Successive Elimination Algorithm

• Successive Elimination Algorithm is a sequential decision procedure that progressively discards candidates based on statistically derived confidence bounds.
• It is applied in stochastic multi-armed bandits, adaptive robotic sensing, and differential privacy to optimize sampling efficiency and reduce regret.
• The approach leverages uniform sampling over active sets and targeted elimination tests to concentrate measurements and achieve optimal performance guarantees.

Searching arXiv for recent and foundational papers on successive elimination in bandits and adaptive sensing. Successive elimination denotes a class of sequential decision procedures that maintain a set of viable candidates, collect additional observations only from the surviving candidates, construct confidence bounds around empirical estimates, and discard any candidate whose upper confidence bound is already dominated by another candidate’s lower confidence bound. In the stochastic multi-armed bandit setting, the standard formulation attributed to Even-Dar et al. repeatedly pulls every remaining arm until only one arm survives; later work adapts the same core logic to adaptive robotic sensing with Poisson measurements and to differentially private bandits with private stopping and elimination rules (Sajed et al., 2019, Rolf et al., 2018).

1. Canonical stochastic-bandit formulation

In the standard non-private stochastic-MAB formulation, one begins with a set of viable arms

$S=\{1,\dots,K\}.$

At round $r=1,2,\dots$, each arm $i\in S$ is pulled once, a reward $X_{i,r}\in[0,1]$ is observed, and the empirical mean $\hat\mu_i(r)$ is updated. A confidence radius is then computed, for example

$$\epsilon_r=\sqrt{\frac{1}{2r}\ln\!\left(\frac{4Kr^2}{\delta}\right)}.$$

Any arm $i$ is eliminated if there exists an arm $j\in S$ such that

$\hat\mu_i(r)+\epsilon_r<\hat\mu_j(r)-\epsilon_r.$

The procedure stops when $|S|=1$, and the remaining arm is returned (Sajed et al., 2019).

This formulation is a pure elimination algorithm rather than an optimism-driven allocation rule in the style of UCB. Its state is the active set $r=1,2,\dots$0, not a global ranking over all previously sampled arms. The computational structure is correspondingly simple: repeated uniform sampling over the current active set, paired with a confidence-based elimination test. In the exposition of Sajed and Sheffet, this standard SE procedure is described as repeatedly pulling all remaining arms until an arm is found to be suboptimal and is then eliminated (Sajed et al., 2019).

The same source states that SE achieves the same instance-dependent regret bound as UCB,

$r=1,2,\dots$1

as well as minimax regret $r=1,2,\dots$2 (Sajed et al., 2019). This places successive elimination within the standard optimality landscape of stochastic bandits, while preserving a particularly transparent elimination logic.

2. Statistical mechanism and elimination logic

The central statistical object in successive elimination is the confidence interval. Elimination occurs only when confidence bounds separate sufficiently to certify that one candidate cannot be optimal. In the classical bandit version, the test is symmetric: an arm is removed when its optimistic estimate, $r=1,2,\dots$3, falls below another arm’s conservative estimate, $r=1,2,\dots$4 (Sajed et al., 2019). This induces a monotone shrinking of the active set.

In the adaptive sensing formulation of AdaSearch, the same logic is expressed through lower-confidence bounds (LCBs) and upper-confidence bounds (UCBs) tailored to Poisson counts. At round $r=1,2,\dots$5, AdaSearch doubles the per-point measurement duration on the current candidate set,

$r=1,2,\dots$6

and uses the per-round confidence allocation

$r=1,2,\dots$7

For Poisson counts, it defines

$r=1,2,\dots$8

and

$r=1,2,\dots$9

Then, for each surviving location $i\in S$0,

$i\in S$1

The algorithm identifies $i\in S$2 and $i\in S$3, accepts $i\in S$4 if its LCB exceeds every competing UCB, and eliminates $i\in S$5 if its UCB is below the maximum competing LCB (Rolf et al., 2018).

For general top-$i\in S$6, the same source states that one compares each $i\in S$7’s UCB to the $i\in S$8-th largest LCB of others to reject, and each $i\in S$9’s LCB to the $X_{i,r}\in[0,1]$0-th largest UCB of others to accept (Rolf et al., 2018). This makes explicit that successive elimination is not restricted to best-arm identification with $X_{i,r}\in[0,1]$1; the elimination criterion extends directly to top-$X_{i,r}\in[0,1]$2 selection.

3. Embodied and adaptive variants

AdaSearch modifies the classical successive-elimination template to address an embodied source-seeking problem in which a mobile robot must identify the top-$X_{i,r}\in[0,1]$3 highest-emission locations in a discretized search space $X_{i,r}\in[0,1]$4. Each point $X_{i,r}\in[0,1]$5 emits counts according to an unknown Poisson rate $X_{i,r}\in[0,1]$6, and the objective is to identify the top-$X_{i,r}\in[0,1]$7 locations with probability at least $X_{i,r}\in[0,1]$8 while minimizing total robot flight time (Rolf et al., 2018).

The algorithm interleaves adaptive sampling with an embodied motion model. Rather than treating samples as costless pulls, AdaSearch assumes that the robot must physically move to each point to sample it. The motion plan is a “snake” trajectory: the robot allocates more dwell time over candidate points in the current active set $X_{i,r}\in[0,1]$9, but only the baseline dwell time $\hat\mu_i(r)$0 over locations in $\hat\mu_i(r)$1, so that the motion remains roughly a space-filling curve (Rolf et al., 2018). This is the principal departure from classical MAB successive elimination.

The same work emphasizes three differences due to embodiment. First, there is travel-time overhead: runtime includes the cost of visiting candidate locations, and the stated runtime bound therefore has an extra $\hat\mu_i(r)$2 term. Second, the observation model is heterogeneous Poisson noise rather than bounded or sub-Gaussian noise, so the analysis uses specialized Poisson-tail inequalities rather than Hoeffding’s inequality. Third, the algorithm incorporates multiple-testing correction through a per-round confidence schedule so that the union over rounds and points remains bounded by $\hat\mu_i(r)$3 (Rolf et al., 2018).

This variant shows that successive elimination is a structural idea rather than a single sampling schedule. The active-set logic remains unchanged, but both the sampling unit and the confidence model are altered to accommodate embodied sensing, dwell-time control, and non-Gaussian count data.

4. Differential privacy and private stopping rules

A distinct extension embeds successive elimination into pure $\hat\mu_i(r)$4-differential privacy. Sajed and Sheffet define neighboring input streams as streams of length $\hat\mu_i(r)$5 that differ in exactly one reward at one timestep, and define a streaming algorithm $\hat\mu_i(r)$6 to be $\hat\mu_i(r)$7-differentially private if, for every neighboring pair $\hat\mu_i(r)$8 and every event $\hat\mu_i(r)$9 in the output stream,

$$\epsilon_r=\sqrt{\frac{1}{2r}\ln\!\left(\frac{4Kr^2}{\delta}\right)}.$$0

The mechanisms used are the Laplace mechanism, which adds $$\epsilon_r=\sqrt{\frac{1}{2r}\ln\!\left(\frac{4Kr^2}{\delta}\right)}.$$1 noise to a statistic $$\epsilon_r=\sqrt{\frac{1}{2r}\ln\!\left(\frac{4Kr^2}{\delta}\right)}.$$2 with $$\epsilon_r=\sqrt{\frac{1}{2r}\ln\!\left(\frac{4Kr^2}{\delta}\right)}.$$3-sensitivity $$\epsilon_r=\sqrt{\frac{1}{2r}\ln\!\left(\frac{4Kr^2}{\delta}\right)}.$$4, and the Sparse Vector Technique, which privately tests a sequence of threshold queries with a one-time privacy cost $$\epsilon_r=\sqrt{\frac{1}{2r}\ln\!\left(\frac{4Kr^2}{\delta}\right)}.$$5 (Sajed et al., 2019).

The private construction is built around a private stopping rule, denoted DP-NAS. The stopping-rule problem takes as input an i.i.d. stream $$\epsilon_r=\sqrt{\frac{1}{2r}\ln\!\left(\frac{4Kr^2}{\delta}\right)}.$$6 from an unknown distribution on $$\epsilon_r=\sqrt{\frac{1}{2r}\ln\!\left(\frac{4Kr^2}{\delta}\right)}.$$7 with mean $$\epsilon_r=\sqrt{\frac{1}{2r}\ln\!\left(\frac{4Kr^2}{\delta}\right)}.$$8, and asks for an $$\epsilon_r=\sqrt{\frac{1}{2r}\ln\!\left(\frac{4Kr^2}{\delta}\right)}.$$9-stopping rule that halts at $i$0 and outputs $i$1 such that

$i$2

while preserving $i$3-DP. The non-private building block halts when $i$4 exceeds a threshold proportional to

$i$5

The private version adds Laplace noise with scales

$i$6

uses an SVT check to decide halting, and upon halting outputs

$i$7

where $i$8 (Sajed et al., 2019).

Private successive elimination (DP-SE) then replaces the roundwise suboptimality test by the private stopping machinery. To avoid paying $i$9 per arm, the algorithm groups pulls into epochs of geometrically increasing length and re-starts fresh each epoch. At epoch $j\in S$0, it sets

$j\in S$1

and

$j\in S$2

After $j\in S$3 pulls per active arm, it adds Laplace noise $j\in S$4 to each arm’s empirical mean, defines

$j\in S$5

and removes every arm $j\in S$6 satisfying

$j\in S$7

The stated privacy theorem is that each reward contributes to exactly one epoch, so adding Laplace noise per epoch yields total $j\in S$8-DP by parallel and sequential composition (Sajed et al., 2019).

5. Complexity guarantees and empirical behavior

The complexity guarantees differ by domain, but all three settings preserve the central elimination intuition: easy-to-reject alternatives disappear quickly, and the remaining sample budget is concentrated on difficult competitors.

For AdaSearch with $j\in S$9, the per-point difficulty is defined as

$\hat\mu_i(r)+\epsilon_r<\hat\mu_j(r)-\epsilon_r.$0

where $\hat\mu_i(r)+\epsilon_r<\hat\mu_j(r)-\epsilon_r.$1. The main theorem states that, with probability at least $\hat\mu_i(r)+\epsilon_r<\hat\mu_j(r)-\epsilon_r.$2, AdaSearch returns the true maximizer $\hat\mu_i(r)+\epsilon_r<\hat\mu_j(r)-\epsilon_r.$3 and uses at most

$\hat\mu_i(r)+\epsilon_r<\hat\mu_j(r)-\epsilon_r.$4

total measurement time; in particular,

$\hat\mu_i(r)+\epsilon_r<\hat\mu_j(r)-\epsilon_r.$5

The same source contrasts this with uniform sampling, which requires

$\hat\mu_i(r)+\epsilon_r<\hat\mu_j(r)-\epsilon_r.$6

and states a matching lower bound

$\hat\mu_i(r)+\epsilon_r<\hat\mu_j(r)-\epsilon_r.$7

for any algorithm (Rolf et al., 2018).

For private stochastic bandits, the stated instance-dependent regret bound for DP-SE is

$\hat\mu_i(r)+\epsilon_r<\hat\mu_j(r)-\epsilon_r.$8

for sufficiently large $\hat\mu_i(r)+\epsilon_r<\hat\mu_j(r)-\epsilon_r.$9. This matches the non-private lower bound $|S|=1$0 and the private lower bound $|S|=1$1. The minimax regret guarantee is

$|S|=1$2

which is described as close to the optimal non-private $|S|=1$3 and private $|S|=1$4 lower bounds (Sajed et al., 2019).

The empirical record reported in the two applications is likewise domain-specific. In AdaSearch simulations on $|S|=1$5 grids with $|S|=1$6 and $|S|=1$7 with a unique maximum, the reported sample/time complexity grows like $|S|=1$8, versus approximately $|S|=1$9 for the naïve uniform strategy, and the observed total time is approximately $r=1,2,\dots$00 versus approximately $r=1,2,\dots$01. In hardware, a quadrotor in an indoor arena with three $r=1,2,\dots$02-sources eliminates the two weaker sources after a few adaptive rounds and achieves end-to-end identification that is approximately $r=1,2,\dots$03 faster than uniform “snake” sampling (Rolf et al., 2018).

For private bandits, Sajed and Sheffet compare DP-SE against DP-UCB on Bernoulli-armed bandits with $r=1,2,\dots$04, gap profiles including uniform, linearly decreasing, convex, and concave, $r=1,2,\dots$05, and $r=1,2,\dots$06. The reported finding is that DP-SE consistently outperforms DP-UCB by large margins, often with $r=1,2,\dots$07–$r=1,2,\dots$08 smaller regret, with the benefit most pronounced when $r=1,2,\dots$09 is small or $r=1,2,\dots$10 is large (Sajed et al., 2019).

6. Scope, misconceptions, and related elimination procedures

The literature makes clear that “successive elimination” is not tied to a single noise model, a single sampling primitive, or a single optimality criterion. In one setting, the observation model is bounded rewards in $r=1,2,\dots$11; in another, it is Poisson counts over a discretized physical environment; in a third, privacy constraints require noisy releases and a private stopping rule (Sajed et al., 2019, Rolf et al., 2018). A common misconception is therefore to treat successive elimination as merely a best-arm identification routine for abstract bandits. The cited work suggests a broader template: maintain a shrinking candidate set, allocate additional measurements to the candidates that remain plausible, and remove candidates once confidence separation is certified.

A separate use of staged elimination appears in linear algebra. Tran, Justino, and van den Berg analyze Gauss–Jordan elimination through its successive intermediate matrices $r=1,2,\dots$12. For a matrix $r=1,2,\dots$13 of rank $r=1,2,\dots$14 over a field $r=1,2,\dots$15, they consider the usual two-step cycle in which, at step $r=1,2,\dots$16, the $r=1,2,\dots$17-st pivot is normalized by dividing row $r=1,2,\dots$18 by $r=1,2,\dots$19, and at step $r=1,2,\dots$20, that normalized pivot is used to eliminate all other entries in column $r=1,2,\dots$21. Under the nonvanishing principal minor assumption

$r=1,2,\dots$22

the matrix after $r=1,2,\dots$23 operations has block form

$r=1,2,\dots$24

and the entries of the upper-right and lower-right blocks are given by explicit quotient-of-minors formulas. The proof proceeds by induction on $r=1,2,\dots$25, using one-row Laplace expansions rather than general multi-term determinant identities (Tran et al., 2020).

This linear-algebraic use is not the stochastic-algorithmic successive elimination of the bandit literature, but it is related at the level of staged reduction: intermediate states are explicitly characterized, pivots or candidates are progressively resolved, and the procedure’s numerical behavior can be analyzed through its elimination steps. The same source remarks that entry-growth can be controlled through bounds on minors, that round-off propagation can be tracked step by step, and that although naïve evaluation of the closed forms is factorial in $r=1,2,\dots$26, the usual row-operation implementation still runs in $r=1,2,\dots$27 time (Tran et al., 2020).