Bandit Scheduler

Why we use epsilon-greedy multi-armed bandits with UCB for compaction scheduling instead of heuristics or round-robin.

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Decision

Use epsilon-greedy multi-armed bandit with Upper Confidence Bound (UCB) to select which slot to compact:

pub struct BanditScheduler {
    arms: Vec<BanditArm>,  // One arm per slot
    epsilon: f64,          // Exploration rate (default: 0.1)
    ucb_c: f64,            // Exploration constant (default: 2.0)
}

pub struct BanditArm {
    slot_id: u32,
    avg_reward: f64,        // EMA of historical rewards
    selection_count: u64,   // Times this slot was selected
}

impl BanditScheduler {
    pub fn select_slot(&mut self) -> u32 {
        if random() < self.epsilon {
            // Exploration: random slot
            random_slot()
        } else {
            // Exploitation: best UCB score
            argmax(|arm| arm.ucb_score(self.total_selections()))
        }
    }
}

impl BanditArm {
    pub fn ucb_score(&self, total_selections: u64) -> f64 {
        self.avg_reward + self.ucb_c * sqrt(ln(total_selections) / self.selection_count)
        // └─exploitation─┘   └──────────exploration bonus──────────┘
    }
}

Reward function:

reward = (predicted_latency_reduction × heat_score) / bytes_written

Parameters: - ε (epsilon) = 0.1 (10% exploration, 90% exploitation) - c (UCB constant) = 2.0 (standard value from bandit literature) - α (reward EMA) = 0.1 (smoothing factor)

---

Alternatives Considered

1. Round-Robin

Approach: Compact slots in order (0 → 1 → 2 → ... → N → 0)

pub fn select_slot(&mut self) -> u32 {
    let slot = self.current_slot;
    self.current_slot = (self.current_slot + 1) % self.num_slots;
    slot
}

Rejected because: - Ignores heat: Hot/cold treated equally - Wasteful: Compacts cold slots unnecessarily - Inflexible: No adaptation to workload - High WA: Compact everything regardless of benefit

Example problem:

Slots:
  0-3: Hot (1000 queries/sec each)
  4-63: Cold (1 query/day each)

Round-robin:
  Compacts all 64 slots equally
  60/64 = 94% of compactions are wasted (on cold data)

Bandit:
  Learns to compact slots 0-3 frequently, 4-63 rarely
  >90% of compactions focus on hot slots

2. Priority Queue (Largest First)

Approach: Compact largest slots first

pub fn select_slot(&mut self) -> u32 {
    self.slots
        .iter()
        .max_by_key(|s| s.total_size())
        .unwrap()
        .slot_id
}

Rejected because: - Size ≠ benefit: Large cold slot has low read benefit - Ignores heat: Doesn't consider access frequency - Starvation: Small hot slots never compacted - No learning: Static heuristic

Example problem:

Slot 0: 10 MB, heat=0.9 (hot, small)
Slot 1: 1 GB, heat=0.1 (cold, large)

Priority queue: Always compacts Slot 1 (large)
  Benefit: Low (rarely queried)
  Cost: High (1 GB rewrite)

Bandit: Learns to compact Slot 0 (high reward/byte)
  Benefit: High (frequently queried)
  Cost: Low (10 MB rewrite)

3. Age-Based (Oldest Runs First)

Approach: Compact slot with oldest run

pub fn select_slot(&mut self) -> u32 {
    self.slots
        .iter()
        .max_by_key(|s| s.oldest_run_age())
        .unwrap()
        .slot_id
}

Rejected because: - Age ≠ benefit: Old cold data doesn't benefit from compaction - Ignores heat: Doesn't consider query patterns - Predictable: No exploration of new strategies - No optimization: Just a heuristic

4. Thompson Sampling

Approach: Bayesian bandit algorithm

pub struct BanditArm {
    alpha: f64,  // Success count (Beta distribution)
    beta: f64,   // Failure count
}

pub fn select_slot(&mut self) -> u32 {
    // Sample from Beta distribution for each arm
    let samples: Vec<f64> = self.arms
        .iter()
        .map(|arm| sample_beta(arm.alpha, arm.beta))
        .collect();

    argmax(samples)
}

Rejected because: - More complex: Beta distributions, sampling - Unclear reward mapping: How to map compaction reward to success/failure? - Implementation overhead: Beta distribution sampling - Epsilon-greedy is sufficient: Proven to work for non-stationary environments

Decision: May revisit for future optimization, but epsilon-greedy is simpler and sufficient

5. Contextual Bandits

Approach: Use system state as context

pub struct Context {
    l0_file_count: usize,
    memory_pressure: f32,
    recent_query_latency: f64,
}

pub fn select_slot(&mut self, context: &Context) -> u32 {
    // Learn reward function: f(slot, context) → reward
    self.model.predict(slot, context)
}

Rejected for now because: - Much more complex: Requires feature engineering, model training - Data requirements: Need large dataset to train model - Unclear benefit: Simple bandits may be sufficient - Future work: Promising direction for optimization

---

Rationale

1. Exploration vs Exploitation

Multi-armed bandit problem: Balance trying new actions (exploration) vs using known-good actions (exploitation)

Epsilon-greedy solution:

With probability ε (0.1):
  → Exploration: Try random slot
  → Benefit: Discover better strategies
  → Cost: 10% suboptimal compactions

With probability 1-ε (0.9):
  → Exploitation: Choose best UCB score
  → Benefit: Use known-good strategy
  → Cost: May miss better strategies

Why 10% exploration? - Standard value in bandit literature - Balances learning vs performance - Non-stationary environments (workloads change) need exploration - Validated through benchmarks

Example:

1000 compaction decisions:
  100 are random (exploration)
  900 use best UCB score (exploitation)

Result:
  Learn about all slots (avoid starvation)
  Mostly compact high-reward slots (good performance)

2. UCB Exploration Bonus

Upper Confidence Bound formula:

ucb_score = avg_reward + c × sqrt(ln(total_selections) / arm_selections)
            └─exploitation─┘   └──────────exploration bonus──────────┘

Exploration bonus properties: - Large when rarely selected: Encourages trying less-explored slots - Small when frequently selected: Don't over-explore well-known slots - Grows with total_selections: More data → more confidence in exploration - Theoretical regret bounds: O(K × log T) regret over T selections

Example:

Slot 0 (hot, frequently compacted):
  avg_reward = 8.5
  selections = 900
  total = 1000

  ucb_score = 8.5 + 2.0 × sqrt(ln(1000) / 900)
            = 8.5 + 2.0 × sqrt(6.9 / 900)
            = 8.5 + 2.0 × 0.088
            = 8.68

Slot 1 (cold, rarely compacted):
  avg_reward = 2.0
  selections = 10
  total = 1000

  ucb_score = 2.0 + 2.0 × sqrt(ln(1000) / 10)
            = 2.0 + 2.0 × sqrt(6.9 / 10)
            = 2.0 + 2.0 × 0.831
            = 3.66

Decision: Compact Slot 0 (higher UCB score)
  But Slot 1 gets exploration bonus (not starved)

3. Reward Function Design

Formula:

reward = (predicted_latency_reduction × heat_score) / bytes_written

Components:

1. Predicted Latency Reduction:

fn predicted_latency_reduction(slot: &Slot) -> f64 {
    // Assume each run adds 100µs to query latency
    let current_latency = slot.runs.len() as f64 * 100.0;

    // After compaction to k_max, latency reduces
    let target_latency = slot.k_max as f64 * 100.0;

    current_latency - target_latency
}

2. Heat Score: Weight by access frequency

heat_score ∈ [0.0, 1.0]  // From EWMA tracking

3. Bytes Written: Cost of compaction

bytes_written = total_size_of_runs_to_merge

Intuition: - High reward: Hot slots with high RA (good targets for leveling) - Low reward: Cold slots with low RA (avoid unnecessary compaction) - Normalized by cost: Prefer cheap compactions over expensive ones

Example:

Slot 0 (hot, 4 runs, 10 MB each):
  latency_reduction = (4 - 1) × 100µs = 300µs
  heat_score = 0.9
  bytes_written = 40 MB

  reward = (300 × 0.9) / 40
         = 270 / 40
         = 6.75

Slot 1 (cold, 4 runs, 100 MB each):
  latency_reduction = (4 - 3) × 100µs = 100µs (k_max=3 for cold)
  heat_score = 0.1
  bytes_written = 400 MB

  reward = (100 × 0.1) / 400
         = 10 / 400
         = 0.025

Decision: Compact Slot 0 (270× higher reward)

4. Non-Stationary Environment

Workloads change over time:

Week 1: Slots 0-3 hot (user traffic)
Week 2: Slots 10-13 hot (analytics job)
Week 3: Slots 0-3 hot again (back to normal)

Epsilon-greedy adapts: - Exploration: Discovers new hot slots (Week 2) - EMA rewards: Recent rewards weigh more (forgets old patterns) - UCB bonus: Encourages trying previously-cold slots

Contrast with static heuristics:

Priority queue (largest first):
  Week 1: Compacts same large slots
  Week 2: Still compacts same large slots (misses new hot slots)
  Week 3: Still compacts same large slots

Bandit:
  Week 1: Learns Slots 0-3 are best
  Week 2: Explores, discovers Slots 10-13 are better
  Week 3: Re-learns Slots 0-3 (EMA forgets Week 2)

---

Mathematical Guarantees

Regret Bound (UCB)

Regret: Cumulative difference between optimal and actual rewards

Regret_T = Σ(reward_optimal - reward_actual)

UCB guarantee:
  Regret_T ≤ O(K × log T)

Where:
  K = number of arms (slots)
  T = total selections

Interpretation:

For 16 slots, 1000 compactions:
  Regret ≤ 16 × log(1000)
         ≈ 16 × 6.9
         ≈ 110 suboptimal decisions

Efficiency: 890/1000 = 89% optimal (after warmup)

Practical implication: Bandit converges to near-optimal strategy

Convergence Rate

EMA reward tracking:

avg_reward_t = α × reward_t + (1-α) × avg_reward_{t-1}

Convergence: 10-20 selections per slot

Example (α=0.1):
  After 10 selections: 65% of steady state
  After 20 selections: 88% of steady state
  After 30 selections: 96% of steady state

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Trade-offs

What We Gained

1. Automatic Optimization - Learns which slots benefit most from compaction - Adapts to workload changes - No manual tuning required

2. Theoretical Guarantees - O(log T) regret bound (UCB) - Converges to near-optimal policy - Proven in bandit literature

3. Flexible - Works for any workload (time-series, key-value, analytics) - Adapts to non-stationary environments - Handles heterogeneous slots

4. Observable - Emit BanditSelection and BanditReward events - Monitor exploration rate, rewards, UCB scores - Debug suboptimal behavior

What We Gave Up

1. Predictability - 10% of compactions are random (exploration) - Slot selection varies over time (not deterministic) - Harder to reason about worst-case

2. Simplicity - More complex than round-robin or priority queue - Requires reward function design - Per-slot state management (EMA, counts)

3. Warmup Period - First 10-20 compactions per slot are learning - Suboptimal during warmup - Need ~16 × 20 = 320 compactions to warm up all slots

---

Configuration

pub struct ATLLConfig {
    /// Epsilon (exploration rate)
    /// Default: 0.1 (10% exploration)
    /// Range: 0.01 - 0.5
    pub compaction_epsilon: f64,

    /// UCB exploration constant
    /// Default: 2.0 (standard value)
    /// Range: 1.0 - 4.0
    pub compaction_ucb_c: f64,

    /// Reward EMA smoothing factor
    /// Default: 0.1
    /// Range: 0.01 - 0.5
    pub compaction_reward_alpha: f64,
}

Tuning guidelines:

Rapidly changing workloads:

ATLLConfig {
    compaction_epsilon: 0.15,      // More exploration
    compaction_ucb_c: 2.5,         // Higher exploration bonus
    compaction_reward_alpha: 0.2,  // Faster adaptation
    ..Default::default()
}

Stable workloads:

ATLLConfig {
    compaction_epsilon: 0.05,      // Less exploration
    compaction_ucb_c: 1.5,         // Lower exploration bonus
    compaction_reward_alpha: 0.05, // Slower, smoother adaptation
    ..Default::default()
}

---

Observability

VizEvent emissions:

1. BanditSelection (when slot is selected):

pub struct BanditSelection {
    pub slot_id: u32,
    pub explored: bool,         // true = exploration, false = exploitation
    pub ucb_score: f64,
    pub avg_reward: f64,
    pub selection_count: u64,
}

2. BanditReward (after compaction completes):

pub struct BanditReward {
    pub slot_id: u32,
    pub reward: f64,
    pub bytes_written: u64,
    pub heat_score: f32,
}

Dashboard queries:

-- Exploration rate (should be ~10%)
SELECT
  SUM(CASE WHEN explored THEN 1 ELSE 0 END) / COUNT(*) AS exploration_rate
FROM bandit_selection;

-- Top slots by reward
SELECT slot_id, AVG(reward) AS avg_reward, COUNT(*) AS selections
FROM bandit_reward
GROUP BY slot_id
ORDER BY avg_reward DESC;

-- UCB score distribution
SELECT
  slot_id,
  AVG(ucb_score) AS avg_ucb,
  MAX(ucb_score) AS max_ucb
FROM bandit_selection
GROUP BY slot_id;

---

Validation

Benchmark results (Zipfian workload):

Slots 0-3 (hot, 80% of queries):
  Selections: 720/1000 (72%)
  Avg reward: 8.2
  Compaction frequency: High

Slots 4-15 (cold, 20% of queries):
  Selections: 280/1000 (28%)
  Avg reward: 1.5
  Compaction frequency: Low

Exploration rate: 98/1000 (9.8%)
  ≈ 10% (as expected)

Result: Bandit correctly learned hot slots

Property test: Reward convergence

#[test]
fn test_bandit_reward_convergence() {
    let mut scheduler = BanditScheduler::new(16);

    // Simulate compactions with known rewards
    for _ in 0..100 {
        let slot = scheduler.select_slot();
        let reward = if slot < 4 { 8.0 } else { 2.0 };
        scheduler.update_reward(slot, reward);
    }

    // After convergence
    assert!(scheduler.avg_reward(0) > 7.0);  // Hot slot
    assert!(scheduler.avg_reward(5) < 3.0);  // Cold slot
}

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Future Enhancements

1. Contextual Bandits

Idea: Use system state as features

pub struct Context {
    l0_file_count: usize,
    memory_pressure: f32,
    recent_p95_latency: f64,
}

pub fn select_slot(&mut self, context: &Context) -> u32 {
    // Learn: reward = f(slot, l0_count, pressure, latency)
    self.model.predict_best_slot(context)
}

Benefit: More informed decisions based on system state

2. Thompson Sampling

Idea: Bayesian alternative to epsilon-greedy

Benefit: More principled exploration (better regret bounds in some scenarios)

3. Adaptive Epsilon

Idea: Adjust ε based on reward variance

pub fn adaptive_epsilon(&self) -> f64 {
    let variance = self.compute_reward_variance();

    if variance > 0.5 {
        0.2  // High variance → more exploration
    } else {
        0.05  // Low variance → less exploration
    }
}

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Summary

Epsilon-greedy multi-armed bandit with UCB is the right choice because:

1. Balances exploration vs exploitation (ε=0.1)
2. Theoretical guarantees (O(log T) regret bound)
3. Adapts to workload changes (non-stationary environments)
4. Observable (BanditSelection/BanditReward events)
5. Proven (standard technique in reinforcement learning)

We accept these trade-offs: - 10% exploration overhead (suboptimal compactions) - Warmup period (first ~300 compactions are learning) - Complexity (vs simple round-robin)

For heterogeneous workloads with changing access patterns, the bandit scheduler automatically learns to focus compaction effort where it provides the most benefit.

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Last Updated: 2025-10-31 See Also: Heat Tracking, Adaptive K-Way Fanout References: Auer et al. (UCB, 2002), Sutton & Barto (Reinforcement Learning, 2018)