SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Chung-En Tsai
Department of Computer Science and Information Engineering
CHUNGENTSAI@NTU.EDU.TW
Ying-Ting Lin
R08922060@NTU.EDU.TW
Yen-Huan Li
Center for Quantum Science and Engineering
YENHUAN.LI@CSIE.NTU.EDU.TW
This work introduces the first small-loss and gradual-variation regret bounds for online portfolio selection, marking the first instances of data-dependent bounds for online convex optimization with non-Lipschitz, non-smooth losses. The algorithms we propose exhibit sublinear regret rates in the worst cases and achieve logarithmic regrets when the data is "easy," with per-round time almost linear in the number of investment alternatives. The regret bounds are derived using novel smoothness characterizations of the logarithmic loss, a local norm-based analysis of following the regularized leader (FTRL) with self-concordant regularizers, which are not necessarily barriers, and an implicit variant of optimistic FTRL with the log-barrier.
Designing an optimal algorithm for online portfolio selection (OPS), with respect to both regret and computational efficiency, has remained a significant open problem in online convex optimization for over three decades<sup>1</sup>. OPS models long-term investment as a multi-round game between two strategic players—the market and the INVESTOR—thereby avoiding the need for hard-to-verify probabilistic models for the market. In addition to its implications for robust long-term investment, OPS is also a generalization of probability forecasting and universal data compression [5].
The primary challenge in OPS stems from the absence of Lipschitzness and smoothness in the loss functions. Consequently, standard online convex optimization algorithms do not directly apply. For example, standard analyses of online mirror descent (OMD) and following the regularized leader (FTRL) bound the regret by the sum of the norms of the gradients (see, e.g., the lecture notes by
<sup>1</sup>Readers are referred to recent papers, such as those by Luo et al. [22], van Erven et al. [35], Mhammedi and Rakhlin [24], Zimmert et al. [38], and Jézquel et al. [17], for reviews on OPS.
Orabona [28] and Hazan [12]). In OPS, the loss functions are not Lipschitz, so these analyses do not yield a sub-linear regret rate. Without Lipschitzness, the self-bounding property of smooth functions enables the derivation of a "small-loss" regret bound for smooth loss functions [33]. However, the loss functions in OPS are not smooth either.
The optimal regret rate of OPS is known to be  $O(d\log T)$ , where  $d$  and  $T$  denote the number of investment alternatives and number of rounds, respectively. This optimal rate is achieved by Universal Portfolio [9, 10]. Nevertheless, the current best implementation of Universal Portfolio requires  $O(d^{4}T^{14})$  per-round time [20], too long for the algorithm to be practical. Subsequent OPS algorithms can be classified into two categories.
- Algorithms in the first category exhibit near-optimal  $\tilde{O}(d)$  per-round time and moderate  $\tilde{O}(\sqrt{dT})$  regret rates. This category includes the barrier subgradient method [26], Soft-Bayes [30], and LB-OMD [34].
- Algorithms in the second category exhibit much faster  $\tilde{O}(\mathrm{poly}(d)\mathrm{polylog}(T))$  regret rates with much longer  $\tilde{O}(\mathrm{poly}(d)\mathrm{poly}(T))$  per-round time, which is, however, significantly shorter than the per-round time of Universal Portfolio. This category includes ADA-BARRONS [22], PAE+DONS [24], BISONS [38], and VB-FTRL [17].
The aforementioned results are worst-case and do not reflect how "easy" the data is. For instance, if the price relatives of the investment alternatives remain constant over rounds, then a small regret is expected. Data-dependent regret bounds refer to regret bounds that maintain acceptable rates in the worst case and much better rates when the data is easy. In this work, we consider three types of data-dependent bounds.
- A small-loss bound bounds the regret by the cumulative loss of the best action in hindsight.
- A gradual-variation bound bounds the regret by the gradual variation (6) of certain characteristics of the loss functions, such as the gradients and price relatives.
- A second-order bound bounds the regret by the variance or other second-order statistics of certain characteristics of the loss functions, such as the gradients and price relatives.
Few studies have explored data-dependent bounds for OPS. These studies rely on the so-called no-junk-bonds assumption [2], requiring that the price relatives of all investment alternatives are bounded from below by a positive constant across all rounds. Given this assumption, it is easily verified that the losses in OPS become Lipschitz and smooth. Consequently, the result of Orabona et al. [29] implies a small-loss regret bound; Chiang et al. [8] established a gradual-variation bound in the price relatives; and Hazan and Kale [14] proved a second-order bound also in the price relatives. These bounds are logarithmic in the number of rounds  $T$  in the worst cases and can be constant when the data is easy.
The no-junk-bonds assumption may not always hold. Hazan and Kale [14] raised the question of whether it is possible to eliminate this assumption. In this work, we take the initial step towards addressing the question. Specifically, we prove Theorem 1.1.
Theorem 1.1. In the absence of the no-junk-bonds assumption, two algorithms exist that possess a gradual-variation bound and a small-loss bound, respectively. Both algorithms attain  $O(d\log (T))$  regret rates in the best cases and  $\tilde{O} (\sqrt{dT})$  regret in the worst cases, with  $\tilde{O}(d)$  per-round time.
Theorem 1.1 represents the first data-dependent bounds for OPS that do not require the no-junk-bonds assumption. To the best of our knowledge, this also marks the first data-dependent bounds for online convex optimization with non-Lipschitz non-smooth losses. In the worst cases, Theorem 1.1 ensures that both algorithms can compete with the OPS algorithms of the first category mentioned above. In the best cases, both algorithms achieve a near-optimal regret with a near-optimal per-round time, surpassing the OPS algorithms of the second category in terms of the computational efficiency. Table 1 in Appendix A presents a detailed summary of existing OPS algorithms in terms of the worst-case regrets, best-case regrets, and per-round time.
We also derived a second-order regret bound by aggregating a variant of optimistic FTRL with different learning rates. The interpretation of the result is not immediately clear. Therefore, we detail the result in Appendix I.
Technical Contributions. The proof of Theorem 1.1 relies on several key technical breakthroughs:
- Theorem 3.2 provides a general regret bound for optimistic FTRL with self-concordant regularizers, which may not be barriers, and time-varying learning rates. The bound generalizes those of Rakhlin and Sridharan [31] and Zimmert et al. [38, Appendix H] and is of independent interest.
- Existing results on small-loss and gradual-variation bounds assume the loss functions are smooth, an assumption that does not hold in OPS. In Section 4, we present Lemma 4.3 and Lemma 4.7, which serve as local-norm counterparts to the Lipschitz gradient condition and the self-bounding property of convex smooth functions, respectively.
- To apply Theorem 3.2, the gradient estimates and iterates in optimistic FTRL must be computed concurrently. Consequently, we introduce Algorithm 2, a variant of optimistic FTRL with the log-barrier and validate its definition and time complexity.
- The gradual-variation and small-loss bounds in Theorem 1.1 are achieved by two novel algorithms, Algorithm 3 and Algorithm 4, respectively. Both are instances of Algorithm 2.
Notations. For any natural number  $N$ , we denote the set  $\{1, \ldots, N\}$  by  $[N]$ . The sets of non-negative and strictly positive numbers are denoted by  $\mathbb{R}_+$  and  $\mathbb{R}_{++}$ , respectively. The  $i$ -th entry of a vector  $v \in \mathbb{R}^d$  is denoted by  $v(i)$ . The probability simplex in  $\mathbb{R}^d$ , the set of entry-wise non-negative vectors of unit  $\ell_1$ -norm, is denoted by  $\Delta_d$ . We often omit the subscript for convenience. The closure and relative interior of a set  $\mathcal{X}$  is denoted by  $\operatorname{cl} \mathcal{X}$  and  $\operatorname{ri} \mathcal{X}$ , respectively. The  $\ell_p$ -norm is denoted by  $\| \cdot \|_p$ . The all-ones vector is denoted by  $e$ . For any two vectors  $u$  and  $v$  in  $\mathbb{R}^d$ , their entry-wise product and division are denoted by  $u \odot v$  and  $u \oslash v$ , respectively. For time-indexed vectors  $a_1, \ldots, a_t \in \mathbb{R}^d$ , we denote the sum  $a_1 + \dots + a_t$  by  $a_{1:t}$ .
2.1 Log-Barrier for Online Portfolio Selection
All algorithms we propose are instances of optimistic FTRL with the log-barrier regularizer. The first use of the log-barrier in OPS can be traced back to the barrier subgradient method proposed by Nesterov [26]. Later, Luo et al. [22] employed a hybrid regularizer, which incorporated the log-barrier, in the development of ADA-BARRONS. This marked the first OPS algorithm with a regret rate polylogarithmic in  $T$  and an acceptable per-round time complexity of  $O(d^{2.5}T)$ . Van Erven et al. [35] conjectured that FTRL with the log-barrier (LB-FTRL) achieves the optimal regret. The conjecture was recently refuted by Zimmert et al. [38], who established a regret lower bound for LB-FTRL. Jézquel et al. [17] combined the log-barrier and volumetric barrier to develop VB-FTRL, the first algorithm with near-optimal regret and an acceptable per-round time complexity of  $O(d^2 T)$ . These regret bounds are worst-case and do not directly imply our results.
2.2 FTRL with Self-Concordant Regularizer
Abernethy et al. [1] showed that when the regularizer is chosen as a self-concordant barrier of the constraint set, the regret of FTRL is bounded by the sum of dual local norms of the gradients. Rakhlin and Sridharan [31] generalized this result for optimistic FTRL.
The requirement for the regularizer to be a barrier is restrictive. For instance, while the log-barrier is self-concordant, it is not a barrier of the probability simplex. To address this issue, van Erven et al. [35], Mhammedi and Rakhlin [24], and Jezquel et al. [17] introduced an affine transformation such that, after the transformation, the log-barrier becomes a self-concordant barrier of the constraint set. Nonetheless, this reparametrization complicates the proofs.
Theorem 3.2 in this paper shows that optimistic FTRL with a self-concordant regularizer, without the barrier requirement, still satisfies a regret bound similar to that by Rakhlin and Sridharan [31]. The proof of Theorem 3.2 aligns with the analyses by Mohri and Yang [25], McMahan [23], and Joulani et al. [19] of FTRL with optimism and adaptivity, as well as the local-norm based analysis by Zimmert et al. [38, Appendix H].
In comparison, Theorem 3.2 generalizes the analysis of Zimmert et al. [38, Appendix H] for optimistic algorithms and time-varying learning rates; Theorem 3.2 differs from the analyses of Mohri and Yang
[25], McMahan [23], and Joulani et al. [19] in that they require the regularizer to be strongly convex, whereas the log-barrier is not.
2.3 Data-Dependent Bounds
The following is a summary of relevant literature on the three types of data-dependent bounds. For a more comprehensive review, readers may refer to, e.g., the lecture notes of Orabona [28].
- Small-loss bounds, also known as  $L^{\star}$  bounds, were first derived by Cesa-Bianchi et al. [6] for online gradient descent for quadratic losses. Exploiting the self-bounding property, Srebro et al. [33] proved a small-loss bound for convex smooth losses. Orabona et al. [29] proved a logarithmic small-loss bound when the loss functions are not only smooth but also Lipschitz and exp-concave.
- Chiang et al. [8] derived the first gradual-variation bound, bounding the regret by the variation of the gradients over the rounds. Rakhlin and Sridharan [31, 32] interpreted the algorithm proposed by Chiang et al. [8] as optimistic online mirror descent and also proposed optimistic FTRL with self-concordant barrier regularizers. Joulani et al. [18] established a gradual-variation bound for optimistic FTRL.
- Cesa-Bianchi et al. [7] initiated the study of second-order regret bounds. Hazan and Kale [13] derived a regret bound characterized by the empirical variance of loss vectors for online linear optimization. In the presence of the no-junk-bonds assumption, Hazan and Kale [14] proved a regret bound for OPS characterized by the empirical variance of price relatives.
Except for those for specific loss functions, these data-dependent bounds assume either smoothness or Lipschitzness of the loss functions. Nevertheless, both assumptions are violated in OPS.
Recently, Hu et al. [16] established small-loss and gradual-variation bounds in the context of Riemannian online convex optimization. We are unaware of any Riemannian structure on the probability simplex that would render the loss functions in OPS geodesically convex and geodesically smooth. For instance, Appendix B shows that the loss functions in OPS are not geodesically convex on the Hessian manifold induced by the log-barrier.
3 Analysis of Optimistic FTRL with Self-Concordant Regularizers
This section presents Theorem 3.2, a general regret bound for optimistic FTRL with regularizers that are self-concordant but not necessarily barriers. This regret bound forms the basis for the analyses in the remainder of the paper and, as detailed in Section 2.2, generalizes the results of Rakhlin and Sridharan [31] and Zimmert et al. [38, Appendix H].
Consider the following online linear optimization problem involving two players, LEARNER and REALITY. Let  $\mathcal{X} \subseteq \mathbb{R}^d$  be a closed convex set. At the  $t$ -th round,
- first, LEARNER announces  $x_{t} \in \mathcal{X}$
- then, REALITY announces a vector  $v_{t} \in \mathbb{R}^{d}$
- finally, LEARNER suffers a loss given by  $\langle v_t, x_t \rangle$ .
For any given time horizon  $T \in \mathbb{N}$ , the regret  $R_{T}(x)$  is defined as the difference between the cumulative loss of LEARNER and that yielded by the action  $x \in \mathcal{X}$ ; that is,
$$
R _ {T} (x) := \sum_ {t = 1} ^ {T} \left\langle v _ {t}, x _ {t} \right\rangle - \sum_ {t = 1} ^ {T} \left\langle v _ {t}, x \right\rangle , \quad \forall x \in \mathcal {X}.
$$
The objective of LEARNER is to achieve a small regret against all  $x \in \mathcal{X}$ . Algorithm 1 provides a strategy for LEARNER, called optimistic FTRL.
We focus on the case where the regularizer  $\varphi$  is a self-concordant function.
Definition 3.1 (Self-concordant functions). A closed convex function  $\varphi : \mathbb{R}^d \to (-\infty, \infty]$  with an open domain  $\operatorname{dom} \varphi$  is said to be  $M$ -self-concordant if it is three-times continuously differentiable on  $\operatorname{dom} \varphi$  and
$$
\left| D ^ {3} \varphi (x) [ u, u, u ] \right| \leq 2 M \left\langle u, \nabla^ {2} \varphi (x) u \right\rangle^ {3 / 2}, \quad \forall x \in \operatorname {d o m} \varphi , u \in \mathbb {R} ^ {d}.
$$
Algorithm 1 Optimistic FTRL for Online Linear Optimization
Input: A sequence of learning rates  $\{\eta_t\} \subset \mathbb{R}_{++}$
1:  $\hat{v}_1\gets 0$
2:  $x_{1}\in \arg \min_{x\in \mathcal{X}}\eta_{0}^{-1}\varphi (x)$
3: for all  $t \in \mathbb{N}$  do
4: Announce  $x_{t}$  and receive  $v_{t} \in \mathbb{R}^{d}$ .
5: Choose an estimate  $\hat{v}_{t + 1}$  for  $v_{t + 1}$ .
6:  $x_{t + 1}\gets \arg \min_{x\in \mathcal{X}}\langle v_{1:t},x\rangle +\langle \hat{v}_{t + 1},x\rangle +\eta_t^{-1}\varphi (x).$
7: end for
Suppose that  $\nabla^2\varphi$  is positive definite at a point  $x$ . The associated local and dual local norms are given by  $\| v\| _x\coloneqq \sqrt{\langle v,\nabla^2\varphi(x)v\rangle}$  and  $\| v\|_{x,*}\coloneqq \sqrt{\langle v,\nabla^{-2}\varphi(x)v\rangle}$ , respectively. Define  $\omega (t)\coloneqq t - \log (1 + t)$ .
Theorem 3.2. Let  $\varphi$  be an  $M$ -self-concordant function such that  $\mathcal{X}$  is contained in the closure of  $\operatorname{dom} \varphi$  and  $\min_{x \in \mathcal{X}} \varphi(x) = 0$ . Suppose that  $\nabla^2 \varphi(x)$  is positive definite for all  $x \in \mathcal{X} \cap \operatorname{dom} \varphi$  and the sequence of learning rates  $\{\eta_t\}$  is non-increasing. Then, Algorithm 1 satisfies
$$
R _ {T} (x) \leq \frac {\varphi (x)}{\eta_ {T}} + \sum_ {t = 1} ^ {T} \left(\langle v _ {t} - \hat {v} _ {t}, x _ {t} - x _ {t + 1} \rangle - \frac {1}{\eta_ {t - 1} M ^ {2}} \omega (M \| x _ {t} - x _ {t + 1} \| _ {x _ {t}})\right).
$$
If in addition,  $\eta_{t - 1}\| v_t - \hat{v}_t\|_{x_t,*}\leq 1 / (2M)$  for all  $t\in \mathbb{N}$ , then Algorithm 1 satisfies
$$
R _ {T} (x) \leq \frac {\varphi (x)}{\eta_ {T}} + \sum_ {t = 1} ^ {T} \eta_ {t - 1} \| v _ {t} - \hat {v} _ {t} \| _ {x _ {t}, *} ^ {2}.
$$
The proof of Theorem 3.2 is deferred to Appendix D. It is worth noting that the crux of the proof lies in Lemma D.1; the remaining steps follow standard procedure.
4 "Smoothness" in Online Portfolio Selection
4.1 Online Portfolio Selection
Online Portfolio Selection (OPS) is a multi-round game between two players, say INVESTOR and MARKET. Suppose there are  $d$  investment alternatives. A portfolio of INVESTOR is represented by a vector in the probability simplex in  $\mathbb{R}^d$ , which indicates the distribution of INVESTOR's wealth among the  $d$  investment alternatives. The price relatives of the investment alternatives at the  $t$ -th round are listed in a vector  $a_t \in \mathbb{R}_+^d$ .
The game has  $T$  rounds. At the  $t$ -th round,
- first, INVESTOR announces a portfolio  $x_{t} \in \Delta \subset \mathbb{R}^{d}$
- then, MARKET announces the price relatives  $a_{t} \in \mathbb{R}_{+}^{d}$ ;
- finally, INVESTOR suffers a loss given by  $f_{t}(x_{t})$ , where the loss function  $f_{t}$  is defined as
$$
f _ {t} (x) := - \log \left\langle a _ {t}, x \right\rangle .
$$
The objective of INVESTOR is to achieve a small regret against all portfolios  $x \in \Delta$ , defined as<sup>2</sup>
$$
R _ {T} (x) := \sum_ {t = 1} ^ {T} f _ {t} \left(x _ {t}\right) - \sum_ {t = 1} ^ {T} f _ {t} (x), \quad \forall x \in \Delta \cap \bigcap_ {t = 1} ^ {T} \operatorname {d o m} f _ {t}.
$$
In the context of OPS, the regret corresponds to the logarithm of the ratio between the wealth growth rate of INVESTOR and that yielded by the constant rebalanced portfolio represented by  $x \in \Delta$ .
Assumption 1. The vector of price relatives  $a_{t}$  is non-zero and satisfies  $\| a_{t}\|_{\infty} = 1$  for all  $t\in \mathbb{N}$ .
2Because the vectors  $a_{t}$  can have zero entries, in general,  $\mathrm{dom}f_t$  does not contain  $\Delta$
The assumption on  $\| a_t\|_{\infty}$  does not restrict the problem's applicability. If the assumption does not hold, then we can consider another OPS game with  $a_{t}$  replaced by  $\tilde{a}_t\coloneqq a_t / \| a_t\|_{\infty}$  and develop algorithms and define the regret with respect to  $\tilde{a}_t$ . It is obvious that the regret values defined with  $a_{t}$  and  $\tilde{a}_t$  are the same.
The following observation, readily verified by direct calculation, will be useful in the proofs.
Lemma 4.1. The vector  $x \odot (-\nabla f_t(x))$  lies in  $\Delta$  for all  $x \in \mathrm{ri} \Delta$  and  $t \in \mathbb{N}$ .
4.2 Log-Barrier
Standard online convex optimization algorithms, such as those in the lecture notes by Orabona [28] and Hazan [12], assume that the loss functions are either Lipschitz or smooth.
Definition 4.2 (Lipschitzness and smoothness). A function  $\varphi$  is said to be Lipschitz with respect to a norm  $\|\cdot\|$  if
$$
| \varphi (y) - \varphi (x) | \leq L \| y - x \|, \quad \forall x, y \in \operatorname {d o m} \varphi
$$
for some  $L > 0$ . It is said to be smooth with respect to the norm  $\| \cdot \|$  if
$$
\left\| \nabla \varphi (y) - \nabla \varphi (x) \right\| _ {*} \leq L ^ {\prime} \| y - x \|, \quad \forall x, y \in \operatorname {d o m} \nabla \varphi \tag {1}
$$
for some  $L^{\prime} > 0$ , where  $\| \cdot \|_{*}$  denotes the dual norm.
Given that  $\langle a_t, x \rangle$  can be arbitrarily close to zero on  $\Delta \cap \operatorname{dom} f_t$  in OPS, it is well known that there does not exist a Lipschitz parameter  $L$  nor a smoothness parameter  $L'$  for all loss functions  $f_t$ . Therefore, standard online convex optimization algorithms do not directly apply.
We define the log-barrier as<sup>3</sup>
$$
h (x) := - d \log d - \sum_ {i = 1} ^ {d} \log x (i), \quad \forall (x (1), \dots , x (d)) \in \mathbb {R} _ {+ +} ^ {d}. \tag {2}
$$
It is easily checked that the local and dual local norms associated with the log-barrier are given by
$$
\left\| u \right\| _ {x} := \left\| u \oslash x \right\| _ {2}, \quad \left\| u \right\| _ {x, *} = \left\| u \odot x \right\| _ {2}. \tag {3}
$$
In the remainder of the paper, we will only consider this pair of local and dual local norms.
Note that Lipschitzness implies boundedness of the gradient. The following observation motivates the use of the log-barrier in OPS, showing that the gradients in OPS are bounded with respect to the dual local norms defined by the log-barrier.
Lemma 4.3. It holds that  $\| \nabla f_t(x)\|_{x,*}\leq 1$  for all  $x\in \mathrm{ri}\Delta$  and  $t\in \mathbb{N}$
A similar result was proved by van Erven et al. [35, (2)]. We provide a proof of Lemma 4.3 in Section E for completeness.
The following fact will be useful.
Lemma 4.4 (Nesterov [27, Example 5.3.1 and Theorem 5.3.2]). The log-barrier  $h$  and loss functions  $f_{t}$  in OPS are both 1-self-concordant.
4.3 "Smoothness" in OPS
Existing results on small-loss and gradual-variation bounds require the loss functions to be smooth. For example, Chiang et al. [8] exploited the definition of smoothness (1) to derive gradual-variation bounds; Srebro et al. [33] and Orabona et al. [29] used the "self-bounding property," a consequence of smoothness, to derive small-loss bounds.
Lemma 4.5 (Self-bounding property [33, Lemma 2.1]). Let  $f: \mathbb{R}^d \to \mathbb{R}$  be an  $L$ -smooth convex function with  $\text{dom } f = \mathbb{R}^d$ . Then,  $\| \nabla f(x) \|_*^2 \leq 2L(f(x) - \min_{y \in \mathbb{R}^d} f(y))$  for all  $x \in \mathbb{R}^d$ .
While the loss functions in OPS are not smooth, we provide two smoothness characterizations of the loss functions in OPS. The first is analogous to the definition of smoothness (1).
3The definition here slightly differs from those typically seen in the literature with an additional  $-d\log d$  term. The additional term helps us remove a log  $d$  term in the regret bounds.
Lemma 4.6. Let  $f(x) = -\log \langle a, x \rangle$  for some  $a \in \mathbb{R}_+^d$ . Under Assumption 1 on the vector  $a$ ,
$$
\left\| \left(x \odot \nabla f (x)\right) - \left(y \odot \nabla f (y)\right) \right\| _ {2} \leq 4 \min  \left\{\| x - y \| _ {x}, \| x - y \| _ {y} \right\}, \quad \forall x, y \in \mathrm {r i} \Delta ,
$$
where  $\odot$  denotes the entrywise product and  $\| \cdot \| _x$  and  $\| \cdot \| _y$  are the local-norms defined by the log-barrier (3).
The second smoothness characterization is analogous to the self-bounding property (Lemma 4.5). For any  $x \in \mathrm{ri} \Delta$  and  $v \in \mathbb{R}^d$ , define
$$
\alpha_ {x} (v) := \frac {- \sum_ {i = 1} ^ {d} x ^ {2} (i) v (i)}{\sum_ {i = 1} ^ {d} x ^ {2} (i)}, \quad \forall x \in \operatorname {r i} \Delta , \tag {4}
$$
where  $x(i)$  and  $v(i)$  denote the  $i$ -th entries of  $x$  and  $v$ , respectively.
Lemma 4.7. Let  $f(x) = -\log \langle a, x \rangle$  for some  $a \in \mathbb{R}_+^d$ . Then, under Assumption 1 on the vector  $a$ , it holds that
$$
\left\| \nabla f (x) + \alpha_ {x} (\nabla f (x)) e \right\| _ {x, *} ^ {2} \leq 4 f (x), \quad \forall x \in \mathrm {r i} \Delta ,
$$
where the notation  $e$  denotes the all-ones vector and  $\| \cdot \|_{x,*}$  denotes the dual local norm defined by the log-barrier (3).
Remark 4.8. The value  $\alpha_{x}(v)$  is indeed chosen to minimize  $\| v + \alpha e\|_{x,*}^{2}$  over all  $\alpha \in \mathbb{R}$ .