SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The proofs of Lemma 4.6 and Lemma 4.7 are deferred to Appendix E.
5 LB-FTRL with Multiplicative-Gradient Optimism
Define  $g_{t} \coloneqq \nabla f_{t}(x_{t})$ . By the convexity of the loss functions, OPS can be reduced to an online linear optimization problem described in Section 3 with  $v_{t} = g_{t}$  and  $\mathcal{X}$  being the probability simplex  $\Delta$ . Set the regularizer  $\varphi$  as the log-barrier (2) in Optimistic FTRL (Algorithm 1). By Lemma 4.4, for the regret guarantee in Theorem 3.2 to be valid, it remains to ensure that  $\hat{g}_{t}$ , the estimate of  $g_{t}$ , is selected to satisfy  $\eta_{t-1}\|g_{t}-\hat{g}_{t}\|_{x_{t,*}} \leq 1/2$  for all  $t$ . However, as  $x_{t}$ , which defines the dual local norm, depends on  $\hat{g}_{t}$  in Algorithm 1, selecting such  $\hat{g}_{t}$  is non-trivial.
To address this issue, we introduce Algorithm 2. This algorithm simultaneously computes the next iterate  $x_{t + 1}$  and the gradient estimate  $\hat{g}_{t + 1}$  by solving a system of nonlinear equations (5). As we estimate  $x_{t + 1} \odot g_{t + 1}$  instead of  $g_{t + 1}$ , we call the algorithm LB-FTRL with Multiplicative-Gradient Optimism. Here, LB indicates that the algorithm adopts the log-barrier as the regularizer.
Algorithm 2 LB-FTRL with Multiplicative-Gradient Optimism for OPS
Input: A sequence of learning rates  $\{\eta_t\} \subseteq \mathbb{R}_{++}$ .
1:  $h(x) \coloneqq -d\log d - \sum_{i = 1}^{d}\log x(i)$ .
2:  $\hat{g}_1\coloneqq 0$
3:  $x_{1}\gets \arg \min_{x\in \Delta}\eta_{0}^{-1}h(x)$
4: for all  $t \in \mathbb{N}$  do
5: Announce  $x_{t}$  and receive  $a_{t}$ .
6:  $g_{t} \coloneqq \nabla f_{t}(x_{t})$ .
7: Choose an estimate  $p_{t+1} \in \mathbb{R}^d$  for  $x_{t+1} \odot g_{t+1}$ .
8: Compute  $x_{t + 1}$  and  $\hat{g}_{t + 1}$  such that
$$
\left\{ \begin{array}{l} x _ {t + 1} \odot \hat {g} _ {t + 1} = p _ {t + 1}, \\ x _ {t + 1} \in \arg \min  _ {x \in \Delta} \langle g _ {1: t}, x \rangle + \langle \hat {g} _ {t + 1}, x \rangle + \eta_ {t} ^ {- 1} h (x). \end{array} \right. \tag {5}
$$
By the definitions of the dual local norm (3) and  $p_t$  (5), we write
$$
\left\| g _ {t} - \hat {g} _ {t} \right\| _ {x _ {t}, *} = \left\| x _ {t} \odot g _ {t} - x _ {t} \odot \hat {g} _ {t} \right\| _ {2} = \left\| x _ {t} \odot g _ {t} - p _ {t} \right\| _ {2}.
$$
It suffices to choose  $p_t$  such that  $\eta_{t-1} \| x_t \odot g_t - p_t \|_2 \leq 1/2$ . Indeed, Algorithm 3 and Algorithm 4 correspond to choosing  $p_t = x_{t-1} \odot g_{t-1}$  and  $p_t = 0$ , respectively. Algorithm 5 in Appendix I corresponds to choosing  $p_t = (1/\eta_{0:t-2}) \sum_{\tau=1}^{t-1} \eta_{\tau-1} x_\tau \odot g_\tau$ .
Theorem 5.1 guarantees that  $x_{t}$  and  $\hat{g}_{t}$  are well-defined and can be efficiently computed. Its proof and the computational details can be found in Appendix F.1.
Theorem 5.1. If  $\eta_t p_{t+1} \in [-1,0]^d$ , then the system of nonlinear equations (5) has a solution. The solution can be computed in  $\tilde{O}(d)$  time.
Algorithm 2 corresponds to Algorithm 1 with  $v_{t} = g_{t}$ ,  $\hat{v}_{t} = \hat{g}_{t} = p_{t} \oslash x_{t}$ , and  $\varphi(x) = h(x)$ . Corollary 5.2 then follows from Theorem 3.2. Its proof can be found in Appendix F.2.
Corollary 5.2. Assume that the sequence  $\{\eta_t\}$  is non-increasing and  $p_t \in (-\infty, 0]^d$  for all  $t \in \mathbb{N}$ . Under Assumption 1, Algorithm 2 satisfies
$$
R _ {T} (x) \leq \frac {d \log T}{\eta_ {T}} + \sum_ {t = 1} ^ {T} \left(\langle g _ {t} - p _ {t} \oslash x _ {t}, x _ {t} - x _ {t + 1} \rangle - \frac {1}{\eta_ {t - 1}} \omega (\| x _ {t} - x _ {t + 1} \| _ {x _ {t}})\right) + 2,
$$
In addition, for any sequence of vectors  $\{u_t\}$  such that  $\eta_{t-1} \| (g_t + u_t) \odot x_t - p_t \|_2 \leq 1/2$  and  $\langle u_t, x_t - x_{t+1} \rangle = 0$  for all  $t \in \mathbb{N}$ , Algorithm 2 satisfies
$$
R _ {T} (x) \leq \frac {d \log T}{\eta_ {T}} + \sum_ {t = 1} ^ {T} \eta_ {t - 1} \| \left(g _ {t} + u _ {t}\right) \odot x _ {t} - p _ {t} \| _ {2} ^ {2} + 2.
$$
Remark 5.3. The vectors  $u_{t}$  are deliberately introduced to derive a small-loss bound for OPS.
6 Data-Dependent Bounds for OPS
6.1 Gradual-Variation Bound
We define the gradual variation as
$$
V _ {T} := \sum_ {t = 2} ^ {T} \| \nabla f _ {t} \left(x _ {t - 1}\right) - \nabla f _ {t - 1} \left(x _ {t - 1}\right) \| _ {x _ {t - 1}, *} ^ {2} \leq \sum_ {t = 2} ^ {T} \max  _ {x \in \Delta} \| \nabla f _ {t} (x) - \nabla f _ {t - 1} (x) \| _ {x, *} ^ {2}, \tag {6}
$$
where  $\| \cdot \|_{*}$  denotes the dual local norm associated with the log-barrier. The definition is a local-norm analog to the existing one [8, 18], defined as  $\sum_{t = 2}^{T}\max_{x\in \Delta}\| \nabla f_t(x) - \nabla f_{t - 1}(x)\|^2$  for a fixed norm  $\| \cdot \|$ . Regarding Lemma 4.3, our definition appears to be a natural extension.
In this sub-section, we introduce Algorithm 3, LB-FTRL with Last-Multiplicative-Gradient Optimism, and Theorem 6.1, which provides the first gradual-variation bound for OPS. Algorithm 3 is an instance of Algorithm 2 with  $p_1 = 0$  and  $p_t = x_{t - 1} \odot g_{t - 1}$  for  $t \geq 2$ . Note that the learning rates specified in Theorem 6.1 do not require the knowledge of  $V_{T}$  in advance and can be computed on the fly.
The proof of Theorem 6.1 can be found in Appendix G.
Theorem 6.1. Let  $\eta_0 = \eta_1 = 1 / (16\sqrt{2})$  and  $\eta_t = \sqrt{d / (512d + 2 + V_t)}$  for  $t \geq 2$ . Then, Algorithm 3 satisfies
$$
R _ {T} (x) \leq (\log T + 8) \sqrt {d V _ {T} + 5 1 2 d ^ {2}} + \sqrt {2 d} \log T + 2 - 1 2 8 \sqrt {2 d}, \quad \forall T \in \mathbb {N}.
$$
By the definition of the dual local norm (3) and Lemma 4.1,
$$
V _ {T} = \sum_ {t = 2} ^ {T} \| x _ {t - 1} \odot \nabla f _ {t} (x _ {t - 1}) - x _ {t - 1} \odot \nabla f _ {t - 1} (x _ {t - 1}) \| _ {2} ^ {2} \leq 2 (T - 1).
$$
As a result, the worst-case regret of Algorithm 3 is  $O(\sqrt{dT}\log T)$ , comparable to the regret bounds of the barrier subgradient method [26], Soft-Bayes [30], and LB-OMD [34] up to logarithmic factors. On the other hand, if the price relatives remain constant over rounds, then  $V_{T} = 0$  and  $R_{T} = O(d\log T)$ .
Algorithm 3 LB-FTRL with Last-Multiplicative-Gradient Optimism for OPS
Input: A sequence of learning rates  $\{\eta_t\} \subseteq \mathbb{R}_{++}$    
1:  $h(x)\coloneqq -d\log d - \sum_{i = 1}^{d}\log x(i).$    
2:  $p_1\gets 0$    
3:  $x_{1}\leftarrow \arg \min_{x\in \mathcal{X}}\eta_{0}^{-1}h(x)$    
4: for all  $t\in \mathbb{N}$  do   
5: Announce  $x_{t}$  and receive  $a_{t}$    
6:  $g_{t}\gets \nabla f_{t}(x_{t})$    
7:  $p_{t + 1}\gets x_t\odot g_t$    
8: Compute  $x_{t + 1}$  and  $\hat{g}_{t + 1}$  such that  $\begin{array}{r}\left\{ \begin{array}{ll}x_{t + 1}\odot \hat{g}_{t + 1} = p_{t + 1},\\ x_{t + 1}\in \arg \min_{x\in \Delta}\langle g_{1:t},x\rangle +\langle \hat{g}_{t + 1},x\rangle +\eta_t^{-1}h(x). \end{array} \right. \end{array}$    
9: end for
Time Complexity. The vectors  $g_{t}$  and  $p_{t + 1}$ , as well as the quantity  $V_{t}$ , can be computed using  $O(d)$  arithmetic operations. By Theorem 5.1, the iterate  $x_{t + 1}$  can be computed in  $\tilde{O}(d)$  arithmetic operations. Therefore, the per-round time of Algorithm 3 is  $\tilde{O}(d)$ .
6.2 Small-Loss Bound
In this sub-section, we introduce Algorithm 4, Adaptive LB-FTRL, and Theorem 6.2, the first small-loss bound for OPS. The algorithm is an instance of Algorithm 2 with  $p_t = 0$ . Then,  $\hat{g}_{t + 1} = 0$  and  $x_{t + 1}$  is directly given by Line 8 of Algorithm 4. Note that Theorem 5.1 still applies.
Algorithm 4 Adaptive LB-FTRL for OPS
1:  $h(x)\coloneqq -d\log d - \sum_{i = 1}^{d}\log x(i).$    
2:  $x_{1}\gets \arg \min_{x\in \Delta}\eta_{0}^{-1}h(x).$    
3: for all  $t\in \mathbb{N}$  do   
4: Announce  $x_{t}$  and receive  $a_t$    
5:  $g_{t}\gets \nabla f_{t}(x_{t}) = -\frac{a_{t}}{\langle a_{t},x_{t}\rangle}.$    
6:  $\alpha_{t}\leftarrow \alpha_{xt}(g_{t})$  (see the definition (4)).   
7:  $\eta_t\gets \frac{\sqrt{d}}{\sqrt{4d + 1 + \sum_{\tau = 1}^t\|g_\tau + \alpha_\tau e\|_{x_\tau,*}^2}}.$    
8:  $x_{t + 1}\gets \arg \min_{x\in \Delta}\langle g_{1:t},x\rangle +\eta_t^{-1}h(x).$    
9: end for
The proof of Theorem 6.2 is provided in Appendix H.
Theorem 6.2. Let  $L_T^\star = \min_{x \in \Delta} \sum_{t=1}^T f_t(x)$ . Then, under Assumption 1, Algorithm 4 satisfies
$$
R _ {T} (x) \leq 2 (\log T + 2) \sqrt {4 d L _ {T} ^ {\star} + 4 d ^ {2} + d} + d (\log T + 2) ^ {2}.
$$
Under Assumption 1,
$$
L _ {T} ^ {\star} = \min _ {x \in \Delta} \sum_ {t = 1} ^ {T} - \log \left\langle a _ {t}, x \right\rangle \leq \sum_ {t = 1} ^ {T} - \log \frac {\| a \| _ {1}}{d} = \sum_ {t = 1} ^ {T} \log d = T \log d.
$$
Assuming  $T > d$ , the worst-case regret bound is  $O(\sqrt{dT\log d}\log T)$ , also comparable to the regret bounds of the barrier subgradient method [26], Soft-Bayes [30], and LB-OMD [34] up to logarithmic factors. On the other hand, suppose that there exists an  $i^{\star} \in [d]$  such that  $a_{t}(i^{\star}) = 1$  for all  $t \in [T]$ . That is, the  $i^{\star}$ -th investment alternative always outperforms all the other investment alternatives. Then,  $L_T^\star = 0$  and  $R_{T} = O(d\log^{2}T)$ .
Assumption 1 does not restrict the applicability of Theorem 6.2, as mentioned earlier. If the assumption does not hold, Theorem 6.2 is applied with respect to the normalized price relatives  $\tilde{a}_t = a_t / \| a_t\|_\infty$ . In this case,  $L_T^\star$  is defined with respect to  $\{\tilde{a}_t\}$ .
Time Complexity. Since  $\| g_t + \alpha_t e\|_{x_{t,*}}^2 = \| x_t\odot g_t + \alpha_tx_t\|_2^2$ , it is obvious that computing  $\alpha_{t}$ ,  $\eta_{t}$ , and  $g_{1:t}$  can be done in  $O(d)$  arithmetic operations. By Theorem 5.1, the iterate  $x_{t + 1}$  can be computed in  $\tilde{O} (d)$  arithmetic operations. Hence, the per-round time of Algorithm 4 is  $\tilde{O} (d)$ .
We have presented Theorem 6.1 and Theorem 6.2, the first gradual-variation and small-loss bounds for OPS that do not require the no-junk-bonds assumption, respectively. The algorithms exhibit sublinear regrets in the worst cases and achieve logarithmic regrets in the best cases, with per-round time almost linear in the dimension. They mark the first data-dependent bounds for non-Lipschitz non-smooth losses.
A potential direction for future research is to extend our analyses for a broader class of online convex optimization problems. In particular, it remains unclear how to extend the two smoothness characterizations (Lemma 4.6 and Lemma 4.7) for other loss functions.
Orabona et al. [29] showed that achieving a regret rate of  $O\left(d^{2} + \log L_{T}^{\star}\right)$  is possible under the no-junk-bonds assumption, where  $L_{T}^{\star}$  denotes the cumulative loss of the best constant rebalanced portfolio. This naturally raises the question: can a similar regret rate be attained without relying on the no-junk-bonds assumption? If so, then the regret rate will be constant in  $T$  in the best cases and logarithmic in  $T$  in the worst cases. However, considering existing results in probability forecasting with the logarithmic loss [5, Chapter 9]—a special case of OPS without the no-junk-bonds assumption—such a data-dependent regret rate seems improbable. Notably, classical rate-optimal algorithms for probability forecasting with the logarithmic loss, such as Shtarkov's minimax-optimal algorithm, the Laplace mixture, and the Krichevsky-Trofimov mixture, all achieve logarithmic regret rates for all possible data sequences [5, Chapter 9].
Zhao et al. [37] showed that for Lipschitz and smooth losses, an algorithm with a gradual-variation bound automatically achieves a small-loss bound. Generalizing their argument for non-Lipschitz non-smooth losses is a natural direction to consider.
The authors are supported by the Young Scholar Fellowship (Einstein Program) of the National Science and Technology Council of Taiwan under grant number NSTC 112-2636-E-002-003, by the 2030 Cross-Generation Young Scholars Program (Excellent Young Scholars) of the National Science and Technology Council of Taiwan under grant number NSTC 112-2628-E-002-019-MY3, and by the research project "Pioneering Research in Forefront Quantum Computing, Learning and Engineering" of National Taiwan University under grant number NTU-CC-112L893406.
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A A Summary of OPS Algorithms
Table 1: A summary of existing algorithms for online portfolio selection without the no-junk-bonds assumption. Assume  $T \gg d$ .
Algorithms	Regret (O)	Per-round time (O)
Best-case	Worst-case
Universal Portfolio [9, 20]	d log T	d4T14
EG [15, 34]	d1/3T2/3	d
BSM [26], Soft-Bayes [30], LB-OMD [34]	√dT	d
ADA-BARRONS [22]	d2 log4T	d2.5T
LB-FTRL without linearized losses [35]	d logd+1T	d2T
PAE+DONS [24]	d2 log5T	d3
BISONS [38]	d2 log2T	d3
VB-FTRL [17]	d logT	d2T
This work (Algorithm 3)	d logT	√dT	d
This work (Algorithm 4)	d log2T	√dT	d
B Loss Functions in OPS are not Geodesically Convex
Let  $h(x) \coloneqq -d\log d - \sum_{i=1}^{d}\log x(i)$  be the log-barrier. Let  $(\mathcal{M}, g)$  be the Hessian manifold induced by the log-barrier, where  $\mathcal{M} = \mathrm{ri}\Delta$  and  $g$  is the Riemannian metric defined by  $\langle u, v \rangle_x \coloneqq \langle u, \nabla^2 h(x)v \rangle$ .
Lemma B.1 (Gzyl and Nielsen [11, Theorem 3.1]). Let  $x,y\in \mathrm{ri}\Delta$ . The geodesic  $\gamma :[0,1]\to \mathrm{ri}\Delta$  connecting  $x,y$  is given by
$$
\gamma (t) (i) = \frac {x (i) ^ {1 - t} y (i) ^ {t}}{\sum_ {j = 1} ^ {d} x (j) ^ {1 - t} y (j) ^ {t}}, \quad \forall i \in [ d ],
$$
where  $\gamma (t)(i)$  denotes the  $i$ -th entry of the vector  $\gamma (t)\in \mathrm{ri}\Delta$
The proposition below shows that the loss functions in OPS are not geodesically convex on  $(\mathcal{M},g)$  in general.
Proposition B.2. Let  $a = (2,1) \in \mathbb{R}_{++}^2$ . The function  $f(x) = -\log \langle a,x\rangle$  is not geodesically convex on  $(\mathcal{M},g)$ .
Proof. It suffices to show that  $f \circ \gamma : [0,1] \to \mathbb{R}$  is not a convex function [4, Definition 11.3]. Let  $x = (1/2, 1/2)$  and  $y = (1/(1 + \mathrm{e}), \mathrm{e}/(1 + \mathrm{e}))$ . By Lemma B.1, the geodesic connecting  $x, y$  is  $\gamma(t) = (1/(1 + \mathrm{e}^t), \mathrm{e}^t/(1 + \mathrm{e}^t))$ . We write
$$
(f \circ \gamma) ^ {\prime \prime} (t) = \frac {(2 - \mathrm {e} ^ {2 t}) \mathrm {e} ^ {t}}{(2 + \mathrm {e} ^ {t}) ^ {2} (1 + \mathrm {e} ^ {t}) ^ {2}} <   0, \quad \forall t > \frac {\log 2}{2},
$$
showing that  $f \circ \gamma$  is not convex. The proposition follows.
C Properties of Self-Concordant Functions
This section reviews properties of self-concordant functions relevant to our proofs. Readers are referred to the book by Nesterov [27] for a complete treatment.
Throughout this section, let  $\varphi$  be a self-concordant function (Definition 3.1) and  $\| \cdot \| _x$  be the associated local norm, i.e.,
$$
\left\| u \right\| _ {x} := \sqrt {\langle u , \nabla^ {2} \varphi (x) u \rangle}, \quad \forall u \in \mathbb {R} ^ {d}, x \in \operatorname {d o m} \varphi .
$$
Self-concordant functions are neither smooth nor strongly convex in general. The following theorem indicates that nevertheless, they are locally smooth and strongly convex.
Theorem C.1 (Nesterov [27, Theorem 5.1.7]). For any  $x,y\in \mathrm{dom}\varphi$  such that  $r\coloneqq \| y - x\| _x < 1 / M$
$$
(1 - M r) ^ {2} \nabla^ {2} \varphi (x) \leq \nabla^ {2} \varphi (y) \leq \frac {1}{(1 - M r) ^ {2}} \nabla^ {2} \varphi (x).
$$
Define  $\omega(t) \coloneqq t - \log(1 + t)$ . Let  $\omega_*$  be its Fenchel conjugate, i.e.,  $\omega_*(t) = -t - \log(1 - t)$ .
Theorem C.2 (Nesterov [27, Theorem 5.1.8 and 5.1.9]). For any  $x,y\in \mathrm{dom}\varphi$  with  $r\coloneqq \| y - x\| _x$
$$
\langle \nabla \varphi (y) - \nabla \varphi (x), y - x \rangle \geq \frac {r ^ {2}}{1 + M r}, \tag {7}
$$
$$
\varphi (y) \geq \varphi (x) + \langle \nabla \varphi (x), y - x \rangle + \frac {1}{M ^ {2}} \omega (M r).
$$
If  $r < 1 / M$ , then
$$
\langle \nabla \varphi (y) - \nabla \varphi (x), y - x \rangle \leq \frac {r ^ {2}}{1 - M r}, \tag {8}
$$
$$
\varphi (y) \leq \varphi (x) + \langle \nabla \varphi (x), y - x \rangle + \frac {1}{M ^ {2}} \omega_ {*} (M r).
$$
The following proposition bounds the growth of  $\omega$  and  $\omega_{*}$
Proposition C.3 (Nesterov [27, Lemma 5.1.5]). For any  $t \geq 0$ ,
$$
\frac {t ^ {2}}{2 (1 + t)} \leq \omega (t) \leq \frac {t ^ {2}}{2 + t}.
$$
For any  $t\in [0,1)$
$$
\frac {t ^ {2}}{2 - t} \leq \omega_ {*} (t) \leq \frac {t ^ {2}}{2 (1 - t)}.
$$
It is easily checked that both the loss functions in OPS and the log-barrier are not only self-concordant but also self-concordant barriers.
Definition C.4. A 1-self-concordant function  $\varphi$  is said to be a  $\nu$ -self-concordant barrier if
$$
\left\langle \nabla \varphi (x), u \right\rangle^ {2} \leq \nu \left\langle u, \nabla^ {2} \varphi (x) u \right\rangle , \quad \forall x \in \operatorname {d o m} \varphi , u \in \mathbb {R} ^ {d}.
$$
Lemma C.5 (Nesterov [27, (5.4.3)]). The log-barrier (2) is a  $d$ -self-concordant barrier.
D Proof of Theorem 3.2 (Regret Analysis of Optimistic FTRL With Self-Concordant Regularizers)
D.1 Proof of Theorem 3.2
The proof of the following stability lemma is deferred to the next subsection.
Lemma D.1. Define  $F_{t + 1}(x) \coloneqq \langle v_{1:t}, x \rangle + \eta_t^{-1} \varphi(x)$ . Let  $\{\eta_t\}$  be a non-increasing sequence of strictly positive numbers such that  $\eta_{t - 1} \| v_t - \hat{v}_t \|_{x_t,*} \leq 1 / (2M)$ . Then, Algorithm 1 satisfies
$$
\begin{array}{l} F _ {t} (x _ {t}) - F _ {t + 1} (x _ {t + 1}) + \langle v _ {t}, x _ {t} \rangle \leq \langle v _ {t} - \hat {v} _ {t}, x _ {t} - x _ {t + 1} \rangle - \frac {1}{\eta_ {t - 1} M ^ {2}} \omega (M \| x _ {t + 1} - x _ {t} \| _ {x _ {t}}) \\ \leq \frac {1}{\eta_ {t - 1} M ^ {2}} \omega_ {*} (\eta_ {t - 1} M \| v _ {t} - \hat {v} _ {t} \| _ {x _ {t}, *}) \\ \leq \eta_ {t - 1} \| v _ {t} - \hat {v} _ {t} \| _ {x _ {t}, *} ^ {2}. \\ \end{array}
$$
Define  $\hat{F}_t(x) \coloneqq F_t(x) + \langle \hat{v}_t, x \rangle$ . Since  $\hat{v}_{T+1}$  has no effect on the regret, we set  $\hat{v}_{T+1} = 0$ . By the strong FTRL lemma [28, Lemma 7.1],
$$
\begin{array}{l} R _ {T} (x) = \sum_ {t = 1} ^ {T} (\langle v _ {t}, x _ {t} \rangle - \langle v _ {t}, x \rangle) \\ = \langle \hat {v} _ {T + 1}, x \rangle + \frac {\varphi (x)}{\eta_ {T}} - \min  _ {x \in \mathcal {X}} \frac {\varphi (x)}{\eta_ {0}} + \sum_ {t = 1} ^ {T} (\hat {F} _ {t} (x _ {t}) - \hat {F} _ {t + 1} (x _ {t + 1}) + \langle v _ {t}, x _ {t} \rangle) \\ + \hat {F} _ {T + 1} (x _ {T + 1}) - \hat {F} _ {T + 1} (x) \\ \leq \frac {\varphi (x)}{\eta_ {T}} + \sum_ {t = 1} ^ {T} \left(\hat {F} _ {t} \left(x _ {t}\right) - \hat {F} _ {t + 1} \left(x _ {t + 1}\right) + \langle v _ {t}, x _ {t} \rangle\right) \\ \leq \frac {\varphi (x)}{\eta_ {T}} + \sum_ {t = 1} ^ {T} \left(F _ {t} \left(x _ {t}\right) - F _ {t + 1} \left(x _ {t + 1}\right) + \langle v _ {t}, x _ {t} \rangle\right). \\ \end{array}
$$
The penultimate line above follows from the assumption that  $\min_{x\in \mathcal{X}}\varphi (x) = 0$  and that  $x_{T + 1}$  minimizes  $\hat{F}_{T + 1}$  on  $\mathcal{X}$ ; the last line above follows from a telescopic sum and  $\hat{v}_1 = \hat{v}_{T + 1} = 0$ . The theorem then follows from Lemma D.1.
D.2 Proof of Lemma D.1
Since  $\varphi$  is  $M$ -self-concordant, the function  $\eta_{t-1}F_t$  is  $M$ -self-concordant. By Theorem C.2,
$$
\begin{array}{l} \eta_ {t - 1} F _ {t} \left(x _ {t + 1}\right) - \eta_ {t - 1} F _ {t} \left(x _ {t}\right) \\ \geq \left\langle \eta_ {t - 1} \nabla F _ {t} \left(x _ {t}\right), x _ {t + 1} - x _ {t} \right\rangle + \frac {1}{M ^ {2}} \omega \left(M \| x _ {t} - x _ {t + 1} \| _ {x _ {t}}\right) \\ = \left\langle \eta_ {t - 1} \left(\nabla F _ {t} \left(x _ {t}\right) + \hat {v} _ {t}\right), x _ {t + 1} - x _ {t} \right\rangle - \eta_ {t - 1} \left\langle \hat {v} _ {t}, x _ {t + 1} - x _ {t} \right\rangle + \frac {1}{M ^ {2}} \omega \left(M \| x _ {t} - x _ {t + 1} \| _ {x _ {t}}\right). \\ \end{array}
$$
Since  $x_{t}$  minimizes  $F_{t}(x) + \langle \hat{v}_{t}, x \rangle$  on  $\mathcal{X}$ , the optimality condition implies
$$
\langle \nabla F _ {t} (x _ {t}) + \hat {v} _ {t}, x _ {t + 1} - x _ {t} \rangle \geq 0.
$$
Then, we obtain
$$
\eta_ {t - 1} F _ {t} \left(x _ {t + 1}\right) - \eta_ {t - 1} F _ {t} \left(x _ {t}\right) \geq - \eta_ {t - 1} \left\langle \hat {v} _ {t}, x _ {t + 1} - x _ {t} \right\rangle + \frac {1}{M ^ {2}} \omega \left(M \| x _ {t + 1} - x _ {t} \| _ {x _ {t}}\right).
$$
Next, by the non-increasing of  $\{\eta_t\}$ ,  $\varphi(x_{t+1}) \geq 0$ , and the last inequality,
$$
\begin{array}{l} F _ {t} \left(x _ {t}\right) - F _ {t + 1} \left(x _ {t + 1}\right) + \left\langle v _ {t}, x _ {t} \right\rangle \\ = F _ {t} \left(x _ {t}\right) - F _ {t} \left(x _ {t + 1}\right) + \left\langle v _ {t}, x _ {t} \right\rangle - \left\langle v _ {t}, x _ {t + 1} \right\rangle + \left(\frac {1}{\eta_ {t - 1}} - \frac {1}{\eta_ {t}}\right) \varphi \left(x _ {t + 1}\right) \\ \leq F _ {t} \left(x _ {t}\right) - F _ {t} \left(x _ {t + 1}\right) + \langle v _ {t}, x _ {t} - x _ {t + 1} \rangle \\ \leq \left\langle v _ {t} - \hat {v} _ {t}, x _ {t} - x _ {t + 1} \right\rangle - \frac {1}{\eta_ {t - 1} M ^ {2}} \omega (M \| x _ {t + 1} - x _ {t} \| _ {x _ {t}}). \\ \end{array}
$$
This proves the first inequality in the lemma.