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# Optimized projection-free algorithms for online learning: construction and worst-case analysis

Julien Weibel, Pierre Gaillard, Wouter M. Koolen, Adrien Taylor

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Julien Weibel, Pierre Gaillard, Wouter M. Koolen, Adrien Taylor. Optimized projection-free algorithms for online learning: construction and worst-case analysis. 2025. hal-05097004

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# Optimized projection-free algorithms for online learning: construction and worst-case analysis

**Julien Weibel**

Inria, D.I. École Normale Supérieure, CNRS
PSL Research University
julien.weibel@inria.fr

**Pierre Gaillard**

Université Grenoble Alpes, Inria, CNRS
Grenoble INP, LJK
pierre.gaillard@inria.fr

**Wouter M. Koolen**

Centrum Wiskunde & Informatica, Amsterdam
Twente University, Enschede
wmkoolen@cwi.nl

**Adrien Taylor**

Inria, D.I. École Normale Supérieure, CNRS
PSL Research University
adrien.taylor@inria.fr

## Abstract

This work studies and develops projection-free algorithms for online learning with linear optimization oracles (a.k.a. Frank-Wolfe) for handling the constraint set. More precisely, this work (i) provides an improved (optimized) variant of an online Frank-Wolfe algorithm along with its conceptually simple potential-based proof, and (ii) shows how to leverage semidefinite programming to jointly design and analyze online Frank-Wolfe-type algorithms numerically in a variety of settings—that include the design of the variant (i). Based on the semidefinite technique, we conclude with strong numerical evidence suggesting that no pure online Frank-Wolfe algorithm within our model class can have a regret guarantee better than $O(T^{3/4})$ ($T$ is the time horizon) without additional assumptions, that the current algorithms do not have optimal constants, that the algorithm benefits from similar *anytime* properties $O(t^{3/4})$ not requiring to know $T$ in advance, and that multiple linear optimization rounds do not generally help to obtain better regret bounds.

## 1 Introduction

This work considers the online learning problem, where we sequentially generate query points $x_1, x_2, \dots, x_T \in \mathcal{K}$. At each time step $t$, we play $x_t$ and incur a cost $\ell_t(x_t)$ which we aim to minimize by convention, and observe a gradient $g_t = \nabla \ell_t(x_t)$. The quality of our guesses is evaluated by comparison with a reference $x_* \in \mathcal{K}$. In this context, we aim to estimate (or bound) the *cumulative regret* incurred by our choices $x_1, \dots, x_T$ as compared to a reference point $x_* \in \mathcal{K}$:

$$R_T(x_1, \dots, x_T; x_*) \triangleq \frac{1}{T} \sum_{t=1}^{T} \left\{ \ell_t(x_t) - \ell_t(x_*) \right\} \le \frac{1}{T} \sup_{x_* \in \mathcal{K}} \left\{ \sum_{t=1}^{T} \ell_t(x_t) - \sum_{t=1}^{T} \ell_t(x) \right\}.$$

In many applications—such as online advertising, sensor networks, or mobile user applications—data is acquired and processed in real time, arriving as a continuous, high-rate flow. This necessitates the adoption of online learning methods, which aim to rapidly integrate large volumes of data as they arrive. Online learning algorithms and regret bounds form essential frameworks for the theoretical study and optimization of reinforcement learning algorithms [17, 31], recommender systems [4], or for forecasting time series using expert advice [6]. These methods find applications in areas like
load forecasting [8], finance and portfolio selection [22], generative adversarial networks [20], and, more recently, large language models [26]. Online learning has been extensively studied, and we refer to the classical work [14] and its references for a comprehensive overview of the topic.

**Related Work.** As discussed in classical textbooks (see, e.g., [14, 25]), a few meta algorithms drive the basis and intuitions behind most online learning schemes, which can often be seen as appropriate approximations to the meta algorithms. Prominent examples include the "follow the leader" (FTL) and "follow the regularized leader" (FTRL) methods, as well as online gradient descent [32]. These algorithms offer favorable regret guarantees of order $O(\sqrt{T})$. However, they rely on projections to manage the constraint set $\mathcal{K}$. In numerous applications, such as matrix completion or recommender systems [14], these projections are computationally expensive and constitute a significant efficiency bottleneck. This challenge has led to the development of projection-free algorithms, which replace projections with more efficient oracles, allowing for faster computations. In offline smooth convex optimization over polyhedral sets, the pioneering projection-free algorithm was introduced by Frank-Wolfe [11], that resort to a linear optimization oracle. In the context of online convex optimization, the first such algorithm was proposed by [18], but it was limited to linear losses. Subsequently, [15] introduced an online version of the Frank-Wolfe algorithm—referred to as Online Frank-Wolfe (OFW)—which guarantees a regret upper bound of order $O(T^{3/4})$ for convex losses. This regret rate is less favorable compared to standard online learning algorithms that permit projections. As a result, considerable effort has been directed toward achieving improved regret guarantees for online projection-free algorithms. Some variants have attained the optimal regret rate of $O(\sqrt{T})$ by using membership oracles, which differ from the linear oracles initially used in Frank-Wolfe and may be less efficient in certain scenarios [21, 23]. For linear oracles, [16] made progress by introducing a stochastic algorithm that achieves an expected regret of $O(T^{2/3})$ for smooth functions.

Despite these advancements, it remains an open question whether Online Frank-Wolfe variants that solely use linear oracle calls are inherently limited to a regret rate of $\Omega(T^{3/4})$ or if improved rates are possible in this setting.

**Assumptions and problem setup.** In this work, we consider the standard situation where the loss functions are convex with bounded gradients $\|g_t\| \le L$ for all $t \ge 0$ for simplicity. Similarly, we assume that the domain $\mathcal{K}$ is a closed convex bounded set of diameter D. Variations around those assumptions are discussed in Section 3 and in the appendix.

**Algorithmic setup.** This work focuses on the specific case of online learning using projection-free algorithms with access to the domain $\mathcal{K}$ only through a linear optimization oracle. We introduce an online procedure that encompasses a family of Online Frank-Wolfe-type algorithms and unfolds as follows. At each round $t$, we play $x_t \in \mathcal{K}$, incur some loss $\ell_t(x_t)$, and observe a gradient $g_t = \nabla \ell_t(x_t)$. To choose the next query point $x_{t+1} \in \mathcal{K}$, we form the intermediate objective function $f_t(x) = \langle \sum_{s=1}^t \eta_{t,s} g_s, x \rangle + \frac{1}{2} \|x - x_1\|^2_2$, approximate it by the linear function $x \mapsto \langle x, \nabla f_t(\bar{x}_t) \rangle$ in the point $\bar{x}_t = x_1 + \sum_{s=1}^{t-1} \beta_{t,s} (v_s - x_1)$ for some sub-probability $\beta_t$, and use the linear optimization oracle to optimize that linear approximation over $\mathcal{K}$. More precisely, we study the scheme

$$ \begin{align} \operatorname{dir}_t &= \sum_{s=1}^{t} \eta_{t,s} g_s + \sum_{s=1}^{t-1} \beta_{t,s} (v_s - x_1) \nonumber \\ v_t &= \operatorname*{arg\,min}_{v \in \mathcal{K}} \langle \operatorname{dir}_t, v \rangle \tag{1} \\ x_{t+1} &= x_1 + \sum_{s=1}^{t} \gamma_{t+1,s} (v_s - x_1), \nonumber \end{align} $$

where the algorithm is parameterized by $\{\eta_{t,s}\}_{1 \le s \le t \le T}$ (to combine previous gradient information), $\{\beta_{t,s}\}_{1 \le s < t \le T}$ (to pick the linearization point $\bar{x}_t$), and $\{\gamma_{t,s}\}_{1 \le s < t \le T}$ to combine the previous atoms $v_t$ (extreme points of $\mathcal{K}$) of the linear optimization procedure. Note that translations of the domain $\mathcal{K}$ and initial point $x_1$ leave the search direction $\operatorname{dir}_t$ invariant, while the oracle responses $v_t$ and iterates $x_t$ translate along.
The case of [15, Algorithm 1 & Theorem 4.4] correspond to the choices $\eta_{t,s} = \frac{D}{2LT^{3/4}}$, $\beta_{t,s} = \gamma_{t,s}$, $\gamma_{t,t-1} = \frac{1}{\sqrt{t}}$ and $\gamma_{t,s} = \gamma_{t-1,s}(1 - \gamma_{t,t-1})$ for $s < t - 1$. As for [14, Algorithm 27 & Theorem 7.3], they correspond to nearly the same choices with $\gamma_{t,t-1} = \min(1, \frac{2}{\sqrt{t}})$ instead.

**Contributions.** We present a particularly simple and tight tuning for OFW. Our tuning is obtained by optimising the regret bound delivered by a certain potential-based proof strategy (see Section 2). Our optimized regret bound, along with its parameter tuning, is obtained through a constructive approach (Section 3.4) based on semidefinite programming and numerical insights. This approach can be adapted to other online learning algorithms, possibly for optimizing their regret bounds.

We present numerical evaluations of the exact worst-case regret of the existing and our new OFW tunings based on semidefinite optimization up to moderate horizon $T = 100$ (Section 3). We further (a) construct a likely minimax optimal Frank-Wolfe algorithm, (b) conclude that our bound for our version of OFW is not tight (best possible bound is about 2/3 of ours), and that that of Hazan is also suboptimal (but still worse than ours). Note that the theoretical bound for our algorithms remains better than the best possible bound for Hazan's OFW. (c) allowing for multiple rounds of linear optimization only helps the constant in the regret bounds; (d) we can obtain optimized variations around the same theme, e.g., using $\beta_{t,s} = 0$ all the way. This allows obtaining parameter-free algorithms, at the cost of worsened regret rates. (e) as a bonus, it allows to obtain numerical bound for an anytime variant of our OFW algorithm (where $T$ replaced by $t$ in parameters), showcasing that its behavior is close to that of the numerical bound depending on $T$.

On the negative side, whereas we provide a constructive approach to all those elements (including that of optimizing over the parameters of the algorithm) using convex optimization, those bounds are for a large part numerical and are often challenging to translate to closed-forms. This issue is discussed in Remark 3.1.

## 2 Tuning and regret bound for online Frank-Wolfe

In this section we study our optimized online Frank-Wolfe algorithm. We analyze the scheme with fixed parameters shown in Algorithm 1. Throughout this section, we fix

$$ \eta = \frac{D}{2L} \left( \frac{3}{T} \right)^{3/4} \quad \text{and} \quad \sigma = \min \left( 1, \sqrt{\frac{3}{T}} \right) \qquad (2) $$

We assume the time horizon is at least $T \ge 3$.

<caption>**Algorithm 1 Online Frank-Wolfe algorithm (fixed $\eta, \sigma$)**</caption>

**Require:** $T \ge 1$, $x_1 \in \mathcal{K}$, $\eta \ge 0$, $\sigma \in (0, 1)$
1: **for** $t = 1$ to $T$ **do**
2: &nbsp;&nbsp;&nbsp;&nbsp;Play $x_t$, pay cost $\ell_t(x_t)$, and observe $g_t = \nabla \ell_t(x_t)$.
3: &nbsp;&nbsp;&nbsp;&nbsp;dir$_t \leftarrow \eta \sum_{s=1}^t g_s + (x_t - x_1)$
4: &nbsp;&nbsp;&nbsp;&nbsp;$v_t \leftarrow \arg \min_{v \in \mathcal{K}} \langle \text{dir}_t, v \rangle$
5: &nbsp;&nbsp;&nbsp;&nbsp;$x_{t+1} \leftarrow (1 - \sigma)x_t + \sigma v_t$
6: **end for**

**Theorem 2.1.** Fix $T \ge 3$. Denote by $\ell_t$ the cost function at time $t \in [1, T]$. Assume that those cost functions are convex and L-Lipschitz, and that the convex closed domain $\mathcal{K}$ of feasible points has a diameter bounded by $D$. Then, for any $x_* \in \mathcal{K}$, the following upper bound on the regret of the Online Frank-Wolfe Algorithm 1, with parameters defined in (2), holds:

$$ R_T \triangleq \sum_{t=1}^{T} \ell_t(x_t) - \ell_t(x_*) \le \frac{2D}{L3^{3/4}T^{1/4}} \sum_{t=1}^{T} \|g_t\|^2 + \frac{L}{D3^{3/4}T^{1/4}} \sum_{t=1}^{T} \|x_t - v_t\|^2 + \frac{LT^{3/4}}{D3^{3/4}} \|x_* - x_1\|^2. $$

In particular, we get that $R_T \le \frac{4}{3^{3/4}}LDT^{3/4} < 1.76LDT^{3/4}$.

Inspired by [2], see also [27, 19], we use a potential-based proof to prove Theorem 2.1, which relies on the next lemma. Motivated by the fact that Algorithm 1 is seen as an approximation of FTRL
(see [14, Chapter 7]), the next lemma will consider a potential that relates the OFW iterates to the iterates of FTRL, which we display in Algorithm 2. To make the connection, it is necessary that OFW and FTRL encounter the same gradients. To ensure that, we formulate the below lemma for linear loss functions.

<caption>**Algorithm 2** Follow The Regularized Leader (FTRL)</caption>

**Require:** $T \ge 1$, $y_1 \in \mathcal{K}$, $\eta \ge 0$

1: **for** $t = 1$ to $T$ **do**

2: Play $y_t$, pay cost $\ell_t(y_t)$, and observe $g_t = \nabla \ell_t(y_t)$.

3: $y_{t+1} \leftarrow \arg \min_{y \in \mathcal{K}} \eta \left\langle \sum_{s=1}^{t} g_s, y \right\rangle + \frac{1}{2} \|y - x_1\|^2$

4: **end for**

**Lemma 2.2.** Let $T \ge 3$. Denote by $\ell_t: x \mapsto \langle g_t, x \rangle$ the linear cost function at time $t \in [1, T]$. Assume that $\|g_t\| \le L$ for all $t \in [1, T]$ and that the convex closed domain $\mathcal{K}$ of feasible points has a diameter bounded by $D$. Define the sequence of potentials $(\phi_t)_{0 \le t \le T}$ as:

$$ \phi_t \triangleq \sum_{s=1}^{t} \langle g_s, x_s - y_{t+1} \rangle + \frac{1}{6\eta} \|x_{t+1} - y_{t+1}\|^2 - \frac{1}{2\eta} \|y_{t+1} - x_1\|^2. $$

Then, Algorithm 1 and 2, run with parameters in (2) and initialized in $y_1 = x_1 \in \mathcal{K}$, satisfy,

$$ \phi_t - \phi_{t-1} \le \frac{2D}{L3^{3/4}T^{1/4}} \|g_t\|^2 + \frac{L}{D3^{3/4}T^{1/4}} \|x_t - v_t\|^2, \quad \forall t \in [1, T]. $$

Let us remark that the potential used in Lemma 2.2 is inspired by the potential $\psi_t$ used to obtain the optimal upper bound on the regret of FTRL (which gives a reformulation of classical FTRL regret proofs; see Appendix A.2 and, e.g., [25, Chapters 6–7]), where $\psi_t$ is defined as:

$$ \psi_t \triangleq \sum_{s=1}^{t} \langle g_s, y_s - y_{t+1} \rangle - \frac{1}{2\eta} \|y_{t+1} - y_1\|^2. \quad (3) $$

The proof of Lemma 2.2 is quite technical and we defer it to Appendix A.1. In Section 3.4, we discuss how we designed the proof of Lemma 2.2 and jointly obtained the optimal choice of the OFW parameters (2) and of the potential function $\phi_t$ used in Lemma 2.2.

Finally, we close this section with the proof of Theorem 2.1 which relies on Lemma 2.2.

*Proof of Theorem 2.1.* First, using convexity of the cost functions $\ell_t$ (recall $g_t = \nabla \ell_t(x_t)$), we get:

$$ R_T \le \sum_{t=1}^{T} \langle g_t, x_t - x_* \rangle. $$

Then, summing the inequalities from Lemma 2.2 applied to the linearized cost functions $\tilde{\ell}_t: x \mapsto \langle g_t, x \rangle$, and observing that $\phi_0 = 0$ as $y_1 = x_1$, we get:

$$ \phi_T \le \frac{2D}{L3^{3/4}T^{1/4}} \sum_{t=1}^{T} \|g_t\|^2 + \frac{L}{D3^{3/4}T^{1/4}} \sum_{t=1}^{T} \|x_t - v_t\|^2. $$

Throughout the proof, we will repeatedly use the fact that the optimum of a constrained convex optimization problem $x_* \in \arg \min_{x \in \mathcal{K}} f(x)$ satisfies $\langle \nabla f(x_*), x_* - x \rangle \le 0$ for all $x \in \mathcal{K}$. Using the optimality from the definition of $y_{T+1}$, we get:

$$ \sum_{t=1}^{T} \langle g_t, x_t - x_* \rangle - \frac{1}{2\eta} \|x_* - x_1\|^2 \le \sum_{t=1}^{T} \langle g_t, x_t - y_{T+1} \rangle - \frac{1}{2\eta} \|y_{T+1} - x_1\|^2 \le \phi_T. $$

Combining those three inequalities (recall $\eta = \frac{3^{3/4}D}{2LT^{3/4}}$), we get:

$$ R_T \le \frac{2D}{L3^{3/4}T^{1/4}} \sum_{t=1}^{T} \|g_t\|^2 + \frac{L}{D3^{3/4}T^{1/4}} \sum_{t=1}^{T} \|x_t - v_t\|^2 + \frac{LT^{3/4}}{D3^{3/4}} \|x_* - x_1\|^2. $$

In particular, as $\|g_t\| \le L$ and $\|x_t - v_t\| \le D$ for all $t \in [1, T]$ and $\|x_* - x_1\| \le D$, we get that $R_T \le \frac{4}{3^{3/4}} LDT^{3/4}$. This concludes the proof of Theorem 2.1. $\square$
# 3 Constructive approaches to optimized regret bounds

In this section we present how to leverage semidefinite programming [30] to jointly design and analyze online Frank-Wolfe-type algorithms (1) numerically in a variety of settings. Section 3.1 is dedicated to the computation of regret bounds, and Section 3.2 focuses on joint parameter optimization. Those techniques are then leveraged in Section 3.3 to provide strong numerical evidence on the regret behaviors of different variations of online Frank-Wolfe-type algorithms. Finally, Section 3.4 explains how to construct proofs such as that of Section 2.

## 3.1 Worst-case regret bounds via semidefinite programming

The core idea for obtaining worst-case regret bounds (and corresponding examples) consists in casting the problem of computing the worst regret for a sequence generated by (1) as an optimization problem, as follows (we omit the $(D, L)$ dependence of $B_T$ for readability):

$$ B_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}) \triangleq \sup_{\substack{\mathcal{K}, \{\ell_t\}_{t=1,\dots,T} \\ x_*, \{x_t\}_{t=1,\dots,T} \\ d \in \mathbb{N}}} R_T(x_1, \dots, x_T; x_*) \\ \text{subject to: } \ell_t \text{ is convex and L-Lipschitz for } t=1, \dots, T, \quad (4) \\ \mathcal{K} \text{ is a non-empty closed convex set of } \mathbb{R}^d, \\ \text{Diam}(\mathcal{K}) \le D, \\ \{x_t\}_{t=1,\dots,T} \text{ is generated by (1).} $$

This kind of problem seeks the worst-case dimension $d \in \mathbb{N}$, convex domain $\mathcal{K}$ with bounded diameter, and sequence of Lipschitz convex functions $\{\ell_t\}_{t=1,\dots,T}$ that together produce the largest possible regret when those losses are evaluated at points $\{x_t\}_{t=1,\dots,T}$ compatible with (1). In the optimization literature, (4) is commonly referred to as a performance estimation problem (PEP, as coined by [10]) which can be reformulated losslessly into a convex semidefinite program [28], as briefly outlined below and detailed in Appendix B.1.

In a nutshell, (4) is a priori an infinite-dimensional problem (e.g., as it includes functional variables such as the losses $\ell_t$). A key idea to reformulate (4) as a tractable problem consists in sampling the losses $\ell_t$ at the query points $x_t$ and $x_*$, and to treat only the responses $(\nabla \ell_t(x_t), \ell_t(x_t))$ and $(\nabla \ell_t(x_\star), \ell_t(x_\star))$ as variables. By appropriately constraining these responses, we can force them to be compatible with some losses $\ell_t$ satisfying the desired assumptions (convexity and Lipschitzness of $\ell_t$). With a bit more details, it suffices to require that the samples are compatible through the subgradient inequalities $\ell_t(x_t) - \ell_t(x_\star) \le \langle g_t, x_t - x_\star \rangle$ for $t=1, \dots, T$ and the Lipschitz bounds $\|\nabla \ell_t(x_t)\| \le L$. As for handling the domain $\mathcal{K}$, one possible approach is to sample its indicator function at the query points $v_t$ and impose similar compatibility constraints (see, e.g., [28, Theorem 3.6]). Finally, the sampled version of (4) can be lifted to a semidefinite program via a standard change of variables: all vectors and gradients appearing in (4) are replaced with their Gram matrix (which, recall, encodes all pairwise inner products between these vectors / gradients). The objective, as well as all constraints, then become linear functions of the entries of this Gram matrix. For simplicity and brevity, a complete exposition is provided in Appendix B.1.

**Remark 3.1 (Obtaining proofs from (4)).** A key feature of (4) is that it enables the construction of algorithm-dependent lower bounds on the worst-case regret by solving semidefinite programs. That is, for given numerical values of $T, L, D$ and the algorithm parameters, one can compute a worst-case example by solving a tractable convex problem. In order to obtain algorithm-dependent upper bounds on the worst-case regret, a natural procedure consists in formulating the Lagrange dual of (4) (which is also a semidefinite program; see, e.g., [30, 5]), whose feasible points naturally corresponds to upper bounds on the regret. In this context, finding a proof consists in finding a feasible point to the dual problem [12]. While algebraic techniques exist for solving such parametric semidefinite programs in closed form, they typically suffer from exponential complexity in the problem size, number of variables/constraints, and number of parameters [3, 24]. This motivates the search for simpler/structured proofs—e.g., potential-based methods [2]—as exemplified in the proof of Theorem 2.1; see Section 3.4 for further discussion.
## 3.2 Jointly optimising algorithm parameters and regret bounds

A natural path forward is to use (4) to obtain worst-case optimal algorithms. That is, by solving

$$ \min_{\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}} \left\{ B_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}) \text{ s.t. } \sum_{s=1}^{t-1} \gamma_{t,s} \le 1, \gamma_{t,s} \ge 0 \right\}. $$

This problem can be formulated as a linear optimization problem with a bilinear matrix inequality constraint (see Appendix B.2), which are unfortunately NP-hard in general [29]. For this reason, we propose a slight relaxation of $B_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s})$ which corresponds to remove a few constraints from (4) (that we numerically observed to be inactive). More precisely, this relaxation is obtained through: (i) we observe that all $x_t$ for $t = 2, \dots, T$ are in the convex body of $x_1, v_1, \dots, v_{T-1}$, and thus the domain constraint for $\mathcal{K}$ are only covered by vectors $x_1, v_1, \dots, v_{T-1}, x_\star$; (ii) we only keep the boundary constraints corresponding to the optimality of $v_t$ compared with $v_{t+1}, \dots, v_{T-1}, x_\star$:

$$ W_T(\{\eta_{t,s}, \beta_{t,s}, \gamma_{t,s}\}_{t,s}) \triangleq \sup_{\substack{\{g_t\}_{t=1,\dots,T}, x_\star \\ \{(x_t, v_t, \text{dir}_t)\}_{t=1,\dots,T} \\ d \in \mathbb{N}}} \sum_{t=1}^T \langle g_t, x_t - x_\star \rangle \\ \text{subject to: } \{(x_t, \text{dir}_t)\}_{t=1,\dots,T} \text{ compatible with (1),} \\ \langle -\text{dir}_t, u - v_t \rangle \le 0 \text{ for all } t = 1, \dots, T-1 \\ \text{and } u \in \{v_{t+1}, \dots, v_{T-1}, x_\star\}, \\ \text{Diam}(\{x_1, x_\star, v_1, \dots, v_{T-1}\}) \le D, \\ \|g_t\| \le L \text{ for } t = 1, \dots, T, $$

where the line “$\{(x_t, \text{dir}_t)\}_{t=1,\dots,T}$ compatible with (1)” means that $x_t$ and $\text{dir}_t$ can be substituted by their expressions in (1), leading to $W_T(\{\eta_{t,s}, \beta_{t,s}, \gamma_{t,s}\}_{t,s}) \ge B_T(\{\eta_{t,s}, \beta_{t,s}, \gamma_{t,s}\}_{t,s})$. For this problem, an appropriate change of variables allows to cast

$$ \min_{\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}} \left\{ W_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}) \text{ s.t. } \sum_{s=1}^{t-1} \gamma_{t,s} \le 1, \gamma_{t,s} \ge 0 \right\} \quad (5) $$

as a convex semidefinite program again (details in Appendix B.2). Numerical results illustrating the performance of the numerically optimized methods are provided in Figure 1; examples of numerically optimized parameters are further provided in Appendix B.3.

## 3.3 Numerical results: tight regret bounds, direct stepsize optimization, multiple linear optimization rounds, unregularized online Frank-Wolfe, and anytime bounds

The numerical experiments of this section rely on the CVXPY [9] modeling language used in combination with the MOSEK semidefinite solver [1]. For computing (4) we directly implemented the online algorithms within the PEPit software [13]. In all those numerical experiments, we used $L = D = 1$. Those numerical experiments were performed on a MacBook Pro 14" with M3 Pro SoC and 36GB of RAM under few tens of minutes for the largest time horizon and under few minutes for other values, note that RAM size was the main limiting factor. Those numerical experiments strongly support the following claims (see Figure 1).

**Comparison of known upper bounds to tight numerical bounds.** Algorithm 1, instantiated with parameters in Equation (2), enjoys a regret guarantee that improves upon the classical bound from [14, Theorem 7.3] (corresponding to Algorithm 27 therein). This improvement also holds when comparing the tight bounds of both algorithms, with a gain of approximately 34% (i.e., a factor of about 0.66). Although the upper bound provided in Theorem 2.1 is not tight, it remains within a constant factor of approximately 1.5 of the numerically computed tight worst-case bound. Moreover, Algorithm 1 (with the parameters from Theorem 2.1) is near-optimal among online Frank-Wolfe algorithms of the form (1), up to a constant multiplicative factor of roughly 1.18.
<figure><img src="image_3.png" /><figcaption>Figure 1: (Top left) Comparison of known upper bounds (respectively from [14, Theorem 7.3] and Theorem 2.1) against tight numerical bounds obtained from (4), for [14, Algorithm 27] and Algorithm 1 (parameters from Theorem 2.1). (Top right) Tight numerical bounds for optimized online Frank-Wolfe with respectively $r \in \{1, 2, 3\}$ linear optimization steps per time round (where (19) is a variant of (5) with (1) replaced by (6), which we detail in Appendix B.5). (Bottom left) Tight numerical regret bounds for optimized online Frank-Wolfe with and without regularization (i.e., (5) with and without $\beta_{t,s} = 0$). (Bottom right) Tight numerical regret bounds for [14, Algorithm 27] and Algorithm 1 (parameters from Theorem 2.1) against their anytime versions.</figcaption></figure>

**Multiple linear optimization rounds per iteration.** A natural extension of (1) consists in performing a fixed number ($r > 1$) of linear optimization steps per iteration by defining $r$ search directions $dir_{t,1}, \dots, dir_{t,r}$ (sequentially) per time step, together with the corresponding atoms
$v_{t,1}, \dots, v_{t,k}:$

$$ \mathrm{dir}_{t,k} = \sum_{s=1}^{t} \eta_{t,k,s} g_s + \sum_{s=1}^{t-1} \sum_{j=1}^{r} \beta_{t,k,s,j} (v_{s,j} - x_1) + \sum_{j=1}^{k-1} \beta_{t,k,t,j} (v_{t,j} - x_1) \quad (6) $$

$$ v_{t,k} = \arg \min_{v \in \mathcal{K}} \langle \mathrm{dir}_{t,k}, v \rangle $$

for choosing the next query point $x_{t+1} = x_1 + \sum_{s=1}^{t} \sum_{k=1}^{r} \gamma_{t+1,s,k} (v_{s,k} - x_1)$. We observe that multiple rounds (fixed in advanced and not a function of the time horizon) of linear optimization oracles do not help improving the regret rates of online Frank-Wolfe algorithms. In all cases, the regret scales in $T^{3/4}$. The numerical regret bounds for optimized online Frank-Wolfe algorithms with fixed number $r > 1$ of linear optimization oracle calls per time round were obtained in a similar way to the method outlined in Section 3.2, see Appendix B.4 for details.

**Unregularized online Frank-Wolfe.** A desirable feature of setting $\beta_{t,s} = 0$ in (1) is that the remaining parameters $\eta_{t,s}$ and $\gamma_{t,s}$ are naturally dimension-independent, that is, they cannot depend on $L$ and $D$ in a meaningful way (as there is no other external quantities, there is no way to have $\eta_{t,s}$ and $\gamma_{t,s}$ being dimension-independent while depending on $L$ and $D$). Unfortunately, this nice feature is counter-balanced by an apparent worse regret rate. More precisely, the optimized Frank-Wolfe algorithms with fixed $\beta_{t,s} = 0$ appear to have their worst-case regrets scaling as $O(T^\alpha)$ for $\alpha \approx 7/8$.

**Anytime variants.** The anytime algorithmic variants presented in Figure 1 correspond to respectively [14, Algorithm 27] and Algorithm 1 (with parameter choices from Theorem 2.1) where we brutally substituted the time-horizon $T$ dependence of algorithm parameters by that of the current time $t$. We observe that the anytime variant of Algorithm 1 has a guarantee close to that of the original Algorithm 1 within a constant multiplicative factor of about 1.17. The anytime variant of [14, Algorithm 27] has a better guarantee than the original [14, Algorithm 27], but still worse than the anytime variant of Algorithm 1 (with a constant multiplicative factor of about 1.24).

## 3.4 How we got the proof of Section 2

As explained in Remark 3.1, (4) can be used to obtain rigorous regret upper bounds and their proofs for Algorithm 1. However, the algebraic structure of the problem to be solved proved itself quite challenging. In particular, we did not manage to extract a simple solution from (4). Therefore, we restricted ourselves to search for optimized algorithms within the set of algorithms with simple structured proofs. A classical template for such proofs is that of relying on the construction of potential (or Lyapunov) functions (see, e.g., [2] for a nice introduction). We thereby adapted (4) to help searching for appropriate potential functions.

The idea of a potential-based proof is to define a sequence of potentials $(\phi_t)_{0 \le t \le T}$ such that (i) $R_T \le \phi_T + A$ and (ii) we can prove inequalities $\phi_t - \phi_{t-1} \le B_t$ for all $t$, where $A$ and $B_t$ are constants which can only depend on $L$, $D$, $t$ and $T$. Then, we immediately obtain an upper bound on the regret $R_T$ as $R_T \le \phi_0 + A + \sum_{t=1}^{T} \phi_t - \phi_{t-1} \le \phi_0 + A + \sum_{t=1}^{T} B_t$. Under some good choice of potentials $\phi_t$, each potential difference $\phi_t - \phi_{t-1}$ depends only on a small number of vectors, gradients and sum of past gradients (for instance, we will use $x_t$, $x_{t+1}$, $y_t$, $y_{t+1}$, $g_t$ and $G_{t-1} = \sum_{s=1}^{T-1} g_s$). In other words, (ii) consists in studying a single iteration of the procedure while (i) ensures that we can combine those one-iteration analyses to form the global bound.

In such proofs, the first step is therefore to understand what information requires to be summarized in $\phi_t$. Here, motivated by the fact that Algorithm 1 is seen as an approximation of FTRL (see [14, Chapter 7]), and by the potential functions used for upper bounding FTRL's regret (see (3), which is a reformulation of classical FTRL regret proofs; see, e.g., [25, Chapters 6–7]), we tried the family of potentials parametrised by $a$ and $b$ as

$$ \phi_t = \sum_{s=1}^{t} \langle g_s, x_s - y_{t+1} \rangle + a \|x_{t+1} - y_{t+1}\|^2 + b \eta \langle G_t, x_{t+1} - y_{t+1} \rangle \\ + \frac{b}{2} (\|x_{t+1} - x_1\|^2 - \|y_{t+1} - x_1\|^2) - \frac{1}{2\eta} \|y_{t+1} - x_1\|^2. \quad (7) $$
Note that $\phi_t - \phi_{t-1}$ is autonomous: it is a function of $t$ only through its dependence in $x_t, x_{t+1}, y_t, y_{t+1}, g_t, G_{t-1}, x_1$. Because of this fact, and as $x_{t+1}, y_t$ and $y_{t+1}$ are obtained as autonomous functions of $x_t, g_t, G_{t-1}$ and $x_1$ using Algorithm 1 and FTRL, it is possible to use the same proof for upper bounding $\phi_t - \phi_{t-1}$ for all iteration times $t$ (then, we get that $B_t$ is also autonomous, which imposes that $B_t = B$ is a constant that depends only on $L, D$ and $T$). Thus, we can use only 1 iteration of upper bounding $\phi_t - \phi_{t-1} \le B$ for an abstract $t$ instead of doing an upper bound for each $t$ individually. We also note that $\phi_0 = 0$ and $R_T - \frac{1}{2\eta}\|x_* - x_1\|^2 \le \phi_T$ for $a \ge 0$ and $b \ge 0$ (using optimality of $y_{T+1}$).

To jointly design the proof of Algorithm 1 (with fixed $\eta$ and $\sigma$) and the potential (7), we use the following 1-iteration variant of (4):

$$ \inf_{a \ge 0, b \ge 0} \sup_{\substack{\mathcal{K}, d \in \mathbb{N} \\ g_t, G_{t-1} \\ x_1, x_t, v_t, x_{t+1} \\ y_t, y_{t+1}}} \phi_t - \phi_{t-1} $$

subject to: $\phi_t - \phi_{t-1}$ is generated from $\{x_t, x_{t+1}, y_t, y_{t+1}, g_t, G_{t-1}, x_1\}$ by (7),
$$ \|g_t\| \le L, \tag{8} $$
$\mathcal{K}$ is a non-empty closed convex set of $\mathbb{R}^d$,
$\text{Diam}(\{x_1, x_t, v_t, x_{t+1}, y_t, y_{t+1}\}) \le D$,
$(x_{t+1}, v_t)$ are generated from $\{x_1, x_t, g_t, G_{t-1}\}$ by Algorithm 1,
$y_t$ and $y_{t+1}$ are generated from $\{x_1, g_t, G_{t-1}\}$ by FTRL.

Note that (8) can be rewritten as a semidefinite program whose size does not depend on $T$, which allows efficient numerical solving even for large values of $T$.

Then, numerically solving (8) for fixed $\eta$ and $\sigma$ gives $b = 0$ and the values of Lagrange multipliers of the constraints gives the proof structure of Lemma 2.2 (with literal values for $\eta, \sigma, a, \lambda$ and $\lambda^g$ the Lagrange multiplier of the norm constraint for $g_t$) up to the final step of verifying that $Q(\lambda)$ is a sum of squares. Hence, we are left with finding the optimal $\eta$ and $\sigma$ minimizing the regret upper bound given by (8) with the constraint that $Q(\lambda)$ is a sum of squares, which is a non-convex problem in $\eta, \sigma, a$ and $\lambda^g$. We then relax this non-convex problem by keeping only the leading order terms in $Q(\lambda)$, which we can then solve algebraically, resulting in the choices

$$ \eta = \frac{D3^{3/4}}{2LT^{3/4}}, \quad \sigma = \frac{\sqrt{3}}{\sqrt{T}}, \quad a = \frac{1}{6\eta}, \quad \text{and,} \quad \lambda^g = 2a\sigma^2 \frac{D^2}{L^2} = \frac{D^2}{\eta T L^2}, $$

thereby concluding the construction of the result of Lemma 2.2. The details for rewriting (8) as a semidefinite program, numerically solving it, and then for obtaining the optimal parameters above can be found in Appendix B.5.

# 4 Conclusion

In this work, we have studied and developed projection-free algorithms for online learning that rely on linear optimization oracles (a.k.a. Frank-Wolfe) for handling the constraint set. More precisely, the contributions of this work were (i) an improved (optimized) variant of an online Frank-Wolfe (OFW) algorithm (Algorithm 1) together with its conceptually simple potential-based proof, and (ii) methods leveraging semidefinite programming to jointly design and analyze of online Frank-Wolfe-type algorithms in a variety of settings. These semidefinite programming methods provide a constructive approach to obtain regret bounds and their corresponding worst-case instances. Algorithms with optimal regret guarantees can then be designed by jointly optimising the algorithm parameters and the regret bound. We then used these semidefinite programming methods to perform numerical experiments strongly supporting (a) near-optimality claims of the proposed parameter choices, and (b) that all OFW-type algorithm have a regret rate of $\Omega(T^{3/4})$. Lastly, we explained how the semidefinite programming methods we presented can be used to obtain conceptually simple proofs with optimal regret bounds, taking as an example our potential-based proof from part (i).

The findings of our numerical experiments motivate the following future work opportunities: (1) can we find a tighter analysis of the OFW algorithm?; (2) can we find a closed-form for the optimal OFW-type method?; and (3) can we find a closed-form solution and the corresponding regret bound for the optimal unregularized OFW-type method?
Acknowledgments and Disclosure of Funding

This work took place in the context of the 4TUNE within CWI-Inria international lab. Julien Weibel and Adrien Taylor are supported by the European Union (ERC grant CASPER 101162889). The French government also partly funded this work under the management of Agence Nationale de la Recherche as part of the “France 2030” program, reference ANR-23-IACL-0008 “PR[AI]RIE-PSAI”.

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which imply that $\langle g_t, y_{t+1} - y_t \rangle \le -\frac{1}{\eta} \|y_t - y_{t+1}\|^2$. From this inequality, we obtain:

$$ \begin{aligned} \langle g_t, x_t - y_{t+1} \rangle &= \langle g_t, x_t - y_{t+1} + \lambda(y_t - y_{t+1}) \rangle + \lambda \langle g_t, y_{t+1} - y_t \rangle \\ &\le \langle g_t, x_t - y_{t+1} + \lambda(y_t - y_{t+1}) \rangle - \frac{\lambda}{\eta} \|y_t - y_{t+1}\|^2 \\ &\le \lambda^g \|g_t\|^2 + \frac{1}{4\lambda^g} \|x_t - y_{t+1} + \lambda(y_t - y_{t+1})\|^2 - \frac{\lambda}{\eta} \|y_t - y_{t+1}\|^2, \end{aligned} $$

for any $\lambda^g > 0$ to be chosen latter. Combining this inequality with Equation (10), we get:

$$ \phi_t - \phi_{t-1} \le \frac{\sigma^2}{6\eta} \|x_t - v_t\|^2 + \lambda^g \|g_t\|^2 + \frac{1}{4\lambda^g} \|x_t - y_{t+1} + \lambda(y_t - y_{t+1})\|^2 \\ - \frac{1+2\lambda}{2\eta} \|y_t - y_{t+1}\|^2 + \frac{1}{6\eta} \left( (1-2\sigma) \|x_t - y_{t+1}\|^2 - \|x_t - y_t\|^2 \right). $$

Hence, to complete the proof for the upper bound on $\phi_t - \phi_{t-1}$, it suffices to show that there exists $\lambda > 0$ such that the following expression is non-negative, which can be done by rewriting it as a sum of squares:

$$ \begin{aligned} Q(\lambda) &\triangleq \frac{1+2\lambda}{2\eta} \|y_t - y_{t+1}\|^2 - \frac{1}{4\lambda^g} \|x_t - y_{t+1} + \lambda(y_t - y_{t+1})\|^2 \\ &\quad - \frac{1}{6\eta} \left( (1-2\sigma) \|x_t - y_{t+1}\|^2 - \|x_t - y_t\|^2 \right). \\ &= \frac{2+3\lambda}{3\eta} \|y_t - y_{t+1}\|^2 + \frac{\sigma}{3\eta} \|x_t - y_{t+1}\|^2 + \frac{1}{3\eta} \langle x_t - y_{t+1}, y_{t+1} - y_t \rangle \\ &\quad - \frac{1}{4\lambda^g} \|x_t - y_{t+1} + \lambda(y_t - y_{t+1})\|^2 \end{aligned} $$

Indeed, substituting our choices of $\eta = \frac{D^{3/4}}{2LT^{3/4}}$, $\sigma = \min(1, \frac{\sqrt{3}}{\sqrt{T}})$ and $\lambda^g = \frac{\sigma^2 D^2}{3\eta L^2}$, whose derivations are detailed in Section 2, and defining $\tilde{T} = T/3$ for conciseness, we obtain:

$$ \begin{aligned} \frac{D}{L} Q(\lambda) &= \frac{4+6\lambda}{3} \tilde{T}^{3/4} \|y_t - y_{t+1}\|^2 + \frac{2\tilde{T}^{1/4}}{3} \|x_t - y_{t+1}\|^2 + \frac{2\tilde{T}^{3/4}}{3} \langle x_t - y_{t+1}, y_{t+1} - y_t \rangle \\ &\quad - \frac{3\tilde{T}^{1/4}}{8} \|x_t - y_{t+1} + \lambda(y_t - y_{t+1})\|^2 \\ &= \left( \frac{4+6\lambda}{3} \tilde{T}^{3/4} - \frac{3\tilde{T}^{1/4}}{8} \lambda^2 \right) \|y_t - y_{t+1}\|^2 + \frac{7\tilde{T}^{1/4}}{24} \|x_t - y_{t+1}\|^2 \\ &\quad + 2 \times \left( \frac{\tilde{T}^{3/4}}{3} + \frac{3\tilde{T}^{1/4}}{8} \lambda \right) \langle x_t - y_{t+1}, y_{t+1} - y_t \rangle. \end{aligned} $$

Thus, using a Schur complement argument, $Q(\lambda)$ is a sum of squares if and only if:

$$ \left( \frac{1}{3} \tilde{T}^{3/4} + \frac{3}{8} \tilde{T}^{1/4} \lambda \right)^2 \le \frac{7}{24} \tilde{T}^{1/4} \left( \frac{4+6\lambda}{3} \tilde{T}^{3/4} - \frac{3}{8} \tilde{T}^{1/4} \lambda^2 \right), $$

which is equivalent to:

$$ \frac{\sqrt{\tilde{T}}}{4} \lambda^2 - \frac{\tilde{T}}{3} \lambda + \left( \frac{\tilde{T}^{3/2}}{9} - \frac{7\tilde{T}}{18} \right) \le 0. $$

Hence, there exist a non-negative $\lambda$ solving this second order equation if and only if $\Delta \ge 0$ where:

$$ \Delta \triangleq \frac{\tilde{T}^2}{9} - 4 \frac{\sqrt{\tilde{T}}}{4} \left( \frac{\tilde{T}^{3/2}}{9} - \frac{7\tilde{T}}{18} \right) = \frac{7\tilde{T}^{3/2}}{18} \ge 0. $$

As a consequence, there exist a choice of $\lambda \ge 0$ such that $Q(\lambda)$ is a sum of squares, which concludes the proof of the upper bound on $\phi_t - \phi_{t-1}$, and thus also concludes the proof of Lemma 2.2. $\square$
## A.2 Potential-based proof for FTRL

We present here the potential-based proof of the optimal upper bound on the regret of FTRL (see Algorithm 2) using the potential $\psi_t$ defined in (3). This potential-based proof is fundamentally a reformulation of classical FTRL regret proofs (see, e.g., [25, Chapters 6–7]). Before presenting the proof, we first re-state this optimal regret upper bound in the following lemma.

**Lemma A.1.** Fix $T \ge 3$. Denote by $\ell_t$ the cost function at time $t \in [1, T]$. Assume that those cost functions are convex and L-Lipschitz, and that the convex closed domain $K$ of feasible points has a diameter bounded by $D$. Then, for any $y_* \in K$, the following upper bound on the regret of the FTRL Algorithm 2 holds:

$$R_T \triangleq \sum_{t=1}^{T} \ell_t(y_t) - \ell_t(y_*) \le \frac{\eta}{2} \sum_{t=1}^{T} \|g_t\|^2 + \frac{1}{2\eta} \|y_* - y_1\|^2.$$

In particular, for $\eta = \frac{D}{L\sqrt{T}}$, we get that $R_T \le LD\sqrt{T}$.

*Proof.* We present a potential-based proof based on the potential $\psi_t$ defined in (3).

We start by upper bounding $\psi_t - \psi_{t-1}$ for fixed $t \in [1, T]$. Using the optimality in the definition of $y_t$ gives:

$$\langle G_{t-1}, y_t - y_{t+1} \rangle + \frac{1}{\eta} \langle y_t - y_1, y_t - y_{t+1} \rangle \le 0.$$

Then, using this equation and the inequality $\langle u, v \rangle \le \frac{\eta}{2}\|u\|^2 + \frac{1}{2\eta}\|v\|^2$, we get the upper bound:

$$\begin{align*}
\psi_t - \psi_{t-1} &= \langle G_t, y_t - y_{t+1} \rangle + \frac{1}{2\eta} (\|y_t - y_1\|^2 - \|y_{t+1} - y_1\|^2) \\
&\le \langle g_t, y_t - y_{t+1} \rangle + \frac{1}{2\eta} (\|y_t - y_1\|^2 + 2\langle y_t - y_1, y_t - y_{t+1} \rangle - \|y_{t+1} - y_1\|^2) \\
&= \langle g_t, y_t - y_{t+1} \rangle - \frac{1}{2\eta} \|y_t - y_{t+1}\|^2 \\
&\le \frac{\eta}{2} \|g_t\|^2.
\end{align*}$$

Now, combining those potential differences, the sum telescops (remark that $\psi_0 = 0$) and we get:

$$\psi_T = \sum_{t=1}^{T} \psi_t - \psi_{t-1} \le \frac{\eta}{2} \sum_{t=1}^{T} \|g_t\|^2.$$

Using the optimality in the definition of $y_{t+1}$ gives:

$$R_T - \frac{1}{2\eta} \|y_* - y_1\|^2 \le \psi_T,$$

which concludes the proof of the main upper bound of $R_T$. The second statement follows directly by using the Lipschitz and diameter bounds. □

# B Detailed semidefinite formulations

## B.1 Tractable formulation of (4)

As explained in Section 3.2, (4) is a priori an infinite-dimensional problem, as it includes functional variables which are the losses $\ell_t$ and the indicator function of the feasible set $K$. However, (4) can be reformulated as finite dimensional linear semidefinite program. In this section, we detail how to obtain this reformulation.

The first step consists in reformulating (4) as a finite-dimensional problem by sampling the losses $\ell_t$ at the query points $x_t$ and $x_*$, and to treat only the responses $(\nabla \ell_t(x_t), \ell_t(x_t))$ and $(\nabla \ell_t(x_*), \ell_t(x_*))$ as variables. By appropriately constraining these responses, we can force them to be compatible with some losses $\ell_t$ satisfying the desired assumptions (convexity and Lipschitzness of $\ell_t$). Using an
interpolation / extension theorem [28, Theorem 3.3 and Equation (7)], it suffices to require that the samples are compatible through the subgradient inequalities $\ell_t(x_t) - \ell_t(x_\star) \le \langle \nabla \ell_t(x_t), x_t - x_\star \rangle$ and $\ell_t(x_\star) - \ell_t(x_t) \le \langle \nabla \ell_t(x_\star), x_\star - x_t \rangle$ and the Lipschitz bounds $\|\nabla \ell_t(x_t)\| \le L$ for $t = 1, \dots, T$. Indeed, as the subgradients $\nabla \ell_t(x_\star)$ are never used in (4) their values do not matter, and thus we can impose $\nabla \ell_t(x_\star) = \nabla \ell_t(x_t)$ without changing the value of the problem (that is, we get a tight reformulation of (4) and not just an upper bound). This allows to assume without loss of generality that the cost functions $\ell_t$ are all linear. This corresponds to upper bound $\ell_t(x_t) - \ell_t(x_\star)$ by $\langle g_t, x_t - x_\star \rangle$ (recall $g_t = \nabla \ell_t(x_t)$) in the objective of (4), and replace the cost functions variables $\ell_t$ by the gradient variables $g_t$. Note that we get a tight reformulation of (4) and not just an upper bound as any point (with linear cost functions) in this new reformulation is still feasible with the same objective value in the initial problem (4).

As for handling the convex domain $\mathcal{K}$, one possible approach is to sample its indicator function at the query points $v_t$ and $x_t$ for $t = 1, \dots, T$ and $x_\star$ and impose similar compatibility constraints. Indeed, denoting by $\iota_{\mathcal{K}}$ the indicator function of the feasible set $\mathcal{K}$, and using [28, Theorem 3.6], those compatibility constraints are: for all $u, v \in \{x_1, \dots, x_T, v_1, \dots, v_{T-1}, x_\star\}$, and $g \in \nabla \iota_{\mathcal{K}}(u)$ a subgradient of $\iota_{\mathcal{K}}$ at $u$, we have $\langle g, v-u \rangle \le 0$. As the points $\{(x_t, v_t, \text{dir}_t)\}_{t=1,\dots,T}$ are generated by (1), we get that $-\text{dir}_t$ is a subgradient of $\iota_{\mathcal{K}}$ at $v_t$ for $t = 1, \dots, T-1$. However, we have no special choice for the subgradients of $\iota_{\mathcal{K}}$ at $x_1, \dots, x_T, x_\star$, and we will choose their subgradients to be 0 (which leads to trivial constraints that we remove from the problem). Those two function sampling steps give us the following finite-dimensional reformulation of (4):

$$
\begin{align*}
B_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}) &= \sup_{
  \begin{subarray}{l}
    \{g_t\}_{t=1,\dots,T}, x_\star \\
    \{(x_t, v_t, \text{dir}_t)\}_{t=1,\dots,T}
  \end{subarray}
} \sum_{t=1}^T \langle g_t, x_t - x_\star \rangle \\
&\quad \text{subject to: } \|g_t\| \le L \text{ for } t=1,\dots,T, \\
&\quad \langle -\text{dir}_t, u - v_t \rangle \le 0 \text{ for } t=1,\dots,T-1 \\
&\quad \text{and } u \in \{x_1, \dots, x_T, v_1, \dots, v_{T-1}, x_\star\}, \\
&\quad \text{Diam}(\{x_1, \dots, x_T, v_1, \dots, v_{T-1}, x_\star\}) \le D, \\
&\quad \{(x_t, v_t, \text{dir}_t)\}_{t=1,\dots,T} \text{ is generated by (1).}
\end{align*}
$$

As $x_2, \dots, x_T$ are convex combination of $x_1, v_1, \dots, v_{T-1}$, we can omit them in the boundary and diameter constraints, leading to the following simpler finite-dimensional reformulation of (4):

$$
\begin{align}
B_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}) &= \sup_{
  \begin{subarray}{l}
    \{g_t\}_{t=1,\dots,T}, x_\star \\
    \{(x_t, v_t, \text{dir}_t)\}_{t=1,\dots,T}
  \end{subarray}
} \sum_{t=1}^T \langle g_t, x_t - x_\star \rangle \nonumber \\
&\quad \text{subject to: } \{(x_t, \text{dir}_t)\}_{t=1,\dots,T} \text{ compatible with (1),} \tag{11} \\
&\quad \langle -\text{dir}_t, u - v_t \rangle \le 0 \text{ for all } t=1,\dots,T-1 \nonumber \\
&\quad \text{and } u \in \{x_1, v_1, \dots, v_{T-1}, x_\star\}, \nonumber \\
&\quad \text{Diam}(\{x_1, v_1, \dots, v_{T-1}, x_\star\}) \le D, \nonumber \\
&\quad \|g_t\| \le L \text{ for } t=1,\dots,T. \nonumber
\end{align}
$$

Finally, this sampled version (11) of (4) can be lifted to a semidefinite program via a standard change of variables: all vectors and gradients appearing in (11) are replaced with their Gram matrix (which, recall, encodes all pairwise inner products between these vectors / gradients). As the problem is invariant by translation of the vectors $x_t$, $v_t$ and $x_\star$ and of the feasible set $\mathcal{K}$, without loss of generality, we can assume that $x_1 = 0$. Let $G$ be the Gram matrix of the gradients $g_1, \dots, g_T$ and the vectors $v_1, \dots, v_{T-1}$ and $x_\star$, that is, $G = P^T P$ with $P = [g_1 | \cdots | g_T | v_1 | \cdots | v_{T-1} | x_\star]$.

Let $e_1, \dots, e_{2T}$ be the standard basis vectors of $\mathbb{R}^{2T}$. Define $\bar{g}_t = e_t$ for $t = 1, \dots, T$, $\bar{v}_t = e_{T+t}$ for $t = 1, \dots, T-1$, $\bar{x}_\star = e_{2T}$ and $\bar{x}_1 = 0$. Thus, we get that $\bar{g}_s^T G \bar{g}_t = \langle g_s, g_t \rangle$ for all $s, t = 1, \dots, T$; and similarly with the other vectors $\bar{v}_t$ for $t = 1, \dots, T-1$, $\bar{x}_\star$ and $\bar{x}_1$. For two vectors $u$ and $v$ in $\mathbb{R}^{2T}$, we define $u \odot v = (uv^T + vu^T)/2$ their symmetric outer product. We denote by tr the trace operator for square matrices.
The objective, as well as all constraints, then become linear functions of the entries of the Gram matrix G. Indeed, note that for all $u, v \in \{x_1, v_1, \dots, v_{T-1}, x_\star\}$ and $t \in [1, T]$, we have:

$$ \|g_t\|^2 = \bar{g}_t^T G \bar{g}_t = \text{tr}(\bar{g}_t \odot \bar{g}_t) G), $$

$$ \|u - v\|^2 = \text{tr}((\bar{u} - \bar{v}) \odot (\bar{u} - \bar{v})) G), $$

$$ \langle g_t, x_t \rangle = \left\langle g_t, \sum_{s=1}^{t-1} \gamma_{t,s} v_s \right\rangle = \sum_{s=1}^{t-1} \text{tr}((\bar{g}_t \odot \bar{v}_s) G), $$

$$ \langle g_t, x_\star \rangle = \text{tr}((\bar{g}_t \odot \bar{x}_\star) G). $$

This gives us the following linear (in G) semidefinite program reformulation of (11):

$$
\begin{align}
B_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}) &= \sup_{G \succcurlyeq 0} \sum_{t=1}^{T} \sum_{s=1}^{t-1} \gamma_{t,s} \text{tr}((\bar{g}_t \odot \bar{v}_s) G) - \sum_{t=1}^{T} \text{tr}((\bar{g}_t \odot \bar{x}_\star) G) \nonumber \\
&\quad \text{subject to: } \bar{\text{dir}}_t = \sum_{s=1}^{t} \eta_{t,s} \bar{g}_s + \sum_{s=1}^{t-1} \beta_{t,s} \bar{v}_s \text{ for } t=1, \dots, T-1, \nonumber \\
&\qquad \text{tr}((\bar{\text{dir}}_t \odot (\bar{v}_t - u)) G) \le 0 \text{ for all } t=1, \dots, T-1 \tag{12} \\
&\qquad \text{and } u \in \{\bar{x}_1, \bar{v}_1, \dots, \bar{v}_{T-1}, \bar{x}_\star\}, \nonumber \\
&\qquad \text{tr}(((u-v) \odot (u-v)) G) \le D^2 \nonumber \\
&\qquad \text{for } u, v \in \{\bar{x}_1, \bar{v}_1, \dots, \bar{v}_{T-1}, \bar{x}_\star\}, \nonumber \\
&\qquad \text{tr}((\bar{g}_t \odot \bar{g}_t) G) \le L^2 \text{ for } t=1, \dots, T. \nonumber
\end{align}
$$

Note that the variable $d \in \mathbb{N}$ for the dimension of the feasible set $\mathcal{K}$ (and of all the gradients and vectors) that appears in (4) and (11) induces a constraint imposing that the rank of the matrix G is at most d. However, this constraint disappears when taking the supremum over all values of $d \in \mathbb{N}$.

As explained in Remark 3.1, for given numerical values of T, L, D and the algorithm parameters, solving the tractable convex problem (12) allows to get worst-case examples giving algorithm-dependent lower bounds on the worst-case regret. In order to obtain algorithm-dependent upper bounds on the worst-case regret, a natural procedure consists in formulating the Lagrange dual of (4) (which is also a semidefinite program; see, e.g., [30, 5]), whose feasible points naturally correspond to upper bounds on the regret. In this context, finding a proof consists in finding a feasible point to the dual problem [12]. In particular, it is useful to reformulate (12) as its Lagrange dual which is the following semidefinite problem:

$$
\begin{align}
B_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}) &= \inf_{\substack{\lambda^{\text{Lip}} \ge 0 \\ \lambda^{\text{Diam}} \ge 0 \\ \lambda^{\text{Brd}} \ge 0}} \sum_{t=1}^{T} \lambda_t^{\text{Lip}} L^2 + \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{v}_1, \dots, \bar{v}_{T-1}, \bar{x}_\star\}} \lambda_{\{u,v\}}^{\text{Diam}} D^2 \nonumber \\
&\quad \text{subject to: } S(\eta, \beta, \gamma; \lambda) \ge 0, \nonumber \\
&\quad \text{where } S(\eta, \beta, \gamma; \lambda) = \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{v}_1, \dots, \bar{v}_{T-1}, \bar{x}_\star\}} \lambda_{\{u,v\}}^{\text{Diam}} ((u-v) \odot (u-v)) \nonumber \\
&\quad \qquad + \sum_{t=1}^{T} \lambda_t^{\text{Lip}} (\bar{g}_t \odot \bar{g}_t) + \sum_{t=1}^{T-1} \sum_{u \in \{\bar{x}_1, \bar{v}_1, \dots, \bar{v}_{T-1}, \bar{x}_\star\}} \lambda_{\bar{v}_t, u}^{\text{Brd}} (\bar{\text{dir}}_t \odot (\bar{v}_t - u)) \tag{13} \\
&\quad \qquad - \sum_{t=1}^{T} \sum_{s=1}^{t-1} \gamma_{t,s} (\bar{g}_t \odot \bar{v}_s) + \sum_{t=1}^{T} (\bar{g}_t \odot \bar{x}_\star), \nonumber
\end{align}
$$

$$ \text{and } \lambda = (\lambda^{\text{Lip}}, \lambda^{\text{Diam}}, \lambda^{\text{Brd}}) \text{ and } \bar{\text{dir}}_t = \sum_{s=1}^{t} \eta_{t,s} \bar{g}_s + \sum_{s=1}^{t-1} \beta_{t,s} \bar{v}_s \text{ for } t=1, \dots, T-1. $$
## B.2 Joint stepsize optimization; semidefinite formulation of (5)

As mention in Section 3.2, a natural path forward is to use (4) to obtain worst-case optimal algorithms. That is, by solving

$$ \min_{\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}} \left\{ B_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}) \text{ s.t. } \sum_{s=1}^{t-1} \gamma_{t,s} \le 1, \gamma_{t,s} \ge 0 \right\}. $$

Using (13), this problem can be reformulated as a linear optimization problem with a bilinear matrix inequality constraint, which are unfortunately NP-hard in general [29]. The bilinearity in the matrix inequality $S(\eta, \beta, \gamma; \lambda) \ge 0$ is due to terms $\lambda_{\bar{v}_t, u}^{\text{Brd}} (\text{dir}_t \odot (\bar{v}_t - u))$ appearing in the definition of $S(\eta, \beta, \gamma; \lambda)$ which are bilinear in $(\eta, \beta)$ and $\lambda^{\text{Brd}}$. A classical approach (see [10]) to circumvent bilinearity in the matrix inequalities consist in using convex relaxation of the problem, which works well in our case as we explained in Section 3.2 and we detail in this section. Note that another possible approach (see [7]) consist in adapting a branch-and-bound algorithm to compute the best possible regret guarantee by: (i) dividing the search space into regions, (ii) computing upper and lower bounds on the best possible regret guarantee for each region via convex relaxations of the problem, (iii) discarding regions whose lower bound is larger than the best (across all regions) current upper bound as those regions cannot contain the optimal point / value, and (iv) repeating steps (i)–(iii) with the remaining regions until convergence. However, this branch-and-bound approach is numerically more costly and is not necessary when the direct convex relaxation method works.

For this reason, we propose a slight relaxation of $B_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s})$ which corresponds to remove a few constraints from (4) (that we numerically observed to be inactive). More precisely, this relaxation is obtained through: (i) we observe that all $x_t$ for $t = 2, \dots, T$ are in the convex body of $x_1, v_1, \dots, v_{T-1}$, and thus the domain constraint for $K$ are only covered by vectors $x_1, v_1, \dots, v_{T-1}, x_\star$; (ii) we only keep the boundary constraints corresponding to the optimality of $v_t$ compared with $v_{t+1}, \dots, v_{T-1}, x_\star$: this leads to the definition of $W_T(\{\eta_{t,s}, \beta_{t,s}, \gamma_{t,s}\}_{t,s})$ in Section 3.2 with $W_T(\{\eta_{t,s}, \beta_{t,s}, \gamma_{t,s}\}_{t,s}) \ge B_T(\{\eta_{t,s}, \beta_{t,s}, \gamma_{t,s}\}_{t,s})$. In terms of the semidefinite program reformulation (13) of (4), this relaxation corresponds to impose $\lambda_{\bar{v}_t, u}^{\text{Brd}} = 0$ for all $t = 1, \dots, T-1$ and $u \in \{\bar{v}_{t+1}, \dots, \bar{v}_{T-1}, \bar{x}_\star\}$, giving us:

$$ W_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}) = \inf_{\substack{\lambda^{\text{Lip}} \ge 0 \\ \lambda^{\text{Diam}} \ge 0 \\ \lambda^{\text{Brd}} \ge 0}} \sum_{t=1}^{T} \lambda_t^{\text{Lip}} L^2 + \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{v}_1, \dots, \bar{v}_{T-1}, \bar{x}_\star\}} \lambda_{\{u,v\}}^{\text{Diam}} D^2 $$

subject to: $S(\eta, \beta, \gamma; \lambda) \ge 0$,

$$ \text{where } S(\eta, \beta, \gamma; \lambda) = \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{v}_1, \dots, \bar{v}_{T-1}, \bar{x}_\star\}} \lambda_{\{u,v\}}^{\text{Diam}} ((u-v) \odot (u-v)) \\ + \sum_{t=1}^{T} \lambda_t^{\text{Lip}} (\bar{g}_t \odot \bar{g}_t) + \sum_{t=1}^{T-1} \sum_{u \in \{\bar{v}_{t+1}, \dots, \bar{v}_{T-1}, \bar{x}_\star\}} \lambda_{\bar{v}_t, u}^{\text{Brd}} (\text{dir}_t \odot (\bar{v}_t - u)) \\ - \sum_{t=1}^{T} \sum_{s=1}^{t-1} \gamma_{t,s} (\bar{g}_t \odot \bar{v}_s) + \sum_{t=1}^{T} (\bar{g}_t \odot \bar{x}_\star), \quad (14) $$

$$ \text{and } \lambda = (\lambda^{\text{Lip}}, \lambda^{\text{Diam}}, \lambda^{\text{Brd}}) \text{ and } \bar{\text{dir}}_t = \sum_{s=1}^{t} \eta_{t,s} \bar{g}_s + \sum_{s=1}^{t-1} \beta_{t,s} \bar{v}_s \text{ for } t = 1, \dots, T-1. $$

Using this problem, the joint minimization problem

$$ \min_{\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}} \left\{ W_T(\{(\eta_{t,s}, \beta_{t,s}, \gamma_{t,s})\}_{t,s}) \text{ s.t. } \sum_{s=1}^{t-1} \gamma_{t,s} \le 1, \gamma_{t,s} \ge 0 \right\} \quad (5) $$

is still a priori a linear optimization problem with a bilinear matrix inequality constraint (which recall are unfortunately NP-hard in general [29]) due to the presence in $S(\eta, \beta, \gamma; \lambda)$ of the bilinear
(in $(\eta, \beta)$ and $\lambda^{\text{Brd}}$) term (where abusing notations, we write $\bar{v}_T = \bar{x}_\star$):

$$ \sum_{t=1}^{T-1} \sum_{u \in \{\bar{v}_{t+1}, \dots, \bar{v}_{T-1}, \bar{x}_\star\}} \lambda_{\bar{v}_t, u}^{\text{Brd}} (\bar{\text{dir}}_t \odot (\bar{v}_t - u)) \\ = \sum_{t=1}^{T-1} \sum_{s=1}^{t} \sum_{j=t+1}^{T} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} \eta_{t,s} (\bar{g}_s \odot (\bar{v}_t - \bar{v}_j)) + \sum_{t=1}^{T-1} \sum_{s=1}^{t-1} \sum_{j=t+1}^{T} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} \beta_{t,s} (\bar{v}_s \odot (\bar{v}_t - \bar{v}_j)). $$

However, we remark that:

$$ \sum_{t=1}^{T-1} \sum_{s=1}^{t} \sum_{j=t+1}^{T} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} \eta_{t,s} (\bar{g}_s \odot (\bar{v}_t - \bar{v}_j)) \\ = \sum_{t=1}^{T-1} \sum_{s=1}^{t} \left( \sum_{j=t+1}^{T} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} \eta_{t,s} \right) (\bar{g}_s \odot \bar{v}_t) - \sum_{j=1}^{T} \sum_{s=1}^{j} \sum_{t=s}^{j-1} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} \eta_{t,s} (\bar{g}_s \odot \bar{v}_j) \\ = \sum_{t=1}^{T-1} \sum_{s=1}^{t} \left( \sum_{j=t+1}^{T} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} \eta_{t,s} - \sum_{j=s}^{t-1} \lambda_{\bar{v}_j, \bar{v}_t}^{\text{Brd}} \eta_{j,s} \right) (\bar{g}_s \odot \bar{v}_t), $$

and:

$$ \sum_{t=1}^{T-1} \sum_{s=1}^{t-1} \sum_{j=t+1}^{T} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} \beta_{t,s} (\bar{v}_s \odot (\bar{v}_t - \bar{v}_j)) \\ = \sum_{t=1}^{T-1} \sum_{s=1}^{t-1} \left( \sum_{j=t+1}^{T} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} \beta_{t,s} \right) (\bar{v}_s \odot \bar{v}_t) - \sum_{j=1}^{T} \sum_{s=1}^{j-1} \sum_{t=s+1}^{j-1} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} \beta_{t,s} (\bar{v}_s \odot \bar{v}_j) \\ = \sum_{t=1}^{T-1} \sum_{s=1}^{t-1} \left( \sum_{j=t+1}^{T} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} \beta_{t,s} - \sum_{j=s+1}^{t-1} \lambda_{\bar{v}_j, \bar{v}_t}^{\text{Brd}} \beta_{j,s} \right) (\bar{v}_s \odot \bar{v}_t). $$

This motivates the following change of variables in (14) which allows to recast (5) as a linear convex semidefinite program again: $\eta, \beta$ and $\lambda^{\text{Brd}}$ are replaced by:

$$ B_{t,s} = \eta_{t,s} \sum_{j=t+1}^{T} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} - \sum_{j=s}^{t-1} \eta_{j,s} \lambda_{\bar{v}_j, \bar{v}_t}^{\text{Brd}} \quad \forall 1 \le s \le t \le T, \qquad (15) $$

$$ C_{t,s} = \beta_{t,s} \sum_{j=t+1}^{T} \lambda_{\bar{v}_t, \bar{v}_j}^{\text{Brd}} - \sum_{j=s+1}^{t-1} \beta_{j,s} \lambda_{\bar{v}_j, \bar{v}_t}^{\text{Brd}} \quad \forall 1 \le s < t \le T. $$

Note that the other variables, that is, $\gamma$, $\lambda^{\text{Lip}}$ and $\lambda^{\text{Diam}}$, are left unchanged. Also note that from the definition of $B_{t,s}$ and $C_{t,s}$, we get that they must satisfy the constraints $\sum_{s=t}^{T} B_{t,s} = 0$ and $\sum_{s=t+1}^{T} B_{t,s} = 0$ for all $t = 1, \dots, T$. Thus, (5) can be reformulated as the following linear convex
semidefinite program:

$$ \inf_{\{(B_{t,s}, C_{t,s}, \gamma_{t,s})\}_{t,s}} \inf_{\substack{\lambda^{\text{Lip}} \ge 0 \\ \lambda^{\text{Diam}} \ge 0 \\ \lambda^{\text{Brd}} \ge 0}} \sum_{t=1}^{T} \lambda_t^{\text{Lip}} L^2 + \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{v}_1, \dots, \bar{v}_{T-1}, \bar{x}_*\}} \lambda_{\{u,v\}}^{\text{Diam}} D^2 $$

subject to: $S(B, C, \gamma; \lambda) \ge 0$,

$$ \sum_{t=s}^{T} B_{t,s} = 0 \quad \text{and} \quad \sum_{t=s+1}^{T} C_{t,s} = 0 \quad \text{for } s=1, \dots, T $$

$$ \text{where } S(B, C, \gamma; \lambda) = \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{v}_1, \dots, \bar{v}_{T-1}, \bar{x}_*\}} \lambda_{\{u,v\}}^{\text{Diam}} ((u-v) \odot (u-v)) \quad (16) $$

$$ \begin{aligned} & + \sum_{t=1}^{T} \lambda_t^{\text{Lip}} (\bar{g}_t \odot \bar{g}_t) \\ & + \sum_{t=1}^{T} \sum_{s=1}^{T} B_{t,s} (\bar{g}_s \odot \bar{v}_t) + \sum_{t=1}^{T} \sum_{s=1}^{t-1} C_{t,s} (\bar{v}_s \odot \bar{v}_t) \\ & - \sum_{t=1}^{T} \sum_{s=1}^{t-1} \gamma_{t,s} (\bar{g}_t \odot \bar{v}_s) + \sum_{t=1}^{T} (\bar{g}_t \odot \bar{x}_*) \end{aligned} $$

and $\lambda = (\lambda^{\text{Lip}}, \lambda^{\text{Diam}}, \lambda^{\text{Brd}})$.

Note that there are $T^2$ of the variables $\{B_{t,s}, C_{t,s}\}_{t,s}$, while there was $(T-1)^2 + T(T-1)/2$ of the variables $\eta, \beta$ and $\lambda^{\text{Brd}}$; this is because problem (14) was overparametrized. Thus, when inverting the change of variables, for given values of $\{B_{t,s}, C_{t,s}\}_{t,s}$, there exist several solutions for $\eta, \beta$ and $\lambda^{\text{Brd}}$ solving (15). We propose one solution for inverting (15) giving simple algorithms: (i) we choose $\eta_{t,s} = 1$ for all $1 \le s \le t \le T$, (ii) we solve for $\lambda^{\text{Brd}}$ using the first line of (15) substituting in the value of $\eta$, (iii) we solve for $\beta$ using the second line of (15) substituting in the value of $\lambda^{\text{Brd}}$; that is:

$$ \eta_{t,s} = 1 \qquad \forall 1 \le s \le t \le T, $$

$$ \lambda_{\bar{v}_s, \bar{v}_t}^{\text{Brd}} = B_{t,s+1} - B_{t,s} \qquad \forall 1 \le s < t \le T, $$

$$ \beta_{t,s} = \frac{1}{B_{t,t}} \left( C_{t,s} + \sum_{j=s+1}^{t-1} \beta_{j,s} \lambda_{\bar{v}_j, \bar{v}_t}^{\text{Brd}} \right) \qquad \forall 1 \le s < t \le T. \quad (17) $$

## B.3 Examples of numerically optimized stepsize patterns

The following list provides numerical examples of optimal parameter values for the online Frank-Wolfe-type algorithm (1) for $T = 2, \dots, 6$, together with their optimal worst-case regret guarantees. (Note that for $T = 1$ the algorithm is necessarily trivial as we can only play the uninformed choice $x_1$.) Those values were obtained by numerically solving the relaxed linear convex semidefinite program (16) with $L = D = 1$. Note that we chose the parameters given by (17) and thus we do not report the values of $\eta_{t,s} = 1$ for $1 \le s \le t \le T-1$. We present the values of the parameters $\{\gamma_{t,s}\}_{1 \le s < t \le T}$ and $\{\beta_{t,s}\}_{1 \le s < t \le T-1}$ as square $T \times T$ and $(T-1) \times (T-1)$ matrices, respectively, where the values for out of range indices are left blank intentionally.

* For $T = 2$, we have $R_T \le B_2 \le 1.7321$ and:

$$ [\gamma_{t,s}] = \begin{bmatrix} 0.5 \end{bmatrix}, \quad [\beta_{t,s}] = \begin{bmatrix} \end{bmatrix}. $$

* For $T = 3$, we have $R_T \le B_3 \le 2.3421$ and:

$$ [\gamma_{t,s}] = \begin{bmatrix} 0.5 & 0.3764 \\ 0.3118 & 0.3764 \end{bmatrix}, \quad [\beta_{t,s}] = \begin{bmatrix} -0.1099 \end{bmatrix}. $$
• For $T = 4$, we have $R_T \le B_4 \le 2.9029$ and:

$$ [\gamma_{t,s}] = \begin{bmatrix} 0.5 \\ 0.3133 & 0.3734 \\ 0.1843 & 0.2197 & 0.4116 \end{bmatrix}, \quad [\beta_{t,s}] = \begin{bmatrix} 0.1961 \\ -0.4249 & 0.1465 \end{bmatrix}. $$

• For $T = 5$, we have $R_T \le B_5 \le 3.4217$ and:

$$ [\gamma_{t,s}] = \begin{bmatrix} 0.5 \\ 0.3067 & 0.3866 \\ 0.2124 & 0.2677 & 0.3075 \\ 0.1201 & 0.1514 & 0.1739 & 0.4345 \end{bmatrix}, $$

$$ [\beta_{t,s}] = \begin{bmatrix} 0.5649 \\ -0.2595 & 0.212 \\ -0.6282 & -0.253 & 0.3473 \end{bmatrix}. $$

• For $T = 6$, we have $R_T \le B_6 \le 3.917$ and:

$$ [\gamma_{t,s}] = \begin{bmatrix} 0.5 \\ 0.3068 & 0.3863 \\ 0.2101 & 0.2646 & 0.3151 \\ 0.1406 & 0.177 & 0.2108 & 0.3309 \\ 0.0784 & 0.0985 & 0.1174 & 0.1842 & 0.4432 \end{bmatrix}, $$

$$ [\beta_{t,s}] = \begin{bmatrix} 0.5856 \\ -0.0481 & 0.4808 \\ -0.5053 & -0.0949 & 0.3922 \\ -0.7675 & -0.4251 & -0.001 & 0.3876 \end{bmatrix}. $$

Note that the structure of the values of {$\gamma_{t,s}$}$_{1 \le s < t \le T}$ implies that we get the update rule $x_{t+1} = \sigma_t v_t + (1 - \sigma_t)x_t$ with $\sigma_t = \gamma_{t+1,t}$ in (1).

## B.4 Joint stepsize optimization with multiple linear optimization rounds per iteration

As explained in Section 3.3, the joint stepsize optimization method for online Frank-Wolfe-type algorithms explained in Section 3.2 and in Appendix B.2 can be easily adapted to the variant algorithm with multiple linear optimization rounds per iteration defined in (6). In this section, we explain the details on how to make this adaptation.

We first define the worst-case regret for (6) with given parameters by simply replacing (1) by (6) in (4):

$$ \tilde{B}_T(\{(\eta_{t,k,s}, \beta_{t,k,s,j}, \gamma_{t,s,j})\}_{t,k,s,j}) \triangleq \sup_{\substack{K, \{\ell_t\}_{t=1,\dots,T} \\ x_*, \{x_t\}_{t=1,\dots,T} \\ d \in \mathbb{N}}} R_T(x_1, \dots, x_T; x_*) $$

subject to: $\ell_t$ is convex and L-Lipschitz for $t = 1, \dots, T$, $K$ is a non-empty closed convex set of $\mathbb{R}^d$, $\text{Diam}(K) \le D$, $\{x_t\}_{t=1,\dots,T}$ is generated by (6).

To adapt the joint minimization problem (5) to (6), we need to replace $W_T(\{\eta_{t,s}, \beta_{t,s}, \gamma_{t,s}\}_{t,s})$ from Section 3.3 by some relaxation of the problem above. More precisely, this relaxation is obtained through: (i) we observe that all $x_t$ for $t = 2, \dots, T$ are in the convex body of $x_1, v_{1,1}, \dots, v_{1,r}, \dots, v_{T-1,r}$, and thus the domain constraint for $K$ are only covered by vectors $x_1, v_{1,1}, \dots, v_{T-1,r}$, $x_*$; (ii) we only keep the boundary constraints corresponding to the optimality
of $v_{t,k}$ compared with $v_{t,k+1}, \dots, v_{t,r}, v_{t+1,1}, \dots, v_{T-1,r}, x_\star$:

$$
\begin{align}
\tilde{W}_T(\{(\eta_{t,k,s}, \beta_{t,k,s,j}, \gamma_{t,s,j})\}_{t,k,s,j}) &\triangleq \sup_{\substack{\{g_t\}_{t=1,\dots,T}, x_\star \\ \{(x_t, v_{t,k}, \text{dir}_{t,k})\}_{t=1,\dots,T, k=1,\dots,r} \\ d \in \mathbb{N}}} \sum_{t=1}^T \langle g_t, x_t - x_\star \rangle \\
&\quad \text{subject to: } \{(x_t, \text{dir}_{t,1}, \dots, \text{dir}_{t,r})\}_{t=1,\dots,T} \text{ compatible with (6),} \nonumber \\
&\quad \langle -\text{dir}_{t,k}, u - v_{t,k} \rangle \le 0 \text{ for all } t=1, \dots, T-1 \text{ and } k=1, \dots, r \nonumber \\
&\quad \text{and } u \in \{v_{t,k+1}, \dots, v_{t,r}, v_{t+1,1}, \dots, v_{T-1,r}, x_\star\}, \nonumber \\
&\quad \text{Diam}(\{x_1, v_{1,1}, \dots, v_{1,r}, \dots, v_{T-1,r}, x_\star\}) \le D, \nonumber \\
&\quad \|g_t\| \le L \text{ for } t=1, \dots, T, \tag{18}
\end{align}
$$

where the line “{$\{(x_t, \text{dir}_{t,1}, \dots, \text{dir}_{t,r})\}_{t=1,\dots,T}$ compatible with (6)” means that $x_t$ and $\text{dir}_{t,k}$ can be substituted by their expressions in (6), leading to $\tilde{W}_T(\{(\eta_{t,k,s}, \beta_{t,k,s,j}, \gamma_{t,s,j})\}_{t,k,s,j}) \ge \tilde{B}_T(\{(\eta_{t,k,s}, \beta_{t,k,s,j}, \gamma_{t,s,j})\}_{t,k,s,j})$.

We can now formulate the adaptation of the joint minimization problem (5) to (6), giving us:

$$
\min_{\{(\eta_{t,k,s}, \beta_{t,k,s,j}, \gamma_{t,s,j})\}_{t,k,s,j}} \left\{ \tilde{W}_T(\{(\eta_{t,k,s}, \beta_{t,k,s,j}, \gamma_{t,s,j})\}_{t,k,s,j}) \text{ s.t. } \sum_{s=1}^{t-1} \sum_{j=1}^{r} \gamma_{t,s,j} \le 1, \gamma_{t,s,j} \ge 0 \right\}. \quad (19)
$$

In the remaining of this section, we will need the following notations. For integers $t, s \in \mathbb{N}$ and $k, j \in [1, r]$, we write $(t, k) \le_{\text{lex}} (s, j)$ if $t < s$, or $t = s$ and $k < j$. We write $(t, k) <_{\text{lex}} (s, j)$ if $(t, k) \le_{\text{lex}} (s, j)$ and $(t, k) \neq (s, j)$.

Then, the method detailed in Appendix B.1 and B.2 can be immediately adapted to reformulate (18)
as a linear convex semidefinite program and then form its Lagrange dual, giving us:

$$
\tilde{W}_T(\{(\eta_{t,k,s}, \beta_{t,k,s,j}, \gamma_{t,s,j})\}_{t,k,s,j}) = \inf_{\substack{\lambda^{\text{Lip}} \ge 0 \\ \lambda^{\text{Diam}} \ge 0 \\ \lambda^{\text{Brd}} \ge 0}} \sum_{t=1}^{T} \lambda_t^{\text{Lip}} L^2 + \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{v}_{1,1}, \dots, \bar{v}_{T-1,r}, \bar{x}_\star\}} \lambda_{\{u,v\}}^{\text{Diam}} D^2 \\
\text{subject to: } S(\eta, \beta, \gamma; \lambda) \ge 0,
$$

$$
\begin{align*}
\text{where } S(\eta, \beta, \gamma; \lambda) &= \sum_{t=1}^{T} \lambda_t^{\text{Lip}} (\bar{g}_t \odot \bar{g}_t) + \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{v}_{1,1}, \dots, \bar{v}_{T-1,r}, \bar{x}_*\}} \lambda_{\{u,v\}}^{\text{Diam}} ((u-v) \odot (u-v)) \\
&\quad + \sum_{t=1}^{T-1} \sum_{k=1}^{r} \sum_{u \in \{\bar{v}_{t,k+1}, \dots, \bar{v}_{T-1,r}, \bar{x}_*\}} \lambda_{\bar{v}_{t,k}, u}^{\text{Brd}} (\bar{\text{dir}}_{t,k} \odot (\bar{v}_{t,k} - u)) \\
&\quad - \sum_{t=1}^{T} \sum_{s=1}^{t-1} \sum_{k=1}^{r} \gamma_{t,s,k} (\bar{g}_t \odot \bar{v}_{s,k}) + \sum_{t=1}^{T} (\bar{g}_t \odot \bar{x}_*),
\end{align*}
$$

$$
\text{and } \lambda = (\lambda^{\text{Lip}}, \lambda^{\text{Diam}}, \lambda^{\text{Brd}}) \text{ and } \bar{\text{dir}}_{t,k} = \sum_{s=1}^{t} \eta_{t,k,s} \bar{g}_s + \sum_{(s,j)<_{\text{lex}}(t,k)} \beta_{t,k,s,j} \bar{v}_{s,j} \text{ for } t=1, \dots, T-1. \tag{20}
$$

Substituting (20) in (19), we get a linear optimization problem with a bilinear matrix inequality constraint, which we reformulate as a linear convex semidefinite program using a change of variables
similar to that of (15): $(\eta, \beta, \lambda^{\text{Brd}})$ are replaced by the variables (with the convention $\bar{v}_{T,1} = \bar{x}_{\star}$):

$$
\begin{align}
B_{t,k,s} &= \eta_{t,k,s} \sum_{(t,k) < \text{lex}(m,j) \le \text{lex}(T,1)} \lambda_{\bar{v}_{t,k}, \bar{v}_{m,j}}^{\text{Brd}} - \sum_{(s,1) \le \text{lex}(m,i) < \text{lex}(t,k)} \eta_{m,i,s} \lambda_{\bar{v}_{m,i}, \bar{v}_{t,k}}^{\text{Brd}} \nonumber \\
&\quad \forall (1,1) \le \text{lex}(s,j) \le \text{lex}(t,k) \le \text{lex}(T,1), \tag{21} \\
C_{t,k,s,j} &= \beta_{t,k,s,j} \sum_{(t,k) < \text{lex}(m,i) \le \text{lex}(T,1)} \lambda_{\bar{v}_{t,k}, \bar{v}_{m,i}}^{\text{Brd}} - \sum_{(s,j) \le \text{lex}(m,i) < \text{lex}(t,k)} \beta_{m,i,s,j} \lambda_{\bar{v}_{m,i}, \bar{v}_{t,k}}^{\text{Brd}} \nonumber \\
&\quad \forall (1,1) \le \text{lex}(s,j) < \text{lex}(t,k) \le \text{lex}(T,1). \nonumber
\end{align}
$$

Finally, this change of variables gives us the following linear convex semidefinite program which is
the adaptation of (16) for (6):

$$
\begin{align*}
& \inf_{\{(B_{t,s,k}, C_{t,s,k}, \gamma_{t,s,k})\}_{t,s,k}} \inf_{\substack{\lambda^{\text{Lip}} \ge 0 \\ \lambda^{\text{Diam}} \ge 0 \\ \lambda^{\text{Brd}} \ge 0}} \sum_{t=1}^{T} \lambda_t^{\text{Lip}} L^2 + \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{v}_{1,1}, \dots, \bar{v}_{T-1,r}, \bar{x}_*\}} \lambda_{\{u,v\}}^{\text{Diam}} D^2 \\
& \text{subject to: } S(B, C, \gamma; \lambda) \ge 0, \\
& \qquad \sum_{(s,1) \le \text{lex}(t,k) \le \text{lex}(T,1)} B_{t,k,s} = 0 \quad \text{for } s = 1, \dots, T, \\
& \qquad \sum_{(s,j) < \text{lex}(t,k) \le \text{lex}(T,1)} C_{t,k,s,j} = 0 \quad \text{for } (1,1) \le \text{lex}(s,j) < \text{lex}(T,1),
\end{align*}
$$

$$
\begin{align*}
\text{where } S(B, C, \gamma; \lambda) &= \frac{1}{2} \sum_{\substack{u, v \in \{\bar{x}_1, \bar{v}_{1,1}, \dots, \bar{v}_{T-1,r}, \bar{x}_*\}}}
  &\lambda_{\{u,v\}}^{\text{Diam}} ((u-v) \odot (u-v)) \\
  &+ \sum_{t=1}^{T} \lambda_t^{\text{Lip}} (\bar{g}_t \odot \bar{g}_t) \\
  &+ \sum_{\substack{(1,1) \le \text{lex}(t,k) \le \text{lex}(T,1) \\ 1 \le s \le t}} B_{t,s} (\bar{g}_s \odot \bar{v}_{t,k}) \\
  &+ \sum_{\substack{(1,1) \le \text{lex}(t,k) \le \text{lex}(T,1) \\ (1,1) \le \text{lex}(s,j) < \text{lex}(t,k)}} C_{t,s} (\bar{v}_{s,j} \odot \bar{v}_{t,k}) \\
  &- \sum_{t=1}^{T} \sum_{s=1}^{t-1} \sum_{k=1}^{r} \gamma_{t,s,k} (\bar{g}_t \odot \bar{v}_{s,k}) + \sum_{t=1}^{T} (\bar{g}_t \odot \bar{x}_*) ,
\end{align*}
$$

and $\lambda = (\lambda^{\text{Lip}}, \lambda^{\text{Diam}}, \lambda^{\text{Brd}})$.

Note that, as in Appendix B.2, this change of variables is not one-to-one: for given values of *B* and *C*
there exist several possible values of $(\eta, \beta, \lambda^{\text{Brd}})$ satisfying (21). Nevertheless, as in Appendix B.2,
imposing $\eta_{t,k,s} = 1$ for all $t, k, s$, we get a unique solution by solving the part of (21) for $\lambda^{\text{Brd}}$, and
then solving the second part of (21) for $\beta$.

B.5 Design and optimization of the proof of Theorem 2.1

As we explained in Section 3.4, we used variants of (4) to design the potential-based proof of Sec-
tion 2 and to jointly optimize the algorithm and potential parameters for this proof. We outlined our
general method in Section 3.4, and here we present the details for rewriting (8) as a semidefinite
program, numerically solving it, and then for obtaining the optimal parameters given at the end of
Section 3.4.
We start by restating the 1-iteration inner maximization problem from (8) which upper bounds the potential increase for an abstract t and for given pontential parameters a and b:

$$
B_t(\eta, \sigma, a, b) \triangleq \sup_{\substack{\mathcal{K}, d \in \mathbb{N} \\ g_t, G_{t-1} \\ x_1, x_t, v_t, x_{t+1} \\ y_t, y_{t+1}}} \phi_t - \phi_{t-1}
$$

subject to: $\phi_t - \phi_{t-1}$ is generated from $\{x_t, x_{t+1}, y_t, y_{t+1}, g_t, G_{t-1}, x_1\}$ by (7),
$$ \|g_t\| \le L, $$
$\mathcal{K}$ is a non-empty closed convex set of $\mathbb{R}^d$,
$\text{Diam}(\{x_1, x_t, v_t, x_{t+1}, y_t, y_{t+1}\}) \le D$,
$(x_{t+1}, v_t)$ are generated from $\{x_1, x_t, g_t, G_{t-1}\}$ by Algorithm 1,
$y_t$ and $y_{t+1}$ are generated from $\{x_1, g_t, G_{t-1}\}$ by FTRL.
(22)

In particular, we have that (8) is equal to the minimum of $B_t(\eta, \sigma, a, b)$ over $a \ge 0$ and $b \ge 0$.

Note that $x_{t+1}$ can be removed from the boundary conditions as it is combination of $v_t$ and $x_t$.
Substituting in the definition of $\phi_t - \phi_{t-1}$ and the optimality condition of $v_t$, $y_t$ and $y_{t+1}$, we can
get rid of $\mathcal{K}$, and we obtain the following self-containing reformulation of (22):

$$
B_t(\eta, \sigma, a, b) = \sup_{\substack{d \in \mathbb{N} \\ g_t, G_{t-1} \\ x_1, x_t, v_t, \bar{x}_{t+1} \\ y_t, y_{t+1}}} \left\{
\begin{aligned}
& \langle g_t, x_t - y_{t+1} \rangle + \langle G_{t-1}, y_t - y_{t+1} \rangle \\
& + a(\|x_{t+1} - y_{t+1}\|^2 - \|x_t - y_t\|^2) \\
& + b \eta (\langle G_{t-1} + g_t, x_{t+1} - y_{t+1} \rangle - \langle G_{t-1}, x_t - y_t \rangle) \\
& + \frac{b}{2} (\|x_{t+1} - x_1\|^2 - \|y_{t+1} - x_1\|^2 \\
& \qquad \qquad - \|x_t - x_1\|^2 + \|y_t - x_1\|^2) \\
& + \frac{1}{2\eta} (\|y_t - x_1\|^2 - \|y_{t+1} - x_1\|^2)
\end{aligned}
\right\}
$$

subject to: $\|g_t\| \le L$,
$$
\text{Diam}(\{x_1, x_t, v_t, y_t, y_{t+1}\}) \le D,
$$
$$
x_{t+1} = \sigma v_t + (1 - \sigma)x_t,
$$
$$
\langle \eta G_{t-1} + (y_t - x_1), y_t - u \rangle \le 0 \text{ for } u \in \{x_t, v_t, y_{t+1}\},
$$
$$
\langle \eta (G_{t-1} + g_t) + (y_{t+1} - x_1), y_{t+1} - u \rangle \le 0 \text{ for } u \in \{x_t, v_t, y_t\},
$$
$$
\langle \eta (G_{t-1} + g_t) + (x_t - x_1), v_t - u \rangle \le 0 \text{ for } u \in \{x_t, y_t, y_{t+1}\}.
\tag{23}
$$

Without loss of generality, we assume that $x_1 = 0$. Then, using the same method as detailed in Appendix B.1, we can reformulate this problem as a linear convex semidefinite program. To this end, we do a change of variables in (23), replacing the gradients $G_{t-1}$ and $g_t$ and the vectors $x_t$, $v_t$, $y_t$ and $y_{t+1}$ by their Gram matrix $H$. Let $e_1, \dots, e_6$ be the standard basis vectors of $\mathbb{R}^6$. Define $\bar{g}_t = e_1$, $\bar{G}_{t-1} = e_2$, $\bar{x}_t = e_3$, $\bar{v}_t = e_4$, $\bar{y}_t = e_5$ and $\bar{y}_{t+1} = e_6$. Thus, we have $\bar{g}_t^T H \bar{x}_t = \langle g_t, x_t \rangle$, and similarly with the others gradients / vectors. Define the matrix $A_{\text{obj}}(\eta, \sigma, a, b)$, which is linear in $(a, b)$ but not in $(\eta, \sigma, a, b)$, as:

$$
\begin{align}
A_{\text{obj}}(\eta, \sigma, a, b) \triangleq & \bar{g}_t \odot (\bar{x}_t - \bar{y}_{t+1}) + \bar{G}_{t-1} \odot (\bar{y}_t - \bar{y}_{t+1}) \nonumber \\
& + a((\bar{x}_{t+1} - \bar{y}_{t+1}) \odot (\bar{x}_{t+1} - \bar{y}_{t+1}) - (\bar{x}_t - \bar{y}_t) \odot (\bar{x}_t - \bar{y}_t)) \nonumber \\
& + b \eta ((\bar{G}_{t-1} + \bar{g}_t) \odot (\bar{x}_{t+1} - \bar{y}_{t+1}) - \bar{G}_{t-1} \odot (\bar{x}_t - \bar{y}_t)) \nonumber \\
& + \frac{b}{2} (\bar{x}_{t+1} \odot \bar{x}_{t+1} - \bar{y}_{t+1} \odot \bar{y}_{t+1} - \bar{x}_t \odot \bar{x}_t + \bar{y}_t \odot \bar{y}_t) \nonumber \\
& + \frac{1}{2\eta} (\bar{y}_t \odot \bar{y}_t - \bar{y}_{t+1} \odot \bar{y}_{t+1}), \tag{24}
\end{align}
$$
where $\bar{x}_{t+1} = \sigma\bar{v}_t + (1 - \sigma)\bar{x}_t$. Also define:

$$\mathrm{dir}_{\bar{v}_t} = \eta \bar{G}_{t-1} + \eta \bar{g}_t + \bar{x}_t,$$

$$\mathrm{dir}_{\bar{y}_t} = \eta \bar{G}_{t-1} + \bar{y}_t,$$

$$\mathrm{dir}_{\bar{y}_{t+1}} = \eta \bar{G}_{t-1} + \eta \bar{g}_t + \bar{y}_{t+1}.$$

With this change of variables, (23) can be reformulated as the following linear convex semidefinite program:

$$B_t(\eta, \sigma, a, b) = \sup_{H \succcurlyeq 0} \mathrm{tr}(A_{\mathrm{obj}} H)$$

$$
\begin{array}{ll}
\text{subject to:} & \mathrm{tr}((\bar{g}_t \odot \bar{g}_t) H) \le L^2, \\
& \mathrm{tr}((u \odot v) H) \le D^2 \quad \text{for } u, v \in \{\bar{x}_1, \bar{x}_t, \bar{v}_t, \bar{x}_{t+1}, \bar{y}_t, \bar{y}_{t+1}\}, \\
& \mathrm{tr}((\mathrm{dir}_u \odot (u-v)) H) \le 0 \quad \text{for } u \in \{\bar{v}_t, \bar{y}_t, \bar{y}_{t+1}\} \\
& \qquad \text{and } v \in \{\bar{x}_1, \bar{x}_t, \bar{v}_t, \bar{y}_t, \bar{y}_{t+1}\} \setminus \{u\}.
\end{array}
\qquad (25)
$$

We then reformulate (25) as its Lagrange dual, giving us:

$$B_t(\eta, \sigma, a, b) = \inf_{\substack{\lambda^{\mathrm{Lip}} \ge 0 \\ \lambda^{\mathrm{Diam}} \ge 0 \\ \lambda^{\mathrm{Brd}} \ge 0}} \lambda^{\mathrm{Lip}} L^2 + \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{x}_t, \bar{v}_t, \bar{y}_t, \bar{y}_{t+1}\}} \lambda_{\{u,v\}}^{\mathrm{Diam}} D^2$$

$$\text{subject to: } S(\eta, \sigma, a, b; \lambda) \succcurlyeq 0,$$

$$
\begin{array}{ll}
\text{where } S(\eta, \sigma, a, b; \lambda) = & \displaystyle \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{x}_t, \bar{v}_t, \bar{y}_t, \bar{y}_{t+1}\}} \lambda_{\{u,v\}}^{\mathrm{Diam}} ((u-v) \odot (u-v)) & (26) \\
& + \displaystyle \sum_{u \in \{\bar{v}_t, \bar{y}_t, \bar{y}_{t+1}\}} \sum_{v \in \{\bar{x}_1, \bar{x}_t, \bar{v}_t, \bar{y}_t, \bar{y}_{t+1}\} \setminus \{u\}} \lambda_{u,v}^{\mathrm{Brd}} (\mathrm{dir}_u \odot (u-v)) \\
& + \lambda^{\mathrm{Lip}} (\bar{g}_t \odot \bar{g}_t) - A_{\mathrm{obj}}(\eta, \sigma, a, b),
\end{array}
$$

$$\text{and } \lambda = (\lambda^{\mathrm{Lip}}, \lambda^{\mathrm{Diam}}, \lambda^{\mathrm{Brd}}).$$

Now, jointly minimizing (26) in $(a, b)$ and $\lambda$ for fixed $(\eta, \sigma)$, we get a reformulation of (8) as the following linear convex semidefinite program:

$$\inf_{\substack{a \ge 0, b \ge 0 \\ \lambda^{\mathrm{Lip}} \ge 0 \\ \lambda^{\mathrm{Diam}} \ge 0 \\ \lambda^{\mathrm{Brd}} \ge 0}} \inf_{\lambda^{\mathrm{Lip}} \ge 0} \lambda^{\mathrm{Lip}} L^2 + \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{x}_t, \bar{v}_t, \bar{y}_t, \bar{y}_{t+1}\}} \lambda_{\{u,v\}}^{\mathrm{Diam}} D^2$$

$$\text{subject to: } S(\eta, \sigma, a, b; \lambda) \succcurlyeq 0,$$

$$
\begin{array}{ll}
\text{where } S(\eta, \sigma, a, b; \lambda) = & \displaystyle \frac{1}{2} \sum_{u, v \in \{\bar{x}_1, \bar{x}_t, \bar{v}_t, \bar{y}_t, \bar{y}_{t+1}\}} \lambda_{\{u,v\}}^{\mathrm{Diam}} ((u-v) \odot (u-v)) & (27) \\
& + \displaystyle \sum_{u \in \{\bar{v}_t, \bar{y}_t, \bar{y}_{t+1}\}} \sum_{v \in \{\bar{x}_1, \bar{x}_t, \bar{v}_t, \bar{y}_t, \bar{y}_{t+1}\} \setminus \{u\}} \lambda_{u,v}^{\mathrm{Brd}} (\mathrm{dir}_u \odot (u-v)) \\
& + \lambda^{\mathrm{Lip}} (\bar{g}_t \odot \bar{g}_t) - A_{\mathrm{obj}}(\eta, \sigma, a, b),
\end{array}
$$

$$\text{and } \lambda = (\lambda^{\mathrm{Lip}}, \lambda^{\mathrm{Diam}}, \lambda^{\mathrm{Brd}}).$$

Note that the size of (27) does not depend on $T$, which allows efficient numerical solving even for large values of $T$. Numerically solving (27) gives us that for jointly optimal $(a, b)$ and $\lambda$ we have $b = 0$, $\lambda_{\{u,v\}}^{\mathrm{Diam}} = 0$ except for $\lambda_{\{\bar{x}_t, \bar{v}_t\}}^{\mathrm{Diam}} = a\sigma^2$, $\lambda_{u,v}^{\mathrm{Brd}} = 0$ except for $\lambda_{\bar{v}_t, \bar{y}_{t+1}}^{\mathrm{Brd}} = \lambda_{\bar{y}_{t+1}, \bar{v}_t}^{\mathrm{Brd}} = 2a\sigma$, $\lambda_{\bar{y}_{t+1}, \bar{y}_t}^{\mathrm{Brd}}$ and $\lambda_{\bar{y}_t, \bar{y}_{t+1}}^{\mathrm{Brd}} = 1 + \lambda_{\bar{y}_{t+1}, \bar{y}_t}^{\mathrm{Brd}}$. Following our notations from the proof of Lemma 2.2 (see Appendix A.1), we write $\lambda^g = \lambda^{\mathrm{Lip}}$ and $\tilde{\lambda} = \lambda_{\bar{y}_{t+1}, \bar{y}_t}^{\mathrm{Brd}}$. Note that the Cholesky decomposition of $S = S(\eta, \sigma, a, b; \lambda)$ encodes for the sum of squares tr($S H$) $\ge 0$ that must be added during the proof, where some of those squares can be interpreted as inequalities. The Cholesky decomposition of $S$
has 2 eigenvectors: the first eigenvector $\sqrt{\lambda^g} \bar{g}_t + \frac{1}{2\sqrt{\lambda^g}} (\bar{x}_t - \bar{y}_{t+1} + \tilde{\lambda}(\bar{y}_t - \bar{y}_{t+1}))$ encodes for the inequality

$$ \langle g_t, x_t - y_{t+1} + \tilde{\lambda}(y_t - y_{t+1}) \rangle \le \lambda^g \|g_t\|^2 + \frac{1}{4\lambda^g} \|x_t - y_{t+1} + \tilde{\lambda}(y_t - y_{t+1})\|^2, $$

while the second eigenvector indicates that the upper bound we get from the proof will contain minus a sum of squares, which is composed of the vectors $x_t, y_t$ and $y_{t+1}$. The values of all those Lagrange multipliers in $\lambda$ and the value of $S$ gives us what inequalities and in which quantity to use in the proof of Lemma 2.2. Thus, from all of this we can deduce the proof up to showing that $Q(\tilde{\lambda})$ is a sum of squares, where:

$$ Q(\tilde{\lambda}) \triangleq \frac{1 + 2\tilde{\lambda}}{2\eta} \|y_t - y_{t+1}\|^2 - \frac{1}{4\lambda^g} \|x_t - y_{t+1} + \tilde{\lambda}(y_t - y_{t+1})\|^2 \\ - a((1 - 2\sigma) \|x_t - y_{t+1}\|^2 - \|x_t - y_t\|^2). $$

Hence, all that is left to do is finding optimal $\eta, \sigma, a$ $\lambda^g$ such that there exists $\tilde{\lambda} \ge 0$ for which $Q(\tilde{\lambda})$ is a sum of squares and that minimizes the upper bound given by the proof on the regret of OFW $R_T$, which is:

$$ \frac{1}{2\eta} \|x_* - x_1\| + \sum_{t=1}^{T} B_t(\eta, \sigma, a, 0) \le \frac{1}{2\eta} D^2 + \lambda^g L^2 T + a\sigma^2 D^2 T. $$

Then, we rewrite $Q(\tilde{\lambda})$ as:

$$ \begin{aligned} Q(\tilde{\lambda}) &= \left( \frac{1 + 2\tilde{\lambda}}{2\eta} + a \right) \|y_t - y_{t+1}\|^2 + 2a\sigma \|x_t - y_{t+1}\|^2 + 2a \langle x_t - y_{t+1}, y_{t+1} - y_t \rangle \\ &\quad - \frac{1}{4\lambda^g} \|x_t - y_{t+1} + \tilde{\lambda}(y_t - y_{t+1})\|^2 \\ &= \left( \frac{1 + 2\tilde{\lambda}}{2\eta} + a - \frac{\tilde{\lambda}^2}{4\lambda^g} \right) \|y_t - y_{t+1}\|^2 + \left( 2a\sigma - \frac{1}{4\lambda^g} \right) \|x_t - y_{t+1}\|^2 \\ &\quad + \left( 2a + \frac{\tilde{\lambda}}{2\lambda^g} \right) \langle x_t - y_{t+1}, y_{t+1} - y_t \rangle. \end{aligned} $$

Thus, we get that $Q(\tilde{\lambda})$ is a sum of squares if and only if:

$$ \left\{ \begin{array}{l} \left( 2a\sigma - \frac{1}{4\lambda^g} \right) \ge 0 \\ \left( a + \frac{\tilde{\lambda}}{4\lambda^g} \right)^2 \le \left( 2a\sigma - \frac{1}{4\lambda^g} \right) \left( \frac{1 + 2\tilde{\lambda}}{2\eta} + a - \frac{\tilde{\lambda}^2}{4\lambda^g} \right), \end{array} \right. $$

where the second line is equivalent to:

$$ \left( \frac{a\sigma}{2\lambda^g} \right) \tilde{\lambda}^2 - \left( \frac{1}{\eta} \left( 2a\sigma - \frac{1}{4\lambda^g} \right) - \frac{a}{2\lambda^g} \right) \tilde{\lambda} + \left( \left( 1 - 2\sigma \right) a^2 + \frac{a}{4\lambda^g} - \frac{1}{2\eta} \left( 2a\sigma - \frac{1}{4\lambda^g} \right) \right) \le 0. \quad (28) $$

An elementary second order polynomial study indicates that this polynomial equation has no non-negative solution $\tilde{\lambda} \ge 0$ when the linear coefficient is positive, that is, we get the necessary condition:

$$ \left( \frac{1}{\eta} \left( 2a\sigma - \frac{1}{4\lambda^g} \right) - \frac{a}{2\lambda^g} \right) \ge 0. \quad (29) $$

When (29) holds, we can upper bound the constant coefficient of (28) by its leading order term $a^2$, giving us the (slightly) more conservative but easier no analyze condition:

$$ \left( \frac{a\sigma}{2\lambda^g} \right) \tilde{\lambda}^2 - \left( \frac{1}{\eta} \left( 2a\sigma - \frac{1}{4\lambda^g} \right) - \frac{a}{2\lambda^g} \right) \tilde{\lambda} + a^2 \le 0. \quad (30) $$
Moreover, when (29) holds, we have that (30) has a non-negative solution $\tilde{\lambda} \ge 0$ if and only if:

$$ \left( \frac{1}{\eta} \left( 2a\sigma - \frac{1}{4\lambda^g} \right) - \frac{a}{2\lambda^g} \right)^2 - 4 \left( \frac{a\sigma}{2\lambda^g} \right) a^2 \ge 0. $$

Hence, combining all of the above, we get that $Q(\tilde{\lambda})$ is a sum of squares whenever:

$$ \left\{ \begin{array}{l} \left( 2a\sigma - \frac{1}{4\lambda^g} \right) \ge 0 \\ \left( \frac{1}{\eta} \left( 2a\sigma - \frac{1}{4\lambda^g} \right) - \frac{a}{2\lambda^g} \right) \ge 0 \\ \left( \frac{1}{\eta} \left( 2a\sigma - \frac{1}{4\lambda^g} \right) - \frac{a}{2\lambda^g} \right)^2 - 4 \left( \frac{a\sigma}{2\lambda^g} \right) a^2 \ge 0. \end{array} \right. \qquad (31) $$

Hence, to find the values of $\eta, \sigma, a, \lambda^g$ giving the optimal upper bound, we must solve the problem:

$$ \inf_{\eta, \sigma, a, \lambda^g \ge 0} \frac{1}{2\eta} D^2 + \lambda^g L^2 T + a\sigma^2 D^2 T $$

subject to: $(\eta, \sigma, a, \lambda^g)$ satisfy (31), and $\sigma \le 1$.

Solving this non-convex problem algebraically via Lagrange relaxation (note that only the last constraint in (31) is saturated at the optimum) we get that the optimal values are:

$$ \eta = \frac{D3^{3/4}}{2LT^{3/4}}, \quad \sigma = \frac{\sqrt{3}}{\sqrt{T}}, \quad a = \frac{1}{6\eta}, \quad \text{and,} \quad \lambda^g = 2a\sigma^2 \frac{D^2}{L^2} = \frac{D^2}{\eta TL^2}. $$

Plugging those values in $Q(\tilde{\lambda})$ then allows to conclude the proof of Lemma 2.2. This concludes the design of the optimal proof of Lemma 2.2.