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BPQP: A Differentiable Convex Optimization Framework for Efficient End-to-End Learning
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Proposes BPQP, which reformulates the backward pass of a differentiable convex optimization layer as a decoupled equality-constrained QP solved with OSQP (ADMM), giving large speedups over implicit-differentiation baselines.
Problem
Differentiating through convex optimization layers via implicit differentiation of KKT conditions requires costly Jacobian/linear-system solves, which scale poorly to large-scale problems with many constraints.
Approach
Rewrite the KKT-based backward linear system as an equality-constrained QP over the active set (decoupled from the forward solver), so any first-order solver such as OSQP can compute gradients without differentiating through the solver or inverting the KKT matrix.
Key results
- On QP 100x20, BPQP total time 35.1s vs 1862.9s for CVXPY and 1030.2s for qpth/OptNet (order-of-magnitude speedup), per the paper's claims of up to 21x.
- On portfolio optimization, BPQP reaches Sharpe 1.28 (±0.43) vs 0.65 for Two-Stage, with 7.7 min/epoch vs 21.2 for qpth/OptNet.
- BPQP backward-pass cosine similarity 0.992 on QP vs 0.989 for qpth/OptNet and 0.924 for CVXPY.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use BPQP instead of qpth/OptNet | QP backward pass, 10×5 · QP 10×5 | runtime: 70.6(±13.3) -> 2.0(±0.5) | Helped |
| Use BPQP instead of qpth/OptNet | QP backward pass, 500×100 · QP 500×100 | runtime: 1872.8(±1254.0) -> 571.6(±130.7) | Helped |
| Use BPQP instead of Exact | large QP · QP 5000×2000 | runtime: 489.4 (±22.5) -> 262.8(±17.0) | Helped |
| Use BPQP instead of Exact | LP backward pass · LP 500×100 | runtime: 2949.1 (±378.5) -> 318.7(±106.4) | Helped |
| Use BPQP instead of Exact | SOCP backward pass · SOCP 500×100 | runtime: 300.2 (±119.5) -> 242.9(±2.7) | Helped |
| Use BPQP instead of CVXPY | simulated QP backward pass · QP | CosSim: 0.924(±0.15) -> 0.992(±0.09) | Helped |
| Train with BPQP rather than Two-Stage | portfolio optimization · Portfolio optimization | Sharpe: 0.65(±0.25) -> 1.28(±0.43) | Helped |
| Train with BPQP rather than qpth/OptNet | portfolio optimization · Portfolio optimization | training time per epoch (minute): 21.2 -> 7.7 | Helped |
| Train with BPQP rather than Two-Stage | portfolio optimization · Portfolio optimization | IC: 0.033(±0.004) -> 0.026(±0.002) | Hurt |
| Train with BPQP rather than Two-Stage | portfolio optimization · Portfolio optimization | Annualized return (%): 9.28(±3.46) -> 17.67(±6.11) | Helped |
| Use BPQP rather than DC3 | portfolio optimization · Portfolio optimization | Sharpe: -0.16 (±0.60) -> 1.28(±0.43) | Helped |
Surprising findings
- DC3, a learn-to-optimize approximate method, is faster than BPQP at large scale (500x100) but yields negative annualized return and Sharpe ratio in portfolio optimization.
- Two-Stage gives the best prediction metrics (IC) yet the worst portfolio decision metrics.
Limitations
- Non-convex extension is only discussed theoretically, with no experiments
- Portfolio experiments exclude transaction costs
- Covariance matrix is held constant rather than learned
Experimental setup
Datasets: simulated QP/LP/SOCP instances Qlib CSI 500 stock data
Benchmarks: simulated QP, LP, SOCP datasets Qlib CSI 500 portfolio optimization
Metrics: runtime (s) cosine similarity of gradients IC ICIR annualized return Sharpe ratio regret training time per epoch
Baselines: CVXPY qpth/OptNet Alt-Diff JAXOpt Exact Two-Stage DC3
Builds on: OptNet OSQP CvxpyLayers Alt-Diff JAXOpt
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