Selective harvesting over networks

Abstract

Active search on graphs focuses on collecting certain labeled nodes (targets) given global knowledge of the network topology and its edge weights (encoding pairwise similarities) under a query budget constraint. However, in most current networks, nodes, network topology, network size, and edge weights are all initially unknown. In this work we introduce selective harvesting, a variant of active search where the next node to be queried must be chosen among the neighbors of the current queried node set; the available training data for deciding which node to query is restricted to the subgraph induced by the queried set (and their node attributes) and their neighbors (without any node or edge attributes). Therefore, selective harvesting is a sequential decision problem, where we must decide which node to query at each step. A classifier trained in this scenario can suffer from what we call a tunnel vision effect: without any recourse to independent sampling, the urge to only query promising nodes forces classifiers to gather increasingly biased training data, which we show significantly hurts the performance of active search methods and standard classifiers. We demonstrate that it is possible to collect a much larger set of targets by using multiple classifiers, not by combining their predictions as a weighted ensemble, but switching between classifiers used at each step, as a way to ease the tunnel vision effect. We discover that switching classifiers collects more targets by (a) diversifying the training data and (b) broadening the choices of nodes that can be queried in the future. This highlights an exploration, exploitation, and diversification trade-off in our problem that goes beyond the exploration and exploitation duality found in classic sequential decision problems. Based on these observations we propose D$^3$TS, a method based on multi-armed bandits for non-stationary stochastic processes that enforces classifier diversity, which outperforms all competing methods on five real network datasets in our evaluation and exhibits comparable performance on the other two.

Appendices

Appendix A: Complementary results

In Sect. 6.2 we presented results obtained when defining the target populations either as in prior work or as the largest subpopulation in the network. We extend these results by running simulations on ten additional datasets derived by taking the two largest subpopulations as targets (other than the original targets) from CiteSeer, DBpedia, Wikipedia, DonorsChoose and Kickstarter. These datasets are indicated by CS, DBP, WK, DC and KS, followed by 1 and 2, respectively. Table 8 shows performance results for five standalone models and for their combinations using round-robin and D$^3$TS. Except for DBP1 and WK1, D$^3$TS consistently figures among the two best performing methods.

Table 8 Simulation results on ten datasets derived from the original data attest

Appendix B: Can we leverage diversity using a single classifier?

Intuitively, when a learning model is fitted to the nodes it chose to query, it tends to specialize in one region of the feature space and the search will consequently only explore similar parts of the graph, which can severely undermine its potential to find target nodes.

One potential way to mitigate this overspecialization would be to sample nodes probabilistically, as opposed to deterministically querying the node with the highest score. Clearly, we should not query nodes uniformly at random all the time. It turns out that querying nodes uniformly at random periodically does not help either, according to the following experiment. We implemented an algorithm for selective harvesting that samples at each step t, with probability p, an uniformly random node from $\mathcal {B}(t)$, and with $1-p$, the best ranked node according to a support vector regression (SVR) model. Table 9 shows the results for $p=2.5$, 5.0, 10, 15 and $20\%$.

Table 9 Results for SVR w/uniformly random queries on CiteSeer (at $t=1500$) averaged over 40 runs

We observe that the performance does not improve significantly for $p \ge 2.5$%, either because the diversity is not increasing in a way that translates into performance improvements or because all gains are offset by the samples wasted when querying nodes at random.

Instead of querying uniformly at random, we could query nodes according to a probability distribution that concentrates most of the mass on the top k nodes w.r.t. model scores. We experimented with several ways of mapping scores to a probability distribution P. In particular, we considered two classes of distributions:

• truncated geometric distribution ($0< q < 1$):

  $$\begin{aligned} P(v) \propto (1-q)^{\pi (v)-1} q, \quad \text {and} \end{aligned}$$

• truncated Zeta distribution ($r \ge 1$):

  $$\begin{aligned} P(v) \propto \pi (v)^{-r}, \end{aligned}$$

where $\pi (v)$ is the rank of v based on the scores given by the model to $v \in \mathcal {B}(t)$. In each experiment, we set q or r at each step in one of nine ways:

• 1. Top 10 have $x\%$ of the probability mass; for $x \in \{70,90,99\}$.

• 2. Top 10% nodes have $x\%$ of the probability mass; for $x \in \{90,99,99.9\}$.

• 3. Top $k(t) = \min \{10\times (1-t/T),1\}$ have $x\%$ of the probability mass; for $x \in \{70,90,99\}$.

None of the mappings was able to substantially increase the search’s performance. In contrast to almost $20\%$ performance improvement seen by SVR under round-robin on CiteSeer at $T=1500$ (Fig. 3), mapping scores to a probability distribution increased the number of targets nodes found by at most 3%.

Appendix C: Evaluation of MAB algorithms applied to Selective Harvesting

We experiment with representative algorithms of each of the following bandit classes:

• Stochastic Bandits: UCB1, Thompson Sampling (TS), $\epsilon$-greedy,

• Adversarial Bandits: Exp3 (Auer et al. 2002),

• Non-stationary stochastic bandits: Dynamic Thompson Sampling (DTS) (Gupta et al. 2011),

• Contextual Bandits: Exp4 (Auer et al. 2002) and Exp4.P (Beygelzimer et al. 2011).

Fig. 9

Comparison between the best parameterizations of each MAB algorithm

UCB1 and TS are parameter-free. For $\epsilon$-greedy, Exp3 and Exp4.P we set the probability of uniformly random pulls, to $\epsilon \in \{0.10,0.20,0.50\}$, $\gamma \in \{0.10,0.20,0.50\}$ and $Kp_{\min } \in \{0.01,0.05,0.10,0.20,0.50\}$ (respectively). We set parameter $\gamma$ in Exp4 as $K_p{\min }$ in Exp4.P. For DTS, we set the cap on the parameter sum $C \in \{5,10, 20,50\}$. Interestingly, for each MAB algorithm, there was always one parameter value that outperformed all the others in almost all seven datasets. In Fig. 9 we show three representative plots of the performance comparison between the best parameterizations of each MAB algorithm. Since Exp4 was slightly outperformed by Exp4.P, Exp4 is not shown. These results corroborate our expectations (Sect. 5) that DTS would outperform other bandits in selective harvesting problems.