SECBOOST( S, T ) // red boxes pinpoint substantial differences with "classical" boosting

Input sample S GLYPH<16> tp x i , y i q , i GLYPH<16> 1 , 2 , ..., m u , number of iterations T , initial p h 0 , v 0 q (constant classification and offset). Step 1 : let H 0 -1 GLYPH<4> h 0 and w 1 GLYPH<16> GLYPH<1> δ v 0 F p h 0 q GLYPH<4> 1 ; // h 0 , v 0 GLYPH<24> 0 chosen s. t. δ v 0 F p h 0 q GLYPH<24> 0 Step 2 : for t GLYPH<16> 1 , 2 , ..., T Step 2.1 : let h t -WL p S t , | w t |q //weak learner call, S t . GLYPH<16> tp x i , y i GLYPH<4> sign p w ti qqu Step 2.2 : let η t -p 1 { m q GLYPH<4> GLYPH<176> i w ti y i h t p x i q //unnormalized edge Step 2.3 : If bound on W 2 ,t available (Section 5.3) otherwise | general procedure pick ε t ¡ 0 , π t P p 0 , 1 q and α t P η t 2 p 1 GLYPH<0> ε t q M 2 t W 2 ,t GLYPH<4> r 1 GLYPH<1> π t , 1 GLYPH<0> π t s ; (6) α t -SOLVE α ( S, w t , h t ) // W 2 ,t ¡ 0 , ε t ¡ 0 , π t P p 0 , 1 q // Theorem 5.8 Step 2.4 : let H t -H t GLYPH<1> 1 GLYPH<0> α t GLYPH<4> h t //classifier update Step 2.5 : if I ti p ε t GLYPH<4> α 2 t M 2 t W 2 ,t q GLYPH<24>H , @ i P r m s then //new offsets for i GLYPH<16> 1 , 2 , ..., m , let v ti -OO p t, i, ε t GLYPH<4> α 2 t M 2 t W 2 ,t q ; else return H t ; Step 2.6 : for i GLYPH<16> 1 , 2 , ..., m , let //weight update w p t GLYPH<0> 1 q i -GLYPH<1> δ v ti F p y i H t p x i qq ; (7) Step 2.7 : if w t GLYPH<0> 1 GLYPH<16> 0 then break; Return H T .

5.1 Algorithm: SECBOOST

5.1.1 General steps

Without further ado, Algorithm SECBOOST presents our approach to boosting without using derivatives information. The key differences with traditional boosting algorithms are red color framed. We summarize its key steps.

Step 1 This is the initialization step. Traditionally in boosting, one would pick h 0 GLYPH<16> 0 . Note that w 1 is not necessarily positive. v 0 is the initial offset (Section 4).

Step 2.1 This step calls the weak learner, as in traditional boosting, using variable "weights" on examples (the coordinate-wise absolute value of w t , denoted | w t | ). The key difference with traditional boosting is that examples labels can switch between iterations as well, which explains that the training sample, S t , is indexed by the iteration number.

Step 2.3 This step computes the leveraging coefficient α t of the weak classifier h t . It involves a quantity, W 2 ,t , which we define as any strictly positive real satisfying

For boosting rate's sake, we should find W 2 ,t as small as possible. We refer to (5) for the e . , ˜ e . notations; v . is the current (set of) offset(s) (Section 4 for their definition). The second-order V -derivative in the LHS plays the same role as the second-order derivative in classical boosting rates, see for example [45, Appendix, eq. 29]. As offsets GLYPH<209> 0 , it converges to a second-order derivative; otherwise, they still share some properties, such as the sign for convex functions.

Lemma 5.2. Suppose F convex. For any a P R , b, c P R GLYPH<6> , δ t b,c u F p a q ¡ 0 .

(Proof in Appendix, Section B.3) We can also see a link with weights variation since, modulo a slight abuse of notation, we have δ t e ti ,v p t GLYPH<1> 1 q i u F p ˜ e p t GLYPH<1> 1 q i q GLYPH<16> δ e ti w ti . A substantial difference with traditional boosting algorithms is that we have two ways to pick the leveraging coefficient α t ; the first one can be used when a convenient W 2 ,t is directly accessible from the loss. Otherwise, there is a simple algorithm that provides parameters (including W 2 ,t ) such that (8) is satisfied. Section 5.3

details those two possibilities and their implementation. In the more favorable case (the former one), α t can be chosen in an interval, furthermore defined by flexible parameters ε t ¡ 0 , π t P p 0 , 1 q . Note that fixing beforehand these parameters is not mandatory: we can also pick any

and then compute choices for the corresponding ε t and π t . ε t is important for the algorithm and both parameters are important for the analysis of the boosting rate. From the boosting standpoint, a smaller ε t yields a larger α t and a smaller π t reduces the interval of values in which we can pick α t ; both cases tend to favor better convergence rates as seen in Theorem 5.3.

Step 2.4 is just the crafting of the final model.

Step 2.5 is new to boosting, the use of a so-called offset oracle, detailed in Section 5.1.2.

Step 2.6 The weight update does not rely on a first-order oracle as in traditional boosting, but uses only loss values through v -derivatives. The finiteness of F implies the finiteness of weights.

Step 2.7 Early stopping happens if all weights are null. While this would never happen with traditional ( e.g. strictly convex) losses, some losses that are unusual in the context of boosting can lead to early stopping. A discussion on early stopping and how to avoid it is in Section 6.

5.1.2 The offset oracle, OO

Let us introduce notation

(see Figure 3 below to visualize I ti p z q for a non-convex F ) The offset oracle is used in Step 2.5, which is new to boosting. It requests the offsets to carry out weight update in (7) to an offset oracle , which achieves the following, for iteration # t , example # i , limit OBI z :

Note that the offset oracle has the freedom to pick the offset in a whole set. Section 5.4 investigates implementations of the offset oracle, so let us make a few essentially graphical remarks here. OO does not need to build the whole I ti p z q to return some v P I ti p z q for Step 2.5 in SECBOOST. In the construction steps of Figure 3, as soon as O GLYPH<24> H , one element of O can be returned. Figure 4 presents more examples of I ti p z q . One can remark that the sign of the offset v ti in Step 2.5 of SECBOOST is the same as the sign of ˜ e p t GLYPH<1> 1 q i GLYPH<1> ˜ e ti GLYPH<16> GLYPH<1> y i α t h t p x i q . Hence, unless F is derivable or all edges y i h t p x i q are of the same sign ( @ i ), the set of offsets returned in Step 2.5 always contain at least two different offsets, one non-negative and one non-positive (Figure 4, (a-b)).

5.2 Convergence of SECBOOST

The offset oracle has a technical importance for boosting: I ti p z q is the set of offsets that limit an OBI for a training example (Definition 4.6). The importance for boosting comes from Lemma 4.7: upperbounding an OBI implies lowerbounding a Bregman Secant divergence, which will also guarantee a sufficient slack between two successive boosting iterations. This is embedded in a blueprint of a proof technique to show boosting-compliant convergence which is not new, see e.g. [45]. We now detail this convergence.

Remark that the expected edge η t in Step 2.2 of SECBOOST is not normalized. We define a normalized version of this edge as:

with ˜ y ti . GLYPH<16> y i GLYPH<4> sign p w ti q , W t . GLYPH<16> GLYPH<176> i | w ti | GLYPH<16> GLYPH<176> i | δ v p t GLYPH<1> 1 q i F p ˜ e p t GLYPH<1> 1 q i q| . Remark that the labels are corrected by the weight sign and thus may switch between iterations. In the particular case where the loss is non-increasing (such as with traditional convex surrogates), the labels do not switch. We need also a quantity which is, in absolute value, the expected weight:

In classical boosting for convex decreasing losses † , weights are non-negative and converge to a minimum (typically 0) as examples get the right class with increasing confidence. Thus, W 1 ,t can be an indicator of when classification becomes "good enough" to stop boosting. In our more general setting, it shall be used in a similar indicator. We are now in a position to show a first result about SECBOOST.

Theorem 5.3. Suppose assumption 5.1 holds. Let F 0 . GLYPH<16> F p S, h 0 q in SECBOOST and z GLYPH<6> any real such that F p z GLYPH<6> q / F 0 . Then we are guaranteed that classifier H T output by SECBOOST satisfies F p S, H T q / F p z GLYPH<6> q when the number of boosting iterations T yields:

where parameters ε t , π t appear in Step 2.3 of SECBOOST .

(proof in Appendix, Section B.4) We observe the tradeoff between the freedom in picking parameters and convergence guarantee as exposed by (14): to get more freedom in picking the leveraging coefficient α t , we typically need π t large (Step 2.3) and to get more freedom in picking the offset v t GLYPH<24> 0 , we typically need ε t large (Step 2.5). However, allowing more freedom in such ways reduces the LHS and thus impairs the guarantee in (14). Therefore, there is a subtle balance between "freedom" of choice and convergence. This balance becomes more clear as boosting compliance formally enters convergence requirement.

Boosting-compliant convergence We characterize convergence in the boosting framework, which shall include the traditional weak learning assumption.

Assumption 5.4. ( γ -Weak Learning Assumption , γ -WLA) We assume the following on the weak learner: D γ ¡ 0 such that @ t ¡ 0 , | ˜ η t | ¥ γ .

As is usually the case in boosting, the weights are normalized in the weak learning assumption (12). So the minimization "potential" of the loss does not depend on the absolute scale of weight. This is not surprising because the loss is "nice" in classical boosting: a large γ guarantees most examples' edges moving to the right of the x -axis after the classifier update which, because the loss is strictly decreasing (exponential loss, logistic loss, etc.), is sufficient to yield a smaller expected loss. In our case it is not true anymore as for example there could be a local bump in the loss that would have it increase after the update. This is not even a pathological example: one may imagine that instead of a single bump the loss jiggles a lot locally. How can we keep boosting operating in such cases ? A sufficient condition takes the form of a second assumption that also integrates weights, ensuring that the variation of weights is locally not too large compared to (unnormalized) weights, which is akin to comparing local first- and second-order variations of the loss in the differentiable case. We encapsulate this notion in what we call a weight regularity assumption.

Assumption 5.5. ( ρ -Weight Regularity Assumption , ρ -WRA) Let ρ t . GLYPH<16> W 2 1 ,t { W 2 ,t . We assume there exists ρ ¡ 0 such that @ t ¥ 1 , ρ t ¡ ρ .

In Figure 2 we present a(n overly) simplified depiction of the cases where W 2 ,t is large for "not nice" losses, and two workarounds on how to keep it small enough for the WRA to hold. Keep in mind that W 1 ,t is an expected local variation of the loss (13), (5), so as it goes to zero, boosting converges to a local minimum and it is reasonable to expect that the WRA breaks. Otherwise, there are two strategies that keep W 2 ,t relatively small enough for WRA to hold: either we pick small enough offsets, which essentially works for most losses but make us converge in general to a local minimum (this is in essence our experimental choice) or we optimize the offset oracle so that it sometimes "passes" local jiggling (Figure 2 (d)). While this eventually requires to tune the weak learner jointly with the offset oracle and fine-tune that latter algorithm on a loss-dependent basis, such a strategy can be used to eventually pass local minima of the loss. To do so, "larger" offsets directly translate into corresponding requests for larger magnitude classification for the next weak classifier, for the related examples. We are now in a position to state a simple corollary to Theorem 5.3.

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Algorithm 2 SOLVE α ( S, w , h )

Input sample S GLYPH<16> tp x i , y i q , i GLYPH<16> 1 , 2 , ..., m u , w P R m , h : X GLYPH<209> R .

Step 1 : find any a ¡ 0 such that

Return sign p η p w , h qq GLYPH<4> a .

Corollary 5.6. Suppose assumptions 5.1, 5.5 and 5.4 hold. Let F 0 . GLYPH<16> F p S, h 0 q in SECBOOST and z any real such that F p z q / F 0 . If SECBOOST is run for a number T of iterations satisfying then F p S, H T q / F p z q .

We remark that the dependency in γ is optimal [4].

5.3 Finding W 2 ,t

There is lots of freedom in the choice of α t in Step 2.3 of SECBOOST, and even more if we look at (9). This, however, requires access to some bound W 2 ,t . In the general case, the quantity it upperbounds in (8) also depends on α t because e ti . GLYPH<16> α t GLYPH<4> y i h t p x i q . So unless we can obtain such a "simple" W 2 ,t that does not depend on α t , (6) - and (9) - provide a system to solve for α t .

W 2 ,t via properties of F Classical assumptions on loss functions for zeroth-order optimization can provide simple expressions for W 2 ,t (Table 1). Consider smoothness: we say that F is β -smooth if it is derivable and its derivative satisfies the Lipschitz condition | F 1 p z 1 q GLYPH<1> F 1 p z q| / β | z 1 GLYPH<1> z | , @ z, z 1 [12]. Notice that this implies the condition on the v -derivative of the derivative: | δ v F 1 p z q| / β, @ z, v . This also provides a straightforward useful expression for W 2 ,t .

Lemma 5.7. Suppose that the loss F is β -smooth. Then we can fix W 2 ,t GLYPH<16> 2 β .

(Proof in Appendix, Section B.5) What the Lemma shows is that a bound on the v -derivative of the derivative implies a bound on order-2 V -derivatives (in the quantity that W 2 ,t bounds (8)). Such a condition on v -derivatives is thus weaker than a condition on derivatives, and it is strictly weaker if we impose a strictly positive lowerbound on the offset's absolute value, which would be sufficient to characterize the boosting convergence of SECBOOST.

A general algorithm for W 2 ,t If we cannot make any assumption on F , there is a simple way to first obtain α t and then W 2 ,t , from which all other parameters of Step 2.3 can be computed. We first need a few definitions. We first generalize the edge notation appearing in Step 2.2:

Offset oracle

Offset oracle

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so that η t . GLYPH<16> η p w t , h t q . Remind the weight update, w ti . GLYPH<16> GLYPH<1> δ v p t GLYPH<1> 1 q i F p y i H t GLYPH<1> 1 p x i qq . We define a "partial" weight update,

(if we were to replace v p t GLYPH<1> 1 q i by v ti and let α . GLYPH<16> α t , then ˜ w ti p α q would be w p t GLYPH<0> 1 q i , hence the partial weight update). Algorithm 2 presents the simple procedure to find α t . Notice that we use ˜ w with sole dependency on the prospective leveraging coefficient; we omit for clarity the dependences in the current ensemble ( H . ), weak classifier ( h . ) and offsets ( v .i ) needed to compute (17).

Theorem 5.8. Suppose Assumptions 5.1 and 5.4 hold and F is continuous at all abscissae t ˜ e p t GLYPH<1> 1 q i . GLYPH<16> y i H t GLYPH<1> 1 p x i q , i P r m su . Then there are always solutions to Step 1 of SOLVE α and if we let α t -SOLVE α ( S, w t , h t ) and then compute

then W 2 ,t satisfies (8) and α t satisfies (6) for some ε t ¡ 0 , π t P p 0 , 1 q .

The proof, in Section B.6, proceeds by reducing condition (9) to (16). The Weak Learning Assumption (5.4) is important for the denominator in the LHS of (16) to be non zero. The continuity assumption at all abscissae is important to have lim a GLYPH<209> 0 η p ˜ w t p a q , h t q GLYPH<16> η t , which ensures the existence of solutions to (16), also easy to find, e.g. by a simple dichotomic search starting from an initial guess for a . Note the necessity of being continuous only at abscissae defined by the training sample, which is finite in size. Hence, if this condition is not satisfied but discontinuities of F are of Lebesgue measure 0, it is easy to add an infinitesimal constant to the current weak classifier, ensuring the conditions of Theorem 5.8 and keeping the boosting rates.

5.4 Implementation of the offset oracle

Figure 3 explains how to build graphically I ti p z q for a general F . While it is not hard to implement a general procedure following the blueprint ( i.e. accepting the loss function as input), it would be far

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from achieving computational optimality: a much better choice consists in specializing it to the (set of) loss(es) at hand via hardcoding specific optimization features of the desired loss(es). This would not prevent "loss oddities" to get absolutely trivial oracles (see Appendix, Section B.7).

6 Discussion

For an efficient implementation, boosting requires specific design choices to make sure the weak learning assumption stands for as long as necessary; experimentally, it is thus a good idea to adapt the weak learner to build more complex models as iterations increase ( e.g. learning deeper trees), keeping Assumption 5.4 valid with its advantage over random guessing parameter γ ¡ 0 . In our more general setting, our algorithm SECBOOST pinpoints two more locations that can make use of specific design choices to keep assumptions stand for a larger number of iterations.

The first is related to handling local minima. When Assumption 5.5 breaks, it means we are close to a local optimum of the loss. One possible way of escaping those local minima is to adapt the offset oracle to output larger offsets (Step 2.5) that get weights computed outside the domain of the local minimum. Such offsets can be used to inform the weak learner of the specific examples that then need to receive larger magnitude in classification, something we have already discussed in Section 5. There is also more: the sign of the weight indicates the polarity of the next edge ( e t. , (5)) needed to decrease the loss in the interval spanned by the last offset . To simplify, suppose a substantial fraction of examples have an edge ˜ e t. in the vicinity of the blue dotted line in Figure 2 (d) so that the loss value is indicated by the big arrow and suppose their current offset GLYPH<16> v t GLYPH<1> 1 so that their weight (positive) signals that to minimize further the loss, the weak learner's next weak classifier has to have a positive edge over these examples. Such is the polarity constraint which essentially comes to satisfy the WLA, but there is a magnitude constraint that comes from the WRA: indeed, if the positive edge is too small so that the loss ends up in the "bump" region, then there is a risk that the WRA breaks because the loss around the bump is quite flat, so the numerator of ρ t in Assumption 5.5 can be small. Passing the bump implies escaping the local minimum at which the loss would otherwise be trapped. Section 5.4 has presented a general blueprint for the offset oracle but more specific implementation designs can be used; some are discussed in the Appendix, Section B.7.

The second is related to handling losses that take on constant values over parts of their domain. To prevent early stopping in Step 2.7 of SECBOOST, one needs w t GLYPH<0> 1 GLYPH<24> 0 . The update rule of w t imposes that the loss must then have non-zero variation for some examples between two successive edges (5). If the loss F is constant, then clearly the algorithm obviously stops without learning anything. If F is piecewise-constant, this constrain the design of the weak learner to make sure that some examples receive a different loss with the new model update H . . As explained in Appendix, Section B.11, this can be efficiently addressed by specific designs on SOLVE α .

In the same way as there is no "1 size fits all" weak learner for all domains in traditional boosting, we expect specific design choices to be instrumental in better handling specific losses in our more general setting. Our theory points two locations further work can focus on.

7 Conclusion

Boosting has rapidly moved to an optimization setting involving first-order information about the loss optimized, rejoining, in terms of information needed, that of the hugely popular (stochastic) gradient descent. But this was not a formal requirement of the initial setting and in this paper, we show that essentially any loss function can be boosted without this requirement. From this standpoint, our results put boosting in a slightly more favorable light than recent development on zeroth-order optimization since, to get boosting-compliant convergence, we do not need the loss to meet any of the assumptions that those analyses usually rely on. Of course, recent advances in zeroth-order optimization have also achieved substantial design tricks for the implementation of such algorithms, something that undoubtedly needs to be adressed in our case, such as for the efficient optimization of the offset oracle. We leave this as an open problem but provide in Appendix some toy experiments that a straightforward implementation achieves, hinting that SECBOOST can indeed optimize very 'exotic' losses.

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Appendix

To differentiate with the numberings in the main file, the numbering of Theorems, etc. is letter-based (A, B, ...).

Table of contents

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A A quick summary of recent zeroth-order optimization approaches

Table 1 summarizes a few dozens of recent approaches that can be related to zeroth-order optimization in various topics of ML. Note that no such approaches focus on boosting.

B Supplementary material on proofs

B.1 Helper results

We now show that the order of the elements of V does not matter to compute the V -derivative as in Definition 4.2. For any σ P t 0 , 1 u n , we let 1 σ . GLYPH<16> GLYPH<176> i σ i .

Lemma B.1. For any z P R , any n P N GLYPH<6> and any V . GLYPH<16> t v 1 , v 2 , ..., v n u GLYPH<128> R ,

Hence, δ V F is invariant to permutations of the elements of V .

Proof. We show the result by induction on the size of V , first noting that

Wethen assume that (18) holds for V n . GLYPH<16> t v 1 , v 2 , ..., v n u and show the result for V n GLYPH<0> 1 . GLYPH<16> V n Yt v n GLYPH<0> 1 u , writing (induction hypothesis used in the second identity):

We also have the following simple Lemma, which is a direct consequence of Lemma B.1.

Lemma B.2. For all z, P R , v, z 1 P R GLYPH<6> , we have

Proof. It comes from Lemma B.1 that δ t z 1 ,v u F p z q GLYPH<16> δ t v,z 1 u F p z q GLYPH<16> p δ v F p z GLYPH<0> z 1 q GLYPH<1> δ v F p z qq{ z 1 (and we reorder terms).

B.2 Removing the GLYPH<24> 0 part in Assumption 5.1

Because everything needs to be encoded, finiteness is not really an assumption. However, the non-zero assumption may be seen as limiting (unless we are happy to use first-order information about the loss (Section 5). There is a simple trick to remove it. Suppose h t zeroes on some training examples. The training sample being finite, there exists an open neighborhood I in 0 such that h 1 t . GLYPH<16> h t GLYPH<0> δ does not zero anymore on training examples, for any δ P I . This changes the advantage γ in the WLA (Definition 5.4) to some γ 1 satisfying (we assume δ ¡ 0 wlog)

from which it is enough to pick δ / ε γ M t {p 1 GLYPH<0> γ q to guarantee advantage γ 1 ¥ p 1 GLYPH<1> ε q γ . If ε is a constant, this translates in a number of boosting iterations in Corollary 5.6 affected by a constant factor that we can choose as close to 1 as desired.

B.3 Proof of Lemma 5.2

We reformulate

Role of 'second-order' secant

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Both µ 1 and µ 2 are averages that can be computed from the midpoints of two secants (respectively):

Also, the midpoints of both secants have the same abscissa (and the ordinates are µ 1 and µ 2 ), so to study the sign of δ t b,c u F p a q , we can study the position of both secants with respect to each other. F being convex, we show that the abscissae of one secant are included in the abscissae of the other, this being sufficient to give the position of both secants with respect to each other. We distinguish four cases.

Case 1 : c ¡ 0 , b ¡ 0 . We have a GLYPH<0> b GLYPH<0> c ¡ max t a GLYPH<0> b, a GLYPH<0> c u and a GLYPH<160> min t a GLYPH<0> b, a GLYPH<0> c u . F being convex, p ∆ 2 q sits above p ∆ 1 q . So, µ 2 ¥ µ 1 and finally δ t b,c u F p a q ¥ 0 .

Case 2 : c GLYPH<160> 0 , b GLYPH<160> 0 . We now have a GLYPH<0> b GLYPH<0> c GLYPH<160> min t a GLYPH<0> b, a GLYPH<0> c u while a ¡ max t a GLYPH<0> b, a GLYPH<0> c u , so p ∆ 2 q sits above p ∆ 1 q . Again, µ 2 ¥ µ 1 and finally δ t b,c u F p a q ¥ 0 .

Case 3 : c ¡ 0 , b GLYPH<160> 0 . We have a GLYPH<0> b GLYPH<160> a and a GLYPH<0> b GLYPH<160> a GLYPH<0> b GLYPH<0> c . Also a GLYPH<0> c ¡ max t a GLYPH<0> b GLYPH<0> c, a u , so this time p ∆ 2 q sits below p ∆ 1 q but cb GLYPH<160> 0 , so δ t b,c u F p a q ¥ 0 again.

Case 4 : c GLYPH<160> 0 , b ¡ 0 . So a GLYPH<0> c GLYPH<160> a GLYPH<160> a GLYPH<0> b and a GLYPH<0> c GLYPH<160> a GLYPH<0> b GLYPH<0> c . So a GLYPH<0> c GLYPH<160> min t a, a GLYPH<0> b GLYPH<0> c u and a GLYPH<0> b ¡ max t a, a GLYPH<0> c u , so p ∆ 2 q sits below p ∆ 1 q . Since cb GLYPH<160> 0 , so δ t b,c u F p a q ¥ 0 again.

B.4 Proof of Theorem 5.3

Let us remind key simplified notations about edges, @ t ¥ 0 :

where Q GLYPH<6> .. is defined in (4). We also split the computation of the leveraging coefficient α t in SECBOOSTin two parts, the first computing a real a t as:

For short, we also let:

Proprietary + Confidential

and then using α t -a t η t . We now use Lemma 4.7 (main file) and get

If we reorganise (28) using the definition of S F | . p . } . q , we get:

(29) - (31) make use of definitions (24) (twice) and (26) as well as the decomposition of the leveraging coefficient in (27).

Looking at (32), we see that we can have a boosting-compliant decrease of the loss if the two quantities depending on ∆ p t GLYPH<0> 1 q . and Q GLYPH<6> p t GLYPH<0> 1 q . can be made small enough compared to a t GLYPH<0> 1 η 2 t GLYPH<0> 1 . This is what we investigate.

Bounding the term depending on ∆ p t GLYPH<0> 1 q . - We use Lemma B.2 with z . GLYPH<16> ˜ e ti , z 1 . GLYPH<16> e p t GLYPH<0> 1 q i , v . GLYPH<16> v t , which yields (also using (24) and the assumption that h t GLYPH<0> 1 p x i q GLYPH<24> 0 ):

and so we get:

Bounding the term depending on Q GLYPH<6> . p t GLYPH<0> 1 q - We immediately get from the value picked in argument of I t GLYPH<0> 1 in step 2.5 of SECBOOST, the definition of I ti p . q in (10) and our decomposition α t -a t η t that Q GLYPH<6> p t GLYPH<0> 1 q i / ε t GLYPH<0> 1 GLYPH<4> a 2 t GLYPH<0> 1 η 2 t GLYPH<0> 1 M 2 t GLYPH<0> 1 GLYPH<4> W 2 ,t GLYPH<0> 1 , @ i P r m s , so that:

Finishing up with the proof - Suppose that we choose ε t GLYPH<0> 1 ¡ 0 , π t GLYPH<0> 1 P p 0 , 1 q and a t GLYPH<0> 1 as in (27). We then get from (32), (34), (35) that for any choice of v ti in Step 2.5 of SECBOOST,

where the last inequality is a consequence of (27). Suppose we pick H 0 . GLYPH<16> h 0 P R a constant and v 0 ¡ 0 such that

The final classifier H T of SECBOOST satisfies:

with F 0 . GLYPH<16> E i GLYPH<18>r m s r F p ˜ e i 0 qs . GLYPH<16> E i GLYPH<18>r m s r F p y i H 0 qs GLYPH<16> E i GLYPH<18>r m s r F p y i h 0 qs . If we want E i GLYPH<18>r m s r F p y i H T p x i qqs / F p z GLYPH<6> q , assuming wlog F p z GLYPH<6> q / F 0 , then it suffices to iterate until:

Remind that the edge η t is not normalized. We have defined a normalized edge,

with ˜ y ti . GLYPH<16> y i GLYPH<4> sign p w ti q and W t . GLYPH<16> GLYPH<176> i | w ti | GLYPH<16> GLYPH<176> i | δ v p t GLYPH<1> 1 q i F p ˜ e p t GLYPH<1> 1 q i q| . We have the simple relationship between η t and ˜ η t :

recalling W 1 ,t . GLYPH<16> GLYPH<7> GLYPH<7> E i GLYPH<18> D GLYPH<16> δ v p t GLYPH<1> 1 q i F p ˜ e p t GLYPH<1> 1 q i q GLYPH<24> GLYPH<7> GLYPH<7> . It comes from (42) that a sufficient condition for (39) to hold is:

which is the statement of Theorem 5.3.

resulting in ( @ t ¥ 1 ),

B.5 Proof of Lemma 5.7

We first observe that for any a P R , b, c P R GLYPH<6> ,

where we used the β -smoothness of F and twice [12, Lemma 3.4]. We can also make a permutation in the expression of δ t b,c u F p a q and instead write

We thus have by the power mean inequality [13, Chapter III, Theorem 2]. Since | h t p x i q| / M t by definition, we thus have

which allows us to fix W 2 ,t GLYPH<16> 2 β and completes the proof of Lemma 5.7.

Remark B.3. Our result is optimal in the sense that if we make one offset (say b ) go to zero, then the ratio in (46) goes to zero and we recover the condition on the v -derivative of the derivative, | δ c F 1 p z q| / β .

B.6 Proof of Theorem 5.8

We consider the upperbound::

(The last identity uses the fact that y i P tGLYPH<1> 1 , 1 u ). Remark that we have extracted α t from the denominator but it is still present in the arguments ˜ e ti . For any classifier h , we introduce notation

and so η t (Step 2.2 in SECBOOST) is also η p w t , h t q , which is guaranteed to be non-zero by the Weak Learning Assumption (5.4). We want, for some ε t ¡ 0 , π t P r 0 , 1 q ,

This says that the sign of α t is the same as the sign of η p w t , h t q GLYPH<16> η t . Since we know its sign, let us look for its absolute value:

From (9) (main file), we can in fact search α t in the union of all such intervals for ε t ¡ 0 , π t P r 0 , 1 q , which amounts to find first:

and then find any ε t ¡ 0 , π t P r 0 , 1 q such that (50) holds. Using (48) and simplifying the external dependency on α t , we then need

under the constraint that the sign of α t be the same as that of η t . But, using notation (17) (main file), we have

and so to get (51) satisfied, it is sufficient that

which is Step 1 in SOLVE α . The Weak Learning Assumption (5.4) guarantees that the denominator is GLYPH<24> 0 so this can always be evaluated. The continuity of F in all ˜ e p t GLYPH<1> 1 q i guarantees lim α t GLYPH<209> 0 η p ˜ w t p α t q , h t q GLYPH<16> η t , and thus guarantees the existence of solutions to (52) for some | α t | ¡ 0 .

To summarize, finding α t can be done in two steps, (i) solve

for some a ¡ 0 and (ii) let α t . GLYPH<16> sign p η t q GLYPH<4> a . This is the output of SOLVE α ( S, w t , h t ), which ends the proof of Theorem 5.8.

B.7 Implementation of the offset oracle: particular cases

Consider the "spring loss" that we define, for r . s denoting the nearest integer, as:

Figure 6 plots this loss, which composes the logistic loss with a " U "-shaped term. This loss would escape all optimization algorithms of Table 1 (Appendix), yet there is a trivial implementation of our offset oracle, as explained in Figure 6:

if the interval I defined by ˜ e p t GLYPH<1> 1 q i and ˜ e ti contains at least one peak, compute the tangence point ( z t ) at the closest local " U " that passes through p ˜ e p t GLYPH<1> 1 q i , F p ˜ e p t GLYPH<1> 1 q i qq ; then if z t P I then v ti -z t GLYPH<1> ˜ e p t GLYPH<1> 1 q i , else v ti -˜ e ti GLYPH<1> ˜ e p t GLYPH<1> 1 q i ;

otherwise F in I is strictly convex and differentiable: a simple dichotomic search can retrieve a feasible v ti (see convex losses below);

Notice that one can alleviate the repetitive dichotomic search by pre-tabulating a feasible v for a set of differences | a GLYPH<1> b | ( a, b belonging to the abscissae of the same " U ") decreasing by a fixed factor, choosing v ti -v of the largest tabulated | a GLYPH<1> b | no larger than | ˜ e ti GLYPH<1> ˜ e p t GLYPH<1> 1 q i | .

Discontinuities discontinuities do not represent issues if the argument z of I ti p z q is large enough, as shown from the following simple Lemma.

Lemma B.4. Define the discontinuity of F as:

For any z ¥ 0 , if disc p F q / z then I ti p z q GLYPH<24>H , @ t ¥ 1 , @ i P r m s .

Figure 4 (c) shows a case where the discontinuity is larger than z . In this case, an issue eventually happens for computing the next weight happens, only when the current edge is at the discontinuity. We note that as iterations increase and the weak learner finds it eventually more difficult to return weak hypotheses with η . large enough, the discontinuities may become an issue for SECBOOST to not stop at Step 2.5. Or one can always use a simple trick to avoid stopping and which relies on the leveraging coefficient α t : this is described in the Appendix, Section B.9.

The case of convex losses If F is convex (not necessarily differentiable nor strictly convex), there is a simple way to find a valid output for the offset oracle, which relies on the following Lemma.

Lemma B.5. Suppose F convex. Then for any z, z 1 P R , v GLYPH<24> 0 ,

(proof in Appendix, Section B.8) By definition, I ti p z 1 q GLYPH<132> I ti p z q for any z 1 / z , so a simple way to implement the offset oracle's output OO p t, i, z q is, for some 0 GLYPH<160> r GLYPH<160> z , to solve the Bregman identity in the RHS of (55) and then return any relevant v . If F is strictly convex, there is just one choice.

If solving the Bregman identity is tedious but F is strictly convex, there a simple dichotomic search that is guaranteed to find a feasible v . It exploits the fact that the abscissa maximizing the difference between any secant of F and F has a simple closed form (see [21, Supplement, Figure 13]) and so the OBI in (1) (Definition 4.6) has a closed form as well. In this case, it is enough, after taking a first non-zero guess for v (either positive or negative), to divide it by a constant ¡ 1 until the corresponding OBI is no larger than the z in the query OO p t, i, z q .

B.8 Proof of Lemma B.5

F being convex, we first want to compute the set

where r is supposed small enough for I z,v,r to be non-empty. There is a simple graphical solution to this which, as Figure 7 explains, consists in finding v solution of

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which yields the statement of the Lemma.

B.9 Handling discontinuities in the offset oracle to prevent stopping in Step 2.5

Theorem 5.3 and Lemma 5.6 require to run SECBOOST for as many iterations are required. This implies not early stopping in Step 2.5. Lemma B.4 shows that early stopping can only be triggered by too large local discontinuities at the edges. This is a weak requirement on running SECBOOST, but there exists a weak assumption on the discontinuities of the loss itself that simply prevent any early stopping and does not degrade the boosting rates. The result exploits the freedom in choosing α t in Step 2.3.

Lemma B.6. Suppose F is any function defined over R discontinuities of zero Lebesgue measure. Then Corollary 5.6 holds for boosting F with its inequality strict while never triggering early stopping in Step 2.5 of SECBOOST .

Proof. To show that we never trigger stopping in Step 2.5, it is sufficient to show that we can run SECBOOSTwhile ensuring F is continuous in an open neighborhood around all edges y i H t p x i q , @ i P r m s , @ t ¥ 0 (by letting H 0 . GLYPH<16> h 0 ). Remind that ˜ e ti . GLYPH<16> ˜ e p t GLYPH<1> 1 q i GLYPH<0> α t GLYPH<4> y t h t p x i q , so changing α t changes all edges. We just have to show that either computing α t ensures such a continuity, or α t can be slightly modified to do so. We have two ways to compute α t :

using a value for W 2 ,t that represents an "absolute" upperbound in the sense of (8) ( e.g. Lemma 5.7) and then compute α t as in Step 2.3 of SECBOOST;

using algorithm SOLVE α .

Because of the assumption on F , we can always ensure that F is continuous in an open neighborhood of all edges (the basis of the induction amounts to a straightforward choice for h 0 ). This proves the Lemma for [2.].

If we rely on [1.] and the α t computed leads to some discontinuities, then we have complete control to change α t : any continuous change of ε t induces a continuous change in α t and thus a continuous change of all edges as well. So, starting from the initial ε t chosen in Step 2.3, we increase it to a value ε GLYPH<6> t ¡ ε t , which we want to keep as small as possible. We can define for each i P r m s an open set p a i , b i q which is the interval spanned by the new ˜ e ti p ε 1 t q using ε 1 t P p ε t , ε GLYPH<6> t q . Since there are only finitely many discontinuities on F , there exists a small ε GLYPH<6> t ¡ ε t such that

This means that @ ε 1 t P p ε t , ε GLYPH<6> t q , we end up with a loss without any discontinuities on the new edges. Now comes the reason why we want ε GLYPH<6> t GLYPH<1> ε t small: we can check that there always exist a small enough ε GLYPH<6> t ¡ ε t such that for any ε 1 t we choose, the boosting rate in Corollary 5.6 is affected by at most 1 additional iteration. Indeed, while we slightly change parameter ε t to land all new edges outside of discontinuities of F , we also increase the contribution of the boosting iteration in the RHS of (15) by a quantity δ ¡ 0 which can be made as small as required - hence we can just replace the inequality in (15) by a strict inequality. This proves the statement of the Lemma if we rely on [1.] above.

This completes the proof of Lemma B.6.

B.10 A boosting pattern that can "survive" above differentiability

Suppose F is strictly convex and strictly decreasing as for classical convex surrogates ( e.g. logistic loss). Assuming wlog all α . ¡ 0 and example i has both y i h t p x i q ¡ 0 and y i h t GLYPH<1> 1 p x i q ¡ 0 , as long as z is small enough, we are guaranteed that any choice v t GLYPH<1> 1 P I p t GLYPH<1> 1 q i p z q and v t P I ti p z q results in 0 GLYPH<160> w p t GLYPH<0> 1 q i GLYPH<160> w ti , which follows the classical boosting pattern that examples receiving the right class by weak hypotheses have their weight decreased (See Figure 8). If z GLYPH<16> z 1 is large enough, then this does not hold anymore as seen from Figure 8.

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Proprietary + Confidential

B.11 The case of piecewise constant losses for SOLVE α

Initialization

Step 2.6, iteration

t

(A)

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Hence, our generic implementation of Algorithms 3 and 4 may completely fail at providing non-zero weights for the next iteration, which makes the algorithm stop in step 2.7. And even when not all weights are zero, there may be just a too small subset of those, that would break the Weak Learning Assumption for boosting compliance of the next iteration (Assumption 5.5).

C Supplementary material on algorithms, implementation tricks and a toy experiment

C.1 Algorithm and implementation of SOLVE α and how to find parameters from Theorem 5.8

As Theorem 5.8 explains, SOLVE α can easily get to not just the leveraging coefficient α t , but also other parameters that are necessary to implement SECBOOST: W 2 ,t and ε t (both used in Step 2.5). We now provide a simple pseudo code on how to implement SOLVE α amnd get, on top of it, the two other parameters. We do not seek π t since it is useful only in the convergence analysis. Also, our proposal implementation is optimized for complexity (because of the geometric updating of δ, W in their respective loops) but much less so for for accuracy. Algorithm SOLVE_extended explains the overall procedure.

Algorithm 3 SOLVE_extended( S, w , h, M )

Input sample S GLYPH<16> tp x i , y i q , i GLYPH<16> 1 , 2 , ..., m u , w P R m , h : X GLYPH<209> R , M GLYPH<24> 0 . ---

// in our case, w w t ; h h t ; M M t (current weights, weak hypothesis and max confidence, see Step 2.3 in SECBOOST and Assumption 5.1) Step 1 : // all initializations

// Step 2 computes the leveraging coefficient α t found_alpha

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There exists a very simple trick to get some adequate offset v to satisfy (11) (main file), explained in Figure 10. In short, we seek the optimally bended secant and check that the OBI is no more than a required z . This can be done via parsing the interval r ˜ e ti , ˜ e p t GLYPH<1> 1 q i s using regularly spaced values. If the OBI is too large, we can start again with a smaller step size. Algorithm OO_simple details the key part of the search.

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Input loss F , two last edges ˜ e , ˜ e , maximal OBI z , precision Z

Step 1 :

Step 2 : do t t GLYPH<1> 1 . // in our case, ˜ e t -˜ e ti ; ˜ e t GLYPH<1> 1 -˜ e p t GLYPH<1> 1 q i ; (for training example index i P r m s

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C.3 A toy experiments

We provide here a few toy experiments using SECBOOST. These are just meant to display that a simple implementation of the algorithm, following the blueprints given above, can indeed manage to optimize various losses. These are not meant to explain how to pick the best hyperparameters ( e.g. (60)) nor how to choose the best loss given a domain, a problem that is far beyond the scope of our paper.

In this implementation, the weak learner learns decision trees and we minimize Matushita's loss at the leaves of decision trees to learn fixed size trees, see [33] for the criterion and induction scheme, which is standard for decision trees. SECBOOST is implemented as is given in the paper, and so are the implementation of SOLVE α and the offset oracle provided above. We have made no optimization whatsoever, with one exception: when numerical approximation errors lead to an offset that is machine 0, we replace it by a small random value to prevent the use of derivatives in SECBOOST.

We have investigated three losses. The first is the well known logistic loss:

The other two are tweaks of the logistic loss. We have investigated a clipped version of the logistic loss,

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with z Q . GLYPH<16> Qz GLYPH<1> 1 { 2 ( r . s is the closest integer), which adds to the logistic loss regularly spaced peaks of variable width. This loss is non-convex, non-differentiable, non-Lipschitz. Figure 11 provides a crop of the clipped logistic loss and spring loss we have used in our test. Notice the 'hardness' that the spring loss intuitively represents for ML.

We provide an experiment on public domain UCI tictactoe [23] (using a 10-fold stratified crossvalidation to estimate test errors). In addition to the three losses, we have crossed them with several other variables: the size of the trees (either they have a single internal node = stumps or at most 20 nodes) and, to give one example of how changing a (key) hyperparameter can change the result, we have tested for a scale of changes on the initial value of δ in (60). Finally, we have crossed all these variables with the existence of symmetric label noise in the training data, following the setup of [37, 39]. We flip each label in the training sample with probability η . Table 12 summarizes the results obtained. One can see that SECBOOST manages to optimize all losses in pretty much all settings, with an eventual early stopping required for the spring loss if δ is too large. Note that the best initial value for δ depends on the loss optimized in these experiments: for δ GLYPH<16> 0 . 1 , test error from the spring loss decreases much faster than for the other losses, yet we remind that the spring loss is just the logistic loss plus regularly spaced peaks. This could signal interesting avenues for the best possible implementation of SECBOOST, or a further understanding of the best formal ways to fix those paramaters, all of which are out of the scope of this paper.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a framework for boosting any (bounded) loss function given access to a weak learner. The authors present how recent developments in boosting have involved turning the boosting problem into an optimization problem, where different combinations of assumptions on the loss such as convexity, lipschitcness, differential, etc. are assumed to be true to obtain bounds on the error. The authors then point to the origin of the boosting idea which was that one had access to a weak learner who given a weighted sample produced a hypothesis that was slightly better than guessing on the weighted sample and by using the access to query this weak learner produce a final classifier which performed arbitrarily well compared to just slightly better than guessing. I.e. there were no assumptions on the loss function itself. Thus the authors want to consider this more general setting where no extra information on the loss $F$ (other than a bound on the loss in the hypothesis returned by the weak learner on the sample is bounded). To construct boosting for such general $F$ they look to a zero'th order optimization (only access to losses) and find that they can't find any previous work on boosting using zero'th order optimization. The authors then present a new boosting algorithm SECBOOST that uses zero'th order optimization to boost any loss function. With the before mentioned boundedness assumption on $F$ they are able to show a in-sample error guarantee, which if SECBOOST is successful and run sufficiently long can be made arbitrarily close to the minimum of the loss function $\\inf_{x\\in R}F(x)$. Noted by the authors this guarantee holds when the number of iterations is large enough - they also comment on which problem can arrise in terms of SECBOOST stopping to early  - and how they can be alleviated.\n\nSTRENGTHS:\nOriginality:\n\nThe authors couldn't find previous work on zero'th order optimzation and boosting so it is original in that sense. The authors also points out how SECBOOST may differ from traditional boosting algorithms by for instance, possibly chaning signs, and using emperical quantaties relating v-first and second order derivatives.  \n\nQuality:\n\n The main doesn't contain any proof so I will not comment on the theoretical analysis soundness and is also my reason for my confidence score. The authors seems to be honest about the limitations of SECBOOST and clearly state the assumptions made in the paper.\n\nClarity: \nThe main is very well written and the authors give an intuitive description of SECBOOST and the different concepts that it used - here also using figures to illustrate the concepts. \n\nSignificance: \n As the problem of combinding boosting and zero'th order optimization is presented as new it seems like a interesting problem to look at. The framework is presented very generally which makes it applicable in many places. Also, this new perspective on using zero'th order optimization in boosting may also inspire new ideas for boosting algorithms in problem-specific settings.\n\nWEAKNESSES:\nAs also asked in the questions, to highlight the applicability of the framework it could be nice with and example of a loss $F$ where one could compare SECBOOST to the best-known algorithms for that specific loss in the boosting setting.\n\nQUESTIONS:\nWhat does $R_{\\star}$ and $N_{\\star}$ denote?\n\n Line 142 can you explain why the sign of any $v$ in $I_{ti}(z)$ is always the sign of $-y_{i}\\alpha_{t}h_{t}(x_i)$? \n\n Figure 2, is the dotted lines $w_{t,1},\\ldots,w_{t,m}$\n \n Corollary 5.4, i guess could also be stated with just the weak learning assumption? so with 5.1 and 5.4?\n \n Why does figure 3 and 4 come before figure 2? And why are the figures not on the pages where they are used to discripe the setup?\n \n Line 176 $\\nu_{t}$ depend on $h_{t }$ thorough $M_t$ is this a problem, when we assume that the weak learner find the hypothesis $h_t$ such that $|\\nu_{t}|\\geq \\gamma$. Why isn't it the case that the weak learner output the hypothesis $h_t$ such that $\\nu_{t}\\geq \\gamma$(no absolute value)? \n \n What if the weak learning gaurantee was given in terms of the the loss funciton  $F$?\n \n Can you come up with a setting of $F$ where the algorithm gives a bound on the number of iterations need and compare it to the best known bound for that setting?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 1, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-06-18"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper investigates the theoretical aspect of boosting algorithms in machine learning. The authors propose a new algorithm, SECBOOST, which aims to optimize any loss function using zeroth-order information. This approach dis different from traditional boosting methods that require first-order information such as gradients. By leveraging tools from quantum calculus, the paper claims to extend the applicability of boosting to loss functions that are not necessarily convex, differentiable, or even continuous. The core contribution is the demonstration that boosting can be effectively performed without relying on derivative information.\n\n--\n\nI have read the rebuttal and other reviews. Rating unchanged.\n\nSTRENGTHS:\nI like this paper which is somewhat different from the majority of boosting related work. Though the outbreak of deep learning methods generally comes with derivative info, a vast variety of problems don't. This paper provides theoretical contributions of designing boosting algorithm for any loss function whose set of discontinuities has zero Lebesgue measure which is a pretty general setting. It feels like a piece missing for boosting algorithm literatures and it's nice to have that eventually.\n\nWEAKNESSES:\nI would connect more to real-world applications. This doesn't necessarily mean to run experiments, but at least providing examples about such kind of loss functions and their importance in real world would be helpful. \n\nIn addition, sometimes the loss function is differentiable yet getting the derivative could be expensive. Some discussion around performance vs. cost will be very helpful. This part is optional but would make the paper much stronger. \n\nSome of the material in appendix III are actually insightful and helpful. Suggest to move some to the main body.\n\nQUESTIONS:\nPlease see the strength/weakness section.\n\nLIMITATIONS:\nn/a", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n-\tThis paper discusses an alternative boost algorithm by using the zeroth-order optimization technique. The key benefit by using such technique is that it does not require the loss function to be convex, differentiable or Lipschitz. They provide theoretical results and validate them with experiments.\n\nSTRENGTHS:\n-\tThe contributions from this paper in terms of the generalizability are clear and interesting.\n-\tIf the authors’ claims are proper and assumptions are not hard to satisfy, their contributions are definitely very important in extending the loss function class to be as general as we know in the loss properties.\n\nWEAKNESSES:\n-\tI do not understand some assumptions well, which makes me a bit confused on the strength of the algorithms and subsequent theoretical contributions. And I think more descriptions related to the condition should be provided:\n\nFirst, Assumption 5.4, is this easy to verify for some given learners? Since it is quite convoluted and from my perspective, it seems more like an intermediate result which requires some efforts to analyze; Furthermore, will some structure properties of the loss function depend on $\\rho$. If the loss is super bad, would not the associated Assumption 5.4 fail? How restricted is such Assumption 5.4?\n\nThen, Assumption 5.5, this is not the standard weak learning assumption in literature, since the standard weak learning condition is the unweighted version with taking expectation for the predictor and labels. And I am not sure why the condition is imposed for all $t$ and requires conditions for the subsequent $h_t$ beyond the initial weak learner. I am not sure whether it is the traditional weak learning assumption in boosting literature. If so, more reference and / or  related discussions should be provided. \n\n-\tThere are some inconsistent notations and definitions that appear here and there in the paper, which causes readers confusion, e.g., 1-order and first-order, zero-order and zeroth-order. And in Page 2, both $F(S, h)$ and $F(\\cdot)$ appear as well as its use subscript or superscript alternatively for different losses, which can be improved. Therefore, it is very confusing in Theorem 5.3 and Corollary 5.6 when both of them appear at the same time. Besides, there are some typos and grammar mistakes around the paper and the paper needs some rewritten to improve the writing.\n\nQUESTIONS:\n-\tWhat are examples of the strict benefits of the boosting for any loss function beyond those requiring gradient information? That is, the true computational / generalizability benefit of SECBOOST over existing first-order boosting methods with function approximations.  I am not super familiar with all first-order boosting methods, however, I guess for those non-smooth loss objectives, we can smooth the original non-smooth loss objectives and do boosting for these surrogate loss. Please correct me if this is not case or quite nontrivial for some general losses.\n\nLIMITATIONS:\nThe authors mention some limitations related to their Assumption 5.4 and 5.5. However, I still feel like more discussions related to the scope of these assumptions in terms of the loss function (like the weakness and question I pointed out) to increase the accessibility of the paper and help readers understand better.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-19"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nBoosting can be regarded as a general optimization problem, and most of the currently popular Boosting techniques tend to do so, and do it by using $1^\\text{st}$ order (gradient) information to minimise a loss function.\nThis work proposes an algorithm to minimise an arbitrary loss function using only $0^\\text{th}$ order information.\nThe authors prove that this method converges for essentially any loss function, which is unprecedented in the field.\n\nSTRENGTHS:\n1. The writing style is quite acceptable. I could understand the local meaning of pretty much every sentence.\n2. The result covers a **very** wide class of functions. I agree with the authors that putting it as \"essentially all functions\" is a fair claim.\n3. The use of quantum calculus is interesting and adds flavour to the work.\n4. The authors are upfront about disregarding generalisation aspects.\n\nWEAKNESSES:\n### **Note on the choice of primary area:** This is much more of an *optimisation* than a *learning theory* paper\nThe choice of primary area for this paper (learning theory) was poor.\nThis work is far better placed within the area of optimisation than in learning theory.\nThis is quite clear throughout the technical parts of the paper, and the authors even write \"We do not investigate the questions of generalisation, which would entail specific design choices about the loss at hand\" (within the paragraph after line 74), so I expect the authors to agree with this assessment.\nGiven that, I honestly fail to see how the authors judged their choice of primary area to be the one that would lead their work toward the most appropriate reviewers.\n\nThis issue could have been significantly mitigated by an especially strong presentation of the work.\nInstead, I found the presentation to be, at the very best, fair (I elaborate on this below).\n\nOverall, this work turned out to be laborious to review.\nIt is not that the technical side is particularly complex, but there are many moving parts and nested concepts, making it laborious to keep track of everything without a clear roadmap.\nEspecially when coming from a learning theory background (which, again, seems to be the authors' target audience), overcoming the weak presentation requires an amount of effort that I am not sure is reasonable for the authors to demand from the reader.\n\nUltimately, this matter ended up lowering my confidence score since, for example, it is definitely \"possible I am unfamiliar with some pieces of related work\".\nStill, within the resources I had at my disposal, I did my best to make my review useful to the authors and the community as a whole.\n\n\n### A technical (reformulated) summary of the work's contribution\n\nTo help clarify my understanding of the work, I will start with an abstract of the paper in more technical terms than what I offered in the \"summary\" field above.\nThe goal is to make it easier for the authors to point out and correct potential misunderstandings.\n(It is based on the definitions from Section 3 starting at line 74)\n\nAs mentioned in the note above, the authors completely disregard generalisation.\nThus, only the training set $S = \\{(\\mathbf{x}_1, y_1), \\ldots, (\\mathbf{x}_m, y_m)\\}$ is relevant to the work.\nIn particular, given a hypothesis $h\\colon \\mathcal{X} \\to \\mathbb{R}$ (not the binary $\\{-1, 1\\}$-range as in the classical setting), we may simply consider\n$$f_i \\coloneqq h(\\mathbf{x}_i),$$\ndefining a vector $\\mathbf{f} = (f_1, \\ldots, f_m) \\in \\mathbb{R}^m$,\nsince, again, **the work is oblivious to any evaluation of hypotheses outside of the $m$ points in the training set**.\n\nLet $\\mathcal{G}$ be the hypothesis class of the weak learner, i.e., the set of all hypotheses that the weak learner can output.\n> NOTE 1: Not to be confused with the set $\\mathcal{H}$ defined by the authors to be that of all classifiers attainable via linear combinations of hypotheses in $\\mathcal{G}$.\nI am highlighting this as it is a bit unusual in the context of classical boosting, where one typically argues in terms of the class of weak hypotheses $\\mathcal{G}$ rather than the class of ensemble classifiers $\\mathcal{H}$.\n\nFor simplicity, let us assume that $\\mathcal{G}$ is finite:\nLet $n = \\lvert \\mathcal{G} \\rvert$ and, employing the notation above, consider the enumeration $\\mathcal{G} = \\{\\mathbf{f}^{(1)}, \\ldots, \\mathbf{f}^{(n)}\\}$.\n\nFinally, consider a generic loss function $F\\colon \\mathbb{R} \\to \\mathbb{R}$.\n\nThe problem attacked by the paper then becomes\n\n**Target Problem (reformulated):**\nGiven **fixed vectors** $\\mathbf{f}^{(1)}, \\ldots, \\mathbf{f}^{(n)} \\in \\mathbb{R}^m$\nand associated labels $y_1, \\ldots, y_m \\in \\{-1, 1\\}$,\nand a loss function $F\\colon \\mathbb{R} \\to \\mathbb{R}$,\nfind coefficients $\\alpha_1, \\ldots, \\alpha_n \\in \\mathbb{R}$ to minimise\n$$\\sum_{i=1}^m F\\left(y_i \\sum_{j=1}^n \\alpha_j f^{(j)}_i\\right).$$\n\n> NOTE 2: In my first question below, I've explicitly asked the authors to confirm that this formulation is an accurate summary of the problem.\n\nNaturally, the difficulty of the problem depends on the properties of $F$.\nThe central claim of the paper is to provide an algorithm to effectively find such a set of coefficients while requiring very little from $F$:\nThat its points of discontinuity form a set of measure zero.\nMoreover, the authors provide guarantees on the sparsity of the solution as at most $T$ coefficients are non-zero, where $T$ is the number of boosting steps performed by the algorithm.\n\n> Notice that I omitted the $\\gamma$-weak learner from the reformulation above because I had some issues with the authors' definition.\nI dedicated a subsection to this matter below.\n\n### Weaknesses\n\n> My goal is to be objective and direct, but I acknowledge this can give the text a bitter-like tone. I apologise if I sound too harsh in the review.\n\n\nThe most pervasive weakness of the work is that, while the writing style itself is acceptable, the overall presentation is poor when considering more global aspects.\nAll the following points stem from this issue to some extent.\n\n1. The authors are not sufficiently clear and explicit in enumerating the claimed contributions.\n    The closest to that would be the paragraph starting at line 38, however, one cannot conclude that the claimed contribution is restricted to the contents of that paragraph.\n\nAfter reading the paper, I suspect that the claimed contribution boils down to (put roughly)\n> We provide the first study on the convergence of $0^\\text{th}$ order optimisation methods for boosting under a general loss function.\nMoreover, we ensure convergence for the widest class of loss functions seen so far in the field of $0^\\text{th}$ order optimisation in the ML setting.\nNamely, we cover all functions whose points of discontinuity form a set of measure zero.\n\nIn particular, my understanding is that\n- The authors do not claim any new techniques used to obtain their results.\n    In fact, very little emphasis is given to the proofs, with all of them being left to the appendices and no sketches being provided.\n- As the text does not provide explicit insights into the difficulty of the problem they are solving, the authors do not claim the problem is particularly challenging under the light of the usual frameworks used for this kind of problem.\n\nThis is a fair setting for a paper and I would lean towards acceptance provided the authors do a good job in\nestablishing the relevance of filling the gap in the literature that they claim to have filled.\nTo make this point clearer, an informal and exaggerated version of it would be \"Totally convincing the reader that others weren't aware of the gap and ignored it out of disinterest\" (I am not suggesting this was the case).\nAlternatively, but preferably both, the authors could solidly establish the significance of the uniquely wide class of loss functions they cover.\nI believe that the authors failed to do either, and this is the main reason why I recommend rejection.\n\nWhile the following points are not minor, the authors can regard them as less critical than what I have discussed above.\n\n2. Theorem 5.3 fails to provide a reasonably self-contained statement that leads to something close to what is claimed in the paragraph starting at line 38.\n    Instead, the authors' phrasing of the statement resembles more that of a lemma, in that it requires multiple logical steps and the reference to other results to properly grasp its significance.\n    I can see a good formal rephrasing of the statement can be hard to achieve, so the authors could consider adding a less formal version of result to the introduction.\n    That would also help greatly with the former point.\n\n3. The authors do not provide an exact statement for the optimisation problem at hand (something in the direction of the *Target problem* above).\n    It is important to know early and with absolute certainty what the authors are attacking.\n\n4. **From the perspective of learning theory**, some of the motivation provided for $0^\\text{th}$ order methods seems misplaced:\n    1. On the theoretical side, $0^\\text{th}$ order optimal weak to strong learners are known to exist: see [1].\n    2. On a more practical side, in general, the performance of boosting methods does not really come from the minimisation of the associated loss function.\n        See, for example, the discussion in [2, Section 7.3: \"Loss Minimization Cannot Explain Generalization\"].\n    3. I recognise that the authors explicitly dismiss matters of generalisation, but I still believe the remarks above are relevant in assessing the motivation for the work.\n    The authors may consider bringing up these points in some form, perhaps by mentioning how the related works compare to their results in this regard.\n\n5. I am confused by the authors' concept of what constitutes \"traditional\" boosting.\n    To me, AdaBoost is the most prototypical and traditional boosting algorithm.\n    However, for example, in line 131 it is clear that the authors believe that \"traditional\" boosting requires first-order information, which is not the case for AdaBoost.\n\n6. The discussion around Assumption 5.4 is too loose to fully justify an assumption that seems to be quite relevant (see the role of $\\rho_*$ in Corollary 5.6).\n    I understand the point made just above the assumption, but I believe that would only suffice to fully justify some special treatment of arbitrary small $\\rho_t\\text{s}$.\n    I expected a formal or at least deeper discussion of the significance of the assumption.\n\n\n### References\n\n[1]: Larsen, K. G. (2023, July). Bagging is an optimal PAC learner. In The Thirty Sixth Annual Conference on Learning Theory (pp. 450-468). PMLR.\\\n[2]: Schapire, R.E. and Freund, Y., 2013. Boosting: Foundations and algorithms. Kybernetes, 42(1), pp.164-166.\n\n\n### Minor issues and suggestions\n\n#### Issues with the definition of $\\gamma$-weak learner\n\n1. The authors, unfortunately, do not define it in Section 3, \"Definitions and notations\", delaying it to page 7 (Assumption 5.5 at line 175).\n2. The definition is not self-contained, as it depends on $\\tilde{\\eta}_t$ which itself depends on other parameters (see Eq. 18).\n3. Honestly, I do not recognise that definition as \"the traditional\" one (see line 174).\n    To me, that would be a $\\gamma$-weak learner that, for any distribution over the training set, outputs a (binary) hypothesis whose average error is at most $1/2 - \\gamma$.\n    Of course, the paper discusses non-binary hypotheses so that a definition in terms of edges is natural.\n    Still, I do not think the equivalence/analogy is immediate enough, and, regardless, the authors should provide a self-contained and formal definition of $\\gamma$-weak learner, ideally in Section 3.\n    Step 2.1 of Algorithm 1 does not make it totally clear what are the inputs and outputs of the $\\gamma$-weak learner.\n4. Assumption 5.5 might be less \"global\" than one could expect.\n    It depends on the definition of $\\tilde{\\eta}\\_t$, which depends on $M\\_t \\coloneqq \\max\\_i \\lvert h\\_t(\\mathbf{x}\\_i) \\rvert$, making Assumption 5.5 have a different strength at each round if one parses Eq. 18 as\n    $$\\tilde{\\eta}\\_t = \\frac{1}{M\\_t} \\sum_{i=1}^m \\frac{\\lvert w\\_{ti} \\rvert}{\\sum\\_i \\lvert w\\_{ti} \\rvert} \\cdot \\tilde{y}\\_{ti} h\\_t(\\mathbf{x}\\_i) = \\frac{1}{M\\_t} \\cdot \\text{``edge of $h_t$ relative to a normalised version of the weights $\\mathbf{w}\\_t$\"}.$$\n\n> NOTE 3: Despite the difficulties, I believe to have eventually understood what the authors meant by $\\gamma$-weak learner. At Step 2.1,\n- the notation $\\lvert \\mathbf{w}\\_t \\rvert$ refers to the vector $(\\lvert w\\_{t1} \\rvert, \\ldots, \\lvert w\\_{tm} \\rvert)$ (I do not recall whether this was defined);\n- the notation $⨉$ denotes the weak learner (why not something clear like $\\mathrm{WeakLeaner}$?);\n- the definition of $\\mathbf{S}\\_t$ in the comment (why there?) sets the pseudo-labels;\n- the weak learner $⨉$ operates on the training points $\\mathbf{x}\\_i$ with the pseudo-labels $\\tilde{y}\\_{ti}$ and weights $w\\_{ti}/\\lVert \\mathbf{w}\\_t \\rVert\\_1$;\n- $⨉$ provides a hypothesis $h_t$ with an edge of at least $\\gamma$, where the concept of edge needs a normalisation to contemplate hypotheses with unbounded range; (The authors chose a normalisation factor that is a function of $h_t$ while in points 3 and 4 above I remark that a global factor could be more natural).\n\n> My point is not that it is impossible to retrieve what the authors had in mind, but that they made it significantly harder to do so than what is necessary.\nAlso, one needs explicit formal definitions to fully appreciate contributions.\n\n\n#### General minor issues and suggestions\n\n- In the technical summary above, one could cover infinite $\\mathcal{G}$ by considering a suitable measure and employing a (Lebesgue) integral in the objective function.\n    I haven't thought much about this, but it's too much of a coincidence that the weak regularity assumption on $F$ that the authors found resembles the conditions for integrability so closely.\n\n- The other theorem statements also suffer from the \"lemma-like\" issue I mentioned above\n\n- Section 4 is somewhat representative of the general issue with the presentation.\n    Ideally, a section like this should go along the lines of\n    - What is the goal;\n    - What are the original concepts;\n    - Why they do not suit the current setting;\n    - How you modified them;\n    - How the change fixes the issue.\n\n    Instead, the definitions introduced are left insufficiently motivated and, indeed, it is not even fully clear how novel the concepts introduced are.\n\n- I suspect you require less from the hypotheses returned by the weak learner than it may appear at line 113 and the subsequent Assumption 5.1.\n    As the next note highlights, you don't really need it to be non-zero.\n    Finally, since the training set is finite, to have that $\\forall i \\in [m],\\, \\lvert h(\\mathbf{x}_i) \\rvert < \\infty$ we only need $h$ to be a real-valued function defined in $\\{\\mathbf{x}_1, \\ldots, \\mathbf{x}_m, \\ldots\\}\\subseteq \\mathcal{X}$.\n    Long story short, I think you only need to consider a weak learner returning functions $h\\colon \\mathcal{X} \\to \\mathbb{R}$, which is a very mild assumption.\n    (The central point here is that real-valued functions are always bounded on finite sets. Unless one is working with the extended reals, something like $f(x) = 1/\\lvert x \\rvert$ is not defined at 0.)\n\n- Consider stating more explicitly that the assumption of non-zero predictions (line 113) sacrifices no generality (I was only sure this was the case after checking Section II.2 from the Appendix)\n\n- [Eq. (4)] Consider using $(F(a)F(b))_\\alpha$\n\n- Number only the equations that are referenced in the text\n\n- Consider a version of Figure 1 with an extra dimension (a surface plot) as, after all, the Bregman Secant distortion behaves as taking 2 arguments\n\n- Avoid starting sentences with mathematical symbols (e.g., line 93)\n\n- [122] the reference to [47, Appendix, Section 4] has a bit too much packed in it. The authors should consider adding some further guidance or explanation. That should be feasible since the reference tags almost all equations in that section.\n\n- Adding hyperlinks to the steps of the algorithm (e.g., Step 2.1) would make it meaningfully more convenient to navigate the text.\n\n- Algorithm 1 can be made significantly tidier. Doing so would be a considerable improvement in the presentation.\n\nQUESTIONS:\n1. Up to the simplifying assumption of a finite weak hypothesis class, do you agree with the reformulation of the problem I provided above (see NOTE 2)? Naturally, conveying all nuances would require more space. Still, the goal was to provide a fair summary.\n2. Why do you believe \"optimisation\" is not a more appropriate primary area for this work than \"learning theory\"?\n3. What happens to Definition 4.1 when considering $v = 0$?\n4. Sorry if I simply missed this one, but what is your actual claim about Assumption 5.4? Is it that you expect it to always hold in practice, that there is an easy way to overcome it, or something else?\n\nLIMITATIONS:\nI am not sure this really applies to the work, but adding something like what I suggested in weakness 4.3 should make some relevant limitations of the work significantly clearer to many readers.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 1, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-25"}