SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Given offline access to the sequence of requests, the optimal solution for the  $k$ -server can be computed in polynomial-time while the optimal static solution of Problem 1 cannot be approximated in polynomial-time with ratio less than  $(1 + 2 / e)$  even under a-priori knowledge of the request sequence (inapproximability of  $k$ -median).
In this chapter of the appendix we briefly discuss the details behind Theorem 2 and show how the results of [10] and [24] hold even for the specific definition of HSTs we have considered in Definition 5.
Traditionally, HSTs are not required to be balanced nor are required to have weights that are specifically powers of 2. In fact, the seminal work of [10], later improved by [24], states that there exists a randomized procedure such that for every weighted graph  $G(V,E,w)$ , it constructs (in polynomial-time) a tree  $\mathcal{T}$  such that:
1. There exists a perfect matching  $\sigma : V \mapsto L(\mathcal{T})$  that maps the vertices of  $G$  to the leaves of  $\mathcal{T}$ .
2. For any vertices  $i,j\in V$  , their corresponding distance on  $\mathcal{T}$  can only increase, i.e.  $d_{G}(i,j)\leq d_{\mathcal{T}}(\sigma (i),\sigma (j))$
3. On expectation, distances between vertices are distorted only by a logarithmic factor, i.e.  $\mathbb{E}\left[d_{\mathcal{T}}(\sigma (i),\sigma (j))\right]\leq \mathcal{O}(\log |V|)\cdot d_G(i,j)$
4. The weight of any edge  $e = (v, u)$  between a vertex  $v \in V(\mathcal{T})$  and its parent vertex  $u$  is precisely  $\text{diam}(G) \cdot 2^{-\text{dpt}(v)}$ .
5. The height of  $\mathcal{T}$  satisfies  $h(\mathcal{T})\leq \lceil \log (diam(G))\rceil$
The purpose of this section is to argue that one can easily transform such a tree  $\mathcal{T}$  to match our notion of HSTs (Definition 5), while maintaining the same guarantees for the distortion of the distances. Recall that we have already assumed that the minimum edge weight of  $G$  is 1, i.e.  $\min_{e\in E}w_e = 1$ . Furthermore, we can also assume without loss of generality that the diameter of  $G$  is a power of 2; if not, simple scaling arguments suffice to transform  $G$  into such a graph by only distorting distances by a constant factor. Thus, we assume that  $diam(G) = 2^d$  for some  $d\geq 0$ .
We start from the tree  $\mathcal{T}$  that the algorithm of [24] generates. Recall that by definition, the weight of an edge  $e = (i,j)$  between some vertex  $i$  and its parent node  $j$  is  $2^{d - dpt(i)}$ . In order to balance the tree, we take each leaf vertex  $u \in L(\mathcal{T})$  at depth  $dpt(u)$  and extend it downwards by adding new vertices until it reaches a new depth  $dpt'(u) = d$ . For every new edge that we add during this process, we maintain that the weight of the edge  $e = (i,j)$  from  $i$  to its parent  $j$  is  $diam(G) \cdot 2^{-dpt(i)}$ .
Let  $\mathcal{T}'$  be used to denote our modified tree. Clearly, the above construction guarantees  $h(\mathcal{T}') = d$ . Since by definition  $h(\mathcal{T}) \leq \lceil \log (diam(G)) \rceil = d$ , we know that all leaves initially lie at depth at most  $d$ , and thus by the end of the above process all leaves will lie at the same level of the tree and have depth  $d$ . Thus, we have indeed constructed a balanced tree. Furthermore, since by definition  $dpt(v) = h(\mathcal{T}) - lev(v)$ , we get that the weight of the edge  $e = (i,j)$  from  $i$  to its parent  $j$  is  $w_{e} = diam(G) \cdot 2^{lev(i) - d} = 2^{lev(i)}$ . So, the constructed tree indeed satisfies all the requirements of Definition 5 and is a valid HST (according to our definition).
We will now argue that  $\mathcal{T}'$  also satisfies all items of Theorem 2. Fist of all, the height of our new tree is precisely  $d$ , and thus it is true that  $h(\mathcal{T}') \leq \lceil \log(diam(G)) \rceil$ . Furthermore, since we only added edges to the initial tree  $\mathcal{T}$ , the distance between any two leaves can only increase. Thus, we get that for any vertices  $i, j \in V$  it holds
$$
d _ {G} (i, j) \leq d _ {\mathcal {T}} (i, j) \leq d _ {\mathcal {T} ^ {\prime}} (i, j)
$$
Finally, it remains to upper bound the expected distortion on  $\mathcal{T}'$ . Recall that by construction of [24], we know that
$$
\mathbb {E} \left[ d _ {\mathcal {T}} (\sigma (i), \sigma (j)) \right] \leq \mathcal {O} (\log | V |) \cdot d _ {G} (i, j)
$$
Since edge lengths decrease by a factor of 2 every time we move down the tree, we know that the total length of the path we added in order to move leaf  $i$  from depth  $dpt(i)$  to depth  $d$  is precisely  $1 + 2 + \ldots 2^{dpt(i) - 1} \leq 2^{dpt(i)}$ . This implies that any distance on  $\mathcal{T}'$  can be at most twice the corresponding distance on  $\mathcal{T}$ , i.e.
$$
d _ {\mathcal {T} ^ {\prime}} (\sigma (i), \sigma (j)) \leq 2 \cdot d _ {\mathcal {T}} (\sigma (i), \sigma (j))
$$
C Proofs of Section 3
In this chapter of the appendix we present all the omitted proofs from Section 3 concerning the basic algorithmic primitives we use in order to establish our main result in Theorem 1.
Roadmap. In section C.1 we establish the connection between Problems 1 and 2 and show that our notion of fractional connection and moving cost collapses with our initial definitions in the case of integral facility placements. Then, in section C.2 we present the proof of Theorem 5 and in section C.3 we present the proof of Theorem 1.
C.1 Establishing the relation between Problems 1 and 2
Fix any HST  $\mathcal{T}$  and let  $\mathcal{FP}(\mathcal{T})$  be the corresponding set of fractional facility placements. In this section, we will establish that in the case of integral facility placements  $y\in \mathcal{FP}(\mathcal{T})\cap \mathbb{N}$ , the notions of fractional connection cost and fractional moving cost (formally stated in Definitions 7 and 8) collapse to the notions of actual connection and moving costs (formally stated in Definitions 1 and 2) respectively.
Let  $y \in \mathcal{FP}(\mathcal{T}) \cap \mathbb{N}$  be an integral facility placement. Then, by definition, for each leaf  $v \in L(\mathcal{T})$  we have  $y_v \in \{0,1\}$  facilities that are placed on it, and the total amount of placed facilities is  $k$ , i.e.  $\sum_{v \in L(\mathcal{T})} y_v = k$ . Thus, we can associate with any integral facility placement  $y$  a corresponding set
$$
F (y) = \left\{v \in L (\mathcal {T}): y _ {v} = 1 \right\}
$$
such that  $|F(y)| = k$ , meaning that  $F(y)$  is a valid facility placement of the leaves of the  $\mathcal{T}$ .
In Claim 1 we will establish that for any set of clients, the connection cost under  $F(y)$  is equal to the fractional connection cost under  $y$ . Then, in Claim 2 we will establish that the fractional moving cost between  $y$  and  $y'$  gives us precisely the moving cost between facility placements  $F(y)$  and  $F(y')$  on  $\mathcal{T}$ .
Claim 1. For any integral facility placement  $y \in \mathcal{FP}(\mathcal{T}) \cap \mathbb{N}$  and any set of clients  $R \subseteq L(\mathcal{T})$ , it holds that
$$
f _ {R} (y) = C _ {R} (F (y))
$$
Proof. Fix any  $y \in \mathcal{FP}(\mathcal{T}) \cap \mathbb{N}$  and any  $R \subseteq L(\mathcal{T})$ . By definition of the connection cost (Definition 1), we have
$$
C _ {R} (F) = \sum_ {j \in R} \min  _ {i \in F (y)} d _ {\mathcal {T}} (i, j)
$$
Let's fix a particular client that lies on some leaf  $j \in L(\mathcal{T})$  of  $\mathcal{T}$ . Let  $i^{*} = \arg \min_{i \in F(y)} d_{\mathcal{T}}(i, j)$  be the leaf closest to  $j$  that  $F(y)$  places a facility into. Since  $\mathcal{T}$  is an HST and distances increase by a factor of 2 as we move up the tree, it is not hard to see that  $i^{*}$  is the leaf in  $F(y)$  whose lowest common ancestor (lca) with  $j$  has the smallest level. Let  $l^{*} = lca(j, i^{*})$ . Equivalently,  $l^{*}$  is the minimum-level vertex in  $P(j, r)$  such that  $y_{l^{*}} \geq 1$ . Since  $\mathcal{T}$  is balanced, we have that the connection cost of client  $j$  under  $F(y)$  is precisely
$$
C _ {\{j \}} (F (y)) = 2 \cdot d _ {\mathcal {T}} (j, l ^ {*}) = 2 \cdot \sum_ {l = 0} ^ {\operatorname {l e v} (l ^ {*}) - 1} 2 ^ {l}
$$
and since by integrality we have that  $y_{v} = 0$  for any  $v \in P(j, l^{*}) \setminus \{l^{*}\}$  and  $y_{v} \geq 1$  for all  $v \in P(l^{*}, r)$ , we have
$$
C _ {\{j \}} (F) = 2 \cdot \sum_ {v \in P (j, r)} 2 ^ {\operatorname {l e v} (v)} \cdot \max  (0, 1 - y _ {v})
$$
Summing over all clients  $j \in R$  we get
$$
C _ {R} (F (y)) = \sum_ {j \in R} \sum_ {v \in P (j, r)} 2 ^ {\operatorname {l e v} (v) + 1} \cdot \max  (0, 1 - y _ {v}) = f _ {R} (y)
$$
Claim 2. For any integral facility placements  $y, y' \in \mathcal{FP}(\mathcal{T}) \cap \mathbb{N}$ , it holds that
$$
\left\| y - y ^ {\prime} \right\| _ {\mathcal {T}} = \gamma \cdot M _ {\mathcal {T}} (F (y), F (y ^ {\prime}))
$$
Proof. Fix any two integral facility placements  $y, y' \in \mathcal{FP}(\mathcal{T}) \cap \mathbb{N}$ . By definition of the moving cost (Definition 2), we have that
$$
M _ {\mathcal {T}} (F (y), F \left(y ^ {\prime}\right)) = \min  _ {\sigma \in \Sigma} \sum_ {i \in F (y)} d _ {\mathcal {T}} (i, \sigma (i))
$$
where  $\Sigma$  is the set of all possible matchings from the facilities in  $F(y)$  to the facilities in  $F(y^{\prime})$ .
In general graphs, the minimum transportation cost can have a very complicated structure and typically requires solving a minimum transportation problem in order to compute it. However, in the special case of HSTs, we are actually able to obtain a very simple expression for this quantity.
Recall that in an HST  $\mathcal{T}$ , edge weights increase by a factor of 2 every time we move up a level on the tree. Thus, it is always in our interest to move facilities between leaves whose lowest common ancestor is as low as possible. In other words, the matching  $\sigma$  that minimizes the transportation cost from  $F(y)$  to  $F(y^{\prime})$  can be obtained by selecting an arbitrary leaf in  $F(y)$ , matching it to the leaf in  $F(y^{\prime})$  with which it shares the lowest lowest common ancestor and then repeating the process for the rest of the leaves.
Now fix any vertex  $v \in V(\mathcal{T})$ . Recall that  $y_{v}$  is equal to the number of facilities in  $F(y)$  that are placed in the descendant leaves of  $v$  (respectively for  $y_{v}'$ ). Thus, if we apply the above (optimal) transportation plan, the number of facilities that will end up traversing the edge from  $v$  to its parent vertex is going to be precisely  $|y_{v} - y_{v}'|$ . Since the weight of this edge is by definition  $2^{\mathrm{lev}(v)}$ , we get that
$$
M _ {\mathcal {T}} (F (y), F (y ^ {\prime})) = \sum_ {v \in V (\mathcal {T})} 2 ^ {\operatorname {l e v} (v)} \cdot | y _ {v} - y _ {v} ^ {\prime} |
$$
and since
$$
| | y - y ^ {\prime} | | _ {\mathcal {T}} = \gamma \cdot \sum_ {v \in V (\mathcal {T})} 2 ^ {\operatorname {l e v} (v)} \cdot | y _ {v} - y _ {v} ^ {\prime} |
$$
we have proven the claim.
C.2 Proof of Theorem 5
We will now formally present the proof of Theorem 5, bounding the expected total cost of Algorithm 1. Fix any sequence of clients  $R_{1}, \ldots, R_{T}$ . Since the random seed  $\alpha$  is selected uniformly at random (Step 3 of Algorithm 1), by Item 1 of Theorem 4 we get that
$$
\mathbb {E} \left[ C _ {R _ {t}} \left(F _ {t}\right) \right] = f _ {R _ {t}} \left(y ^ {t}\right)
$$
Moreover since the same random seed  $\alpha$  is used at all rounds  $t\geq 1$ , Item 2 of Theorem 4 implies that
$$
\gamma \cdot \mathbb {E} \left[ M _ {\mathcal {T}} \left(F _ {t + 1}, F _ {t}\right) \right] \leq 4 \cdot \left| \left| y ^ {t + 1} - y ^ {t} \right| \right| \tau
$$
$$
\begin{array}{l} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} C _ {R _ {t}} \left(F _ {t}\right) + \gamma \cdot \sum_ {t = 2} ^ {T} M _ {\mathcal {T}} \left(F _ {t}, F _ {t - 1}\right) \right] \leq 4 \cdot \left(\sum_ {t = 1} ^ {T} f _ {R _ {t}} \left(y ^ {t}\right) + \sum_ {t = 2} ^ {T} | | y ^ {t} - y ^ {t - 1} | | _ {\mathcal {T}}\right) \\ \leq 6 \cdot \min  _ {y ^ {*} \in \mathcal {F P}} \sum_ {t = 1} ^ {T} f _ {R _ {t}} (y ^ {*}) + \beta \cdot \sqrt {T} \\ \end{array}
$$
where the last inequality follows by Theorem 3 for  $\beta = \mathcal{O}\left(k\cdot |L(\mathcal{T})|^{3 / 2}\cdot D_{\mathcal{T}}\cdot \max (\gamma ,1)\right)$ . The proof is concluded by the fact that
$$
\min  _ {y ^ {*} \in \mathcal {F P}} \sum_ {t = 1} ^ {T} f _ {R _ {t}} (y ^ {*}) \leq \min  _ {| F ^ {*} | = k} \sum_ {t = 1} ^ {T} C _ {R _ {t}} (F ^ {*})
$$
which is established in Claim 1 of Appendix C.1, stating that for any placement of  $k$ -facilities  $F \subseteq L(\mathcal{T})$  there exists a corresponding  $y \in \mathcal{FP}(\mathcal{T})$  whose fractional connection cost is equal to  $F$ 's under any client request.
We will now formally present the proof of Theorem 1, bounding the regret of Algorithm 2.
Let  $\mathcal{T}$  be the HST that we randomly embed our graph  $G(V,E,w)$  into. Since  $V = L(\mathcal{T})$  , we slightly abuse notation and use  $u$  to refer both to some vertex of  $G$  and to the corresponding leaf of  $\mathcal{T}$  . From Theorem 5, we know that the output of Algorithm 1 satisfies
$$
\begin{array}{l} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} C _ {R _ {t}} ^ {\mathcal {T}} (F _ {t}) + \gamma \cdot \sum_ {t = 2} ^ {T} M _ {\mathcal {T}} (F _ {t}, F _ {t - 1}) \right] \leq 6 \cdot \min  _ {| F ^ {*} | = k} \sum_ {t = 1} ^ {T} C _ {R _ {t}} ^ {\mathcal {T}} (F ^ {*}) \\ + \mathcal {O} \left(k \cdot | L (\mathcal {T}) | ^ {3 / 2} \cdot D _ {\mathcal {T}} \cdot \max  (1, \gamma)\right) \cdot \sqrt {T} \\ \end{array}
$$
where we use  $\mathcal{T}$  in the connection and moving cost to indicate that all distances are measured on the HST. Here, the expectation is taken over the random choices of Algorithm 1.
Next, notice that both the connection cost and the moving cost are defined as sum of distances. Thus, the results of Theorem 2 about the distance distortion from  $G$  to  $\mathcal{T}$  clearly apply for these quantities as well, namely
$$
C _ {R _ {t}} ^ {G} (F _ {t}) \leq C _ {R _ {t}} ^ {\mathcal {T}} (F _ {t}) \text {a n d} \mathbb {E} \left[ C _ {R _ {t}} ^ {\mathcal {T}} (F _ {t}) \right] \leq \mathcal {O} (\log | V |) \cdot C _ {R _ {t}} ^ {G} (F _ {t})
$$
$$
M _ {G} \left(F _ {t}, F _ {t - 1}\right) \leq M _ {\mathcal {T}} \left(F _ {t}, F _ {t - 1}\right) \text {a n d} \mathbb {E} \left[ M _ {\mathcal {T}} \left(F _ {t}, F _ {t - 1}\right) \right] \leq \mathcal {O} (\log | V |) \cdot M _ {G} \left(F _ {t}, F _ {t - 1}\right)
$$
Thus, taking an expectation over the randomness of  $\mathcal{T}$ , we finally get that
$$
\begin{array}{l} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} C _ {R _ {t}} ^ {G} (F _ {t}) + \gamma \cdot \sum_ {t = 2} ^ {T} M _ {G} (F _ {t}, F _ {t - 1}) \right] \leq \mathcal {O} (\log | V |) \cdot \min  _ {| F ^ {*} | = k} \sum_ {t = 1} ^ {T} C _ {R _ {t}} ^ {G} (F ^ {*}) \\ + \mathcal {O} \left(k \cdot | L (\mathcal {T}) | ^ {3 / 2} \cdot D _ {\mathcal {T}} \cdot \max  (1, \gamma)\right) \cdot \sqrt {T} \\ \end{array}
$$
Let  $n = |V|$  and  $D = \text{diam}(G)$ . From the above, we get that Algorithm 2 is indeed  $\alpha$ -regret for  $\alpha = \mathcal{O}(\log n)$ . Furthermore, we have that  $|L(\mathcal{T})| = |V| = n$ , and  $D_{\mathcal{T}} = 2 \cdot (2^{h(\mathcal{T})} - 1) \leq 4D$  since  $h(\mathcal{T}) \leq \lceil \log D \rceil$ . Thus, setting  $\beta = \mathcal{O}(k \cdot n^{3/2} \cdot D \cdot \max(1, \gamma))$ , we get that Algorithm 2 has  $\beta$ -additive regret, completing the proof of Theorem 1.
D Analysis of FTRL (Proofs of Section 4)
In this chapter of the appendix we present all the omitted proofs from Section 4 concerning our analysis of the Follow the Regularized Leader (FTRL) algorithm (Algorithm 3). To avoid repetition, from now on we fix an arbitrary HST  $\mathcal{T}$  and use  $\mathcal{FP}(\mathcal{T})$  to denote the set of all fractional placements of  $k$  facilities on the leaves of  $\mathcal{T}$ . We use  $n = |L(\mathcal{T})|$  to denote the number of leaves of  $\mathcal{T}$ ,  $h = h(\mathcal{T})$  to denote its height and  $D = \text{diam}(\mathcal{T})$  to denote its diameter. Since  $\mathcal{T}$  is an HST, we know that its diameter  $D$ , i.e. the maximum distance between any two leaves, is precisely  $D = 2 \cdot (2^h - 1)$ .
To ease notation, let  $w_{v} = 2^{\mathrm{lev}(v)}$ . For convenience, we remind the reader that our regularizer function  $R_{\mathcal{T}}: \mathcal{FP}(\mathcal{T}) \mapsto \mathbb{R}$  is defined as
$$
R _ {\mathcal {T}} (y) = \sum_ {v \neq r} w _ {v} \cdot (y _ {v} + \delta_ {v}) \cdot \ln \left(\frac {y _ {v} + \delta_ {v}}{y _ {p (v)} + \delta_ {p (v)}}\right)
$$
where  $\delta_v = k \cdot |L(\mathcal{T}) \cap T(v)| / |L(\mathcal{T})|$  is the percentage of leaves that lie on the sub-tree rooted at vertex  $v$  multiplied by  $k$  and  $p(v)$  is the parent of node  $v$ . Also, recall that for any  $y \in \mathcal{FP}(\mathcal{T})$  we have defined the norm
$$
| | y | | _ {\mathcal {T}} = \gamma \cdot \sum_ {v \in V (\mathcal {T})} w _ {v} | y _ {v} |
$$
Roadmap. In Section D.1 we prove Lemma 1, namely the strong convexity of  $R_{\mathcal{T}}$  with respect to  $||\cdot||_{\mathcal{T}}$ . Then, in Section D.2 we bound the moving cost of FTRL, proving Lemma 2. Next, in Section D.3 we bound the connection cost of FTRL, proving Lemma 3. Finally, in Section D.4 we account for approximation errors in the computation of the regularized leader, proving Lemma 4.
D.1 Strong Convexity (Proof of Lemma 1)
The objective of this section is to prove Lemma 1, specifically that for any fractional facility placements  $y, y' \in \mathcal{FP}(\mathcal{T})$  it holds that
$$
R _ {\mathcal {T}} \left(y ^ {\prime}\right) \geq R _ {\mathcal {T}} (y) + \left\langle \nabla R _ {\mathcal {T}} (y), y ^ {\prime} - y \right\rangle + \alpha \left\| y - y ^ {\prime} \right\| _ {\mathcal {T}} ^ {2}
$$
where  $\alpha = (8kD\gamma^{2})^{-1}$
We begin by computing the gradient of  $R_{\mathcal{T}}$  on any fractional facility placement  $y \in \mathcal{FP}(\mathcal{T})$ .
Claim 3. The partial derivatives of  $R_{\mathcal{T}}$  on any point  $y \in \mathcal{FP}(\mathcal{T})$  are given by
$$
\frac {\partial R _ {\mathcal {T}} (y)}{\partial y _ {v}} = \left\{ \begin{array}{l l} - \frac {w _ {v}}{2} & \text {f o r v = r} \\ w _ {v} \cdot \ln \left(\frac {y _ {v} + \delta_ {v}}{y _ {p (v)} + \delta_ {p (v)}}\right) + w _ {v} & \text {f o r v \in L (\mathcal {T})} \\ w _ {v} \cdot \ln \left(\frac {y _ {v} + \delta_ {v}}{y _ {p (v)} + \delta_ {p (v)}}\right) + \frac {w _ {v}}{2} & \text {f o r v \notin L (\mathcal {T}) \cup \{r \}} \end{array} \right.
$$
Proof. Clearly,  $R_{\mathcal{T}}$  is well-defined and differentiable on  $\mathcal{FP}(\mathcal{T})$ . For any  $v \neq r$ , we compute the partial derivatives of  $R_{\mathcal{T}}(y)$  to obtain
$$
\frac {\partial R _ {\mathcal {T}} (y)}{\partial y _ {v}} = w _ {v} \cdot \ln \left(\frac {y _ {v} + \delta_ {v}}{y _ {p (v)} + \delta_ {p (v)}}\right) + w _ {v} - \sum_ {v \in \operatorname {c l d} (u)} w _ {u} \cdot \frac {y _ {u} + \delta_ {u}}{y _ {v} + \delta_ {v}}
$$
Since  $y \in \mathcal{FP}(\mathcal{T})$ , we know  $y_v = \sum_{u \in \operatorname{cld}(v)} y_u$  and by definition,  $\delta_v = \sum_{u \in \operatorname{cld}(v)} \delta_u$ . Finally, recall that  $w_u = w_v / 2$  for any  $u \in \operatorname{cld}(v)$ . By plugging everything in we get
$$
\frac {\partial R _ {\mathcal {T}} (y)}{\partial y _ {v}} = w _ {v} \cdot \ln \left(\frac {y _ {v} + \delta_ {v}}{y _ {p (v)} + \delta_ {p (v)}}\right) + w _ {v} - \frac {w _ {v}}{2} \cdot \mathbb {1} [ v \notin L (\mathcal {T}) ]
$$
for any  $v \neq r$ . For the root vertex, using similar arguments we get
$$
\frac {\partial R _ {\mathcal {T}} (y)}{\partial y _ {r}} = - \frac {w _ {r}}{2}
$$
Now that we have calculated the gradient of  $R_{\mathcal{T}}$ , we can substitute it into the definition of strong convexity. Specifically, by Claim 3, Lemma 1 states that
$$
\sum_ {v \neq r} w _ {v} \cdot \left(y _ {v} ^ {\prime} + \delta_ {v}\right) \cdot \ln \left(\frac {\frac {y _ {v} ^ {\prime} + \delta_ {v}}{y _ {p (v)} ^ {\prime} + \delta_ {p (v)}}}{\frac {y _ {v} + \delta_ {v}}{y _ {p (v)} + \delta_ {p (v)}}}\right) \geq \frac {1}{8 k D \gamma^ {2}} \cdot | | y ^ {\prime} - y | | _ {\mathcal {T}} ^ {2} \tag {1}
$$
To ease the presentation, we define quantities
$$
f \left(y ^ {\prime}, y\right) = \sum_ {v \neq r} w _ {v} \cdot \left(y _ {v} ^ {\prime} + \delta_ {v}\right) \cdot \ln \left(\frac {\frac {y _ {v} ^ {\prime} + \delta_ {v}}{y _ {p (v)} ^ {\prime} + \delta_ {p (v)}}}{\frac {y _ {v} + \delta_ {v}}{y _ {p (v)} + \delta_ {p (v)}}}\right)
$$
$$
h (y ^ {\prime}, y) = \sum_ {v \neq r} w _ {v} \cdot (y _ {p (v)} + \delta_ {p (v)}) \cdot | \frac {y _ {v} ^ {\prime} + \delta_ {v}}{y _ {p (v)} ^ {\prime} + \delta_ {p (v)}} - \frac {y _ {v} + \delta_ {v}}{y _ {p (v)} + \delta_ {p (v)}} |
$$
We will prove that  $f(y', y) \geq (1/2kD) \cdot h^2(y', y)$  and that  $h(y', y) \geq (1/2\gamma) \cdot ||y' - y||_{\mathcal{T}}$  in Claims 4 and 5 respectively. Combining these claims, equation (1) clearly holds, completing the proof of Lemma 1.
Claim 4. For any  $y, y' \in \mathcal{FP}(\mathcal{T})$ , it holds that  $f(y', y) \geq \frac{1}{2kD} \cdot (h(y', y))^2$ .
Proof. We begin by establishing some notation. For any  $v \neq r$ , let
$$
\mu_ {v} ^ {\prime} = w _ {v} \cdot (y _ {p (v)} ^ {\prime} + \delta_ {p (v)}) \cdot \frac {y _ {v} ^ {\prime} + \delta_ {v}}{y _ {p (v)} ^ {\prime} + \delta_ {p (v)}}
$$
$$
\mu_ {v} = w _ {v} \cdot \left(y _ {p (v)} ^ {\prime} + \delta_ {p (v)}\right) \cdot \frac {y _ {v} + \delta_ {v}}{y _ {p (v)} + \delta_ {p (v)}}.
$$
Then, we have that
$$
\begin{array}{l} f \left(y ^ {\prime}, y\right) = \sum_ {v \neq r} \mu_ {v} ^ {\prime} \cdot \ln \left(\frac {\mu_ {v} ^ {\prime}}{\mu_ {v}}\right) \\ = \sum_ {v \in I} \mu_ {v} ^ {\prime} \cdot \ln \left(\frac {\mu_ {v} ^ {\prime}}{\mu_ {v}}\right) + \sum_ {v \in I ^ {\prime}} \mu_ {v} ^ {\prime} \cdot \ln \left(\frac {\mu_ {v} ^ {\prime}}{\mu_ {v}}\right) \\ \end{array}
$$
where  $I = \{v\neq r:\mu_v^\prime \geq \mu_v\}$  and  $I^{\prime} = \{v\neq r:\mu_v^\prime <  \mu_v\}$ . By applying the log-sum inequality in both of these terms, we obtain
$$
f (y ^ {\prime}, y) \geq \left(\sum_ {v \in I} \mu_ {v} ^ {\prime}\right) \cdot \ln \left(\frac {\sum_ {v \in I} \mu_ {v} ^ {\prime}}{\sum_ {v \in I} \mu_ {v}}\right) + \left(\sum_ {v \in I ^ {\prime}} \mu_ {v} ^ {\prime}\right) \cdot \ln \left(\frac {\sum_ {v \in I ^ {\prime}} \mu_ {v} ^ {\prime}}{\sum_ {v \in I ^ {\prime}} \mu_ {v}}\right)
$$
Next, observe that
$$
\sum_ {v \neq r} \mu_ {v} ^ {\prime} = \sum_ {v \neq r} w _ {v} \cdot \left(y _ {v} ^ {\prime} + \delta_ {v}\right) = 2 k \cdot \left(2 ^ {h} - 1\right) = k \cdot D
$$
$$
\begin{array}{l} \sum_ {v \neq r} \mu_ {v} = \sum_ {v \neq r} w _ {v} \cdot (y _ {p (v)} ^ {\prime} + \delta_ {p (v)}) \cdot \frac {y _ {v} + \delta_ {v}}{y _ {p (v)} + \delta_ {p (v)}} \\ = \sum_ {v \notin L (\mathcal {T})} \left(\frac {w _ {v}}{2} \cdot \left(y _ {v} ^ {\prime} + \delta_ {v}\right) \cdot \sum_ {u \in c l d (v)} \frac {y _ {u} + \delta_ {u}}{y _ {v} + \delta_ {v}}\right) \\ = \sum_ {v \notin L (\mathcal {T})} \frac {w _ {v}}{2} \cdot \left(y _ {v} ^ {\prime} + \delta_ {v}\right) \\ = \frac {1}{2} \cdot 2 k \cdot (2 ^ {h + 1} - 2) \\ = k \cdot D. \\ \end{array}
$$
Let  $B' = \sum_{v \in I} \mu_v'$  and  $B = \sum_{v \in I} \mu_v$ . Then, we have shown that
$$
f \left(y ^ {\prime}, y\right) \geq B ^ {\prime} \cdot \ln \left(\frac {B ^ {\prime}}{B}\right) + \left(k D - B ^ {\prime}\right) \cdot \ln \left(\frac {k D - B ^ {\prime}}{k D - B}\right) \tag {2}
$$
Our next step is to apply Pinsker's inequality to the above expression. Pinsker's inequality states that for any  $p, q \in (0,1)$ , it holds that
$$
p \cdot \ln \left(\frac {p}{q}\right) + (1 - p) \cdot \ln \left(\frac {1 - p}{1 - q}\right) \geq 2 \cdot (p - q) ^ {2}
$$
Since  $B \leq kD$  and  $B' \leq kD$ , we can scale everything in inequality 2 and apply Pinsker's inequality to obtain
$$
f \left(y ^ {\prime}, y\right) \geq \frac {2}{k D} \cdot \left(B - B ^ {\prime}\right) ^ {2} \tag {3}
$$
To complete the proof, we substitute