CVaR training for Binary VQE and QAOA

Track: binary asset selection. Setting: training_objective="cvar".

CVaR changes how circuit outcomes are aggregated during training. Instead of minimizing the mean penalized cost, minimize the average of the lowest-cost cvar_alpha fraction of probability mass. The circuits, portfolio objective, cardinality checks, and final candidate evaluation remain the same.

For bitstrings sorted by penalized cost \(E_1\leq\cdots\leq E_m\), with probabilities \(p_i\), the lower-tail objective is

\[ C_a = \frac{1}{a}\sum_i E_i\min\left(p_i, \max\left(a-\sum_{j<i}p_j,0\right)\right),\qquad 0<a\leq1. \]

The observation crossing the quantile contributes only the mass needed to reach a. Finite-shot training applies the same rule to equally weighted observations; it does not round the tail size up to a whole number of shots. Infeasible bitstrings retain their squared-cardinality penalty during training.

This is the lower tail of the optimization-cost distribution over measured bitstrings, not the upper tail of financial losses across market scenarios. It does not change mean–variance selection into financial CVaR optimization.

Execution and optimization

  • shots_train=None: compute all basis-state probabilities, sort constant costs once, and differentiate the probability-weighted tail with Adam. CVaR is piecewise differentiable; quantile boundaries are nonsmooth. This path uses O(n * 2**n) setup storage and is intended for small simulator problems.

  • Positive shots_train and cvar_alpha < 1: sample bitstrings, score them classically, and use SPSA. No gradient is taken through sampling or sorting. Each update uses two perturbed cost evaluations; logging and best-iterate selection require additional evaluations. The perturbation magnitude is 0.1/(t+1)**0.101; the learning rate is stepsize/(t+1)**0.602/(1+stepsize_decay*t) at zero-based update t. QAOA has no additional stepsize_decay setting and uses zero decay.

  • cvar_alpha=1: use the existing mean-cost QNode and Adam, preserving the expectation-training trajectory for the same configuration and seed.

The default remains training_objective="expectation". The tail fraction defaults to 0.25 and must be finite and in (0, 1]. Small tails can flatten the analytic objective once enough mass reaches the best costs, and increase noise in sampled training. There is no guarantee of improved portfolio quality. Use 10%, 25%, and 100% as experiment settings, with matched execution budgets.

CVaR applies to both QAOA mixers and both solvers’ lambda sweeps. Binary VQE normalization, best-iterate selection, and restart selection use the configured cost. energy_trace/cost_trace and restart_energies record CVaR costs in original penalized objective units when enabled. The historical field names are retained. Results identify training_objective, cvar_alpha (null for expectation), and training_optimizer (adam or spsa).

Budgeted execution accepts the same configuration and includes CVaR setup, optimization and sampling costs. Finite-shot CVaR training avoids enumerating all bitstrings; QAOA’s existing final marginal/probability reporting still constructs the full probability vector.

Examples

Use the shared inputs.

from vqe_portfolio import BinaryVQEConfig, QAOAConfig, run_binary_vqe, run_qaoa

config = BinaryVQEConfig(
    k=2, lam=4.0, alpha="auto", steps=25,
    training_objective="cvar", cvar_alpha=0.25,
)
result = run_binary_vqe(mu, Sigma, config)

config = QAOAConfig(
    k=2, lam=4.0, mixer="xy", xy_initial_state="dicke", p=1,
    steps=25, stepsize=0.05, shots_train=1024,
    training_objective="cvar", cvar_alpha=0.25,
)
result = run_qaoa(mu, Sigma, config)

vqe-portfolio binary --mu "0.1,0.2,0.15" \
  --sigma "0.1,0,0;0,0.2,0;0,0,0.1" --k 2 --lam 4 \
  --training-objective cvar --cvar-alpha 0.25

vqe-portfolio qaoa --mu "0.1,0.2,0.15" \
  --sigma "0.1,0,0;0,0.2,0;0,0,0.1" --k 2 --lam 4 \
  --mixer xy --xy-initial-state dicke --stepsize 0.05 \
  --training-objective cvar --cvar-alpha 0.25 --shots-train 1024

The same flags are available on binary-data and qaoa-data; add --budget-seconds to the solver commands for budgeted execution. CVaR is a training setting, not a separate method ID, and is not applied to the variational allocator.

Reference: Barkoutsos et al., Improving Variational Quantum Optimization using CVaR.