SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We shall refer to the special case of the problem where  $k = m$  simply as the Sequential Spanning Problem.
Proposition 3.4 (From SSP to Self-Directed Learning). Let  $X = \{x^{(1)},\ldots ,x^{(m)}\} \subseteq \mathbb{R}^d$  be a set of examples. For every self-directed linear regression algorithm  $\mathcal{A}$  over  $X$ , with expected learning loss  $\mathcal{L}(\mathcal{A},\mathbb{S}^d)$ , we can use it to get a randomized algorithm  $\mathcal{A}'$  for SSP over  $X$  with expected cost  $\mathbf{E}(C(\sigma ,m))\leq d\mathcal{L}(\mathcal{A},\mathbb{S}^d)$ . Moreover, given a randomized algorithm  $\mathcal{A}'$  for SSP over  $X$ , we can get a self-directed linear regression algorithm  $\mathcal{A}$  over  $X$ , with expected learning loss  $\mathcal{L}(\mathcal{A},\mathbb{S}^d) = \mathbf{E}(C(\sigma ,m)) / d$ . In particular, the construction can be done efficiently.
With the intuition above, we only need to show it is ETH-hard to approximate the SSP. The key technical result we obtain is the following proposition.
Proposition 3.5 (From DkS to SSP). For every function  $\alpha(d, m): R^{+} \times R^{+} \to R^{+}$ , if there is an  $\alpha(d, m)$ -approximate algorithm for the SSP, then there is a  $64\alpha^{2}(n^{3}, n^{3})$ -approximate algorithm for the DkS problem.
As a direct corollary of Proposition 3.5, we obtain the computational hardness of SSP.
Corollary 3.6 (ETH-hardness of SSP). Assuming the Exponential Time Hypothesis (ETH) is true, then there is no algorithm that outputs an  $d^{1 / \log \log^c d}$ -approximate solution to the SSP in poly(d,m) time, where  $c > 0$  is a universal constant.
Proof of Corollary 3.6. By Corollary 1.3 in [Man17] if the Exponential Time Hypothesis (ETH) is true, then there is no polynomial time  $n^{1 / \log \log^c n}$ -approximate algorithm for  $k$ -densest subgraph problem. By Proposition 3.5, if there is a polynomial time  $d^{1 / \mathrm{poly}\log \log d}$ -approximate algorithm for SSP, we can obtain an efficient  $n^{1 / \log \log^c n}$  approximate algorithm for  $k$ -densest subgraph problem. Thus, if ETH is true it is hard to approximate SSP within a  $d^{1 / \mathrm{poly}\log \log d}$  factor in polynomial time.
In the rest of the section, we introduce the high-level of the proof of Proposition 3.5 by breaking it down into two steps.
From  $k$ -Edge Packing to SSP In the first step, we introduce an intermediate problem called  $k$ -Edge Packing. We want to show if we can approximate SSP efficiently then we can approximate  $k$ -Edge Packing efficiently.
Definition 3.7 ( $k$ -Edge Packing). Let  $G = (V, E)$  be an undirected graph with  $n$  vertices and  $m$  edges, and let  $k \in [n]$ . The goal is to find a subset of  $k$  edges  $S$  such that the number of vertices covered by  $S$ ,  $|V(S)|$  is minimized, where  $V(S)$  denotes the set of endpoints of edges in  $S$ . We define opt to be the minimum number of vertices covered by any possible subsets of  $k$  edges. Given  $G$  and  $k$ , an  $\alpha$ -approximate algorithm for the  $k$ -edge packing problem outputs a subset of  $k$  edges  $S$  such that  $V(S) \leq \alpha \text{opt}$  in polynomial time.
The key idea in this step is to show Lemma 3.8, if we can approximate  $k$ -SSP efficiently, then we can approximate  $k$ -edge packing efficiently.
Lemma 3.8. For every function  $\alpha : R^{+} \times R^{+} \to R^{+}$ , if there is an  $\alpha(d, m)$ -approximate algorithm for  $k$ -SSP problem, then there is a  $2\alpha(n, m)$ -approximate algorithm for  $k$ -edge packing problem
To build a connection between a geometric problem and a combinatorial problem, we have the following construction. Given a graph  $G = (V, E)$  with  $n$  vertices and  $m$  edges, we will map every edge  $(u, v)$  to a sparse  $n$ -dimensional vector  $x_{uv}$  such that the only non-zero entries are  $x_u = 1$ ,  $x_v = -1$ . In this way, we obtain a dataset  $X$  as the input of  $\mathcal{A}$ , the approximate algorithm for  $k$ -SSP. Intuitively, if we select  $k$  edges in  $G$  that are disjoint, then no matter how to order the corresponding vectors of these  $k$  edges, the spanning cost is  $\Omega(k)$ . But if we select  $k$  edges that form a clique, the corresponding vectors span a subspace of dimension  $O(\sqrt{k})$ , so it is easy to order these vectors to get a very small spanning cost. With such an intuition, the key structure result we use here is that if a solution to the  $k$ -SSP problem has a sufficiently small cost, then the corresponding edges to these  $k$
vectors must be sufficiently connected to each other and thus cover a sufficiently small number of vertices.
However, now we are only able to approximate the  $k$ -edge packing problem with an algorithm that approximately solves the  $k$ -SSP problem. To finish this step, we should also be able to use an algorithm for SSP as a subroutine to solve the  $k$ -SSP problem. We show this is possible in Lemma 3.9.
Lemma 3.9. Assume that an algorithm for SSP in  $d$  dimensions exists that finds an  $\alpha(d, m)$ -approximate solution in  $\mathrm{poly}(d, m)$  time for some function  $\alpha: R^{+} \times R^{+} \to R^{+}$ . Then an algorithm that finds a  $4\alpha(m(d + m), m(d + m))$ -approximate solution for  $k$ -SSP for every value of  $k$  in  $\mathrm{poly}(d, m)$  time exists.
The intuition here is that if the dataset  $X$  is in general position (every set of  $d$  examples are linearly independent) then SSP is actually a  $d$ -SSP. Given any dataset  $X$  with  $m$  points, we are able to map  $X$  to a dataset  $X'$  in  $f(k, d, m) > d$  dimension that is in the general position. Such a map is done by making multiple copies for each example, lifting them to high dimension and adding tiny structured noise to each of them. We will show that such a transformation well preserves the information in the original dataset so that if we run  $\mathcal{A}$  over  $X'$ , we can extract a good approximate solution to the original  $k$ -SSP from the first  $f(k, d, m)$  terms of the output solution.
From  $\mathbf{D}k\mathbf{S}$  to  $k$ -Edge Packing So far we have shown how to use an algorithm for SSP to solve the  $k$ -edge packing problem. Our final step is to show Lemma 3.10, which implies that if we can approximately solve the  $k$ -edge packing problem then we can also solve the  $k$  densest subgraph problem.
Lemma 3.10 ( $k$ -edge packing and DkS). For  $\alpha > 0$ , if there is an  $\alpha$ -approximate algorithm for  $k$ -edge packing problem, then there is an  $\alpha^2$ -approximate algorithm for DkS problem.
Notice that  $k$ -edge packing is seeking  $k$  edges that cover as few vertices as possible, which can be thought as a dual problem of DkS. If we are able to find the largest number  $f(k)$  such that  $f(k)$  edges can cover at most  $k$  vertices, then these  $f(k)$  edges induce the densest subgraph with  $k$ -vertices. The idea behind Lemma 3.10 is that we can approximately find such  $f(k)$  using an approximate algorithm for  $k$ -edge packing and thus can approximately find the  $k$ -densest subgraph.
With the above three lemmas, we can prove Proposition 3.5.
Proof of Proposition 3.5 Assume we have an  $\alpha(d, m)$ -approximate algorithm for SSP problem, then by Lemma 3.9, we get an efficient  $4\alpha((d + m)m, (d + m)m)$ -algorithm for  $k$ -SSP problem. By Lemma 3.8, we get an efficient  $8\alpha((m + n)m, (m + n)m)$ -approximate algorithm for  $k$ -edge packing problem. Since  $(m + n)m \leq n^3$  always holds, we get an efficient  $8\alpha(n^3, n^3)$ -approximate algorithm for  $k$ -edge packing problem. By Lemma 3.10, this gives us an efficient  $64\alpha^2(n^3, n^3)$ -approximate algorithm for  $k$ -densest subgraph problem.
4 A  $O(\log d)$ -Approximation for Spherical Data
Theorem 1.5, suggests that without any assumption on the structure of the data, it is hard to obtain an efficient self-directed learners that approximates the best possible improvement over learning in random order. This motivates us to study the learning problem over datasets with natural structures. In this section, we consider perhaps the most natural setting, where the dataset  $X$  is drawn i.i.d. from  $\mathbb{S}^d$ . Our main result in this section shows a simple greedy heuristic, Algorithm 1, which selects the "easiest" example in each round, and has a nearly (i.e., off by a log  $d$ -factor) optimal gain of ordering.
We first bound the Gain of Ordering achieved by the greedy heuristic described in Algorithm 1.
Proposition 4.1. Let  $X$  be a set of  $n \geq \mathrm{poly}(d)$  examples drawn i.i.d. from  $\mathbb{S}^d$  and let  $\mathcal{A}$  denote Algorithm 1, then over the randomness of the dataset  $X$ , in expectation,  $\mathcal{G}(\mathcal{A}, X, \mathbb{S}^d)^{-1} \leq O(1 / d + n^{-2 / d})$ .
Algorithm 1 SELFDIRECTEDLINEARREGRESSION(X)
Randomly partition  $X$  into  $d$  subsets  $X_{1},\ldots ,X_{d}$  such that  $X_{i}$  contains  $O(n / (d - i)^{2})$  examples.
for  $i = 1,\dots ,d$  do
Set  $L_{i - 1} = \operatorname {span}\{x^{(1)},\ldots ,x^{(i - 1)}\}$
Find  $w^{(i)} \in L_{i-1}$  consistent with  $(x^{(1)}, y^{(1)}), \ldots, (x^{(i-1)}, y^{(i-1)})$ .
Select  $x^{(i)} = \operatorname{argmin}_{x\in X_i}\left\| \operatorname{proj}_{L_{i - 1}}(x)\right\|$
Predict  $\hat{y}^{(i)} = w^{(i)}\cdot x^{(i)}$  and receive  $y^{(i)}$
Label all unlabeled examples in  $X$  using  $w^{(d)}$ .
We start with some intuition for Algorithm 1. Since the observed labels  $y^{(i)}$  are consistent with  $w^*$ , after selecting examples  $x^{(1)}, \ldots, x^{(i)}$ , our guess  $w^{(i)}$  at step  $i$  coincides with the projection  $w_{L_i}^*$  of  $w^*$  onto the subspace  $L_i$  spanned by these examples. If we choose any  $x^{(i+1)}$  and predict  $w_{L_i}^* \cdot x^{(i+1)}$ , then we will pay  $\ell_i^2 = (w_{L_i^\perp}^* \cdot x^{(i+1)})^2 = (w_{L_i^\perp}^* \cdot x_{L_i^\perp}^{(i+1)})^2$ . Since we make a random partition of the data in advance, we are able to show that in expectation  $\ell_i^2 / (\left\| w_{L_i^\perp}^* \right\|^2 \left\| x_{L_i^\perp}^{(i+1)} \right\|^2) = 1/(d - i)$ .
This suggests the cost we pay in each round is proportional to  $\left\| x_{L_i^{\pm}}^{(i + 1)} \right\|^2$  and we should choose the example that is closest to  $L_{i}$  greedily. Our key technical lemma shows that in the  $i + 1$ -th round, for a greedily chosen example and a randomly chosen example, the expected ratio of the loss is  $O(n^{-2 / (d - i)})$ .
We next show that the optimal Gain from Ordering can only be  $O(\log d)$  times larger than that of Algorithm 1.
Proposition 4.2 (Bounding the Optimal Gain). Let  $X \subseteq S^d$  be a set of  $n > \mathrm{poly}(d)$  examples drawn i.i.d. uniformly from  $S^d$ . For every  $\delta \in (3/n, 1)$ , with probability at least  $99\%$ ,  $\mathcal{G}^*(X, \mathbb{S}^d)^{-1} \geq \Omega(1/d + \delta n^{-(2+2\delta)/d})$ .
Recall that in Section 3, we showed that if  $w^{*} \sim \mathbb{S}^{d}$ , then the expected learning cost is proportional to the optimal spanning cost of a sequence of examples. This implies analyzing the best learner is equivalent to analyzing the sequence of examples in  $X$  with the smallest spanning cost. The key of our proof is to show that the distribution of  $\mathbb{S}^{d}$  is very concentrated and unless the size of  $X$  is very large there is no sequence of examples whose spanning cost is much smaller than the average spanning cost.
As a direct corollary, we are able to show Algorithm 1 has a gaining that is at most  $O(1 / \log d) \mathcal{G}^*(X, \mathbb{S}^d)$ , which is an exponential improvement of the hardness of approximation obtained by Theorem 1.5.
Proof of Theorem 1.6 By Proposition 4.1 and Proposition 4.2, we know that with probability at least  $99\%$ ,  $\mathcal{G}(\mathcal{A}, X, \mathbb{S}^d)^{-1} \leq 1 / d + n^{-2 / d}$  and  $\mathcal{G}^*(X, \mathbb{S}^d)^{-1} \geq 1 / d + \delta n^{-(2 + 2\delta) / d}$ . Here we ignore the hidden constant within the bound obtained from Proposition 4.1 and Proposition 4.2 because it will only add some multiplicative constant factor in our final result. It remains to tune the parameter  $\delta$  to show that  $\frac{1 / d + n^{-2 / d}}{1 / d + \delta n^{-(2 + 2\delta) / d}}$  is at most  $O(\log d)$ . To do this, we write  $s^{d / 2}$  and we consider different ranges for  $s$ . We consider the following two cases.
Case 1: If  $s \geq d$ , then we have  $1 / d + n^{-2 / d} \leq 2 / d \leq 2(1 / d + \delta n^{-(2 + 2\delta) / d})$ , which implies  $\mathcal{G}(\mathcal{A}, X, \mathbb{S}^d) \geq \Omega(1)\mathcal{G}^*(X, \mathbb{S}^d)$ .
Case 2: If  $s < d$ , then  $n^{-2/d} \geq 1/d$  and  $1/d + \delta n^{-(2+2\delta)/d} \geq \delta n^{-(2+2\delta)/d}$ . This implies  $\frac{1/d+n^{-2/d}}{1/d+\delta n^{-(2+2\delta)/d}} \leq \frac{2}{\delta} n^{2\delta/d} = \frac{2}{\delta} s^{\delta}$ . We set  $\delta = 1/\log s$  and we obtain that  $\frac{1}{\delta} s^{\delta} = s^{1/\log s} \log s = O(\log s) \leq O(\log d)$ . This implies that  $\mathcal{G}(\mathcal{A}, X, \mathbb{S}^d) \geq \Omega(1/\log d) \mathcal{G}^*(X, \mathbb{S}^d)$ .
Remark 4.3. We remark that although the statement of Proposition 4.1 and Proposition 4.2 are about the average performance of the algorithm, the same results also hold under the worst-case setting because, with a dataset from a spherical distribution, the learning cost of  $\mathcal{A}^{\mathrm{random}}$  under
these two settings are asymptotically the same in expectation. A detailed discussion about this and the proof of Proposition 4.1 and Proposition 4.2 are deferred to Appendix B.
Remark 4.4. Although according to Theorem 1.6, Algorithm 1 approximates the optimal gaining within a  $O(\log d)$  factor, when  $n \leq s^d$  for some constant  $s$  that doesn't depend on  $d$  or  $n \geq d^{d/2}$ . Algorithm 1 approximates the optimal gaining within a constant factor and thus is nearly optimal.
5 Self-Directed ReLU Regression
Finally, we study the problem of self-directed ReLU regression, which shares a similar spirit to the one of self-directed linear regression. According to Proposition 4.1, given a dataset  $X$  that is drawn from  $\mathbb{S}^d$ , we are able to design an efficient learner with gaining  $\mathcal{G}(\mathcal{A}, X) \geq \min(\Omega(d), \Omega(n^{2/d}))$  if each  $x$  is labeled by  $y(x) = w^* \cdot x$ . In fact, the bound of  $\min(\Omega(d), \Omega(n^{2/d}))$  can also be obtained by the following Algorithm 2, when each  $x$  is labeled by a ReLU function  $\mathrm{Relu}(w^* \cdot x)$ . However, an interesting phenomenon we found is that if we give Algorithm 2 some reference example  $(x^{(0)}, y^{(0)})$  as a warm start such that  $\theta(x^{(0)}, w^*) = \theta_0 < \pi/2$ , then we are able to improve the gaining to  $\min(\Omega(d), \Omega(n^{2/d}/\log d + \log n/\log d))$ , which is a huge improvement when the dimension of the problem is large.
Algorithm 2 SELFIRECTEDRELUREgression(X)
Let  $(x^{(0)},y^{(0)})$  be a pair of reference example such that  $\theta (x^{(0)},w^{*}) = \theta_{0}\leq \pi /2$
(Assume  $(x^{(0)},y^{(0)}) = (0,0)$  if there is no such a warm start)
Randomly partition  $X$  into  $d$  subsets  $X_{1},\ldots ,X_{d}$  such that  $X_{i}$  contains  $n / d$  examples.
Set  $L_{i-1} = \operatorname{span}\{x^{(0)}, \ldots, x^{(i-1)}\}$ , where  $x^{(j)}$  is example that has been selected with positive label for  $j \in \{0, \ldots, i-1\}$ .
Find  $w^{(i)} \in L_{i-1}$  consistent with  $(x^{(0)}, y^{(0)}), \ldots, (x^{(i-1)}, y^{(i-1)})$ .
Keep selecting  $x \in X_{i}$  such that  $w^{(i)} \cdot x \in \operatorname{argmin}_{x' \in X_{i}} w^{(i)} \cdot x'$  and predict  $\mathrm{Relu}(w^{(i)} \cdot x)$  until we see some  $(x^{(i)}, y^{(i)})$  such that  $y^{(i)} > 0$ .
Label all unlabeled examples in  $X$  using  $w^{(d + 1)}$ .
Theorem 5.1. Let  $X \subseteq \mathbb{R}^d$  be a set of  $n \geq \mathrm{poly}(d)$  examples drawn i.i.d. from  $\mathbb{S}^d$ . Let  $\mathcal{G}(\mathcal{A}, X)$  be the gain from ordering of Algorithm 2 for the self-directed ReLU regression problem over  $X$ . Then
$$
\mathcal {G} (\mathcal {A}, X) ^ {- 1} \leq O (\frac {1}{d}) + \left\{ \begin{array}{l l} \min \{O (\frac {\tan^ {2} \theta_ {0} \log d}{d}), O (\frac {\log d}{n ^ {2 / d}}) \} & \quad i f \frac {n \theta_ {0}}{4 \pi d \log d} > \exp (\frac {d}{8}), \\ \min \{O (\frac {\tan^ {2} \theta_ {0} \log d}{\log (n \theta_ {0})}), O (\frac {\log d}{n ^ {2 / d}}) \} & \quad i f 1 \leq \frac {n \theta_ {0}}{4 \pi d \log d} \leq \exp (\frac {d}{8}) \\ O (\frac {1}{n}) & \quad i f \frac {n \theta_ {0}}{4 \pi d \log d} <   1. \end{array} \right.
$$
The main difference between ReLU regression and linear regression is that in ReLU regression, with a good warm start, we are able to start from the region that is labeled 0 by the target to seek the decision boundary. Before we see a positive example, we pay nothing. The first time we see a positive example  $x$ , we pay  $(w^{*} \cdot x)^{2}$ . The key technical lemma, Lemma 5.2 shows that as we keep selecting the most "negative" example with respect to our current guess, the first positive example we see must have a very small margin with respect to  $w^{*}$ . In this way, Algorithm 2 learns  $w^{*}$  with a very small cost. We refer to Appendix C for more details and proofs.
Lemma 5.2 (Geometry Technical Lemma). Let  $w^{*} \in S^{d}$  be a target vector and let  $w \in S^{d}$  be an arbitrary vector such that  $\theta = \theta(w^{*}, w) < \pi/2$ . Denote by  $C = \{x \in S^{d} \mid w^{*}x \leq 0, w \cdot x \geq 0\}$ . For every  $a, b \in (0,1)$ , denote by  $K_{a} := \{x \in C \mid w \cdot x \geq a\sin \theta\}$  and  $K_{a,b} := \{x \in C \mid w \cdot x \geq a\sin \theta, w^{*}x \leq -b\sin \theta\}$ . Let  $x$  be a point uniformly drawn from  $S^{d}$ . There is some absolute constant  $c > 1$  such that if  $a/b \geq c$  then  $\mathbf{Pr}(x \in K_{a,b} \mid x \in K_{a}) \leq 2\exp\left(-\frac{d}{3(1-a^{2})}(b^{2} + 2ab\cos\theta)\right)$ .
In this work, we study the self-directed learning problem for linear regression. Our work presents novel results both computational and information-theoretic. Our first result shows that approxi-
mating the optimal self-directed regret within a  $d^{1/poly\log \log d}$  factor is ETH-hard under arbitrary datasets. Our second result yields a novel characterization of the Gain of Ordering for data that are uniformly distributed on the sphere and gives an efficient approximation algorithm that bypasses the aforementioned hardness result and achieves close to optimal regret by exploiting the structure of the data. A limitation of our work is that our algorithms can currently handle linear (or ReLU regression). Moreover, our presented results assume that the labels are realizable - even in this fundamental setting, nothing was known prior to our work. Generalizing our results to other concept classes and investigating the effect of adding noise to the labels are natural questions for future investigation. Designing robust self-directed learning algorithms for broader concept classes under broader distributional assumptions are interesting direction for future work.
This work was supported by the NSF Award CCF-2144298 (CAREER).
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