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DisCo-DSO: Coupling Discrete and Continuous Optimization for Efficient Generative Design in Hybrid Spaces
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Proposes DisCo-DSO, which uses an autoregressive RNN trained with a risk-seeking policy gradient to jointly sample discrete tokens and continuous parameters, enabling one-evaluation-per-candidate optimization in hybrid variable-length spaces.
Problem
Black-box optimization over hybrid discrete-continuous, variable-length design spaces with prefix-dependent positional constraints (e.g., decision tree policies and symbolic regression), where existing methods decouple discrete skeleton search from continuous parameter fitting.
Approach
Extend deep symbolic optimization so an autoregressive generative model emits, for each token, both a discrete token and a continuous parameter (sampled from a token-conditioned distribution), trained jointly with a quantile-based risk-seeking policy gradient, so each sample is a complete candidate needing one objective evaluation.
Key results
- On decision tree policy tasks, DisCo-DSO achieves mean reward -76.58 on Acrobot-v1 and 99.24 on LunarLander-v2 with 18 and 23 parameters, outperforming the listed literature baselines on those environments.
- On symbolic regression, DisCo-DSO reaches average test reward 0.7045 over all benchmarks versus 0.6400 for Decoupled-RL-BFGS and 0.4953 for Decoupled-GP-BFGS.
- On the Parameterized Bitstring task, DisCo-DSO is more sample efficient than decoupled approaches, with the advantage largest when the discrete skeleton dominates the reward.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Jointly optimize discrete and continuous variables instead of optimizing the continuous parameters after sampling a discrete skeleton | DisCo-DSO vs decoupled methods · Decision tree policies across Acrobot-v1, CartPole-v1, LunarLander-v2, and MountainCar-v0 | mean reward: 4.22; 260.00; 74.80; 7.80 | Helped |
| Jointly optimize discrete and continuous variables instead of using Decoupled-RL-evo | DisCo-DSO · Decision tree policies across four environments | mean reward: 6.52; 0.00; 80.60; 2.60 | Helped |
| Jointly optimize discrete and continuous variables instead of using Decoupled-GP-BFGS | DisCo-DSO · Decision tree policies across four environments | mean reward: 4.72; 265.00; 64.80; 7.70 | Helped |
| Jointly optimize discrete and continuous variables instead of using Decoupled-GP-evo | DisCo-DSO · Decision tree policies across four environments | mean reward: 7.92; 0.00; 54.60; 2.80 | Helped |
| Jointly optimize discrete and continuous variables instead of using Decoupled-GP-anneal | DisCo-DSO · Decision tree policies across four environments | mean reward: 4.52; 0.00; 43.20; 6.10 | Helped |
| Jointly optimize discrete and continuous variables instead of Decoupled-RL-BFGS | DisCo-DSO · Symbolic regression, Dim ≥ 1 | average test reward: 0.6400 ± 0.3684 -> 0.7045 ± 0.3007 | Helped |
| Jointly optimize discrete and continuous variables instead of Decoupled-GP-BFGS | DisCo-DSO · Symbolic regression, Dim ≥ 1 | average test reward: 0.4953 ± 0.4344 -> 0.7045 ± 0.3007 | Helped |
| Use DisCo-DSO rather than Decoupled-RL methods | Parameterized Bitstring · f1, α = 0.5; f1, α = 0.9; f2, α = 0.5; f2, α = 0.9 | reward difference: 0.1263; 0.0123; 0.1433; 0.0253 | Helped |
| Use DisCo-DSO rather than Decoupled-RL-evo | Parameterized Bitstring · f1, α = 0.5; f1, α = 0.9; f2, α = 0.5; f2, α = 0.9 | reward difference: 0.0970; 0.0100; 0.0833; 0.0187 | Helped |
| Use DisCo-DSO rather than Decoupled-RL-anneal | Parameterized Bitstring · f1, α = 0.5; f1, α = 0.9; f2, α = 0.5; f2, α = 0.9 | reward difference: 0.1211; 0.0095; 0.1000; 0.0140 | Helped |
| Use LSTM rather than GRU autoregressive cells (ablation) | DisCo-DSO · Symbolic regression | mean reward: 0.7377; 0.7092; 0.7236; 0.7385; 0.7391; 0.7302 | Mixed |
| Vary autoregressive model size across 16, 32, and 64 hidden units (ablation) | DisCo-DSO · Symbolic regression | mean reward: 0.7377; 0.7092; 0.7236; 0.7385; 0.7391; 0.7302 | No clear effect |
Surprising findings
- Decoupled-GP methods overfit heavily on training data, producing traversals concentrated at maximum length (32) and poor test-set generalization.
- On CartPole-v1, a decoupled baseline (Decoupled-RL-BFGS) reportedly reached a higher reward than DisCo-DSO in the gap table is not reported; the paper notes DisCo-DSO matches the best on CartPole with reward 500.
Limitations
- Relies on domain-specific ranges for continuous variables; performance may degrade when such estimates are unavailable.
- Decision tree experiments restricted to univariate (axis-aligned) trees, not oblique trees.
- Gains over baselines are less clear on linear symbolic regression benchmarks; the paper leaves that comparison to future work.
Experimental setup
Datasets: Parameterized Bitstring (synthetic) OpenAI Gym MountainCar-v0 CartPole-v1 Acrobot-v1 LunarLander-v2 Livermore, Jin, Neat/Korn, Nguyen-style Constant symbolic regression benchmarks (45 total)
Benchmarks: Parameterized Bitstring MountainCar-v0 CartPole-v1 Acrobot-v1 LunarLander-v2 Symbolic regression benchmarks (45 datasets)
Metrics: best reward vs function evaluations mean reward on 1,000 random seeds parameter count of tree test-set reward (1/(1+NMSE))
Baselines: Decoupled-RL-BFGS Decoupled-RL-anneal Decoupled-RL-evo Decoupled-GP-BFGS Decoupled-GP-anneal Decoupled-GP-evo Bayesian Optimization Evolutionary DTs Cascading DTs Interpretable DDTs
Builds on: Deep Symbolic Regression risk-seeking policy gradient deep symbolic optimization
Closest papers by meaning
- Symbolic Regression via Deep Reinforcement Learning Enhanced Genetic Programming Seeding (NeurIPS 2021 · similarity 0.87)
- SYMBOL: Generating Flexible Black-Box Optimizers through Symbolic Equation Learning (ICLR 2024 · similarity 0.87)
- SMILE: Bridging Continuous Optimization and Discrete Symbolic Recovery (NeurIPS 2026 · similarity 0.85)
- Understanding the Challenges in Iterative Generative Optimization with LLMs (NeurIPS 2026 · similarity 0.84)
- MetaBox-v2: A Unified Benchmark Platform for Meta-Black-Box Optimization (NeurIPS 2025 · similarity 0.84)
- Autonomous Option Invention for Continual Hierarchical Reinforcement Learning and Planning (AAAI 2025 · similarity 0.84)
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