We illustrate the approach with application to the protection of quantum gates performed on i) a central spin qubit coupling to bath spins through isotropic Heisenberg interactions, ii) superconducting transmon qubits coupling to environmental two-level-systems (TLS) through dipole-dipole interactions, and iii) qubits coupled to both TLS and a Lindblad bath. The control field is classical and acts only on the system qubits. We use reinforcement learning with policy gradient (PG) to optimize the Hamiltonian switching control protocols, using a fidelity objective defined with respect to specific target quantum gates. We use this approach to demonstrate effective suppression of both coherent and dissipative noise, with numerical studies achieving target gate implementations with fidelities over 0.9999 (four nines) in the majority of our test cases and showing improvement beyond this to values of 0.999999999 (nine nines) upon a subsequent optimization by Gradient Ascent Pulse Engineering (GRAPE). We analyze how the control depth, total evolution time, number of environmental TLS, and choice of optimization method affect the fidelity achieved by the optimal protocols and reveal some critical behaviors of bipartite control of quantum gates.